Wolfgang Lück

L^{2}-Invariants: Theory and Applications to Geometry and K-Theory

SPIN Springer’s internal project number, if known

Mathematics – Monograph (English)

July 1, 2002

Springer-Verlag

Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Preface

There is the general principle to consider a classical invariant of a closed Riemannian manifold M and to define its analog for the universal covering M̃ taking the action of the fundamental group π = π_{1}(M) on M̃ into account. Prominent examples are the Euler characteristic and the signature of M, which lead to Wall’s finiteness obstruction and to all kinds of surgery obstructions such as the symmetric signature or higher signatures. The pth L^{2}-Betti number b_{p}^{(2)}(M̃) arises from this principle applied to the p-th Betti number b_{p}(M). Some effort is necessary to define L^{2}-Betti numbers in the case where π is infinite. Typical problems for infinite π are that M̃ is not compact and that the complex group ring ℂπ is a complicated ring, in general not Noetherian. Therefore some new technical input is needed from operator theory, namely, the group von Neumann algebra and its trace. Analytically Atiyah defined L^{2}-Betti numbers in terms of the heat kernel on M̃. There also is an equivalent combinatorial approach based on the cellular ℂπ-chain complex of M̃. It is one of the main important and useful features of L^{2}-invariants that they can be defined both analytically and combinatorially. There are two further types of L^{2}-invariants. L^{2}-torsion generalizes the classical notion of Reidemeister torsion from finite to infinite π, whereas Novikov-Shubin invariants do not have a classical counterpart.

A very intriguing and important property of L^{2}-invariants is that they have relations to many other fields. From their construction it is clear that they have connections to operator theory, in particular to von Neumann algebras, and to the spectral theory of the Laplacian on M̃. For instance Atiyah’s motivation to consider L^{2}-Betti numbers was to establish his L^{2}- index theorem.

More suprising is the appearance of algebraic K-theory. In all examples where L^{2}-Betti numbers have been computed explicitly, the values turn out to be rational numbers whose denominators are linked to the orders of finite subgroups of π. This is very suprising in view of the actual definition of L^{2}-Betti numbers. This phenomenon is linked to questions in algebraic Ktheory such as whether any finitely generated projective ℂπ-module M is obtained by induction from a finitely generated projective ℂH-module for a finite subgroup H ⊂ π. This leads to the version of the so called Atiyah Conjecture that the L^{2}-Betti numbers are always integers if π is torsionfree.

It turns out that this conjecture implies the Kaplansky Conjecture that ℂπ contains no non-trivial zero-divisors if π is torsionfree. For many groups π the Kaplansky Conjecture was not known until the Atiyah Conjecture was proved. We will investigate interactions between L^{2}-invariants and K-theory and applications of them in both directions throughout this book.

Next we explain a connection to geometry. Provided that M is aspherical, all computations lead to the result that b_{p}^{(2)}(M̃) = 0 holds for 2p ≠ dim(M) and that b_{n}^{(2)}(M̃) = (−1)^{n} · χ(M) is true for the Euler characteristic χ(M) if dim(M) = 2n is even. In particular (−1)^{n} · χ(M) ≥ 0 in the case dim(M) = 2n, since each L^{2}-Betti number is larger or equal to zero by definition. This phenomenon seems to be typical and will be investigated in this book. Recall that M is aspherical if it carries a Riemannian metric with non-positive sectional curvature, but that the converse is not true. If dim(M) = 2n and M carries a Riemannian metric with negative sectional curvature, then all computations yield b_{n}^{(2)}(M̃) = (−1)^{n} · χ(M) > 0. Hence L^{2}-Betti numbers are linked to the Hopf Conjecture which predicts (−1)^{n} · χ(M) ≥ 0 if the 2n-dimensional closed manifold M carries a Riemannian metric with non-positive sectional curvature, and (−1)^{n} · χ(M) > 0 if M carries a Riemannian metric with negative sectional curvature. Further connections between L^{2}-invariants and geometry and group theory will be presented in this book.

Why Study L^{2}-Invariants?

From the author’s point of view there are certain criteria which decide whether a topic or an area in modern mathematics is worth studying or worth further development. Among them are the following:

• The topic has relations to other fields. There is a fruitful exchange of results and techniques with other areas which leads to solutions of problems and to innovations in both the topic of focus and other topics;

• There are some hard open problems which are challenging and promising. They create interesting activity and partial solutions and techniques for their proof already have applications to other problems;

• The topic is accessible with a reasonable amount of effort. In particular talented students are able to learn the basics of the topic within an appropriate period of time and while doing so get a broad basic education in mathematics.

The purpose of this book is to convince the reader that L^{2}-invariants do satisfy these criteria and to give a comprehensible and detailed approach to them which includes the most recent developments.

A User’s Guide

We have tried to write this book in a way which enables the reader to pick out his favourite topic and to find the result she or he is interested in quickly and without being forced to go through other material. The various chapters are kept as independent of one another as possible. In the introduction of each chapter we state what input is needed from the previous chapters, which is in most cases not much, and how to browse through the chapter itself. It may also be worthwhile to go through the last section “Miscellaneous” in each chapter which contains some additional information. In general a first impression can be gained by just reading through the definitions and theorems themselves. Of course one can also read the book linearly.

Each chapter includes exercises. Some of them are easy, but some of them are rather difficult. Hints to their solutions can be found in Chapter 16. The exercises contain interesting additional material which could not be presented in detail in the text. The text contains some (mini) surveys about input from related material such as amenable groups, the Bass Conjecture, deficiency of groups, Isomorphism Conjectures in K-theory, 3-manifolds, Ore localization, residually finite groups, simplicial volume and bounded cohomology, symmetric spaces, unbounded operators, and von Neumann regular rings, which may be useful by themselves. (They are listed in the index under “survey”. One can also find a list of all conjectures, questions and main theorems in the index.)

If one wants to run a seminar on the book, one should begin with Sections 1.1 and 1.2. Then one can continue depending on the own interest. For instance if one is algebraically oriented and not interested in the analysis, one may directly pass to Chapter 6, whereas an analyst may be interested in the rest of Chapter 1 and then pass to Chapter 2. Chapters 9, 10, 11, 12, 13 and 14 are independent of one another. One may directly approach these chapters and come back to the previous material when it is cited there.

We require that the reader is familiar with basic notions in topology (CW -complexes, chain complexes, homology, manifolds, differential forms, coverings), functional analysis (Hilbert spaces, bounded operators), differential geometry (Riemannian metric, sectional curvature) and algebra (groups, modules, elementary homological algebra).

Acknowledgements

I want to thank heartily the present and former members of the topology group in Münster who read through the manuscript and made a lot of useful

VIII Preface

comments, corrections and suggestions. These are Arthur Bartels, Clemens Bratzler, Clement de Seguins Pazzis, Eckehard Hess, Michael Joachim, Michel Matthey, David Meintrup, Morten Pohlers, Holger Reich, Juliane Sauer, Roman Sauer, Thomas Schick, Marco Schmidt, Roland Stamm, Marco Varisco, Julius Verrel and Christian Wegner.

I want to thank the Deutsche Forschungsgemeinschaft which has been and is financing the Sonderforschungsbereich 478 – “Geometrische Strukturen in der Mathematik” and the Graduiertenkolleg “Analytische Topologie und Metageometrie”. These institutions made it possible to invite guests and run workshops on the topics of the book. This was very important for its writing. In particular I had fruitful discussions with and obtained a lot of information from Ulrich Bunke, Marc Burger, Dan Burghelea, Mike Davis, Jozef Dodziuk, Benno Eckmann, Michael Farber, Tom Farrell, Damien Gaboriau, Thomas Kappeler, Ian Leary, Peter Linnell, John Lott, Varghese Mathai, Guido Mislin, Nicolas Monod, Martin Olbrich, Pierre Pansu, Holger Reich, Thomas Schick, Michael Shubin, Alain Valette and Shmuel Weinberger.

I thank the Max-Planck Institute for Mathematics in Bonn for its hospitality during my stay in January and February 2001 while parts of this book were written.

Finally I want to express my deep gratitude to my wife Sibylle and our children Christian, Isabel, Tobias and Severina for everything.

Münster, February 2002 Wolfgang Lück

Contents

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.1 What are L^{2}-Invariants? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 Some Applications of L^{2}-Invariants . . . . . . . . . . . . . . . . . . . . . . . 2 0.3 Some Open Problems Concerning L^{2}-Invariants . . . . . . . . . . . . 3 0.4 L^{2}-Invariants and Heat Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0.5 L^{2}-Invariants and Cellular Chain Complexes . . . . . . . . . . . . . . . 6 0.6 L^{2}-Betti Numbers and Betti Numbers . . . . . . . . . . . . . . . . . . . . . 7 0.7 L^{2}-Invariants and Ring-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 0.8 L^{2}-Invariants and K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 0.9 L^{2}-Invariants and Aspherical Manifolds . . . . . . . . . . . . . . . . . . . 11 0.10 L^{2}-Invariants and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1. L^{2}-Betti Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1 Group von Neumann Algebras and Hilbert Modules. . . . . . . . . 14

1.1.1 Group von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . 14 1.1.2 Hilbert Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.1.3 Dimension Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.1.4 Hilbert Chain Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.1.5 Induction for Group von Neumann Algebras . . . . . . . . . 29

1.2 Cellular L^{2}-Betti Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.2.1 Survey on G-CW -Complexes . . . . . . . . . . . . . . . . . . . . . . 31 1.2.2 The Cellular L^{2}-Chain Complex . . . . . . . . . . . . . . . . . . . . 33 1.2.3 Basic Properties of Cellular L^{2}-Betti Numbers . . . . . . . 37 1.2.4 L^{2}-Betti Numbers and Aspherical Spaces . . . . . . . . . . . . 45

1.3 Analytic L^{2}-Betti Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.3.1 The Classical Hodge-de Rham Theorem . . . . . . . . . . . . . 49 1.3.2 Analytic Definition of L^{2}-Betti Numbers . . . . . . . . . . . . 52

1.4 Comparison of Analytic and Cellular L^{2}-Betti Numbers . . . . . 54 1.4.1 Survey on Unbounded Operators and Spectral Families 55 1.4.2 L^{2}-Hodge-de Rham Theorem . . . . . . . . . . . . . . . . . . . . . . 58

1.5 L^{2}-Betti Numbers of Manifolds with Boundary . . . . . . . . . . . . . 63 1.6 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2. Novikov-Shubin Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.1 Spectral Density Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.1.1 Spectral Density Functions of Morphisms . . . . . . . . . . . . 72 2.1.2 Spectral Density Functions of Hilbert

Chain Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.1.3 Product Formula for Novikov-Shubin Invariants . . . . . . 86 2.1.4 The Laplacian in Dimension Zero . . . . . . . . . . . . . . . . . . . 93

2.2 Cellular Novikov-Shubin Invariants . . . . . . . . . . . . . . . . . . . . . . . 96 2.3 Analytic Novikov-Shubin Invariants . . . . . . . . . . . . . . . . . . . . . . . 103 2.4 Comparison of Analytic and Cellular Novikov-Shubin

Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.5 On the Positivity and Rationality of the Novikov-Shubin

Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2.6 Novikov-Shubin Invariants of Manifolds with Boundary . . . . . . 114 2.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3. L^{2}-Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.1 Survey on Torsion Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.1.1 Whitehead Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.1.2 Whitehead Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.1.3 Reidemeister Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3.2 Fuglede-Kadison Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.3 L^{2}-Torsion of Hilbert Chain Complexes . . . . . . . . . . . . . . . . . . . 139

3.3.1 Basic Definitions and Properties of L^{2}-Torsion . . . . . . . 140 3.3.2 L^{2}-Torsion and Chain Contractions . . . . . . . . . . . . . . . . . 144 3.3.3 Proofs of the Basic Properties of L^{2}-Torsion . . . . . . . . . 148

3.4 Cellular L^{2}-Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.4.1 Cellular L^{2}-Torsion in the Weakly-Acyclic Case . . . . . . 160 3.4.2 Cellular L^{2}-Torsion in the Weakly-Acyclic and

Aspherical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.4.3 Topological L^{2}-Torsion for Riemannian Manifolds …. 176

3.5 Analytic L^{2}-Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 3.5.1 Definition of Analytic L^{2}-Torsion . . . . . . . . . . . . . . . . . . . 178 3.5.2 The Laplace Transform of a Density Function . . . . . . . . 181 3.5.3 Comparison of Topological and Analytic L^{2}-Torsion . . 186 3.5.4 Analytic L^{2}-Torsion for Hyperbolic Manifolds . . . . . . . . 186

3.6 L^{2}-Torsion of Manifolds with Boundary . . . . . . . . . . . . . . . . . . . 190 3.7 Combinatorial Computations of L^{2}-Invariants . . . . . . . . . . . . . . 193 3.8 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 4. L^{2}-Invariants of 3-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 4.1 Survey on 3-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 4.2 L^{2}-Invariants of 3-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 4.3 L^{2}-Invariants of Knot Complements . . . . . . . . . . . . . . . . . . . . . . 217 4.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

5. L^{2}-Invariants of Symmetric Spaces. . . . . . . . . . . . . . . . . . . . . . . . 223 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5.1 Survey on Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5.2 L^{2}-Invariants of Symmetric Spaces of Non-Compact Type … 227 5.3 L^{2}-Invariants of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . 229 5.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

6. L^{2}-Invariants for General Spaces with Group Action . . . . . . 235 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 6.1 Dimension Theory for Arbitrary Modules . . . . . . . . . . . . . . . . . . 237 6.2 Comparison of Modules and Hilbert Modules . . . . . . . . . . . . . . 246 6.3 Induction and the Extended von Neumann Dimension . . . . . . . 253 6.4 The Extended Dimension Function and Amenable Groups … 255

6.4.1 Survey on Amenable Groups . . . . . . . . . . . . . . . . . . . . . . . 256 6.4.2 Amenability and the Coinvariants of the Group

von Neumann Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 6.4.3 Amenability and Flatness Properties of the Group

von Neumann Algebra over the Group Ring . . . . . . . . . 258

6.5 L^{2}-Betti Numbers for General Spaces with Group Actions … 263 6.6 L^{2}-Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

6.6.1 Definition and Basic Properties of L^{2}-Euler

Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 6.6.2 L^{2}-Euler Characteristic, Equivariant Euler

Characteristic and the Burnside Group . . . . . . . . . . . . . . 280

6.7 Finitely Presented Torsion Modules and Novikov-Shubin

Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 6.8 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

7. Applications to Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 7.1 Groups with Vanishing L^{2}-Betti Numbers . . . . . . . . . . . . . . . . . 293

7.1.1 General Criterions for the Vanishing of the L^{2}-Betti

Numbers of a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 7.1.2 The Vanishing of the L^{2}-Betti Numbers of

Thompson’s Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 7.2 Euler Characteristics of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 298 7.3 Deficiency of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

7.3.1 Survey on Deficiency of Groups . . . . . . . . . . . . . . . . . . . . 299 7.3.2 Applications of L^{2}-Betti Numbers to Deficiency and

to Signatures of 4-Manifolds . . . . . . . . . . . . . . . . . . . . . . . 303

7.4 Group Automorphisms and L^{2}-Torsion . . . . . . . . . . . . . . . . . . . . 304

7.4.1 Automorphisms of Groups G with Finite Models

for BG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 7.4.2 Automorphisms of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 307 7.4.3 A Combinatorial Approach for the L^{2}-Torsion of an

Automorphism of a Finitely Generated Free Group … 308 7.4.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

7.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

8. The Algebra of Affiliated Operators . . . . . . . . . . . . . . . . . . . . . . 317 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 8.1 The Algebra of Affiliated Operators . . . . . . . . . . . . . . . . . . . . . . . 318 8.2 Basic Properties of the Algebra of Affiliated Operators . . . . . . 323

8.2.1 Survey on Ore Localization . . . . . . . . . . . . . . . . . . . . . . . . 323 8.2.2 Survey on von Neumann Regular Rings . . . . . . . . . . . . . 325 8.2.3 Basic Properties of the Algebra of Affiliated Operators 327

8.3 Dimension Theory and L^{2}-Betti Numbers over the Algebra

of Affiliated Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 8.4 Various Notions of Torsion Modules over a Group

von Neumann Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 8.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

9. Middle Algebraic K-Theory and L-Theory of von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 9.1 Survey on von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . . . 336

9.1.1 Definition of a von Neumann Algebra . . . . . . . . . . . . . . . 336 9.1.2 Types and the Decomposition of von Neumann

Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 9.1.3 Finite von Neumann Algebras and Traces . . . . . . . . . . . 338 9.1.4 Extending Results for Group von Neumann Algebras

to Finite von Neumann Algebras . . . . . . . . . . . . . . . . . . . 340

9.2 Middle K-Theory of a Neumann Algebra . . . . . . . . . . . . . . . . . . 341 9.2.1 K_{0} of a von Neumann Algebra . . . . . . . . . . . . . . . . . . . . . 341 9.2.2 K_{1} of a von Neumann Algebra . . . . . . . . . . . . . . . . . . . . . 343

9.3 Middle K-Theory of the Algebra of Affiliated Operators . . . . . 345 9.4 L-Theory of a von Neumann Algebra and the Algebra

of Affiliated Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

Contents XIII

9.5 Application to Middle K- and G-Theory of Group Rings …. 354 9.5.1 Detecting Elements in K_{1} of a Complex Group Ring . . 354 9.5.2 Survey on the Isomorphism Conjecture for K_{0}

of Complex Group Rings, the Bass Conjecture

and the Hattori-Stallings Rank . . . . . . . . . . . . . . . . . . . . . 357 9.5.3 G-Theory of Complex Group Rings . . . . . . . . . . . . . . . . . 363

9.6 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

10. The Atiyah Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 10.1 Survey on the Atiyah Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 372

10.1.1 Various Formulations of the Atiyah Conjecture . . . . . . . 372 10.1.2 Relations of the Atiyah Conjecture to Other

Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 10.1.3 Survey on Positive Results about the Atiyah

Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 10.1.4 A Counterexample to the Strong Atiyah Conjecture . . 381

10.2 A Strategy for the Proof of the Atiyah Conjecture . . . . . . . . . . 383 10.2.1 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 10.2.2 Survey on Universal Localizations and Division and

Rational Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 10.2.3 The Strategy for the Proof of Linnell’s Theorem . . . . . . 389

10.3 The Proof of Linnell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 393 10.3.1 The Proof of Atiyah’s Conjecture for Free Groups …. 393 10.3.2 Survey on Crossed Products . . . . . . . . . . . . . . . . . . . . . . . 398 10.3.3 Property (R) Ascends to Finite Extensions . . . . . . . . . . 401 10.3.4 Property (R) Ascends to Extensions by Infinite Cyclic

Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 10.3.5 Property (K) and Extensions by Virtually Finitely

Generated Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . 404 10.3.6 Property (K) holds for Virtually Free Groups . . . . . . . . 407 10.3.7 The Induction Step for Directed Unions . . . . . . . . . . . . . 411

10.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

11. The Singer Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 11.1 Survey on Positive Results about the Singer Conjecture . . . . . 422

11.1.1 Low-Dimensional Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 422 11.1.2 Pinched Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 11.1.3 Aspherical Manifolds and Locally Symmetric Spaces . . 425

XIV Contents

11.2 The Singer Conjecture for Kähler Manifolds . . . . . . . . . . . . . . . 425 11.2.1 Hodge Theory on Kähler manifolds . . . . . . . . . . . . . . . . . 428 11.2.2 The L^{2}-Lefschetz Theorem . . . . . . . . . . . . . . . . . . . . . . . . 430 11.2.3 Novikov-Shubin Invariants for Kähler Hyperbolic

Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 11.2.4 Non-Vanishing of the Middle L^{2}-Betti Number for

Kähler Hyperbolic Manifolds . . . . . . . . . . . . . . . . . . . . . . . 435

11.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

12. The Zero-in-the-Spectrum Conjecture . . . . . . . . . . . . . . . . . . . . 441 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 12.1 An Algebraic Formulation of the Zero-in-the-Spectrum

Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 12.2 Survey on Positive Results about the Zero-in-the-Spectrum

Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 12.2.1 The Zero-in-the-Spectrum Conjecture for

Low-Dimensional Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 444 12.2.2 The Zero-in-the-Spectrum Conjecture for Locally

Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 12.2.3 The Zero-in-the-Spectrum Conjecture for Kähler

Hyperbolic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 12.2.4 The Zero-in-the-Spectrum Conjecture for

HyperEuclidean Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 446 12.2.5 The Zero-in-the-Spectrum Conjecture and the Strong

Novikov Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 12.2.6 The Zero-in-the-Spectrum Conjecture and Finite

Asymptotic Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

12.3 Counterexamples to the Zero-in-the-Spectrum Conjecture in

the Non-Aspherical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 12.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

13. The Approximation Conjecture and the Determinant Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 13.1 Survey on the Approximation Conjecture and Determinant

Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 13.1.1 Survey on Positive Results about the Approximation

Conjecture and the Determinant Conjecture . . . . . . . . . 458 13.1.2 Relations to Other Conjectures . . . . . . . . . . . . . . . . . . . . . 459 13.1.3 A Class of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

13.2 The Proof of the Approximation Conjecture and the Determinant Conjecture in Special Cases . . . . . . . . . . . . . . . . . . 463 13.2.1 The General Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 13.2.2 Limits of Inverse Systems . . . . . . . . . . . . . . . . . . . . . . . . . 469 13.2.3 Colimits of Directed Systems . . . . . . . . . . . . . . . . . . . . . . 471 13.2.4 Amenable Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 13.2.5 Quotients with Finite Kernels . . . . . . . . . . . . . . . . . . . . . . 477

13.3 Variations of the Approximation Results . . . . . . . . . . . . . . . . . . 479 13.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

14. L^{2}-Invariants and the Simplicial Volume . . . . . . . . . . . . . . . . . . 489 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 14.1 Survey on Simplicial Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

14.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 14.1.2 Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 14.1.3 Bounded Cohomology and Amenable Groups . . . . . . . . 493 14.1.4 The Simplicial Volume of Hyperbolic and

Low-Dimensional Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 493 14.1.5 Volume and Simplicial Volume . . . . . . . . . . . . . . . . . . . . 495 14.1.6 Simplicial Volume and Betti Numbers . . . . . . . . . . . . . . . 498 14.1.7 Simplicial Volume and S^{1}-Actions . . . . . . . . . . . . . . . . . . 500 14.1.8 Some Information about the Second Bounded

Cohomology of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 14.1.9 Further Properties of the Simplicial Volume . . . . . . . . . 500

14.2 Simplicial Volume and L^{2}-Invariants of Universal Coverings of Closed Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 14.2.1 Hyperbolic Manifolds and 3-Manifolds . . . . . . . . . . . . . . 501 14.2.2 S^{1}-Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 14.2.3 Amenable Fundamental Groups . . . . . . . . . . . . . . . . . . . . 502 14.2.4 Selfmaps of Degree Different from −1, 0 and 1 . . . . . . . 503 14.2.5 Negative Sectional Curvature and Locally Symmetric

Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 14.2.6 Simplicial Volume and L^{2}-Invariants . . . . . . . . . . . . . . . . 505

14.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

15. Survey on Other Topics Related to L^{2}-Invariants . . . . . . . . . 511 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 15.1 L^{2}-Index Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 15.2 L^{p}-Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 15.3 Intersection Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 15.4 Knot Concordance and L^{2}-signature . . . . . . . . . . . . . . . . . . . . . . 513

16. Solutions of the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

0. Introduction

0.1 What are L^{2}-Invariants?

There is the classical notion of the p-th Betti number b_{p}(X) of a finite CW - complex X, for instance a closed manifold, which is the dimension of the complex vector space H_{p}(X; ℂ). Consider a G-covering p: \overline{X} → X. If G is infinite, the p-th Betti number of \overline{X} may be infinite and hence useless. Using some input from functional analysis involving Hilbert spaces, group von Neumann algebras and traces one can define the p-th L^{2}-Betti number b_{p}^{(2)}(\overline{X}; N (G)) of the total space \overline{X} as the non-negative real number given by the von Neumann dimension of the (reduced) L^{2}-homology of \overline{X}. (Often we briefly write b_{p}^{(2)}(\overline{X}) if G is clear from the context.) If G is finite, b_{p}^{(2)}(\overline{X}) = |G|^{−1} · b_{p}(X) and we get nothing new. But L^{2}-Betti numbers carry new information and have interesting applications in the case where G is infinite. In general b_{p}^{(2)}(\overline{X}) of the total space \overline{X} and b_{p}(X) of the base space X have no relations except for the Euler-Poincaré formula, namely,

χ(X) = ∑_{p≥0}(−1)^{p} · b_{p}(X) = ∑_{p≥0}(−1)^{p} · b_{p}^{(2)}(\overline{X}),  (0.1)

where χ(X) is the Euler characteristic of X (see Section 0.6).

The notion of the classical Reidemeister torsion of \overline{X} for finite groups G will be generalized to the notion of L^{2}-torsion ρ^{(2)}(\overline{X}) ∈ ℝ in the case that G is infinite.

There is a third class of L^{2}-invariants, the Novikov-Shubin invariants α_{p}(\overline{X}), which carry no information if G is finite.

All these types of L^{2}-invariants on the one hand have analytic definitions in terms of the heat kernel on \overline{X}, but on the other hand can be defined combinatorially in terms of the cellular ℂG-chain complex of \overline{X}. These two approaches are equivalent. In the analytic context X must be a compact Riemannian manifold. For the combinatorial definition of L^{2}-Betti numbers and Novikov-Shubin invariants it suffices to require that the base space X is of finite type, i.e. each skeleton of X is finite, but X may be infinite-dimensional.

0.2 Some Applications of L^{2}-Invariants

In order to convince the reader about the potential of L^{2}-invariants we state some results which seem to have nothing to do with L^{2}-invariants but whose proofs — as we will see — use L^{2}-methods. The selection below consists of some easy to formulate examples and is not meant to represent the most important results about L^{2}-invariants. There are plenty of other very interesting and important theorems about L^{2}-invariants, a lot of which will be presented in this book. For simplicity we often will not state the most general formulations in this introduction. All notions appearing in the list of theorems below will be explained in the relevant chapters. The results below are due to Chang-Weinberger, Cheeger-Gromov, Cochran-Orr-Teichner, Dodziuk, Gaboriau, Gromov and Lück.

Theorem 0.2 (see Theorem 1.35 (2) and Corollary 6.75). Let G be a group which contains a normal infinite amenable subgroup. Suppose that there is a finite CW -model for its classifying space BG. Then its Euler characteristic vanishes, i.e.

χ(G) := χ(BG) = 0.

Theorem 0.3 (see Theorem 1.62 and Theorem 11.6). Let M be a closed manifold of even dimension 2m. Suppose that M is hyperbolic, or more generally, that its sectional curvature satisfies −1 ≤ sec(M) < −(1 − \frac{1}{m})^{2} .Then

(−1)^{m} · χ(M) > 0.

Theorem 0.4 (see Theorem 11.14 and Theorem 11.15). Let M be a closed Kähler manifold of (real) dimension 2m. Suppose that M is homotopy equivalent to a closed Riemannian manifold with negative sectional curvature. Then

(−1)^{m} · χ(M) > 0.

Moreover, M is a projective algebraic variety and is Moishezon and Hodge.

Theorem 0.5 (see Theorem 7.25). Let 1 → H → G → K → 1 be an extension of infinite groups such that H is finitely generated and G is finitely presented. Then

(1) The deficiency of G satisfies def(G) ≤ 1;

(2) If M is a closed connected oriented 4-manifold with π_{1}(M) ≅ G, then we get for its signature sign(M) and its Euler characteristic χ(M)

| sign(M)| ≤ χ(M).

0.3 Some Open Problems Concerning L^{2}-Invariants 3

Theorem 0.6 (see Theorem 9.38). Let i: H → G be the inclusion of a normal finite subgroup H into an arbitrary group G. Then the maps coming from i and the conjugation action of G on H

ℤ ⊗_{ℤG} Wh(H) → Wh(G);
Wh(H)^{G} → Wh(G)

have finite kernel, where Wh denotes the Whitehead group.

Theorem 0.7 (see Theorem 9.66). Let G be a group and ℂG be its complex group ring. Let G_{0}(ℂG) be the Grothendieck group of finitely generated (not necessarily projective) ℂG-modules. Then

(1) If G is amenable, the class [ℂG] ∈ G_{0}(ℂG) is an element of infinite order;

(2) If G contains the free group ℤ ∗ ℤ of rank two, then [ℂG] = 0 in G_{0}(ℂG).

Theorem 0.8 (see Section 15.4). There are non-slice knots in 3-space whose Casson-Gordon invariants are all trivial.

Theorem 0.9 (see Section 7.5). There are finitely generated groups which are quasi-isometric but not measurably equivalent.

Theorem 0.10 (see Section 15.1). Let M^{4k+3} be a closed oriented smooth manifold for k ≥ 1 whose fundamental group has torsion. Then there are infinitely many smooth manifolds which are homotopy equivalent to M (and even simply and tangentially homotopy equivalent to M) but not homeomorphic to M.

0.3 Some Open Problems Concerning L^{2}-Invariants

The following conjectures will be treated in detail in Section 2.5 and Chapters 10, 11, 12, 13 and 14. They have created a lot of activity. This book contains proofs of these conjectures in special cases which rely on general methods and give some structural insight or consist of explicit computations. Recall that a free G-CW -complex X is the same as the total space of a G-covering X → G\X with a CW -complex G\X as base space, and that X is called finite or of finite type if the CW -complex G\X is finite or of finite type.

Conjecture 0.11 (Strong Atiyah Conjecture). Let X be a free G-CW - complex of finite type. Denote by \frac{1}{|FIN (G)|}ℤ the additive subgroup of ℝ generated by the set of rational numbers |H|^{−1}, where H runs through the finite subgroups of G. Then we get for the L^{2}-Betti numbers of X

b_{p}^{(2)}(X) ∈ \frac{1}{|FIN (G)|}ℤ.

In Subsection 10.1.4 we will explain that there are counterexamples to the strong Atiyah Conjecture 0.11 due to Grigorchuk and Żuk, but no counterexample is known to the author at the time of writing if one replaces \frac{1}{|FIN (G)|}ℤ by ℚ or if one assumes that there is an upper bound for the orders of finite subgroups of G. The author is not aware of a counterexample to the following conjectures at the time of writing.

Conjecture 0.12. (Positivity and rationality of Novikov-Shubin invariants). Let X be a free G-CW -complex of finite type. Then its Novikov-Shubin invariants α_{p}(X) are positive rational numbers unless they are ∞ or ∞^{+}.

Conjecture 0.13 (Singer Conjecture). Let M be an aspherical closed manifold. Then the L^{2}-Betti numbers of the universal covering M̃ satisfy

b_{p}^{(2)}(M̃) = 0 if 2p ≠ dim(M)

and (−1)^{m} · χ(M) ≥ 0 if dim(M) = 2m is even.

Let M be a closed connected Riemannian manifold with negative sectional curvature. Then

b_{p}^{(2)}(M̃) \{ \begin{matrix}= 0 & if 2p ≠ dim(M); \\ > 0 & if 2p = dim(M), \end{matrix}

and (−1)^{m} · χ(M) > 0 if dim(M) = 2m is even.

Conjecture 0.14 (L^{2}-torsion for aspherical manifolds). If M is an aspherical closed manifold of odd dimension 2m + 1, then the L^{2}-torsion of its universal covering satisfies

(−1)^{m} · ρ^{(2)}(M̃) ≥ 0.

If M is a closed connected Riemannian manifold of odd dimension 2m + 1 with negative sectional curvature, then

(−1)^{m} · ρ^{(2)}(M̃) > 0.

If M is an aspherical closed manifold whose fundamental group contains an amenable infinite normal subgroup, then

ρ^{(2)}(M̃) = 0.

Conjecture 0.15 (Zero-in-the-spectrum Conjecture). Let M̃ be the universal covering of an aspherical closed Riemannian manifold M. Then for some p ≥ 0 zero is in the spectrum of the minimal closure

(Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(M̃) → L^{2}Ω^{p}(M̃)

of the Laplacian acting on smooth p-forms on M̃.

0.4 L^{2}-Invariants and Heat Kernels 5

Conjecture 0.16 (Approximation Conjecture). Let G be a group. Let {G_{i} | i ∈ I} be an inverse system of normal subgroups of G directed by inclusion over the directed set I. Suppose that ∩_{i∈I}G_{i} = {1}. Let X be a free G-CW -complex of finite type. Then G_{i}\X is a free G/G_{i}-CW -complex of finite type and

b_{p}^{(2)}(X; N (G)) = lim_{i∈I} b_{p}^{(2)}(G_{i}\X; N (G/G_{i})).

Conjecture 0.17 (Simplicial volume and L^{2}-invariants). Let M be an aspherical closed orientable manifold of dimension ≥ 1. Suppose that its simplicial volume ||M|| vanishes. Then all the L^{2}-Betti numbers and the L^{2}- torsion of the universal covering M̃ vanish, i.e.

b_{p}^{(2)}(M̃) = 0 for p ≥ 0;
ρ^{(2)}(M̃) = 0.

0.4 L^{2}-Invariants and Heat Kernels

The p-th L^{2}-Betti number b_{p}^{(2)}(\overline{M}) of a G-covering p: \overline{M} → M of a closed Riemannian manifold M was first defined by Atiyah [9, page 71] in connection with his L^{2}-index theorem. By means of a Laplace transform, Atiyah’s original definition agrees with the one given by the non-negative real number

b_{p}^{(2)}(\overline{M}) = lim_{t→∞} ∫_{F} tr_{ℂ}(e^{−tΔ_{p}}(x, x)) dvol .  (0.18)

Here F is a fundamental domain for the G-action on \overline{M} and e^{−tΔ_{p}}(x, y) is the heat kernel on p-forms on \overline{M}. The p-th L^{2}-Betti number b_{p}^{(2)}(\overline{M}) measures the size of the kernel of the Laplacian acting on smooth p-forms on \overline{M}. If G is trivial, then b_{p}^{(2)}(\overline{M}) is the same as the ordinary Betti number b_{p}(M) which is the real dimension of the p-th singular cohomology with real coefficients of M. One important consequence of the L^{2}-index theorem is the Euler-Poincaré formula (0.1) (see Theorem 1.35 (2)).

The p-th Novikov-Shubin invariant α_{p}^{Δ}(\overline{M}) measures how fast the expression ∫_{F} tr_{ℂ}(e^{−tΔ_{p}}(x, x)) dvol approaches its limit b_{p}^{(2)}(\overline{M}) for t → ∞ (see (0.18)). The larger α_{p}^{Δ}(\overline{M}) is, the “thinner” is the spectrum of the p-th Laplacian on \overline{M} at zero.

Notice that the L^{2}-Betti numbers and the Novikov-Shubin invariants are invariants of the large time asymptotics of the heat kernel and hence invariants of the global geometry, in contrast to invariants of the small time asymptotics, such as indices of operators, which are of local nature. For instance the Novikov-Shubin invariant associated to the Laplacian acting on 0-forms of the universal covering of a closed Riemannian manifold M is determined by group theoretic properties of the fundamental group π_{1}(M) such as its growth rate or the question whether it is amenable (see Theorem 2.55 (5)).

In view of the definitions of the L^{2}-Betti numbers and Novikov-Shubin invariants, the strong Atiyah Conjecture 0.11 and the Conjecture 0.12 about the positivity and rationality of Novikov-Shubin invariants are very surprising. Some explanation for the strong Atiyah Conjecture 0.11 comes from connections with algebraic K-theory, whereas the only evidence for the Conjecture 0.12 about the positivity and rationality of Novikov-Shubin invariants is based on computations, and no conceptual reasons are known.

The third important L^{2}-invariant is the L^{2}-torsion ρ^{(2)}(\overline{M}) which was introduced by Carey-Mathai, Lott, Lück-Rothenberg, Mathai and Novikov-Shubin. It is only defined under a certain technical assumption, namely, that \overline{M} is of determinant class. This condition is conjecturally always satisfied and we will suppress it in this discussion. If all L^{2}-Betti numbers of \overline{M} vanish, the L^{2}-torsion ρ^{(2)}(\overline{M}) is independent of the Riemannian metric and depends only on the simple homotopy type. Actually, there is the conjecture that it depends only on the homotopy type (see Conjecture 3.94). Its analytic definition is complicated.

This analytic approach via the heat kernel is important in the following situations. One can compute the L^{2}-Betti numbers of the universal covering M̃ of a closed Riemannian manifold M if M is hyperbolic (see Theorem 1.62), or, more generally, satisfies certain pinching conditions (see Theorem 11.4, Theorem 11.5 and Theorem 11.6). There are explicit computations of the L^{2}-Betti numbers, the Novikov-Shubin invariants and the L^{2}-torsion of the universal covering of a closed manifold M if M is a locally symmetric space (see Theorem 5.12 and Section 5.4). The proof of the Proportionality Principle 3.183 relies on the analytic description. The proofs of these facts do not have combinatorial counterparts.

0.5 L^{2}-Invariants and Cellular Chain Complexes

One important feature of all these L^{2}-invariants is that they can also be defined for a G-covering p: \overline{X} → X of a finite CW -complex X in terms of the cellular ℤG-chain complex C_{∗}(\overline{X}). For L^{2}-Betti numbers and Novikov-Shubin invariants it suffices to require that X is of finite type. The associated L^{2}-chain complex C_{∗}^{(2)}(\overline{X}) is defined by l^{2}(G)⊗_{ℤG}C_{∗}(\overline{X}). Each chain module C_{∗}^{(2)}(\overline{X}) is a Hilbert space with isometric G-action of the special form l^{2}(G)^{n}, where l^{2}(G)^{n} is the n-fold sum of the Hilbert space l^{2}(G). Each differential c_{p}^{(2)} is a bounded G-equivariant operator. The p-th L^{2}-homology H_{p}^{(2)}(\overline{X}) is defined to be the quotient of the kernel of c_{p}^{(2)} by the closure of the image of c_{p+1}^{(2)}. Dividing out the closure of the image has the effect that H_{p}^{(2)}(\overline{X}) is again

0.6 L^{2}-Betti Numbers and Betti Numbers 7

a Hilbert space with isometric G-action. It actually comes with the structure of a finitely generated Hilbert N (G)-module, where N (G) denotes the von Neumann algebra of the group G. This additional structure allows to define the von Neumann dimension of H_{p}^{(2)}(\overline{X}). Dodziuk has shown that this nonnegative real number agrees with b_{p}^{(2)}(\overline{X}) as defined in (0.18) (see Theorem 1.59 and (1.60)). One can also read off the Novikov-Shubin invariants and the L^{2}-torsion from C_{∗}^{(2)}(\overline{X}) by results of Efremov (see Theorem 2.68) and Burghelea-Friedlander-Kappeler-McDonald (see Theorem 3.149). The p-th Novikov-Shubin invariant α_{p}(\overline{X}) measures the difference between the image of c_{p}^{(2)} and the closure of the image of c_{p}^{(2)}.

The point of this cellular description is that it is much easier to handle and calculate than the analytic counterpart. For instance one can show homotopy invariance of L^{2}-Betti numbers, Novikov-Shubin invariants and L^{2}-torsion and prove some very useful formulas like sum formulas, product formulas, fibration formulas and so on using the combinatorial approach (see Theorem 1.35, Theorem 2.55, Theorem 3.93, Theorem 3.96 and Theorem 3.100). The combinatorial approach allows to show for an aspherical closed manifold M that all L^{2}-Betti numbers and the L^{2}-torsion of its universal covering vanish provided M carries a non-trivial S^{1}-action (see Theorem 3.105). There exists a combinatorial proof that all L^{2}-Betti numbers of the universal covering of a mapping torus of a self map of a CW -complex of finite type vanish (see Theorem 1.39). No analytic proofs or no simpler analytic proofs of these results are known to the author. The combination of the analytic and combinatorial methods yields a computation of the L^{2}-invariants of the universal covering of a compact 3-manifold provided Thurston’s Geometrization Conjecture holds for the pieces appearing in the prime decomposition of M (see Theorem 4.1, Theorem 4.2 and Theorem 4.3).

For a kind of algorithmic computation of L^{2}-invariants based on the combinatorial approach we refer to Theorem 3.172.

The possibility to take both an analytic and a combinatorial point of view is one of the main reasons why L^{2}-invariants are so powerful.

0.6 L^{2}-Betti Numbers and Betti Numbers

Let X̃ → X be the universal covering of a connected CW -complex X of finite type. Then the L^{2}-Betti numbers b_{p}^{(2)}(X̃) of X̃ and the (classical) Betti numbers b_{p}(X) share some basic properties such as homotopy invariance, the Euler-Poincaré formula, Poincaré duality, Morse inequalities, Künneth formulas and so on, just replace in the corresponding statement for the classical Betti numbers b_{p}(X) by b_{p}^{(2)}(X̃) everywhere (see Theorem 1.35). There is also an L^{2}-Hodge de Rham Theorem 1.59 which is one important input in the proof of Theorem 0.3.

But there are also differences. One important extra feature of the L^{2}-Betti numbers is that they are multiplicative under finite coverings in the following sense. If p: Y → X is a finite d-sheeted covering, then b_{p}^{(2)}(Ỹ) = d · b_{p}^{(2)}(X̃) (see Theorem 1.35 (9)). This implies for instance b_{p}^{(2)}(S̃^{1}) = 0 for all p ≥ 0 since there is a d-sheeted covering S^{1} → S^{1} for d ≥ 2. The corresponding statement is not true for the Betti numbers. This is one reason why L^{2}-Betti numbers more often tend to be zero than the classical Betti numbers. Often this is the key phenomenon for applications. Another reason for it is the fact that b_{0}^{(2)}(X̃) is 0 if π_{1}(X) is infinite and is |π_{1}(X)|^{−1} if π_{1}(X) is finite (see Theorem 1.35 (8)), whereas b_{0}(X) is always 1.

If π_{1}(X) is finite, then b_{p}^{(2)}(X̃) = |π_{1}(X)|^{−1} · b_{p}(X̃). If π_{1}(X) is infinite, the only general relation between the L^{2}-Betti numbers of X̃ and the Betti numbers of X is the Euler-Poincaré formula (0.1). Given an integer l ≥ 1 and a sequence r_{1}, r_{2}, …, r_{l} of non-negative rational numbers, we construct in Example 1.38 a group G such that BG is of finite type and

b_{p}^{(2)}(G) := b_{p}^{(2)}(EG) = \{ \begin{matrix}r_{p} & for 1 ≤ p ≤ l; \\ 0 & for l + 1 ≤ p; \end{matrix}
b_{p}(G) := b_{p}(BG) = 0  for p ≥ 1.

On the other hand we can construct for any sequence n_{1}, n_{2}, … of nonnegative integers a CW -complex X of finite type such that b_{p}(X) = n_{p} and b_{p}^{(2)}(X̃) = 0 hold for p ≥ 1.

However, there is an asymptotic relation between the L^{2}-Betti numbers of X̃ and the Betti numbers of X. Recall that the Betti numbers are not multiplicative. One may try to force multiplicativity of the Betti numbers by stabilizing under finite coverings as follows. Suppose that π_{1}(X) possesses a nested sequence of normal subgroups of finite index

π_{1}(X) = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ G_{3} ⊃ …

with ∩_{i=0}^{∞}G_{i} = {1}. Then G_{i}\X̃ is a CW -complex of finite type and there is a [G : G_{i}]-sheeted covering G_{i}\X̃ → X. One may consider lim_{i→∞} \frac{b_{p}(G_{i}\X̃)}{[G:G_{i}]}. This expression is automatically multiplicative if the limit exists and is independent of the nested sequence. Actually it turns out that this is true and

lim_{i→∞} \frac{b_{p}(G_{i}\X̃)}{[G : G_{i}]} = b_{p}^{(2)}(X̃).

This result is a special case of the Approximation Conjecture 0.16 which will be investigated in Chapter 13.

0.7 L^{2}-Invariants and Ring-Theory

A more algebraic approach will be presented in Chapter 6. It will enable us to define L^{2}-Betti numbers for arbitrary G-spaces and in particular for groups

0.7 L^{2}-Invariants and Ring-Theory 9

without any restrictions on BG. This allows to apply standard techniques of algebraic topology and homological algebra directly to L^{2}-Betti numbers. The idea is to view the group von Neumann algebra N (G) just as a ring forgetting the functional analysis and the topology. The von Neumann algebra N (G) has zero-divisors and is not Noetherian unless G is finite. This makes N (G) complicated as a ring. But it has one very nice property, it is semihereditary, i.e. any finitely generated submodule of a projective module is itself projective (see Theorem 6.5 and Theorem 6.7 (1)). This justifies the slogan that N (G) behaves like the ring ℤ if one ignores the facts that ℤ has no zero-divisors and is Noetherian. The main input for the ring-theoretic approach is the construction of a dimension function for arbitrary modules over the group von Neumann algebra N (G) (Theorem 6.7). It is uniquely characterized by the condition that it satisfies Additivity, Continuity and Cofinality and extends the classical dimension function for finitely generated projective modules which is defined in terms of the von Neumann trace of idempotents in M_{n}(N (G)). One applies it to the N (G)-modules H_{p}(N (G) ⊗_{ℤG} C_{∗}^{sing}(X)) for a G-space X and gets an extension of the notion of L^{2}-Betti numbers to arbitrary G-spaces if one allows the value ∞. The second key result is that for amenable G the von Neumann algebra N (G) looks like a flat ℂG-module from the point of view of dimension theory (see Theorem 6.37).

In Chapter 8 we introduce the algebra U(G) of operators affiliated to the group von Neumann algebra. From an algebraic point of view U(G) can be described as the Ore localization of N (G) with respect to the multiplicative set of non-zero divisors. The main ring theoretic property of U(G) is that it is von Neumann regular (see Theorem 8.22 (3)) which is a stronger property than to be semihereditary. The dimension theory of N (G) extends to U(G) (see Theorem 8.29). The relation of U(G) to N (G) is analogous to the relation of ℚ to ℤ.

From the point of view of representation theory of finite groups the passage from ℂG to N (G) is the natural one for infinite groups. Namely, two finitely generated projective N (G)-modules P and Q are N (G)-isomorphic if and only if their center valued von Neumann dimensions dim_{N (G)}^{u}(P) and dim_{N (G)}^{u}(Q) agree (see Theorem 9.13). If G is finite, this reduces to the wellknown theorem that two complex finite-dimensional G-representations are isomorphic if and only if they have the same character.

This algebraic approach may be preferred by algebraists who do not have much background in (functional) analysis.

Linnell’s Theorem 10.19 says that the strong Atiyah Conjecture 0.11 is true for a class of groups C which contains all extensions of free groups with elementary amenable groups as quotients, provided that there is an upper bound on the orders of finite subgroups. Its proof is based on techniques from ring theory, in particular localization techniques, and from K-theory. The following square of inclusions of rings plays an important role as explained below

\begin{matrix}ℂG & ⟶ & N (G) \\ ↓ & & ↓ \\ D(G) & ⟶ & U(G) \end{matrix}  (0.19)

where D(G) denotes the division closure of ℂG in U(G).

0.8 L^{2}-Invariants and K-Theory

The strong Atiyah Conjecture 0.11 is related to K-theory in the following way. It is equivalent to the statement that for any finitely presented ℂGmodule M the generalized dimension dim_{N (G)}(N (G) ⊗_{ℂG} M) (see Theorem 6.5 and Theorem 6.7 (1)) of the N (G)-module N (G) ⊗_{ℂG} M takes values in \frac{1}{|FIN (G)|}ℤ (see Lemma 10.7). Notice that any non-negative real number occurs as dim_{N (G)}(P) for a finitely generated projective N (G)-module P, if G contains ℤ as subgroup (see Example 1.11, Theorem 6.24 (4) and Theorem 6.29 (2)). So the point is to understand the passage from ℂG to N (G), not only to investigate modules over N (G).

One may first consider the weaker statement that for any finitely generated projective ℂG-module M the generalized dimension dim_{N (G)}(N (G)⊗_{ℂG} M) takes values in \frac{1}{|FIN (G)|}ℤ. This is equivalent to the statement that

the composition K_{0}(ℂG) ⟶^{i} K_{0}(N (G)) ⟶^{dim_{N (G)}} ℝ must have its image in \frac{1}{|FIN (G)|}ℤ, where i is the change of rings map. This is certainly true for the composition

⨁_{\begin{matrix}H⊂G \\ |H|<∞ \end{matrix}} K_{0}(ℂH) ⟶^{a} K_{0}(ℂG) ⟶^{i} K_{0}(N (G)) ⟶^{dim_{N (G)}} ℝ

where a is the sum of the various change of rings maps. The Isomorphism Conjecture 9.40 for K_{0}(ℂG) implies that a is surjective and hence that the

image of K_{0}(ℂG) ⟶^{i} K_{0}(N (G)) ⟶^{dim_{N (G)}} ℝ is contained in \frac{1}{|FIN (G)|}ℤ.

The proof of Linnell’s Theorem 10.19 can be split into two parts, a ringtheoretic one and a K-theoretic one. Namely, one proves that any finitely presented ℂG-module becomes finitely generated projective over the ring D(G) (see (0.19)) and that the composition

⨁_{\begin{matrix}H⊂G \\ |H|<∞ \end{matrix}} K_{0}(ℂH) ⟶^{a} K_{0}(ℂG) ⟶^{j} K_{0}(D(G))

for j the change of rings map is surjective (see Section 10.2). Then the claim follows from (0.19) and the facts that the change of rings homomorphism

0.9 L^{2}-Invariants and Aspherical Manifolds 11

K_{0}(N (G)) → K_{0}(U(G)) is bijective (see Theorem 9.20 (1)) and that the dimension function dim_{N (G)} for N (G) extends to a dimension function dim_{U(G)} for U(G) satisfying dim_{U(G)}(U(G) ⊗_{N (G)} M) = dim_{N (G)}(M) for any N (G)- module M (see Theorem 8.29).

The extension of the dimension function to arbitrary modules has some applications to G-theory of ℂG as already mentioned in Theorem 0.7 (see Subsection 9.5.3). Computations of the middle K-theory and of the L-theory of von Neumann algebras and the associated algebras of affiliated operators are presented in Chapter 9. L^{2}-methods also lead to results about the Whitehead group Wh(G) (see Theorem 0.6) and some information about the Bass Conjecture (see Subsection 9.5.2). The question whether the L^{2}-torsion in the L^{2}-acyclic case is a homotopy invariant is equivalent to the question whether the map induced by the Fuglede-Kadison determinant Wh(G) → ℝ is trivial (see Conjecture 3.94). This question is related to the Approximation Conjecture 0.16 by the Determinant Conjecture 13.2 (see Lemma 13.6 and Theorem 13.3 (1)). The Approximation Conjecture 0.16 also plays a role in proving that the class of groups for which the strong Atiyah Conjecture 0.11 is true is closed under direct and inverse limits (see Theorem 10.20).

0.9 L^{2}-Invariants and Aspherical Manifolds

Let M be an aspherical closed manifold, for instance a closed Riemannian manifold with non-positive sectional curvature. Then the Singer Conjecture 0.13, Conjecture 0.14 about L^{2}-torsion for aspherical manifolds and the zeroin-the-spectrum Conjecture 0.15 put some restrictions on the L^{2}-invariants of its universal covering. There are special cases where these conjectures have been proved by computations. For instance if M is a compact 3-manifold (see Chapter 4), a locally symmetric space (see Corollary 5.16) or carries a Riemannian metric whose sectional curvature satisfies certain pinching conditions (see Theorem 11.4, Theorem 11.5 and Theorem 11.6). They also have been proved under additional assumptions like the existence of a non-trivial S^{1}-action (see Theorem 3.105), the existence of the structure of a Kähler hyperbolic manifold (see Theorem 11.14) or the existence of a normal infinite (elementary) amenable subgroup of π_{1}(X) (see Theorem 3.113 and Theorem 7.2). But it is still very mysterious why Poincaré duality together with asphericity may have such implications, or what kind of mechanism is responsible for these phenomenons. The status of Conjecture 0.17 about simplicial volume and L^{2}-invariants is similar. Conjectures 0.13, 0.14, 0.15 and 0.17 become false if one drops the condition that M is aspherical. Without this assumption it is easy to construct counterexamples to all but the zero-in-the-spectrum Conjecture 0.15. Counterexamples in the non-aspherical case to the zero-in-the-spectrum Conjecture 0.15 are presented by Farber-Weinberger [187] (see also [258]). We will deal with them in Section 12.3.

0.10 L^{2}-Invariants and Groups

L^{2}-Betti numbers b_{p}^{(2)}(G) (and also Novikov-Shubin invariants α_{p}(G)) can be defined for arbitrary (discrete) groups if one allows the value ∞. In Chapter 7 the L^{2}-Betti numbers of groups are investigated and in particular the question when they vanish is studied. The vanishing of all L^{2}-Betti numbers of G implies the vanishing of the L^{2}-Euler characteristic χ^{(2)}(G) of G. The notion of L^{2}-Euler characteristic agrees with the classical notion of Euler characteristic χ(BG) (or more generally the virtual Euler characteristic) if the latter is defined. Actually Theorem 0.2 is proved by showing that all L^{2}-Betti numbers of a group G vanish if G contains a normal infinite amenable subgroup. This example shows that it is important to extend the definition of L^{2}-Betti numbers from those groups for which BG is finite to arbitrary groups even if one may only be interested in groups with finite BG. Namely, if G has a finite model for BG, this does not mean that a normal subgroup H ⊂ G has a model of finite type for BH. The vanishing of the first L^{2}-Betti number b_{1}^{(2)}(G) has consequences for the deficiency of the group. The hard part of the proof of Theorem 0.5 is to show the vanishing of b_{1}^{(2)}(G), then the claim follows by elementary considerations.

We show in Theorem 7.10 that all L^{2}-Betti numbers of Thompson’s group F vanish. This is a necessary condition for F to be amenable. The group F cannot be elementary amenable and does not contain ℤ ∗ ℤ as subgroup but (at the time of writing) it is not known whether F is amenable or not.

In Section 7.4 a number ρ^{(2)}(f) ∈ ℝ is associated to an automorphism f: G → G of a group G provided that BG has a finite model. One also needs the technical assumption of det ≥ 1-class which is conjecturally always true and proved for a large class of groups and will be suppressed in the following discussion. This invariant has nice properties such as the trace property ρ^{(2)}(g ∘f) = ρ^{(2)}(f ∘g) and multiplicativity ρ^{(2)}(f^{n}) = n·ρ^{(2)}(f) and satisfies a sum formula ρ^{(2)}(f_{1} ∗_{f_{0}} f_{2}) = ρ^{(2)}(f_{1}) + ρ^{(2)}(f_{2}) − ρ^{(2)}(f_{0}) (see Theorem 7.27). If f = π_{1}(g) for an automorphism g: F → F of a compact orientable 2-dimensional manifold F different from S^{2}, D^{2} and T^{2}, then ρ^{(2)}(f) is, up to a constant, the sum of the volumes of the hyperbolic pieces appearing in the Jaco-Shalen-Johannson-Thurston decomposition of the mapping torus of g along tori into Seifert pieces and hyperbolic pieces (see Theorem 7.28). If F is closed and g is irreducible, then ρ^{(2)}(g) = 0 if and only if g is periodic, and ρ^{(2)}(g) ≠ 0 if and only if g is pseudo-Anosov.

In Section 7.5 the question is discussed whether or not the L^{2}-Betti numbers, Novikov-Shubin invariants and the L^{2}-torsion are quasi-isometry invariants or invariants of the measure equivalence class of a countable group G. Theorem 0.9 is one of the main applications of L^{2}-Betti numbers to measurable equivalence.

1. L^{2}-Betti Numbers

Introduction

In this chapter we introduce and study L^{2}-(co-)homology and L^{2}-Betti numbers for Hilbert chain complexes and for regular coverings of CW -complexes of finite type or of compact manifolds.

We follow the general strategy that a good invariant for a finite CW - complex X often has a refined version which is defined in terms of the universal covering X̃ and the action of the fundamental group π = π_{1}(X). For instance the Euler characteristic yields Wall’s finiteness obstruction, and the signature yields all kinds of surgery obstructions under this passage from X to the π-space X̃. The L^{2}-Betti numbers are derived from the Betti numbers in this way. Recall that the p-th Betti number b_{p}(X) is defined by dim_{ℂ}(H_{p}(X; ℂ)). In a naive approach one might try to define improved Betti numbers for a reasonable notion of dim_{ℂπ} by dim_{ℂπ}(H_{p}(X̃; ℂ)), for instance for dim_{ℂπ}(H_{p}(X̃; ℂ)) := dim_{ℂ}(ℂ ⊗_{ℂπ} H_{p}(X̃; ℂ)). The problem is that ℂπ is in general not Noetherian and hence this number is not necessarily finite. The basic idea is to pass to the group von Neumann algebra N (π) and use its standard trace to define the notion of von Neumann dimension which is better behaved than dim_{ℂπ}. Since one needs a Hilbert space setting, one completes the cellular ℂπ-chain complex of X̃ to the cellular L^{2}-chain complex C_{∗}^{(2)}(X̃) and defines the L^{2}-homology H_{p}^{(2)}(X̃) by the quotient of the kernel of the p-th differential by the closure of the image of the (p+1)-th differential. Then the p-th L^{2}-Betti number b_{p}^{(2)}(X̃) of the π-space X̃ is the von Neumann dimension of H_{p}^{(2)}(X̃).

We will introduce the necessary input about von Neumann algebras and Hilbert modules in Section 1.1. The precise definitions of C_{∗}^{(2)}(X̃), H_{p}^{(2)}(X̃) and b_{p}^{(2)}(X̃) and their main properties are given in Section 1.2. The standard properties of Betti numbers such as homotopy invariance, the expression of the Euler characteristic as alternating sum of Betti numbers, Poincaré duality, Künneth formula, and Morse inequalities carry over to L^{2}-Betti numbers. L^{2}- Betti numbers have additional interesting new properties which Betti numbers do not have. For instance they are multiplicative under finite coverings. In general the values of b_{p}(X) and b_{p}^{(2)}(X̃) are not related. A priori L^{2}-Betti numbers can take any non-negative real number as value, but possibly they are always rational. L^{2}-Betti numbers tend to vanish more often than Betti numbers. For instance b_{p}^{(2)}(X̃) vanishes for p ≥ 0 if X is a mapping torus of a selfmap of a CW -complex of finite type, if X is an aspherical closed manifold with a non-trivial S^{1}-action or if X is a hyperbolic closed manifold of dimension n and 2p ≠ n.

In Section 1.3 we extend the classical relation of Betti numbers and the dimension of the space of harmonic forms given by the Hodge-de Rham decomposition and the de Rham isomorphism to the L^{2}-setting. Actually the original definition of L^{2}-Betti numbers of Atiyah in the context of his L^{2}- Index Theorem is analytic, namely, by an expression in the Schwartz kernels of the projections appearing in the spectral family of the Laplacian on the universal covering of a closed Riemannian manifold. The fact, proved in Section 1.4, that the analytic and the cellular version of the L^{2}-Betti numbers agree is one of the important features. It is also interesting to notice that some of the properties of the L^{2}-Betti numbers are proved analytically or topologically respectively, and often there are no topological or analytic respectively proofs available.

If one wants to get a quick impression of L^{2}-Betti numbers, one may ignore their definition and pass directly to Subsections 1.2.3 and 1.3.2.

For the remainder of this chapter G is a discrete group with the exception of Subsection 1.2.1. Manifolds are always smooth.

1.1 Group von Neumann Algebras and Hilbert Modules

In this section we deal with group von Neumann algebras and Hilbert modules. We defer the treatment of the more general notion of a finite von Neumann algebra to Section 9.1 since for the next chapters we will only need the concept of a group von Neumann algebra. We introduce the notions of von Neumann trace and von Neumann dimension and use them to define and study L^{2}-Betti numbers for Hilbert chain complexes.

1.1.1 Group von Neumann Algebras

Let G be a discrete group. Denote by l^{2}(G) the Hilbert space of squaresummable formal sums over G with complex coefficients. This is the same as the Hilbert space completion of the complex group ring ℂG with respect to the pre-Hilbert space structure for which G is an orthonormal basis. An element in l^{2}(G) is represented by a formal sum ∑_{g∈G} λ_{g} · g for complex numbers λ_{g} such that ∑_{g∈G} |λ_{g}|^{2} < ∞. The scalar product is defined by

⟨∑_{g∈G} λ_{g} · g, ∑_{g∈G} μ_{g} · g⟩ := ∑_{g∈G} λ_{g} · \overline{μ_{g}}.

Notice that left multiplication with elements in G induces an isometric Gaction on l^{2}(G). Given a Hilbert space H, denote by B(H) the C^{∗}-algebra of bounded (linear) operators from H to itself, where the norm is the operator norm.

Definition 1.1 (Group von Neumann algebra). The group von Neumann algebra N (G) of the group G is defined as the algebra of G-equivariant bounded operators from l^{2}(G) to l^{2}(G)

N (G) := B(l^{2}(G))^{G}.

An important feature of the group von Neumann algebra is its standard trace.

Definition 1.2 (Von Neumann trace). The von Neumann trace on N (G) is defined by

tr_{N (G)} : N (G) → ℂ, f ↦ ⟨f(e), e⟩_{l^{2}(G)},

where e ∈ G ⊂ l^{2}(G) is the unit element.

Example 1.3. If G is finite, then ℂG = l^{2}(G) = N (G). The trace tr_{N (G)} assigns to ∑_{g∈G} λ_{g} · g the coefficient λ_{e} of the unit element e ∈ G.

The next example is a key example. We recommend the reader to consider and check all the results for group von Neumann algebras in the following special case to get the right intuition.

Example 1.4. If G is ℤ^{n}, there is the following model for the group von Neumann algebra N (ℤ^{n}). Let L^{2}(T^{n}) be the Hilbert space of equivalence classes of L^{2}-integrable complex-valued functions on the n-dimensional torus T^{n}, where two such functions are called equivalent if they differ only on a subset of measure zero. Define the Banach space L^{∞}(T^{n}) by equivalence classes of essentially bounded measurable functions f: T^{n} → ℂ ∐{∞}, where essentially bounded means that there is a constant C > 0 such that the set {x ∈ T^{n} | |f(x)| ≥ C} has measure zero. An element (k_{1}, …, k_{n}) in ℤ^{n} acts isometrically on L^{2}(T^{n}) by pointwise multiplication with the function T^{n} → ℂ which maps (z_{1}, z_{2}, …, z_{n}) to z_{1}^{k_{1}} · . . . · z_{n}^{k_{n}}. Fourier transform yields an isometric ℤ^{n}-equivariant isomorphism l^{2}(ℤ^{n}) ⟶^{≅} L^{2}(T^{n}). Hence N (ℤ^{n}) = B(L^{2}(T^{n}))^{ℤ^{n}}. We obtain an isomorphism

L^{∞}(T^{n}) ⟶^{≅} N (ℤ^{n})

by sending f ∈ L^{∞}(T^{n}) to the ℤ^{n}-equivariant operator M_{f}: L^{2}(T^{n}) → L^{2}(T^{n}), g ↦ g · f where g · f(x) is defined by g(x) · f(x). Under this identification the trace becomes

tr_{N (ℤ^{n})} : L^{∞}(T^{n}) → ℂ, f ↦ ∫_{T^{n}} fdμ.

1.1.2 Hilbert Modules

Definition 1.5 (Hilbert module). A Hilbert N (G)-module V is a Hilbert space V together with a linear isometric G-action such that there exists a Hilbert space H and an isometric linear G-embedding of V into the tensor product of Hilbert spaces H ⊗ l^{2}(G) with the obvious G-action. A map of Hilbert N (G)-modules f: V → W is a bounded G-equivariant operator. We call a Hilbert N (G)-module V finitely generated if there is a non-negative integer n and a surjective map ⨁_{i=1}^{n} l^{2}(G) → V of Hilbert N (G)-modules.

The embedding of V into H ⊗ l^{2}(G) is not part of the structure, only its existence is required. We emphasize that a map of Hilbert N (G)-modules is not required to be isometric. A Hilbert N (G)-module V is finitely generated if and only if there is an isometric linear G-embedding of V into ⨁_{i=1}^{n} l^{2}(G) for some non-negative integer n. If G is finite a finitely generated Hilbert N (G)-module is the same as a finite dimensional unitary representation of G.

Definition 1.6 (Weak exactness). We call a sequence of Hilbert N (G)- modules U ⟶^{i} V ⟶^{p} W weakly exact at V if the kernel ker(p) of p and the closure clos(im(i)) of the image im(i) of i agree. A map of Hilbert N (G)- modules f: V → W is a weak isomorphism if it is injective and has dense image.

The following assertions are equivalent for a map f: V → W of Hilbert N (G)-modules: (1) f is a weak isomorphism; (2) 0 → V ⟶^{f} W → 0 is weakly exact; (3) f and f^{∗} have dense image. An example of a weak isomorphism of N (G)-modules which is not an isomorphism is M_{z−1} : L^{2}(S^{1}) → L^{2}(S^{1}) given by multiplication with (z −1) ∈ L^{∞}(S^{1}) = N (ℤ), where (z −1) denotes the function S^{1} → ℂ, z ↦ (z − 1) (see Example 1.4). A map f: U → V of Hilbert N (G)-modules is an isomorphism if and only if it is bijective. This follows from the Inverse Mapping Theorem [434, Theorem III.11 on page 83] which ensures that f^{−1} is continuous for a bijective bounded operator of Hilbert spaces. If two Hilbert N (G)-modules V and W are weakly isomorphic, then they are even isometrically isomorphic by Polar Decomposition [434, Theorem VI.10 on page 197]. Namely, if f: V → W is a weak isomorphism, the unitary part of its polar decomposition is an isometric isomorphism of Hilbert N (G)-modules V → W. More generally, if 0 → U → V → W → 0 is a weakly exact sequence of Hilbert N (G)-modules, then there is an isometric isomorphism of Hilbert N (G)-modules U ⊕W → V. Notice that a short exact sequence of Hilbert N (G)-modules splits, but a weakly exact sequence does not split in general.

1.1.3 Dimension Theory

A bounded operator f: H → H of Hilbert spaces is called positive if ⟨f(v), v⟩ is a real number and ≥ 0 for all v ∈ H. This is equivalent to the condition that there is a bounded operator g: H → H with f = g^{∗}g since any positive operator f has a square root f^{1/2}, i.e. a positive operator with f = f^{1/2} ∘ f^{1/2} [434, Theorem VI.9 on page 196]. In particular a positive operator is selfadjoint. Given two bounded operators f_{1}, f_{2} : H → H, we write

f_{1} ≤ f_{2} ⇔ f_{2} − f_{1} is positive.  (1.7)

Definition 1.8. Let f: V → V be a positive endomorphism of a Hilbert N (G)-module. Choose a Hilbert space H, a Hilbert basis {b_{i} | i ∈ I} for H (sometimes also called complete orthonormal system, see [434, II.3]), a G-equivariant projection pr: H ⊗ l^{2}(G) → H ⊗ l^{2}(G) and an isometric Gisomorphism u: im(pr) ⟶^{≅} V. Let \overline{f}: H ⊗ l^{2}(G) → H ⊗ l^{2}(G) be the positive operator given by the composition

\overline{f}: H ⊗ l^{2}(G) ⟶^{pr} im(pr) ⟶^{u} V ⟶^{f} V ⟶^{u^{−1}} im(pr) ↪ H ⊗ l^{2}(G).

Define the von Neumann trace of f: V → V by

tr_{N (G)}(f) := ∑_{i∈I}⟨\overline{f}(b_{i} ⊗ e), b_{i} ⊗ e⟩  ∈ [0, ∞],

where e ∈ G ⊂ l^{2}(G) is the unit element.

This definition is independent of the choices of H, {b_{i} | i ∈ I}, pr and u. At least we give the proof for the independence of the Hilbert basis. If {c_{j} | j ∈ J} is a second Hilbert basis, it follows from the following calculation where all terms in the sums are non-negative and hence interchanging is allowed.

∑_{i∈I}⟨\overline{f}(b_{i} ⊗ e), b_{i} ⊗ e⟩ = ∑_{i∈I} ||\overline{f}^{1/2}(b_{i} ⊗ e)||^{2}
= ∑_{i∈I} ∑_{j∈J} ∑_{g∈G} |⟨\overline{f}^{1/2}(b_{i} ⊗ e), c_{j} ⊗ g⟩|^{2}

= ∑_{i∈I} ∑_{j∈J} ∑_{g∈G} |⟨b_{i} ⊗ g^{−1}, \overline{f}^{1/2}(c_{j} ⊗ e)⟩|^{2}

= ∑_{j∈J} ∑_{i∈I} ∑_{g∈G} |⟨\overline{f}^{1/2}(c_{j} ⊗ e), b_{i} ⊗ g⟩|^{2}

= ∑_{j∈J} ||\overline{f}^{1/2}(c_{j} ⊗ e)||^{2}
= ∑_{j∈J}⟨\overline{f}(c_{j} ⊗ e), c_{j} ⊗ e⟩.

A directed set I is a non-empty set with a partial ordering ≤ such that for two elements i_{0} and i_{1} there exists an element i with i_{0} ≤ i and i_{1} ≤ i. A net (x_{i})_{i∈I} in a topological space is a map from a directed set to the topological space. The net (x_{i})_{i∈I} converges to x if for any neighborhood U of x there is an index i(U) ∈ I such that x_{i} ∈ U for each i ∈ I with i(U) ≤ i. A net (f_{i})_{i∈I} in B(H) converges strongly to f ∈ B(H) if for any v ∈ H the net (f_{i}(v))_{i∈I} converges to f(v) in H. A net (f_{i})_{i∈I} in B(H) converges weakly to f ∈ B(H) if for any v, w ∈ H the net (⟨f_{i}(v), w⟩)_{i∈I} converges to ⟨f(v), w⟩ in ℂ. A net (f_{i})_{i∈I} in B(H) converges ultra-weakly if for any two sequences (x_{n})_{n≥0} and (y_{n})_{n≥0} of elements in H with ∑_{n≥0} ||x_{n}||^{2} < ∞ and ∑_{n≥0} ||y_{n}||^{2} < ∞ the net ∑_{n≥0} |⟨f_{i}(x_{n}), y_{n}⟩| converges to ∑_{n≥0} |⟨f(x_{n}), y_{n}⟩|. Obviously normconvergence implies both ultra-weak convergence and strong convergence, ultra-weak convergence implies weak convergence, strong convergence implies weak convergence. In general these implications cannot be reversed and there is no relation between ultra-weak convergence and strong convergence [144, I.3.1 and I.3.2].

Theorem 1.9 (Von Neumann trace). Let U, V and W be Hilbert N (G)- modules.

(1) If f, g: V → V are positive endomorphisms, then

f ≤ g ⇒ tr_{N (G)}(f) ≤ tr_{N (G)}(g);

(2) If (f_{i})_{i∈I} is a directed system of positive endomorphisms f_{i}: V → V, directed by the order relation ≤ for positive operators, and the net converges weakly to the endomorphism f: V → V, then f is positive and

tr_{N (G)}(f) = sup{tr_{N (G)}(f_{i}) | i ∈ I};

(3) We have for a positive endomorphism f: V → V

tr_{N (G)}(f) = 0 ⇔ f = 0;

(4) We have for positive endomorphisms f, g: V → V and a real number λ ≥ 0

tr_{N (G)}(f + λ · g) = tr_{N (G)}(f) + λ · tr_{N (G)}(g);

(5) If the following diagram of endomorphisms of Hilbert N (G)-modules commutes, has exact rows and positive operators as vertical maps

0 ⟶ U ⟶^{i} V ⟶^{p} W ⟶ 0
f↓  ^{g}↓  ^{h}↓
0 ⟶ U ⟶^{i} V ⟶^{p} W ⟶ 0

then

tr_{N (G)}(g) = tr_{N (G)}(f) + tr_{N (G)}(h);

(6) We get for a map f: V → W

tr_{N (G)}(f^{∗}f) = tr_{N (G)}(ff^{∗});

(7) Let U_{1} be a Hilbert N (G)-module and let U_{2} be a Hilbert N (H)-module. Let f_{i}: U_{i} → U_{i} be a positive endomorphism for i = 1, 2. Then the Hilbert tensor product f_{1} ⊗ f_{2} : U_{1} ⊗ U_{2} → U_{1} ⊗ U_{2} is a positive endomorphism of Hilbert N (G × H)-modules and

tr_{N (G×H)}(f_{1} ⊗ f_{2}) = tr_{N (G)}(f_{1}) · tr_{N (H)}(f_{2}),

where we use the convention that 0 · ∞ = 0 and r · ∞ = ∞ for r ∈ (0, ∞];

(8) Let H ⊂ G be a subgroup of finite index [G : H]. Let f: V → V be a positive endomorphism of a Hilbert N (G)-module. Let res(V) be the restriction of V to N (H) which is a Hilbert N (H)-module. If V is finitely generated, then res V is finitely generated. We have

tr_{N (H)}(res(f)) = [G : H] · tr_{N (G)}(f),

where we use the convention [G : H] · ∞ = ∞.

Proof. (1) follows directly from the definitions.

(2) Since each f_{i} is selfadjoint the same is true for f. We first show f_{i} ≤ f for all i ∈ I. Fix v ∈ V and i ∈ I. Given ϵ > 0 there is an index i(ϵ) with i ≤ i(ϵ) and ⟨f_{i(ϵ)}(v), v⟩−ϵ ≤ ⟨f(v), v⟩. Since f_{i} ≤ f_{i(ϵ)}, we get ⟨f_{i}(v), v⟩−ϵ ≤ ⟨f(v), v⟩ for all ϵ > 0. This implies ⟨f_{i}(v), v⟩ ≤ ⟨f(v), v⟩ and hence f_{i} ≤ f. We conclude that f is positive and from (1)

tr_{N (G)}(f) ≥ sup{tr_{N (G)}(f_{i}) | i ∈ I}.

It remains to prove the reverse inequality. In the notation of Definition 1.8 the net (\overline{f_{i}})_{i∈I} converges weakly to \overline{f}. Hence we can assume without loss of generality V = H ⊗ l^{2}(G).

Let {b_{λ} | λ ∈ Λ} be a Hilbert basis for H. Fix ϵ > 0. Choose a finite subset Λ(ϵ) ⊂ Λ such that

tr_{N (G)}(f) ≤ ϵ/2 + ∑_{λ∈Λ(ϵ)}⟨f(b_{λ} ⊗ e), b_{λ} ⊗ e⟩.

Since the net (f_{i})_{i∈I} converges weakly to f and Λ(ϵ) is finite and I is directed, there is an index i(ϵ) ∈ I such that for all λ ∈ Λ(ϵ)

⟨f(b_{λ} ⊗ e), b_{λ} ⊗ e⟩ ≤ \frac{ϵ}{2 · |Λ(ϵ)|} + ⟨f_{i(ϵ)}(b_{λ} ⊗ e), b_{λ} ⊗ e⟩

holds. We conclude

tr_{N (G)}(f) ≤ ϵ/2 + ∑_{λ∈Λ(ϵ)}⟨f(b_{λ} ⊗ e), b_{λ} ⊗ e⟩
≤ ϵ/2 + ϵ/2 + ∑_{λ∈Λ(ϵ)}⟨f_{i(ϵ)}(b_{λ} ⊗ e), b_{λ} ⊗ e⟩
≤ ϵ + tr_{N (G)}(f_{i(ϵ)})
≤ ϵ + sup{tr_{N (G)}(f_{i}) | i ∈ I}.

Since this holds for all ϵ > 0, the claim follows.

(3) Suppose tr_{N (G)}(f) = 0 for f: V → V. Then we get for \overline{f}: H ⊗ l^{2}(G) → H ⊗ l^{2}(G) (using the notation of Definition 1.8)

0 = ∑_{i∈I}⟨\overline{f}(b_{i} ⊗ e), b_{i} ⊗ e⟩ = ∑_{i∈I} ||\overline{f}^{1/2}(b_{i} ⊗ e)||^{2}.

This implies \overline{f}^{1/2}(b_{i} ⊗ e) = 0 for all i ∈ I. Since \overline{f}^{1/2} is G-equivariant, we get \overline{f}^{1/2}(b_{i} ⊗ g) = 0 for all i ∈ I and g ∈ G. This implies \overline{f}^{1/2} = 0 and hence \overline{f} = 0 and f = 0.

(4) follows directly from the definitions.

(5) By the Polar Decomposition and the Inverse Mapping Theorem we obtain a unitary isomorphism U ⊕W ⟶^{≅} V which induces together with the identity on U and W an isomorphism of the given exact sequence with the standard exact sequence 0 → U → U ⊕ W → W → 0. One easily checks for a positive operator

( \begin{matrix}f & u \\ 0 & g \end{matrix} ) : U ⊕ W → U ⊕ W

that

tr_{N (G)} ( \begin{matrix}f & u \\ 0 & g \end{matrix} ) = tr_{N (G)}(f) + tr_{N (G)}(g)

holds. Hence it remains to show for a positive G-operator h: V → V and a unitary G-operator v: V → W that tr_{N (G)}(vhv^{−1}) = tr_{N (G)}(h) holds. This follows from the proof that Definition 1.8 is independent of the various choices.

(6) Let f = u|f| be the Polar Decomposition of f. We conclude

tr_{N (G)}(ff^{∗}) = tr_{N (G)}(u|f|^{2}u^{−1}) = tr_{N (G)}(|f|^{2}) = tr_{N (G)}(f^{∗}f).

(7) Obviously l^{2}(G) ⊗ l^{2}(H) is isometrically G × H-isomorphic to l^{2}(G × H). Hence U_{1} ⊗ U_{2} is a Hilbert N (G × H)-module and f_{1} ⊗ f_{2} is a positive endomorphism. Because of assertion (5) it suffices to treat the case where U_{1} = H_{1} ⊗ l^{2}(G) and U_{2} = H_{2} ⊗ l^{2}(H) for appropriate Hilbert spaces H_{1} and H_{2}. Then the claim follows from the obvious formula

⟨f_{1} ⊗ f_{2}(u_{1} ⊗ u_{2}), v_{1} ⊗ v_{2}⟩_{U_{1}⊗U_{2}} = ⟨f_{1}(u_{1}), v_{1}⟩_{U_{1}} · ⟨f_{2}(u_{2}), v_{2}⟩_{U_{2}}

for u_{1}, v_{1} ∈ U_{1} and u_{2}, v_{2} ∈ U_{2}.

(8) It suffices to treat the case V = l^{2}(G). If s: G/H → G is a set-theoretic section of the projection G → G/H, we obtain an isometric H-isomorphism

⨁_{gH∈G/H} l^{2}(H) ⟶^{≅} res l^{2}(G), {u_{gH} | gH ∈ G/H} ↦ ∑_{gH∈G/H} u_{gH} · s(gH).

This finishes the proof of Theorem 1.9. ⊓ ⊔

Definition 1.10 (Von Neumann dimension). Define the von Neumann dimension of a Hilbert N (G)-module V

dim_{N (G)}(V) := tr_{N (G)}(id: V → V)  ∈ [0, ∞].

If G is finite, then dim_{N (G)}(V) is \frac{1}{|G|}-times the complex dimension of the underlying complex vector space V. The next example shows that dim_{N (G)}(V) can take any non-negative real number or ∞ as value.

Example 1.11. Let X ⊂ T^{n} be any measurable set and χ_{X} ∈ L^{∞}(T^{n}) be its characteristic function. Denote by M_{χ_{X}}: L^{2}(T^{n}) → L^{2}(T^{n}) the ℤ^{n}- equivariant unitary projection given by multiplication with χ_{X}. Its image V is a Hilbert N (ℤ^{n})-module with dim_{N (ℤ^{n})}(V) = vol(X) (see Example 1.4).

Theorem 1.12 (von Neumann dimension). (1) We have for a Hilbert N (G)-module V

V = 0 ⇔ dim_{N (G)}(V) = 0;

(2) If 0 → U → V → W → 0 is a weakly exact sequence of Hilbert N (G)- modules, then

dim_{N (G)}(U) + dim_{N (G)}(W) = dim_{N (G)}(V);

(3) Let {V_{i} | i ∈ I} be a directed system of Hilbert N (G)- submodules of V, directed by ⊂. Then

dim_{N (G)} (clos (⋃_{i∈I} V_{i})) = sup{dim_{N (G)}(V_{i}) | i ∈ I};

(4) Let V be an N (G)-Hilbert module with dim_{N (G)}(V) < ∞ and {V_{i} | i ∈ I} be a directed system of Hilbert N (G)-submodules, directed by ⊃. Then

dim_{N (G)} (⋂_{i∈I} V_{i}) = inf{dim_{N (G)}(V_{i}) | i ∈ I};

(5) Let U_{1} be a Hilbert N (G)-module and let U_{2} be a Hilbert N (H)-module. Then the Hilbert tensor product U_{1} ⊗ U_{2} is a Hilbert N (G × H)-module and

dim_{N (G×H)}(U_{1} ⊗ U_{2}) = dim_{N (G)}(U_{1}) · dim_{N (H)}(U_{2}),

where we use the convention that 0 · ∞ = 0 and r · ∞ = ∞ for r ∈ (0, ∞];

(6) Let H ⊂ G be a subgroup of finite index [G : H]. Let V be Hilbert N (G)- module. Let res(V) be the restriction of V to N (H) which is a Hilbert N (H)-module. If V is finitely generated, then res V is finitely generated. We have

dim_{N (H)}(res(V)) = [G : H] · dim_{N (G)}(V),

where we use the convention [G : H] · ∞ = ∞.

Proof. (1) This follows from Theorem 1.9 (3).

(2) We conclude from the weak exactness and the Polar Decomposition that U ⊕ W is isometrically G-isomorphic to V. Now apply Theorem 1.9 (5).

(3) Let pr: V → V and pr_{i}: V → V be the projections onto clos (⋃_{i∈I} V_{i}) and V_{i}. Next we show that the net (pr_{i})_{i∈I}, directed by the order relation for positive operators, converges strongly (and hence in particular weakly) to pr. Given v ∈ V and ϵ > 0, there is an index i(ϵ) ∈ I and v_{i(ϵ)} ∈ V_{i(ϵ)} with

| pr(v) − v_{i(ϵ)}| ≤ ϵ/2.

We conclude for i ≥ i(ϵ)

| pr(v) − pr_{i}(v)| ≤ | pr(v) − v_{i(ϵ)}| + |v_{i(ϵ)} − pr_{i}(v)|
= | pr(v) − v_{i(ϵ)}| + | pr_{i}(v_{i(ϵ)} − pr(v))|
≤ | pr(v) − v_{i(ϵ)}| + || pr_{i} || · | pr(v) − v_{i(ϵ)}|
≤ ϵ/2 + 1 · ϵ/2
= ϵ.

We conclude from Theorem 1.9 (2)

tr_{N (G)}(pr) = sup{tr_{N (G)}(pr_{i}) | i ∈ I}.

Since dim_{N (G)} (clos (⋃_{i∈I} V_{i})) = tr_{N (G)}(pr) and dim_{N (G)}(V_{i}) = tr_{N (G)}(pr_{i}), the claim follows.

(4) We obtain a directed system (V_{i}^{⊥})_{i∈I}, directed by ⊂. We have

clos (⋃_{i∈I} V_{i}^{⊥})^{⊥} = ⋂_{i∈I} V_{i};
dim_{N (G)}(V_{i}^{⊥}) = dim_{N (G)}(V) − dim_{N (G)}(V_{i});
dim_{N (G)} (clos (⋃_{i∈I} V_{i}^{⊥})^{⊥}) = dim_{N (G)}(V) − dim_{N (G)} (clos (⋃_{i∈I} V_{i}^{⊥})) .

Now the claim follows from (3).

(5) This follows from Theorem 1.9 (7).

(6) This follows from Theorem 1.9 (8). ⊓ ⊔ We conclude from Theorem 1.12 (1) and (2)

Lemma 1.13. Let f: U → V be a morphism of Hilbert N (G)-modules whose von Neumann dimension is finite. Then the following statements are equivalent:

(1) f is a weak isomorphism;

(2) The unitary part of the polar decomposition of f is an isomorphism;

(3) f^{∗} is a weak isomorphism;

(4) f is injective and dim_{N (G)}(U) = dim_{N (G)}(V);

(5) f has dense image and dim_{N (G)}(U) = dim_{N (G)}(V).

Example 1.14. In this example we present a Hilbert N (ℤ)-module U which is not finitely generated but has finite von Neumann dimension. In the sequel we use the identification of Example 1.4. For an interval I ⊂ [0, 1] let χ_{I} ∈ N (ℤ) = L^{∞}(S^{1}) be the characteristic function of the subset {exp(2πit) | t ∈ I}. Define two Hilbert N (ℤ)-modules by the orthogonal Hilbert sums

U = ⨁_{n=1}^{∞} im(χ_{[0,2^{−n}]});

V = ⨁_{n=1}^{∞} im(χ_{[1/(n+1),1/n]}),

where im(χ_{I}) is the direct summand in l^{2}(ℤ) = L^{2}(S^{1}) given by the projection χ_{I} ∈ N (ℤ). Theorem 1.12 and Example 1.11 imply

dim_{N (ℤ)}(U) = dim_{N (ℤ)}(V) = 1.

We want to show that U is not finitely generated. This is not obvious, for instance, V is isomorphic to the Hilbert N (ℤ)-module l^{2}(ℤ) and in particular finitely generated, although it is defined as an infinite Hilbert sum of nontrivial Hilbert N (ℤ)-modules.

To show that U is not finitely generated, we use the center valued dimension function which assigns to a finitely generated Hilbert N (ℤ)-module P an element

dim_{N (G)}^{c}(P) ∈ L^{∞}(S^{1}) = N (ℤ).

Its definition is the same as the definition of dim_{N (G)}(P) with the exception that one replaces the standard trace tr_{N (ℤ)} : N (ℤ) → ℂ by the identity id: N (ℤ) → N (ℤ). (More details will be given in Subsection 9.1.3 and 9.2.1.) Suppose that U is finitely generated. Then U is a direct summand in l^{2}(ℤ)^{k} for some k ≥ 0. Since ⨁_{n=1}^{k+1} im(χ_{[0,2^{−n}]}) is a direct summand in U and hence in l^{2}(ℤ)^{k}, we conclude

∑_{n=1}^{k+1} χ_{[0,2^{−n}]} = dim_{N (ℤ)}^{c} (⨁_{n=1}^{k+1} im(χ_{[0,2^{−n}]})) ≤ dim_{N (ℤ)}^{c}(l^{2}(ℤ)^{k}) = c_{k},

where c_{k}: S^{1} → ℂ is the constant function with value k. But this is a contradiction, since ∑_{n=1}^{k+1} χ_{[0,2^{−n}]} is equal to k + 1 on the subset {exp(2πit) | t ∈ [0, 2^{−k−1}]} whose measure is different from zero.

1.1.4 Hilbert Chain Complexes

Definition 1.15 (Hilbert chain complex). A Hilbert N (G)-chain complex C_{∗} is a sequence indexed by p ∈ ℤ of maps of Hilbert N (G)-modules

… ⟶^{c_{p+2}} C_{p+1} ⟶^{c_{p+1}} C_{p} ⟶^{c_{p}} C_{p−1} ⟶^{c_{p−1}} …

such that c_{p} ∘ c_{p+1} = 0 for all p ∈ ℤ. We call C_{∗} positive if C_{p} vanishes for p < 0. We say it is finitely generated if each C_{p} is a finitely generated Hilbert N (G)-module.

It is d-dimensional if C_{p} vanishes for |p| > d. It is finite dimensional if it is d-dimensional for some d ∈ ℕ. We call C_{∗} finite if C_{∗} is finitely generated and finite dimensional.

There are obvious notions of chain maps and chain homotopies.

Definition 1.16 (L^{2}-homology and L^{2}-Betti numbers). Define the (reduced) p-th L^{2}-homology and the p-th L^{2}-Betti number of a Hilbert N (G)- chain complex C_{∗} by

H_{p}^{(2)}(C_{∗}) := ker(c_{p})/ clos(im(c_{p+1}));
b_{p}^{(2)}(C_{∗}) := dim_{N (G)}(H_{p}^{(2)}(C_{∗})).

Notice that we divide by the closure of the image and not by the image of c_{p+1}. This has the effect that H_{p}^{(2)}(C_{∗}) inherits the structure of a Hilbert space and a G-action from C_{p}. This is indeed a Hilbert N (G)- structure on H_{p}^{(2)}(C_{∗}). Namely, H_{p}^{(2)}(C_{∗}) is isometrically G-isomorphic to ker(c_{p}) ∩ im(c_{p+1})^{⊥}. The Laplace operator is defined by

Δ_{p} = c_{p+1}c_{p+1}^{∗} + c_{p}^{∗}c_{p}: C_{p} → C_{p}.  (1.17)

We have the following “baby”-version of the L^{2}-Hodge-de Rham Theorem 1.57.

Lemma 1.18. Let C_{∗} be a Hilbert chain complex. Then we get the orthogonal decomposition of Hilbert N (G)-modules

C_{p} = ker(Δ_{p}) ⊕ clos(im(c_{p+1})) ⊕ clos(im(c_{p}^{∗}))

and the natural map

i: ker(c_{p}) ∩ ker(c_{p+1}^{∗}) = ker(Δ_{p}) → H_{p}^{(2)}(C_{∗})

is an isometric G-isomorphism.

Proof. We have the obvious orthogonal decomposition

C_{p} = (ker(c_{p}) ∩ im(c_{p+1})^{⊥}) ⊕ clos(im(c_{p+1})) ⊕ ker(c_{p})^{⊥}.

Since ker(c_{p})^{⊥} = clos(im(c_{p}^{∗})) and im(c_{p+1})^{⊥} = ker(c_{p+1}^{∗}), it remains to show

ker(Δ_{p}) = ker(c_{p}) ∩ ker(c_{p+1}^{∗}).

This follows from the calculation

⟨Δ_{p}(v), v⟩ = |c_{p}(v)|^{2} + |c_{p+1}^{∗}(v)|^{2}. ⊓ ⊔

One important tool in the theory of chain complexes is the long exact homology sequence. This does not go through directly, but in a weak sense under certain finiteness conditions. Let 0 → C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{p_{∗}} E_{∗} → 0 be an exact sequence of Hilbert N (G)-chain complexes. The maps i_{∗} and p_{∗} induce maps on the homology groups denoted by H_{n}^{(2)}(i_{∗}) and H_{n}^{(2)}(p_{∗}). Next we want to define a natural boundary operator

∂_{n}: H_{n}^{(2)}(E_{∗}) → H_{n−1}^{(2)}(C_{∗}).

As a linear G-equivariant map it is defined in the usual way. Namely, let z ∈ ker(e_{n}) be a representative of [z] ∈ H_{n}^{(2)}(E_{∗}). Choose y ∈ D_{n} and x ∈ C_{n−1} with the properties that p_{n}(y) = z and i_{n−1}(x) = d_{n}(y). We can find such y since p_{n} is surjective and such x since p_{n−1} ∘ d_{n}(y) = e_{n} ∘ p_{n}(y) = e_{n}(z) = 0. We have x ∈ ker(c_{n−1}) since i_{n−2} is injective and

i_{n−2} ∘ c_{n−1}(x) = d_{n−1} ∘ i_{n−1}(x) = d_{n−1} ∘ d_{n}(y) = 0.

We define

∂_{n}([z]) := [x].

It is easy to check that this is independent of the choice of the representative z for [z] and that this map is linear and G-equivariant. In order to prove that it is a bounded operator, we give a different description. Notice for the sequel that a bijective bounded operator of Hilbert spaces has a bounded inverse by the Inverse Mapping Theorem. Consider the following composition of maps of Hilbert N (G)-modules.

ker(e_{n}) → E_{n} ⟶^{(p_{n}|_{ker(pn)^{⊥}})^{−1}} ker(p_{n})^{⊥} ⟶^{d_{n}|_{ker(pn)^{⊥}}} D_{n−1}

One easily checks that its image lies in ker(p_{n−1}). Hence we can compose it with the isomorphism of Hilbert N (G)-modules i_{n−1}^{−1} : ker(p_{n−1}) → C_{n−1}. Again one easily checks that the image of this map lies in ker(c_{n−1}) so that we have defined a map of Hilbert N (G)-modules

∂′_{n}: ker(e_{n}) → ker(c_{n−1}).

Since it maps im(e_{n+1}) to im(c_{n}), it induces a map on L^{2}-homology which is the desired map ∂_{n}.

Although the boundary operator can always be defined and one gets a long homology sequence, this long homology sequence does not have to be weakly exact as the following example shows.

Example 1.19. Let G be the trivial group. Then a Hilbert N (G)-module is just a Hilbert space and a map of Hilbert N (G)-modules is just a bounded operator. Denote by H the Hilbert space {(a_{n} ∈ ℂ)_{n∈ℕ} | ∑ |a_{n}|^{2} < ∞} with the inner product ⟨(a_{n}), (b_{n})⟩ = ∑ a_{n} · \overline{b_{n}}. Define a linear bounded operator f: H → H by sending (a_{n}) to (1/n · a_{n}). Obviously f is injective and has dense image. But f is not surjective because u := (1/n)_{n} ∈ H cannot have a preimage. Let pr: H → span_{ℂ}(u)^{⊥} be the projection where span_{ℂ}(u) is the one-dimensional subspace generated by u. Then pr ∘f is injective with dense image. Hence f and pr ∘f are weak isomorphisms of Hilbert spaces but pr is not. View the following diagram as a short exact sequence of Hilbert chain complexes which are concentrated in dimensions 0 and 1

0 ⟶  0  ⟶ H ⟶^{id}  H  ⟶ 0
↓  ^{f}↓  ^{pr ∘f}↓
0 ⟶ span_{ℂ}(u) ⟶ H ⟶^{pr} span_{ℂ}(u)^{⊥} ⟶ 0

All the L^{2}-homology groups are trivial except the zero-th L^{2}-homology group of the left Hilbert chain complex 0 → span_{ℂ}(u). Hence there cannot be a long weakly exact homology sequence.

Notice that in the example above the dimension of the Hilbert space H is infinite.

Definition 1.20. A morphism f: U → V of Hilbert N (G)-modules is called Fredholm if for some λ > 0 we have dim_{N (G)}(im(E_{λ}^{f^{∗}f})) < ∞, where {E_{λ}^{f^{∗}f} | λ ∈ ℝ} is the (right continuous) spectral family associated to the positive operator f^{∗}f (see Definition 1.68). A Hilbert N (G)-chain complex C_{∗} is called Fredholm at p if the induced morphism \overline{c_{p}}: C_{p}/ clos(im(c_{p+1})) → C_{p−1} is Fredholm. We call C_{∗} Fredholm if it is Fredholm at p for all p ∈ ℤ.

For this chapter it suffices to know that a Hilbert N (G)-chain complex C_{∗}, for which C_{p} is finitely generated or satisfies dim_{N (G)}(C_{p}) < ∞, is automatically Fredholm at p. The next result is due to Cheeger and Gromov [105, Theorem 2.1].

Theorem 1.21 (Weakly exact long L^{2}-homology sequence). Let 0 → C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{p_{∗}} E_{∗} → 0 be an exact sequence of Hilbert N (G)-chain complexes which are Fredholm. Then the long homology sequence

… ⟶^{H_{n+1}^{(2)}(p_{∗})} H_{n+1}^{(2)}(E_{∗}) ⟶^{∂_{n+1}} H_{n}^{(2)}(C_{∗}) ⟶^{H_{n}^{(2)}(i_{∗})} H_{n}^{(2)}(D_{∗})

⟶^{H_{n}^{(2)}(p_{∗})} H_{n}^{(2)}(E_{∗}) ⟶^{∂_{n}} …

is weakly exact.

Proof. We only prove weak exactness at H_{n}^{(2)}(D_{∗}), the other cases are similar. Since p_{∗} ∘ i_{∗} = 0, we get clos(im(H_{n}^{(2)}(i_{∗}))) ⊂ ker(H_{n}^{(2)}(p_{∗})). It remains to prove equality. Let U be the orthogonal complement of clos(im(H_{n}^{(2)}(i_{∗}))) in ker(H_{n}^{(2)}(p_{∗})). Let V ⊂ ker(d_{n+1}^{∗}) ∩ ker(d_{n}) be the subspace corresponding to U under the isomorphism of Lemma 1.18

ker(d_{n+1}^{∗}) ∩ ker(d_{n}) → H_{n}^{(2)}(D_{∗}).

In view of Theorem 1.12 (1) it remains to show

dim_{N (G)}(V) = 0.

We have

p_{n}(V) ⊂ clos(im(e_{n+1}))

since H_{n}^{(2)}(p_{∗})(U) is zero. The operator e_{n+1} ∘ e_{n+1}^{∗}: E_{n} → E_{n} is positive. Let {E_{λ} | λ ∈ ℝ} be its (right continuous) spectral family (see Definition 1.68) of projections E_{λ}: E_{n} → E_{n}. Since E_{λ} commutes with e_{n+1} ∘ e_{n+1}^{∗}, it sends clos(im(e_{n+1} ∘ e_{n+1}^{∗})) to itself. We have

clos(im(e_{n+1}∘e_{n+1}^{∗})) = ker(e_{n+1}∘e_{n+1}^{∗})^{⊥} = ker(e_{n+1}^{∗})^{⊥} = clos(im(e_{n+1})),

where the second equality follows from ⟨e_{n+1} ∘ e_{n+1}^{∗}(v), v⟩ = |e_{n+1}^{∗}(v)|^{2}. This implies

E_{λ} ∘ p_{n}(V) ⊂ clos(im(e_{n+1})).

Next we show that E_{λ} ∘ p_{n} is injective on V for λ > 0. Consider an element v ∈ V with E_{λ} ∘ p_{n}(v) = 0. We get p_{n}(v) ∈ im(E_{λ})^{⊥}. Notice that e_{n+1} ∘ e_{n+1}^{∗} induces an invertible operator from im(E_{λ})^{⊥} to itself for λ > 0, the inverse is given by

∫_{λ}^{∞} \frac{1}{μ}dE_{μ}.

Hence we can find w ∈ E_{n+1} satisfying

p_{n}(v) = e_{n+1}(w).

Since p_{n+1} is surjective and ker(p_{n}) = im(i_{n}), we can find x ∈ D_{n+1} and y ∈ C_{n} satisfying

i_{n}(y) = d_{n+1}(x) + v.

This implies that c_{n}(y) = 0. Hence we get [y] ∈ H_{n}^{(2)}(C) whose image under H_{n}^{(2)}(i) is the element in U which corresponds to v ∈ V. Since U is orthogonal to im(H_{n}^{(2)}(i)), we conclude v = 0. We have shown that E_{λ} ∘ p_{n} is injective on V for λ > 0. Now we conclude from Theorem 1.12 and the facts that E_{λ} is right continuous in λ and E_{∗} is Fredholm by assumption

dim_{N (G)}(V) = dim_{N (G)} (clos(E_{λ} ∘ p_{n}(V)))
= dim_{N (G)} (clos(E_{λ} ∘ p_{n}(V)) ∩ clos(im(e_{n+1})))
= lim_{λ→0+} dim_{N (G)} (clos(E_{λ} ∘ p_{n}(V)) ∩ clos(im(e_{n+1})))
= dim_{N (G)} ( ⋂_{λ>0} clos(E_{λ} ∘ p_{n}(V)) ∩ clos(im(e_{n+1})))

≤ dim_{N (G)} ( ⋂_{λ>0} im(E_{λ}) ∩ clos(im(e_{n+1})))
= dim_{N (G)} (im(E_{0}) ∩ clos(im(e_{n+1})))
= dim_{N (G)} (ker(e_{n+1} ∘ e_{n+1}^{∗}) ∩ clos(im(e_{n+1})))
= dim_{N (G)} (ker(e_{n+1}^{∗}) ∩ clos(im(e_{n+1})))
= dim_{N (G)}(0)
= 0.

This finishes the proof of Theorem 1.21. ⊓ ⊔

Let G and H be groups. Let U be a Hilbert N (G)-module and V be a Hilbert N (H)-module. The Hilbert space tensor product U ⊗ V with the obvious G × H-operation is a Hilbert N (G × H)-module because l^{2}(G) ⊗ l^{2}(H) is isometrically G × H-isomorphic to l^{2}(G × H). Let C_{∗} be a Hilbert N (G)-chain complex and D_{∗} be a Hilbert N (H)-chain complex. Their tensor product is the Hilbert N (G×H)-chain complex C_{∗}⊗D_{∗} with n-th differential e_{n} which is given by

(C_{∗} ⊗ D_{∗})_{n} := ⨁_{i=0}^{n} C_{i} ⊗ D_{n−i};
e_{n}|_{C_{i}⊗D_{n−i}} := c_{i} ⊗ id +(−1)^{i} · id ⊗d_{n−i}.

Lemma 1.22. Let C_{∗} be a Hilbert N (G)-chain complex and let D_{∗} be a Hilbert N (H)-chain complex. Then there is a natural isomorphism of Hilbert N (G × H)-modules for n ≥ 0

K_{n}: ⨁_{p+q=n} H_{p}^{(2)}(C_{∗}) ⊗ H_{q}^{(2)}(D_{∗}) ⟶^{≅} H_{n}^{(2)}(C_{∗} ⊗ D_{∗}).

Proof. Given an element u ∈ ker(c_{p}) and v ∈ ker(d_{q}), define K_{n}([u] ⊗ [v]) by the class of u_{p} ⊗ v_{q} ∈ ker ((c_{∗} ⊗ d_{∗})_{n}). Notice that the Laplacian Δ_{n} on (C_{∗} ⊗ D_{∗})_{n} = ⨁_{p+q=n} C_{p} ⊗ D_{q} is given by ⨁_{p+q=n} Δ_{p}^{C} ⊗ id_{D_{q}} + id_{C_{p}} ⊗Δ_{q}^{D}, where Δ_{∗}^{C} and Δ_{∗}^{D} denote the Laplacians on C_{∗} and D_{∗}. To show that K_{n} is an isomorphism, it suffices to prove because of Lemma 1.18 that the kernel of Δ_{p}^{C} ⊗ id_{D_{q}} + id_{C_{p}} ⊗Δ_{q}^{D} is ker(Δ_{p}^{C}) ⊗ ker(Δ_{q}^{D}). Because of the orthogonal decompositions C_{p} = ker(Δ_{p}^{C}) ⊕ ker(Δ_{p}^{C})^{⊥} and D_{q} = ker(Δ_{q}^{D}) ⊕ ker(Δ_{q}^{D})^{⊥} it remains to show that the three induced positive endomorphisms

(Δ_{p}^{C})^{⊥} ⊗ id: ker(Δ_{p}^{C})^{⊥} ⊗ ker(Δ_{q}^{D}) → ker(Δ_{p}^{C})^{⊥} ⊗ ker(Δ_{q}^{D});
id ⊗(Δ_{q}^{D})^{⊥}: ker(Δ_{p}^{C}) ⊗ ker(Δ_{q}^{D})^{⊥} → ker(Δ_{p}^{C}) ⊗ ker(Δ_{q}^{D})^{⊥};
(Δ_{p}^{C})^{⊥} ⊗ id + id ⊗(Δ_{q}^{D})^{⊥}: ker(Δ_{p}^{C})^{⊥} ⊗ ker(Δ_{q}^{D})^{⊥} → ker(Δ_{p}^{C})^{⊥} ⊗ ker(Δ_{q}^{D})^{⊥}

are injective. But this follows from the facts that a selfadjoint endomorphism is injective if and only if it has dense image, the sum of two injective positive operators is injective again and from the injectivity of (Δ_{p}^{C})^{⊥} and (Δ_{q}^{D})^{⊥}. ⊓ ⊔

Of course everything in this section has an obvious analog for Hilbert N (G)-cochain complexes.

1.1.5 Induction for Group von Neumann Algebras

Next we investigate how group von Neumann algebras and Hilbert modules behave under induction.

Let i: H → G be an injective group homomorphism. Let M be a Hilbert N (H)-module. There is an obvious pre-Hilbert structure on ℂG ⊗_{ℂH} M for which G acts by isometries since ℂG ⊗_{ℂH} M as a complex vector space can be identified with ⨁_{G/H} M. Its Hilbert space completion is a Hilbert N (G)- module and denoted by i_{∗}M. A map of Hilbert N (H)-modules f: M → N induces a map of Hilbert N (G)-modules i_{∗}f: i_{∗}M → i_{∗}N. Thus we obtain a covariant functor from the category of Hilbert N (H)-modules to the category of Hilbert N (G)-modules. Obviously i_{∗}l^{2}(H) = l^{2}(G) and i_{∗} is compatible with direct sums. If M is finitely generated, i_{∗}M is finitely generated.

Definition 1.23 (Induction). Let i : H ⟶ G be an injective group homomorphism. The Hilbert N (G)-module i_{∗}M is called the induction with i of M. In particular i induces a ring homomorphism

i: N (H) → N (G).

Lemma 1.24. Let i: H → G be an injective group homomorphism. Then

(1) We have tr_{N (H)} = tr_{N (G)} ∘i: N (H) → ℂ;

(2) We get for a Hilbert N (H)-module M

dim_{N (G)}(i_{∗}M) = dim_{N (H)}(M);

(3) If 0 → U → V → W → 0 is a sequence of Hilbert N (H)-modules which is exact or weakly exact respectively, then the induced sequence 0 → i_{∗}U → i_{∗}V → i_{∗}W → 0 is a sequence of Hilbert N (G)-modules which is exact or weakly exact respectively;

(4) Let C_{∗} be a Hilbert N (H)-chain complex. Then i_{∗}C_{∗} is a Hilbert N (G)- chain complex and for all p ∈ ℤ

i_{∗}H_{p}^{(2)}(C_{∗}) = H_{p}^{(2)}(i_{∗}C_{∗});
dim_{N (H)}(H_{p}^{(2)}(C_{∗})) = dim_{N (G)}(H_{p}^{(2)}(i_{∗}C_{∗})).

Proof. (1) follows directly from the definitions.

(2) Let pr: V ⊗ l^{2}(H) → V ⊗ l^{2}(H) be an H-equivariant unitary projection describing M for some Hilbert space V. Then i_{∗}(pr) describes i_{∗}M.

(3) Since any exact sequence splits and i_{∗} is compatible with direct sums, the claim follows for exact sequences. In order to treat the weakly exact case it suffices to show that for a weak isomorphism f: U → V of Hilbert N (H)- modules the induced map i_{∗}f: i_{∗}U → i_{∗}V is a weak isomorphism. By the Polar Decomposition we can assume without loss of generality that U = V and f is positive. Since the kernel of a positive operator is the orthogonal complement of its image and i_{∗}f is positive it remains to show that the image of i_{∗}f is dense if the image of f is dense. This follows from the definition of i_{∗}f since ℂG is dense in l^{2}(G).

(4) This follows from (2) and (3). ⊓ ⊔

1.2 Cellular L^{2}-Betti Numbers

In this section we apply the material of Section 1.1 to regular coverings of CW -complexes of finite type (i.e. all skeleta are finite) and thus define and study cellular L^{2}-Betti numbers for them.

1.2 Cellular L^{2}-Betti Numbers 31

Since regular coverings of CW -complexes are special cases of G-CW - complexes and we will need the notion of a G-CW -complex in its full generality later, we collect some basic facts about G-CW -complexes. However, in order to read this and the next chapters one may skip subsection 1.2.1 and just keep in mind the following two facts: (1) A free finite G-CW -complex X or free G-CW -complex X of finite type respectively is the same as a Gspace X such that the projection X → G\X is a regular covering and G\X is a finite CW -complex or CW -complex of finite type respectively. (2) The cellular ℤG-chain complex of a free G-CW -complex is free and has a cellular ℤG-basis which is unique up to permutation and multiplication with trivial units ±g ∈ ℤG for g ∈ G.

1.2.1 Survey on G-CW -Complexes

In this subsection G can be any topological Hausdorff group. We recall some basic facts about G-CW -complexes. More informations and proofs can be found for instance in [495, Sections II.1 and II.2], [326, Sections 1 and 2], [365]. Throughout this book we will always work in the category of compactly generated spaces (see [482], [521, I.4]).

Definition 1.25 (G-CW -complex). A G-CW -complex X is a G-space together with a G-invariant filtration

∅ = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ … ⊂ X_{n} ⊂ … ⋃_{n≥0} X_{n} = X

such that X carries the colimit topology with respect to this filtration (i.e. a set C ⊂ X is closed if and only if C∩X_{n} is closed in X_{n} for all n ≥ 0) and X_{n} is obtained from X_{n−1} for each n ≥ 0 by attaching equivariant n-dimensional cells, i.e. there exists a G-pushout

∐_{i∈I_{n}} G/H_{i} × S^{n−1} \underrightarrow{∐_{i∈In} q_{i}} X_{n−1}
↓  ↓
∐_{i∈I_{n}} G/H_{i} × D^{n} \underrightarrow{∐_{i∈In} Q_{i}} X_{n}

Provided that G is discrete, a G-CW -complex X is the same as a CW - complex X with G-action such that for any open cell e ⊂ X and g ∈ G with ge ∩ e ≠ ∅ left multiplication with g induces the identity on e.

The space X_{n} is called the n-skeleton of X. Notice that only the filtration by skeleta belongs to the G-CW -structure but not the G-pushouts, only their existence is required. An equivariant open n-dimensional cell is a Gcomponent of X_{n} − X_{n−1}, i.e. the preimage of a path component of G\(X_{n} − X_{n−1}). The closure of an equivariant open n-dimensional cell is called an equivariant closed n-dimensional cell. If one has chosen the G-pushouts in Definition 1.25, then the equivariant closed n-dimensional cells are just the G-subspaces Q_{i}(G/H_{i} × D^{n}).

If X is a G-CW -complex, then G\X is a CW -complex. If G is discrete or if G is a Lie group and H ⊂ G is compact, then the H-fixed point set X^{H} inherits a WH-CW -complex structure. Here and in the sequel NH = {g ∈ G | gHg^{−1} = H} is the normalizer of H in G and WH denotes the Weyl group NH/H of H in G. A G-space X is called proper if for each pair of points x and y in X there are open neighborhoods V_{x} of x and W_{y} of y in X such that the closure of the subset {g ∈ G | gV_{x} ∩ W_{y} ≠ ∅} of G is compact. There are various slightly different definitions of proper G-space in the literature but for all these notions of properness a G-CW -complex X is proper if and only if all its isotropy groups are compact. In particular a free G-CW -complex is always proper. However, not every free G-space is proper. A G-space is called cocompact if G\X is compact. A G-CW -complex X is finite if X has only finitely many equivariant cells. A G-CW -complex is finite if and only if it is cocompact. A G-CW -complex X is of finite type if each n-skeleton is finite. It is called of dimension ≤ n if X = X_{n} and finite dimensional if it is of dimension ≤ n for some integer n. A free G-CW -complex X is the same as a regular covering X → Y of a CW -complex Y with G as group of deck transformations.

A G-map f: X → Y of G-CW -complexes is a G-homotopy equivalence if and only if for any subgroup H ⊂ G which occurs as isotropy group of X or Y the induced map f^{H}: X^{H} → Y^{H} is a weak homotopy equivalence, i.e. induces a bijection on π_{n} for all base points and n ≥ 0. A G-map of G-CW - complexes f: X → Y is cellular if f(X_{n}) ⊂ Y_{n} holds for all n ≥ 0. There is an equivariant version of the Cellular Approximation Theorem, namely, each G-map of G-CW -complexes is G-homotopic to a cellular one. If X is a G-CW - complex and Y is an H-CW -complex, then X × Y is a G × H-CW -complex. Notice that one of the advantages of working in the category of compactly generated spaces is that this is true without any further assumptions on the topology of X or Y such as being locally compact.

Now suppose that G is discrete. The cellular ℤG-chain complex C_{∗}(X) of a G-CW -complex has as n-th chain group the singular homology H_{n}(X_{n}, X_{n−1}) and its n-th differential is the boundary homomorphism associated to the triple (X_{n}, X_{n−1}, X_{n−2}). If one has chosen a G-pushout as in Definition 1.25, then there is a preferred ℤG-isomorphism

⨁_{i∈I_{n}} ℤ[G/H_{i}] ⟶^{≅} C_{n}(X).  (1.26)

If we choose a different G-pushout, we obtain another isomorphism, but the two differ only by the composition of an automorphism which permutes the summands appearing in the direct sum and an automorphism of the shape

⨁_{i∈I_{n}} ℤ[G/H_{i}] \underrightarrow{⨁_{i∈In} ϵ_{i}·r_{g_{i}}} ⨁_{i∈I_{n}} ℤ[G/H_{i}]  (1.27)

where g_{i} ∈ G, ϵ_{i} ∈ {±1} and ϵ_{i} · r_{g_{i}} sends gH_{i} to ϵ_{i} · gg_{i}H_{i}. In particular we obtain for a free G-CW -complex X a cellular ℤG-basis B_{n} for C_{n}(X) which is unique up to permutation and multiplication with trivial units in ℤG, i.e. elements of the shape ±g ∈ ℤG for g ∈ G.

If G is a Lie group and M is a (smooth) proper G-manifold, then an equivariant smooth triangulation induces a G-CW -structure on M. For equivariant smooth triangulations we refer to [271], [272], [273]. There are obvious notions of pairs of G-CW -complexes (X, A) and of a relative G-CW -complex (X, A). In the latter case replace X_{−1} = ∅ by X_{−1} = A for an arbitrary (compactly generated) G-space A.

There are some special G-CW -complexes. A family of subgroups F is a non-empty set of subgroups of G which is closed under conjugation and taking finite intersections. Examples are the trivial family T R consisting of the trivial subgroup and the family FIN of finite subgroups.

Definition 1.28. A classifying space E(G, F) for F is a G-CW -complex such that E(G, F)^{H} is contractible for H ∈ F and all isotropy groups of E(G, F) belong to F.

If F is the family T R or FIN, we abbreviate

EG = E(G, T R);
BG = G\EG;
\underline{E}G = E(G, FIN).

The existence of E(G, F) and proofs of their main property, namely, that for any G-CW -complex X whose isotropy groups belong to F there is up to G-homotopy precisely one G-map from X to E(G, F) and thus that two such classifying spaces are G-homotopy equivalent, are presented in [493],[495, I.6]. A functorial “bar-type” construction is given in [128, section 7]. Notice that G → EG → BG is a model for the universal G-principal bundle. If L is a Lie group with finitely many components and K ⊂ L is a maximal compact subgroup, then for any discrete subgroup G ⊂ L we get a model for E(G; FIN) by L/K (see [1, Corollary 4.14]). The Rips complex of a wordhyperbolic group G is also a model for E(G; FIN) (see [370]). The space E(G; FIN) for a discrete group G is also called the classifying space for proper G-spaces since for any numerably proper G-space X there is up to Ghomotopy precisely one G-map from X to E(G, FIN). For more information about E(G, FIN) we refer for instance to [27], [128, section 7], [495, section I.6], [336] and [342].

1.2.2 The Cellular L^{2}-Chain Complex

Definition 1.29. Let X be a free G-CW -complex of finite type. Define its cellular L^{2}-chain complex and its cellular L^{2}-cochain complex by

C_{∗}^{(2)}(X) := l^{2}(G) ⊗_{ℤG} C_{∗}(X);
C_{(2)}^{∗}(X) := hom_{ℤG}(C_{∗}(X), l^{2}(G)),

where C_{∗}(X) is the cellular ℤG-chain complex.

If we fix a cellular basis for C_{n}(X) we obtain explicit isomorphisms

C_{n}^{(2)}(X) ≅ C_{(2)}^{n}(X) ≅ ⨁_{i=1}^{k} l^{2}(G)

for some non-negative integer k and this induces the structure of a finitely generated Hilbert N (G)-module on C_{n}^{(2)}(X) and C_{(2)}^{n}(X). One easily checks that the choice of cellular basis does not affect this structure. Hence the cellular L^{2}-chain complex and the cellular L^{2}-cochain complex come with preferred structures of finitely generated Hilbert N (G)-chain complexes, provided that X is a free G-CW -complex of finite type.

Definition 1.30 (L^{2}-homology and L^{2}-Betti numbers). Let X be a free G-CW -complex of finite type. Define its (reduced) p-th L^{2}-homology, (reduced) p-th L^{2}-cohomology and p-th L^{2}-Betti number by the corresponding notions of the cellular L^{2}-(co)chain complexes (see Definition 1.16)

H_{p}^{(2)}(X; N (G)) := H_{p}^{(2)}(C_{∗}^{(2)}(X));
H_{(2)}^{p}(X; N (G)) := H_{(2)}^{p}(C_{(2)}^{∗}(X));
b_{p}^{(2)}(X; N (G)) := b_{p}^{(2)}(C_{∗}^{(2)}(X)).

If the group G and its action are clear from the context, we omit N (G) in the notation above. For instance for a connected CW -complex Y of finite type we denote by Ỹ its universal covering, G is understood to be π_{1}(Y) and we abbreviate b_{p}^{(2)}(Ỹ) = b_{p}^{(2)}(Ỹ; N (π_{1}(Y))).

If M is a cocompact free proper G-manifold, define its p-th L^{2}-Betti number by the corresponding notion for any equivariant smooth triangulation.

Remark 1.31. The Hilbert N (G)-modules H_{p}^{(2)}(X) and H_{(2)}^{p}(X) are isometrically G-isomorphic because of Lemma 1.18 since the p-th Laplace operator of the chain complex and the cochain complex are the same. In particular the cohomological and homological L^{2}-Betti numbers are the same.

Since we will prove in Theorem 1.35 (1) that b_{p}^{(2)}(X) depends only on the G-homotopy type of X and two equivariant smooth triangulations of a cocompact free proper G-manifold M are G-homotopy equivalent, the definition of the p-th L^{2}-Betti number for M makes sense.

All these notions extend in the obvious way to pairs or more generally to relative free G-CW -complexes of finite type (X, A).

Example 1.32. If G is finite and X is a free G-CW -complex of finite type, then b_{p}^{(2)}(X) is the classical p-th Betti number of X multiplied with \frac{1}{|G|}.

Let f_{∗}: C_{∗} → D_{∗} be a chain map of Hilbert chain complexes or chain complexes of modules over a ring. Define cyl_{∗}(f_{∗}) to be the chain complex with n-th differential

C_{n−1} ⊕ C_{n} ⊕ D_{n} \underrightarrow{( \begin{matrix}−c_{n−1} & 0 & 0 \\ − id & c_{n} & 0 \\ f_{n−1} & 0 & d_{n} \end{matrix} )} C_{n−2} ⊕ C_{n−1} ⊕ D_{n−1}.

Define cone_{∗}(f_{∗}) to be the quotient of cyl_{∗}(f_{∗}) by the obvious copy of C_{∗}. Hence the n-th differential of cone_{∗}(f_{∗}) is

C_{n−1} ⊕ D_{n} \underrightarrow{( \begin{matrix}−c_{n−1} & 0 \\ f_{n−1} & d_{n} \end{matrix} )} C_{n−2} ⊕ D_{n−1}.

Given a chain complex C_{∗}, define ΣC_{∗} to be the quotient of cone_{∗}(id_{C_{∗}}) by the obvious copy of D_{∗}, i.e. the chain complex with n-th differential

C_{n−1} ⟶^{−c_{n−1}} C_{n−2}.

Definition 1.33. We call cyl_{∗}(f_{∗}) the mapping cylinder, cone_{∗}(f_{∗}) the mapping cone of the chain map f_{∗} and ΣC_{∗} the suspension of the chain complex C_{∗}.

Notice that with these definitions the cellular chain complex of a mapping cylinder of a cellular map f: X → Y of G-CW -complexes is the mapping cylinder of the induced chain map C_{∗}(f): C_{∗}(X) → C_{∗}(Y). Analogous statements holds for the mapping cone and the suspension (relative to the cone and suspension points). For the next lemma see also [115, section 5], [152],

[169] and [331, Example 4.3].

Lemma 1.34. (1) Let C_{∗} be a free ℂ[ℤ^{n}]-chain complex of finite type with some basis. Denote by ℂ[ℤ^{n}]^{(0)} the quotient field of the integral domain ℂ[ℤ^{n}]. Then

b_{p}^{(2)} (l^{2}(ℤ^{n}) ⊗_{ℂ[ℤ^{n}]} C_{∗}) = dim_{ℂ[ℤ^{n}]^{(0)}} (ℂ[ℤ^{n}]^{(0)} ⊗_{ℂ[ℤ^{n}]} H_{p}(C_{∗})) .

(2) Let X be a free ℤ^{n}-CW -complex of finite type. Then

b_{p}^{(2)}(X) = dim_{ℂ[ℤ^{n}]^{(0)}} (ℂ[ℤ^{n}]^{(0)} ⊗_{ℤ[ℤ^{n}]} H_{p}(X)) .

Proof. (1) We abbreviate

C_{∗}^{(0)} := ℂ[ℤ^{n}]^{(0)} ⊗_{ℂ[ℤ^{n}]} C_{∗};
C_{∗}^{(2)} := l^{2}(ℤ^{n}) ⊗_{ℂ[ℤ^{n}]} C_{∗}.

We have to show

dim_{ℂ[ℤ^{n}]^{(0)}} (ℂ[ℤ^{n}]^{(0)} ⊗_{ℂ[ℤ^{n}]} H_{p}(C_{∗})) = dim_{N (ℤ^{n})} (H_{p}^{(2)}(C_{∗}^{(2)})) .

In the sequel we can assume without loss of generality that C_{∗} is finite dimensional. We first treat the case where C_{∗}^{(0)} has trivial homology. Then we can find a ℂ[ℤ^{n}]^{(0)}-chain contraction γ′_{∗} for C_{∗}^{(0)}. Choose an element u ∈ ℂ[ℤ^{n}] and ℂ[ℤ^{n}]-homomorphisms γ_{p}: C_{p} → C_{p+1} such that u ≠ 0 and l_{u} ∘ γ′_{p} = (γ_{p})^{(0)} holds for all p where l_{u} is multiplication with u. Then γ_{∗} is a chain homotopy of ℂ[ℤ^{n}]-chain maps l_{u} ≃ 0: C_{∗} → C_{∗}. This induces a chain homotopy of chain maps of finite Hilbert N (ℤ^{n})-chain complexes l_{u} ≃ 0: C_{∗}^{(2)} → C_{∗}^{(2)}. Hence multiplication with u induces the zero map on the L^{2}-homology of C_{∗}^{(2)}. This is only possible if the L^{2}-homology is trivial and hence all L^{2}-Betti numbers of C_{∗}^{(2)} vanish because l_{u}: l^{2}(ℤ^{n})^{k} → l^{2}(ℤ^{n})^{k} is injective for any non-negative integer k (use the Fourier transform in Example 1.4).

Next we treat the general case. Put

b_{p} = dim_{ℂ[ℤ^{n}]^{(0)}} (ℂ[ℤ^{n}]^{(0)} ⊗_{ℂ[ℤ^{n}]} H_{p}(C_{∗})) .

Notice that ℂ[ℤ^{n}]^{(0)} is flat over ℂ[ℤ^{n}]. There is a ℂ[ℤ^{n}]^{(0)}-isomorphism

⨁_{i=1}^{b_{p}} ℂ[ℤ^{n}]^{(0)} → H_{p}(C_{∗}^{(0)}) = ℂ[ℤ^{n}]^{(0)} ⊗_{ℂ[ℤ^{n}]} H_{p}(C_{∗}).

By composing it with a map given by multiplication with a suitable element in ℂ[ℤ^{n}] one can construct a ℂ[ℤ^{n}]-map

i_{p}: ⨁_{i=1}^{b_{p}} ℂ[ℤ^{n}] → H_{p}(C_{∗})

such that (i_{p})^{(0)} is a ℂ[ℤ^{n}]^{(0)}-isomorphism. Let D_{∗} be the finite free ℂ[ℤ^{n}]- chain complex whose p-th chain module is D_{p} = ⨁_{i=1}^{b_{p}} ℂ[ℤ^{n}] and whose differentials are all trivial. Choose a ℂ[ℤ^{n}]-chain map j_{∗}: D_{∗} → C_{∗} which induces on the p-th homology the map i_{p}. Let cone_{∗}(j_{∗}) be its mapping cone. There is a canonical exact sequence of ℂ[ℤ^{n}]-chain complexes 0 → C_{∗} → cone_{∗}(j_{∗}) → ΣD_{∗} → 0. Since it is split-exact in each dimension, it remains exact under the passage from C_{∗} to C_{∗}^{(0)} or C_{∗}^{(2)}. We conclude from the long exact homology sequence that cone_{∗}(j_{∗})^{(0)} is acyclic since the boundary operator can be identified with the map induced by j_{∗}^{(0)}. Hence the L^{2}-homology of cone_{∗}(j_{∗})^{(2)} is trivial by the first step. We conclude from the long weakly exact L^{2}-homology sequence (see Theorem 1.21) and additivity of the von Neumann dimension (see Theorem 1.12 (2))

b_{p}^{(2)}(C_{∗}^{(2)}) = b_{p}^{(2)}(D_{∗}^{(2)}) = b_{p} = dim_{ℂ[ℤ^{n}]^{(0)}} (ℂ[ℤ^{n}]^{(0)} ⊗_{ℂ[ℤ^{n}]} H_{p}(C_{∗})) .

(2) This follows from (1) applied to C_{∗}(X) ⊗_{ℤ} ℂ. ⊓ ⊔

Notice that Example 1.32 and Lemma 1.34 (2) imply that the L^{2}-Betti numbers of a free G-complex of finite type are always rational if G is finite and integral if G is ℤ^{n}. We will investigate the question, what values these can take for general G, in detail in Chapter 10.

1.2.3 Basic Properties of Cellular L^{2}-Betti Numbers

Theorem 1.35 (L^{2}-Betti numbers). (1) Homotopy invariance Let f: X → Y be a G-map of free G-CW -complexes of finite type. If the map induced on homology with complex coefficients H_{p}(f; ℂ): H_{p}(X; ℂ) → H_{p}(Y; ℂ) is bijective for p ≤ d − 1 and surjective for p = d, then

b_{p}^{(2)}(X) = b_{p}^{(2)}(Y)  for p < d;
b_{d}^{(2)}(X) ≥ b_{d}^{(2)}(Y).

In particular we get for all p ≥ 0 if f is a weak homotopy equivalence

b_{p}^{(2)}(X) = b_{p}^{(2)}(Y);

(2) Euler-Poincaré formula Let X be a free finite G-CW -complex. Let χ(G\X) be the Euler characteristic of the finite CW -complex G\X, i.e.

χ(G\X) := ∑_{p≥0}(−1)^{p} · β_{p}(G\X)  ∈ ℤ,

where β_{p}(G\X) is the number of p-cells of G\X. Then

χ(G\X) = ∑_{p≥0}(−1)^{p} · b_{p}^{(2)}(X);

(3) Poincaré duality Let M be a cocompact free proper G-manifold of dimension n which is orientable. Then

b_{p}^{(2)}(M) = b_{n−p}^{(2)}(M, ∂M);

(4) Künneth formula Let X be a free G-CW -complex of finite type and Y be a free H-CW - complex of finite type. Then X ×Y is a free G×H-CW -complex of finite type and we get for all n ≥ 0

b_{n}^{(2)}(X × Y) = ∑_{p+q=n} b_{p}^{(2)}(X) · b_{q}^{(2)}(Y);

(5) Wedges Let X_{1}, X_{2}, …, X_{r} be connected (pointed) CW -complexes of finite type and X = ⋁_{i=1}^{r} X_{i} be their wedge. Then

b_{1}^{(2)}(X̃) − b_{0}^{(2)}(X̃) = r − 1 + ∑_{j=1}^{r} (b_{1}^{(2)}(X̃_{j}) − b_{0}^{(2)}(X̃_{j}));

b_{p}^{(2)}(X̃) = ∑_{j=1}^{r} b_{p}^{(2)}(X̃_{j})  for 2 ≤ p;

(6) Connected sums Let M_{1}, M_{2}, …, M_{r} be compact connected m-dimensional manifolds with m ≥ 3. Let M be their connected sum M_{1}# … #M_{r}. Then

b_{1}^{(2)}(M̃) − b_{0}^{(2)}(M̃) = r − 1 + ∑_{j=1}^{r} (b_{1}^{(2)}(M̃_{j}) − b_{0}^{(2)}(M̃_{j}));

b_{p}^{(2)}(M̃) = ∑_{j=1}^{r} b_{p}^{(2)}(M̃_{j})  for 2 ≤ p ≤ m − 2;

(7) Morse inequalities Let X be a free G-CW -complex of finite type. Let β_{p}(G\X) be the number of p-cells in G\X. Then we get for n ≥ 0

∑_{p=0}^{n}(−1)^{n−p} · b_{p}^{(2)}(X) ≤ ∑_{p=0}^{n}(−1)^{n−p} · β_{p}(G\X);

(8) Zero-th L^{2}-Betti number Let X be a connected free G-CW -complex of finite type. Then

b_{0}^{(2)}(X) = \frac{1}{|G|},

where \frac{1}{|G|} is to be understood to be zero if the order |G| of G is infinite;

(9) Restriction Let X be a free G-CW -complex of finite type and let H ⊂ G be a subgroup of finite index [G : H]. Let res_{G}^{H} X be the H-space obtained from X by restricting the G-action to an H-action. This is a free H-CW -complex of finite type. Then we get for p ≥ 0

[G : H] · b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(res_{G}^{H} X; N (H));

(10) Induction Let H be a subgroup of G and let X be a free H-CW -complex of finite type. Then G ×_{H} X is a G-CW -complex of finite type and

b_{p}^{(2)}(G ×_{H} X; N (G)) = b_{p}^{(2)}(X; N (H)).

Proof. (1) We can assume that f is cellular. Let C_{∗}(f; ℂ) be the map induced on the cellular ℂG-chain complexes. The homology H_{p}(cone_{∗}(C_{∗}(f; ℂ))) is trivial for p ≤ d. Since cone_{∗}(C_{∗}(f; ℂ)) is a finitely generated free ℂG-chain complex, it is ℂG-chain homotopy equivalent to a finitely generated free ℂGchain complex which is trivial in dimensions ≤ d. Hence cone_{∗}(C_{∗}^{(2)}(f) : C_{∗}^{(2)}(X) → C_{∗}^{(2)}(Y)) is chain homotopy equivalent to a Hilbert N (G)-chain complex which is trivial in dimensions ≤ d and in particular its L^{2}-homology is trivial in dimensions ≤ d. Now the claim follows from the long weakly exact L^{2}-homology sequence associated to C_{∗}^{(2)}(f) (see Theorem 1.21) and additivity of the von Neumann dimension (see Theorem 1.12 (2)).

(2) Analogous to the classical situation where χ(X) is expressed in terms of (ordinary) Betti numbers the claim follows from the fact that the von Neumann dimension is additive (see Theorem 1.12 (2)).

(3) There is a subgroup G_{0} ⊂ G of finite index which acts orientation preserving on M. Since b_{p}^{(2)}(res_{G}^{G_{0}}(M); N (G_{0})) = [G : G_{0}] · b_{p}^{(2)}(M; N (G)) follows from assertion (9) and similarly for (M, ∂M), we can assume without loss of generality that G = G_{0}, i.e. G\M is orientable. Then the Poincaré ℤG-chain homotopy equivalence [510, Theorem 2.1 on page 23]

∩[G\M]: C^{n−∗}(M, ∂M) → C_{∗}(M)

induces a homotopy equivalence of finitely generated Hilbert N (G)-chain complexes C_{(2)}^{n−∗}(M, ∂M) → C_{∗}^{(2)}(M). Now the assertion follows because

dim(H_{p}(C_{(2)}^{n−∗}(M, ∂M))) = b_{n−p}^{(2)}(M; ∂M) holds by Lemma 1.18.

(4) The obvious isomorphism of ℤ[G×H]-chain complexes C_{∗}(X)⊗_{ℤ}C_{∗}(Y) → C_{∗}(X × Y) induces an isomorphism of Hilbert N (G × H)-chain complexes C_{∗}^{(2)}(X)⊗C_{∗}^{(2)}(Y) → C_{∗}^{(2)}(X ×Y). Now apply Theorem 1.12 (5) and Lemma 1.22. (see also [530, Corollary 2.36 on page 181]).

(5) We may assume without loss of generality that r = 2. We obtain an exact sequence of Hilbert N (π)-chain complexes for π = π_{1}(X_{1} ∨ X_{2}) = π_{1}(X_{1}) ∗ π_{1}(X_{2}) and i_{k}: π_{1}(X_{k}) → π and i_{0} : {1} → π the obvious inclusions

0 → (i_{0})_{∗}C_{∗}^{(2)}(\{∗\}) → (i_{1})_{∗}C_{∗}^{(2)}(X̃_{1}) ⊕ (i_{2})_{∗}C_{∗}^{(2)}(X̃_{2}) → C_{∗}^{(2)}({ \atop X_{1} ∨ X_{2}}) → 0.

Notice that (i_{0})_{∗}C_{∗}^{(2)}({∗}) is concentrated in dimension zero. Now the claim follows from Theorem 1.12 (2), Theorem 1.21 and Lemma 1.24 (2).

(6) We may assume without loss of generality that r = 2. The connected sum M_{1}#M_{2} is obtained by glueing M_{1}\ int(D^{m}) and M_{2}\ int(D^{m}) together along ∂D^{m}. Since ∂D^{m} → D^{m} is (m − 1)-connected the inclusion

M_{1}\ int(D^{m}) ∪_{∂D^{m}} M_{2}\ int(D^{m}) → M_{1} ∪_{D^{m}} M_{2}

is d-connected for d = m − 1, i.e. induces an isomorphism on π_{n} for n ≤ d − 1 and an epimorphism on π_{d} for all base points. Obviously M_{1} ∪_{D^{m}} M_{2} is homotopy equivalent to the wedge M_{1} ∨ M_{2}. Because of assertion (1) it suffices to prove the claims for M_{1} ∨ M_{2} which has already been done.

(7) analogous to (2).

(8) If G is finite, the claim follows from Example 1.32. It remains to show b_{0}^{(2)}(X) = 0 if G is infinite. Since X → G\X is a covering with a pathconnected total space, there is a group epimorphism π_{1}(G\X) → G. Since G\X is of finite type, G is finitely generated. Let S be a finite set of generators. Let D_{∗} be the ℤG-chain complex which is concentrated in dimension 0 and 1 and has as first differential

⨁_{s∈S} ℤG \underrightarrow{⨁_{s∈S} r_{s−1}} ℤG,

where r_{s−1} is right multiplication with s − 1. There is a ℤG-chain map f_{∗}: D_{∗} → C_{∗}(X) which is 0-connected. Hence it suffices because of the argument in the proof of (1) to show b_{0}^{(2)}(l^{2}(G) ⊗_{ℤG} D_{∗}) = 0. We have already seen that this is the same as the zero-th L^{2}-Betti number of the associated Hilbert N (G)-cochain complex hom_{ℤG}(D_{∗}, l^{2}(G)) which is the dimension of l^{2}(G)^{G}. Since l^{2}(G)^{G} = 0 for infinite G, the claim follows.

(9) This follows from Lemma 1.9 (8).

(10) This follows from Lemma 1.24 (4) and i_{∗}(C_{∗}^{(2)}(X)) = C_{∗}^{(2)}(G ×_{H} X) for the inclusion i : H → G. ⊓ ⊔

Example 1.36. We give the values of the L^{2}-Betti numbers for the universal coverings of all compact connected 1- and 2-manifolds.

In dimension 1 there are only S^{1} and the unit interval I. We get from Theorem 1.35 (9) or Lemma 1.34 (2) that b_{p}^{(2)}(S̃^{1}) = 0 for all p ≥ 0. As I is contractible, we have b_{0}^{(2)}(Ĩ) = b_{0}(I) = 1 and b_{p}^{(2)}(Ĩ) = 0 for p ≥ 1.

A manifold is called closed if it is compact and has no boundary. Let F_{g}^{d} be the orientable closed surface of genus g with d embedded 2-disks removed. (As any nonorientable compact surface is finitely-covered by an orientable surface, Theorem 1.35 (9) shows that it is enough to handle the orientable case.) From Theorem 1.35 (2), (3) and (8) and the fact that a compact surface with boundary is homotopy equivalent to a bouquet of circles, we conclude

b_{0}^{(2)}(F̃_{g}^{d}) = \{ \begin{matrix}1 & if g = 0, d = 0, 1 \\ 0 & otherwise \end{matrix} ;
b_{1}^{(2)}(F̃_{g}^{d}) = \{ \begin{matrix}0 & if g = 0, d = 0, 1 \\ d + 2 · (g − 1) & otherwise \end{matrix} ;
b_{2}^{(2)}(F̃_{g}^{d}) = \{ \begin{matrix}1 & if g = 0, d = 0 \\ 0 & otherwise \end{matrix} .

Of course b_{p}^{(2)}(F̃_{g}^{d}) = 0 for p ≥ 3.

Example 1.37. Let X → Y be a finite covering with d sheets of connected CW -complexes of finite type. Then Theorem 1.35 (9) implies

b_{p}^{(2)}(X̃) = d · b_{p}^{(2)}(Ỹ).

In particular we get for a connected CW -complex X of finite type for which there is a selfcovering X → X with d sheets for some integer d ≥ 2 that b_{p}^{(2)}(X̃) = 0 for all p ≥ 0.

Example 1.38. The following examples show that in general there are hardly any relations between the ordinary Betti numbers b_{p}(X) and the L^{2}- Betti numbers b_{p}^{(2)}(X̃) for a connected CW -complex X of finite type.

Given a group G such that BG is of finite type, define its p-th L^{2}-Betti number and its p-th Betti number by

b_{p}^{(2)}(G) := b_{p}^{(2)}(EG; N (G));
b_{p}(G) := b_{p}(BG).

We get from Theorem 1.35 (4), (5) and (9) for r ≥ 2 and non-trivial groups G_{1}, G_{2}, …, G_{r} whose classifying spaces BG_{i} are of finite type

b_{1}^{(2)}(∗_{i=1}^{r}G_{i}) = r − 1 + ∑_{i=1}^{r} (b_{1}^{(2)}(G_{i}) − \frac{1}{|G_{i}|});
b_{0}^{(2)}(∗_{i=1}^{r}G_{i}) = 0;
b_{p}^{(2)}(∗_{i=1}^{r}G_{i}) = ∑_{i=1}^{r} b_{p}^{(2)}(G_{i})  for p ≥ 2;

b_{p}(∗_{i=1}^{r}G_{i}) = ∑_{i=1}^{r} b_{p}(G_{i})  for p ≥ 1;

b_{p}^{(2)}(G_{1} × G_{2}) = ∑_{i=0}^{p} b_{i}^{(2)}(G_{1}) · b_{p−i}^{(2)}(G_{2});

b_{p}(G_{1} × G_{2}) = ∑_{i=0}^{p} b_{i}(G_{1}) · b_{p−i}(G_{2});

b_{0}^{(2)}(ℤ/n) = \frac{1}{n};
b_{p}^{(2)}(ℤ/n) = 0  for p ≥ 1;
b_{p}(ℤ/n) = 0  for p ≥ 1.

From this one easily verifies for any integers m ≥ 0, n ≥ 1 and i ≥ 1 that for the group

G_{i}(m, n) = ℤ/n × (∗_{k=1}^{2m+2}ℤ/2) × (∏_{j=1}^{i−1} ∗_{l=1}^{4}ℤ/2)

its classifying space BG_{i}(m, n) is of finite type and

b_{i}^{(2)}(G_{i}(m, n)) = \frac{m}{n};
b_{p}^{(2)}(G_{i}(m, n)) = 0  for p ≠ i;
b_{p}(G_{i}(m, n)) = 0  for p ≥ 1.

Given an integer l ≥ 1 and a sequence r_{1}, r_{2}, …, r_{l} of non-negative rational numbers, we can construct a group G such that BG is of finite type and

b_{p}^{(2)}(G) = \{ \begin{matrix}r_{p} & for 1 ≤ p ≤ l; \\ 0 & for l + 1 ≤ p; \end{matrix}
b_{p}(G) = 0  for p ≥ 1

holds as follows. For l = 1 we have already done this, so assume l ≥ 2 in the sequel. Choose integers n ≥ 1 and k ≥ l with r_{1} = \frac{k−2}{n}. Fix for i = 2, 3 …, k integers m_{i} ≥ 0 and n_{i} ≥ 1 such that \frac{m_{i}}{n·n_{i}} = r_{i} holds for 1 ≤ i ≤ l and m_{i} = 0 holds for i > l. Put

G = ℤ/n × ∗_{i=2}^{k}G_{i}(m_{i}, n_{i}).

On the other hand we can construct for any sequence n_{1}, n_{2}, … of nonnegative integers a CW -complex X of finite type such that b_{p}(X) = n_{p} and b_{p}^{(2)}(X̃) = 0 holds for p ≥ 1, namely take

X = B(ℤ/2 ∗ ℤ/2) × ⋁_{p=1}^{∞} (⋁_{i=1}^{n_{p}} S^{p}) .

Let f: X → X be a selfmap. Its mapping torus T_{f} is obtained from the cylinder X × [0, 1] by glueing the bottom of the cylinder X × [0, 1] to the top by the identification (x, 1) = (f(x), 0). There is a canonical map p: T_{f} → S^{1} which sends (x, t) to exp(2πit). It induces a canonical epimorphism π_{1}(T_{f}) → ℤ = π_{1}(S^{1}) if X is path-connected.

The next theorem will be generalized in Theorem 6.63 and will play a key role in the proof of the vanishing of all L^{2}-Betti numbers of Thompson’s group F in Subsection 7.1.2 (see also [329, Theorem 2.1], [331, Theorem 0.8]). It has been conjectured in [237, page 229] for an aspherical closed manifold M which fibers over the circle S^{1}.

Theorem 1.39 (Vanishing of L^{2}-Betti numbers of mapping tori). Let f: X → X be a cellular selfmap of a connected CW -complex X of finite type and π_{1}(T_{f}) ⟶^{ϕ} G ⟶^{ψ} ℤ be a factorization of the canonical epimorphism into epimorphisms ϕ and ψ. Let \overline{T_{f}} be the covering of T_{f} associated to ϕ which is a free G-CW -complex of finite type. Then we get for all p ≥ 0

b_{p}^{(2)}(\overline{T_{f}}) = 0.

Proof. Fix integers p ≥ 0 and n ≥ 1. Let G_{n} ⊂ G be the preimage of the subgroup n · ℤ ⊂ ℤ under ψ: G → ℤ. This is a subgroup of index n. We get from Theorem 1.35 (9)

b_{p}^{(2)}(\overline{T_{f}}) = \frac{1}{n} · b_{p}^{(2)}(res_{G}^{G_{n}} \overline{T_{f}}).

There is a homotopy equivalence h: T_{f^{n}} → G_{n}\\overline{T_{f}}, where f^{n} is the nfold composition of f. Let \overline{T_{f^{n}}} be the G_{n}-space obtained by the pullback of \overline{T_{f}} → G_{n}\\overline{T_{f}} with h. Then h induces a G_{n}-homotopy equivalence \overline{h}: \overline{T_{f^{n}}} → res_{G}^{G_{n}} \overline{T_{f}}. From homotopy invariance of the L^{2}-Betti numbers of Theorem 1.35 (1) we conclude

b_{p}^{(2)}(res_{G}^{G_{n}} \overline{T_{f}}) = b_{p}^{(2)}(\overline{T_{f^{n}}}).

Let β_{p} be the number of p-cells in X. Then T_{f^{n}} has a CW -structure with β_{p} + β_{p−1} cells of dimension p. Hence the von Neumann dimension of the cellular Hilbert N (G_{n})-chain module C_{p}(\overline{T_{f^{n}}}) is β_{p} + β_{p−1}. This implies by the additivity of the von Neumann dimension of Theorem 1.12 (2) that b_{p}^{(2)}(\overline{T_{f^{n}}}) ≤ β_{p} + β_{p−1}. We have shown

0 ≤ b_{p}^{(2)}(\overline{T_{f}}) ≤ \frac{β_{p} + β_{p−1}}{n} .

Since β_{p} + β_{p−1} is independent of n, the claim follows by taking the limit for n → ∞. ⊓ ⊔

A map f: E → B is called a fibration if it has the homotopy lifting property, i.e. for any homotopy h: X × [0, 1] → B and map f: X → E with p∘f = h_{0} there is a homotopy H: X ×[0, 1] → E with H_{0} = f and p∘H = h (see [488, Definition 4.2 on page 53]). This is a weaker notion than the notion of a (locally trivial) fiber bundle with typical fiber F [269, chapter 4, section 5]. For instance the fiber F of a fibration is only well-defined up to homotopy equivalence, the one of a fiber bundle up to homeomorphism. The notion of a fibration is more general and more flexible than that of a fiber bundle. For example a group extension 1 → Δ → Γ → π → 1 such that Bπ and BΔ are finite CW -complexes yields a fibration BΓ → Bπ with fiber BΔ such that BΓ has the homotopy type of a finite CW -complex but in general one cannot expect the existence of a fiber bundle BΔ → BΓ → Bπ of finite CW -complexes.

If f: F → F is a homotopy equivalence, T_{f} is homotopy equivalent to the total space of a fibration over S^{1} with fiber F. Conversely, the total space of such a fibration is homotopy equivalent to the mapping torus of the selfhomotopy equivalence of F given by the fiber transport with a generator of π_{1}(S^{1}). Therefore L^{2}-Betti numbers are obstructions for a closed manifold to fiber over the circle S^{1}. This problem has been treated by Farrell [189]. The same idea of proof as in Theorem 1.39 yields Novikov-type inequalities for Morse 1-forms [185].

Theorem 1.40 (L^{2}-Betti numbers and S^{1}-actions). Let X be a connected S^{1}-CW -complex of finite type, for instance a connected compact manifold with smooth S^{1}-action. Suppose that for one orbit S^{1}/H (and hence for all orbits) the inclusion into X induces a map on π_{1} with infinite image. (In particular the S^{1}-action has no fixed points.) Let X̃ be the universal covering of X with the canonical π_{1}(X)-action. Then we get for all p ≥ 0

b_{p}^{(2)}(X̃) = 0.

Proof. We show for any S^{1}-CW -complex Y of finite type together with an S^{1}-map f: Y → X that all the L^{2}-Betti numbers b_{p}^{(2)}(f^{∗}X̃; N (π)) are trivial where f^{∗}X̃ is the pullback of the universal covering of X and π = π_{1}(X). Since the p-th L^{2}-Betti number only depends on the (p + 1)-skeleton, we can assume without loss of generality that Y is a finite S^{1}-CW -complex. We use induction over the dimension n of Y. The beginning n = −1 is trivial, the induction step from n − 1 to n ≥ 0 is done as follows.

Choose an equivariant S^{1}-pushout with dim(Z) = n − 1.

∐_{i∈I} S^{1}/H_{i} × S^{n−1} \underrightarrow{∐_{i∈I} q_{i}} Z
↓  ^{j}↓
∐_{i∈I} S^{1}/H_{i} × D^{n} \underrightarrow{∐_{i∈I} Q_{i}} Y

This induces a π-equivariant pushout

∐_{i∈I} q_{i}^{∗}j^{∗}f^{∗}X̃ ⟶ j^{∗}f^{∗}X̃
↓  ↓
∐_{i∈I} Q_{i}^{∗}f^{∗}X̃ ⟶ f^{∗}X̃

It induces a short exact sequence of finitely generated Hilbert N (π)-chain complexes

0 → C_{∗}^{(2)}(j^{∗}f^{∗}X̃) → C_{∗}^{(2)}(f^{∗}X̃) → ⨁_{i∈I} C_{∗}^{(2)}(Q_{i}^{∗}f^{∗}X̃, q_{i}^{∗}j^{∗}f^{∗}X̃) → 0.

Because of Theorem 1.21 and Theorem 1.12 (2) it suffices to prove

dim_{N (π)} (H_{p}^{(2)}(j^{∗}f^{∗}X̃; N (π))) = 0;
dim_{N (π)} (H_{p}^{(2)}(C_{∗}(Q_{i}^{∗}f^{∗}X̃, q_{i}^{∗}j^{∗}f^{∗}X̃))) = 0.

This follows for the first equation by the induction hypothesis. The pair (Q_{i}^{∗}f^{∗}X̃, q_{i}^{∗}j^{∗}f^{∗}X̃) is π-homeomorphic to π ×_{ℤ} S̃^{1} × (D^{n}, S^{n−1}) for an appropriate subgroup ℤ ⊂ π since S^{1}/H_{i} is homeomorphic to S^{1} and the inclusion of an orbit into X induces an injection on the fundamental groups by assumption. Hence C_{∗}(Q_{i}^{∗}f^{∗}X̃, q_{i}^{∗}j^{∗}f^{∗}X̃) is ℤπ-isomorphic to the n-fold suspension of C_{∗}(π ×_{ℤ} S̃^{1}) for a subgroup ℤ ⊂ π. We conclude from Theorem 1.35 (10)

dim_{N (π)} (H_{p}^{(2)}(C_{∗}(Q_{i}^{∗}f^{∗}X̃, q_{i}^{∗}j^{∗}f^{∗}X̃))) = b_{p−n}^{(2)}(S̃^{1}).

We conclude b_{p}^{(2)}(S̃^{1}) = 0 for p ≥ 0 from Theorem 1.35 (9) or Lemma 1.34 (2). ⊓ ⊔

Theorem 1.40 will be generalized in Theorem 6.65 and a more general version of Lemma 1.41 below will be proved in Lemma 6.66.

Lemma 1.41. Let F → E → B be a fibration of connected CW -complexes of finite type such that the inclusion induces an injection π_{1}(F) → π_{1}(E). Suppose that the L^{2}-Betti numbers b_{p}^{(2)}(F̃) of the universal covering F̃ (with the canonical π_{1}(F)-action) vanish for all p ≥ 0. Then the same is true for the universal covering Ẽ (with the canonical π_{1}(E)-action) of E.

1.2.4 L^{2}-Betti Numbers and Aspherical Spaces

A CW -complex X is aspherical if it is connected and its universal covering is contractible. This is equivalent to the condition that π_{n}(X, x) is trivial for n ≠ 1 and x ∈ X. The universal covering X̃ → X of an aspherical CW -complex X with fundamental group π is a model for the universal principal π-bundle Eπ → Bπ. Given a group π, an aspherical CW -complex X with base point x ∈ X together with an isomorphism π_{1}(X, x) ⟶^{≅} π is called an Eilenberg-MacLane space of type (π, 1). Examples for aspherical CW -complexes are closed Riemannian manifolds with non-positive sectional curvature since their universal coverings are diffeomorphic to ℝ^{n} by the Hadamard-Cartan Theorem [218, Theorem 3.87 on page 134].

Assertion (1) of the next Lemma 1.42 is a variation of [124, Lemma 5.1 on page 242] and assertion (2) appears in [232, page 95], where the result is attributed to unpublished work of Sullivan.

Lemma 1.42. (1) Let X be a connected finite dimensional S^{1}-CW -complex with empty fixed point set X^{S^{1}}. Suppose that the universal covering X̃ of X satisfies H^{∗}(X̃; ℚ) = H^{∗}({∗}; ℚ). Then the inclusion of any orbit into X induces an injection on the fundamental group.

(2) Let M be a connected closed oriented n-dimensional manifold with fundamental group π and fundamental class [M] ∈ H_{n}(M; ℤ). Let f: M → Bπ be the classifying map of the universal covering M̃ → M. Suppose that M carries a non-trivial (smooth) S^{1}-action with M^{S^{1}} ≠ ∅. Then the image of the fundamental class in the homology of Bπ with rational coefficients f_{∗}[M] ∈ H_{n}(Bπ; ℚ) is trivial.

Proof. (1) We first show for any fixed point free S^{1}-CW -complex Z that the canonical projection p: ES^{1} ×_{S^{1}} Z → S^{1}\Z is a rational cohomology equivalence where S^{1} → ES^{1} → BS^{1} is a model for the universal principal S^{1}-bundle. Recall that ES^{1} is a free S^{1}-CW -complex which is (nonequivariantly) contractible. It suffices to prove the claim for each k-skeleton of Z because of the Five-Lemma and the short exact sequence due to Milnor [521, Theorem XIII.1.3 on page 605] for Z

0 → lim_{k→∞}^{1}H^{n−1}(Z_{k}; ℚ) → H^{n}(Z; ℚ) → lim_{k→∞}H^{n}(Z_{k}; ℚ) → 0

and the corresponding one for ES^{1} ×_{S^{1}} Z. We use induction over k. The beginning k = −1 is trivial, the induction step from k − 1 to k ≥ 0 is done as follows. Suppose that the k-skeleton Z_{k} is obtained from the (k − 1)-skeleton Z_{k−1} by attaching equivariant cells

∐_{i∈I} S^{1}/H_{i} × S^{k−1} ⟶ Z_{k−1}
↓  ↓
∐_{i∈I} S^{1}/H_{i} × D^{k} ⟶ Z_{k}

where H_{i} ⊂ S^{1} is a finite subgroup for each i ∈ I. Then ES^{1} ×_{S^{1}} Z_{k} is the pushout

∐_{i∈I} ES^{1} ×_{S^{1}} (S^{1}/H_{i} × S^{k−1}) ⟶ ES^{1} ×_{S^{1}} Z_{k−1}
↓  ↓
∐_{i∈I} ES^{1} ×_{S^{1}} (S^{1}/H_{i} × D^{k}) ⟶ ES^{1} ×_{S^{1}} Z_{k}

and S^{1}\Z_{k} is the pushout:

∐_{i∈I} S^{k−1} ⟶ S^{1}\Z_{k−1}
↓  ↓
∐_{i∈I} D^{k} ⟶ S^{1}\Z_{k}

The projections ES^{1} ×_{S^{1}} Y → S^{1}\Y for Y = Z_{k−1}, ∐_{i∈I} S^{1}/H_{i} × S^{k−1} and ∐_{i∈I} S^{1}/H_{i} × D^{k} are rational cohomology equivalences by the induction hypothesis and because BH_{i} → {∗} is one for each i ∈ I. By a Mayer-Vietoris argument the projection for Y = Z_{k} is also a rational cohomology equivalence.

Let X̃ be the universal covering of X and π = π_{1}(X). There is an extension of Lie groups 1 → π → L → S^{1} → 1 and an action of L on X̃ such that the L-action extends the π-action on X̃ and covers the S^{1}-action on X. Moreover, there is an L-CW -structure on X̃ such that the induced S^{1}-CW -structure on X = π\X̃ is the given one [326, Theorem 8.1 on page 138, Lemma 8.9 on page 140]. The basic idea of this construction is to define L as the group of pairs (f, g), where g is an element in S^{1} and f: X̃ → X̃ is map which covers the map l_{g}: X → X given by multiplication with g.

The map ∂ in the exact sequence of homotopy groups associated to the regular π-covering L → S^{1}

{1} → π_{1}(L) → π_{1}(S^{1}) →^{∂} π → π_{0}(L) → {1}

agrees with ev_{∗}: π_{1}(S^{1}) → π_{1}(X) induced by evaluating the S^{1}-action at the base point. We want to show that this map is injective. Suppose it is not true. Then π_{1}(L) is an infinite cyclic group. As L is a 1-dimensional Lie group, its component of the identity is just S^{1}. Obviously each isotropy group under the L-action on X̃ is finite as the S^{1}-action on X has only finite isotropy groups. Hence X̃ carries a fixed point free S^{1}-action. The Gysin sequence [521, Theorem VII.5.12 on page 356] of S^{1} → ES^{1} × X̃ → ES^{1} ×_{S^{1}} X̃ looks like:

… → H^{p}(ES^{1} ×_{S^{1}} X̃; ℚ) → H^{p+2}(ES^{1} ×_{S^{1}} X̃; ℚ) →
H^{p+2}(ES^{1} × X̃; ℚ) → H^{p+1}(ES^{1} ×_{S^{1}} X̃; ℚ) → …

We have H^{p}(ES^{1} × X̃; ℚ) = H^{p}(X̃; ℚ) = 0 for p ≠ 0 by assumption. Hence we get H^{2p}(ES^{1} ×_{S^{1}} X̃; ℚ) ≠ {0} for p ≥ 0. As the projection ES^{1} ×_{S^{1}} X̃ → S^{1}\X̃ induces an isomorphism on rational cohomology, S^{1}\X̃ has non-trivial rational cohomology in all even dimensions. Since X has a finite dimensional S^{1}-CW -structure, S^{1}\X̃ has a finite dimensional CW - structure, a contradiction. This proves assertion (1).

(2) Let ρ: S^{1} × M → M be the given S^{1}-operation and let p: M̃ → M be the universal covering. Choose x ∈ M^{S^{1}} with a lift x̃ ∈ M̃. Notice that the map induced by evaluation π_{1}(S^{1}, e) → π = π_{1}(M, x) is trivial. Hence there is precisely one map ρ̃: S^{1} × M̃ → M̃ which satisfies p ∘ ρ̃ = ρ ∘ (id_{S^{1}} ×p) and ρ̃(e, x̃) = x̃. One easily checks that ρ̃ defines an S^{1}-action on M̃ which commutes with the canonical π = π_{1}(M, x)-action on M̃ with respect to x̃ and covers the S^{1}-action on M. In particular S^{1}\M̃ inherits a π-action and the projection p̃r: M̃ → S^{1}\M̃ is π-equivariant.

Next we want to show for ỹ ∈ M̃ that the isotropy group under the πaction of S^{1}ỹ ∈ S^{1}\M̃ is finite. Let u be the projection to M of any path ũ joining x̃ and ỹ in M̃. Then w · ỹ for w ∈ π is by definition { \atop u^{−} ∗ w ∗ u}(1) for any path { \atop u^{−} ∗ w ∗ u} with initial point ỹ whose projection under p is the loop u^{−} ∗ w ∗ u in M. Suppose that w · S^{1}ỹ = S^{1}ỹ. Then we can find z ∈ S^{1} with z ·ỹ = w ·ỹ. Let v be any path in S^{1} from e to z. Put y = p(ỹ). Let vỹ and vy be the paths obtained from v and the S^{1}-actions. The paths { \atop u^{−} ∗ w ∗ u} and vỹ are homotopic relative endpoints. This implies z ∈ S_{y}^{1} for S_{y}^{1} the isotropy group of y ∈ M under the S^{1}-action and that vy and u^{−} ∗w∗u are homotopic relative endpoints. Hence the isotropy group of S^{1}ỹ under the π = π_{1}(M, x) action can be identified with the image of π_{1}(S^{1}/S_{y}^{1}, eS_{y}^{1}) → π_{1}(M, y) ⟶^{c_{u}} π = π_{1}(M, x) where c_{u} is given by conjugation with u. The composition of this map with the obvious map π_{1}(S^{1}, e) → π_{1}(S^{1}/S_{y}^{1}, eS_{y}^{1}) is trivial since the obvious map π_{1}(S^{1}, e) → π_{1}(M, x) is trivial because of x ∈ M^{S^{1}}. Hence all isotropy subgroups of the π-action on S^{1}\M̃ are finite.

We have introduced \underline{E}π = E(π; FIN) in Definition 1.28. Choose π-maps f̃: M̃ → Eπ, g̃: Eπ → \underline{E}π and h̃: S^{1}\M̃ → \underline{E}π. Then g̃ ∘ f̃ and h̃ ∘ p̃r are π-homotopy equivalent. By passing to the quotients under the π-action we obtain an up to homotopy commutative diagram

M ⟶^{pr} S^{1}\M
f↓  ^{h}↓
Bπ ⟶^{g} π\\underline{E}π

where f: M → Bπ is a classifying map. Hence it suffices to show that H_{n}(S^{1}\M; ℤ) = 0 holds and that g_{∗}: H_{p}(Bπ; ℚ) → H_{p}(π\\underline{E}π; ℚ) is an isomorphism for each p ≥ 0.

Since M is connected and closed and M ≠ M^{S^{1}} by assumption and each component C of M^{S^{1}} is a connected closed submanifold [64, Corollary VI.2.4 on page 308], the dimension of each component C is ≤ n − 1. Since M is obtained from M^{S^{1}} by attaching cells of the type S^{1}/H × D^{k} for finite H ⊂ S^{1} and k ≤ n − 1, the dimension of the CW -complex S^{1}\M is ≤ n − 1 and hence H_{n}(S^{1}\M; ℤ) = 0.

Hence it remains to prove that H_{∗}(g; ℚ): H_{∗}(Bπ; ℚ) → H_{p}(π\\underline{E}π; ℚ) is bijective for all p ≥ 0. Notice that C_{p}(\underline{E}G)⊗_{ℤ}ℚ is projective over ℚG for each p ≥ 0 since it is a direct sum of ℚG-modules of the shape ℚ[G/H] for finite H ⊂ G. The canonical map EG → \underline{E}G induces a homology equivalence of projective ℚG-chain complexes C_{∗}(EG) ⊗_{ℤ} ℚ → C_{∗}(\underline{E}G) ⊗_{ℤ} ℚ. We conclude that this chain map is a ℚG-chain homotopy equivalence. Hence it induces a chain homotopy equivalence C_{∗}(EG)⊗_{ℤG} ℚ → C_{∗}(\underline{E}G)⊗_{ℤG} ℚ which induces on homology H_{∗}(g; ℚ). ⊓ ⊔

Corollary 1.43. Let M be an aspherical closed manifold with non-trivial S^{1}-action. Then the action has no fixed points and the inclusion of any orbit into X induces an injection on the fundamental groups. All L^{2}-Betti numbers b_{p}^{(2)}(M̃) are trivial and χ(M) = 0.

Proof. This follows from Theorem 1.35 (2), Theorem 1.40 and Lemma 1.42 provided M is orientable. Recall that the orientation covering is the double covering \overline{M} = SΛ^{dim(M)}T M → M and has the property that its total space \overline{M} is orientable and it is trivial if and only if M is orientable. The nonorientable case then follows from Theorem 1.35 (9) since \overline{M} is orientable and satisfies the same hypothesis as M. ⊓ ⊔

1.3 Analytic L^{2}-Betti Numbers 49

We will show in Corollary 3.111 that under the conditions of Corollary 1.43 all Novikov-Shubin invariants of M̃ are less or equal to 1 and that the L^{2}-torsion of M̃ is trivial.

The next result will be a special case of a more general result, namely Theorem 7.2 (1) and (2)

Theorem 1.44 (L^{2}-Betti numbers and aspherical CW -complexes). Let X be an aspherical CW -complex of finite type. Suppose that π_{1}(X) contains an amenable infinite normal subgroup. Then b_{p}^{(2)}(X̃) = 0 for all p ≥ 0.

1.3 Analytic L^{2}-Betti Numbers

In this section we state the L^{2}-Hodge-de Rham Theorem 1.59 and give an analytic interpretation of the L^{2}-Betti numbers.

1.3.1 The Classical Hodge-de Rham Theorem

In this subsection we recall the classical Hodge-de Rham Theorem since we need it and the necessary notations later and it motivates the L^{2}-version.

Let M be a manifold without boundary (which is not necessarily compact). Denote by Ω^{p}(M) the space of smooth p-forms on M, i.e. the space of smooth sections of the bundle Alt^{p}(ℂ ⊗_{ℝ} T M) of alternating p-linear ℂ-forms associated to the complexification ℂ ⊗_{ℝ} T M of the tangent bundle T M. Hence a p-form ω assigns to each x ∈ M an alternating p-linear ℂ-form ω_{x}: ℂ ⊗_{ℝ} T_{x}M × … × ℂ ⊗_{ℝ} T_{x}M → ℂ on the complexified tangent space ℂ ⊗_{ℝ} T_{x}M. This is the same as an alternating p-linear ℝ-form ω_{x}: T_{x}M × … × T_{x}M → res_{ℂ}^{ℝ} ℂ. Notice that Ω^{p}(M) inherits the structure of a complex vector space. One can identify Ω^{p}(M) with the space of alternating p-forms on the C^{∞}(M)-module C^{∞}(ℂ⊗_{ℝ} T M), where C^{∞}(M) = Ω^{0}(M) is the ℂ-algebra of smooth ℂ-valued functions on M and C^{∞}(ℂ ⊗_{ℝ} T M) is the C^{∞}(M)-module of smooth sections of the complexified tangent bundle ℂ ⊗_{ℝ} T M. Denote by

∧ : Ω^{p}(M) ⊗ Ω^{q}(M) → Ω^{p+q}(M)  (1.45)

the ∧-product of smooth forms. Recall that the ∧-product is defined by the corresponding notion for finite dimensional vector spaces applied fiberwise. Denote by

d^{p}: Ω^{p}(M) → Ω^{p+1}(M)  (1.46)

the p-exterior differential. Recall that the exterior differentials are uniquely determined by the following properties:

(1) d^{p} is ℂ-linear; (2) (d^{0}f)(X) = Xf for f ∈ Ω^{0}(M) = C^{∞}(M), X a vector field on M and

Xf the derivative of f along X;

(3) d^{p+q}(ω ∧ η) = (d^{p}ω) ∧ η + (−1)^{p}ω ∧ d^{q}η for ω ∈ Ω^{p}(M) and η ∈ Ω^{q}(M);

(4) d^{p+1} ∘ d^{p} = 0.

The cohomology of the de Rham cochain complex

… ⟶^{d^{p−2}} Ω^{p−1}(M) ⟶^{d^{p−1}} Ω^{p}(M) ⟶^{d^{p}} Ω^{p+1}(M) ⟶^{d^{p+1}} …

is the de Rham cohomology of M and denoted by H_{dR}^{p}(M).

Let C_{∗}^{sing}(M) be the ℤ-chain complex given by singular simplices Δ_{p} → M which are continuous. Let C_{∗}^{sing,C^{∞}}(M) be the ℤ-chain complex given by singular simplices Δ_{p} → M which are smooth. (Here Δ_{p} is the standard p-simplex and not the Laplace operator). We denote by C_{sing}^{∗}(M; ℂ) and C_{sing,C^{∞}}^{∗}(M; ℂ) the corresponding cochain complexes hom_{ℤ}(C_{∗}^{sing}(M), ℂ) and hom_{ℤ}(C_{∗}^{sing,C^{∞}}(M), ℂ). Let H_{sing}^{∗}(M; ℂ) := H^{∗}(C_{sing}^{∗}(M; ℂ)) be the singular cohomology of M. The obvious inclusion induces a cochain map

i^{∗}: C_{sing}^{∗}(M; ℂ) → C_{sing,C^{∞}}^{∗}(M; ℂ).

There is a well-defined cochain map

I^{∗}: Ω^{∗}(M) → C_{sing,C^{∞}}^{∗}(M; ℂ)

which sends ω ∈ Ω^{p}(M) to the cochain I^{p}(ω): C_{∗}^{sing,C^{∞}}(M) → ℂ which maps σ: Δ_{p} → M to ∫_{Δ_{p}} σ^{∗}ω. The proof of the de Rham Theorem 1.47 can be found for instance in [63, Section V.9.], [134], [156, Theorem 1.5 on page 11 and Theorem 2.4 on page 20], [357, Theorem A.31 on page 413].

Theorem 1.47 (De Rham Isomorphism Theorem). Let M be a manifold. Then the cochain maps i^{∗} and I^{∗} above induce a natural isomorphisms for p ≥ 0

A^{p}: H_{dR}^{p}(M) → H_{sing}^{p}(M; ℂ).

They are compatible with the multiplicative structures given by the ∧-product and the ∪-product.

Now suppose that M comes with a Riemannian metric and an orientation. Let n be the dimension of M. Denote by

∗^{p} : Ω^{p}(M) → Ω^{n−p}(M)  (1.48)

the Hodge star-operator which is defined by the corresponding notion for oriented finite dimensional Hermitian vector spaces applied fiberwise. It is uniquely characterized by the property

∫_{M} ω ∧ ∗^{p}η = ∫_{M}⟨ω_{x}, η_{x}⟩_{Alt^{p}(T_{x}M)} dvol,  (1.49)

where ω and η are p-forms and ⟨ω_{x}, η_{x}⟩_{Alt^{p}(T_{x}M)} is the inner product on Alt^{p}(T_{x}M) which is induced by the inner product on T_{x}M given by the Riemannian metric.

Define the adjoint of the exterior differential

δ^{p} = (−1)^{np+n+1} ∗^{n−p+1} ∘d^{n−p} ∘ ∗^{p}: Ω^{p}(M) → Ω^{p−1}(M). (1.50)

Notice that in the definition of δ^{p} the Hodge star-operator appears twice and the definition is local. Hence we can define δ^{p} without using an orientation of M, only the Riemannian metric is needed. This is also true for the Laplace operator which is defined by

Δ_{p} = d^{p−1} ∘ δ^{p} + δ^{p+1} ∘ d^{p}: Ω^{p}(M) → Ω^{p}(M).  (1.51)

Let Ω_{c}^{p}(M) ⊂ Ω^{p}(M) be the space of smooth p-forms with compact support. There is the following inner product and norm on it

⟨ω, η⟩_{L^{2}} := ∫_{M} ω ∧ ∗^{p}η = ∫_{M}⟨ω_{x}, η_{x}⟩_{Alt^{p}(T_{x}M)} dvol;  (1.52)
||ω||_{L^{2}} := \sqrt{⟨ω, ω⟩_{L^{2}}}.  (1.53)

Notice that d^{p} and δ^{p} are formally adjoint in the sense that we have for ω ∈ Ω_{c}^{p}(M) and η ∈ Ω_{c}^{p+1}(M)

⟨d^{p}(ω), η⟩_{L^{2}} = ⟨ω, δ^{p+1}(η)⟩_{L^{2}}.  (1.54)

Let L^{2}Ω^{p}(M) be the Hilbert space completion of Ω_{c}^{p}(M). Define the space of L^{2}-integrable harmonic smooth p-forms

H_{(2)}^{p}(M) := {ω ∈ Ω^{p}(M) | Δ_{p}(ω) = 0, ∫_{M} ω ∧ ∗ω < ∞}. (1.55)

Recall that a Riemannian manifold M is complete if each path component of M equipped with the metric induced by the Riemannian metric is a complete metric space. By the Hopf-Rinow Theorem the following statements are equivalent provided that M has no boundary: (1) M is complete, (2) the exponential map is defined for any point x ∈ M everywhere on T_{x}M, (3) any geodesic of M can be extended to a geodesic defined on ℝ (see [218, page 94 and 95]). Completeness enters in a crucial way, namely, it will allow us to integrate by parts [217].

Lemma 1.56. Let M be a complete Riemannian manifold. Let ω ∈ Ω^{p}(M) and η ∈ Ω^{p+1}(M) be smooth forms such that ω, d^{p}ω, η and δ^{p+1}η are squareintegrable. Then

⟨d^{p}ω, η⟩_{L^{2}} = ⟨ω, δ^{p+1}η⟩_{L^{2}}.

Proof. Completeness ensures the existence of a sequence f_{n}: M → [0, 1] of smooth functions with compact support such that M is the union of the compact sets {x ∈ M | f_{n}(x) = 1} and ||df_{n}||_{∞} := sup{||(df_{n})_{x}||_{x} | x ∈ M\} < \frac{1}{n} holds. With the help of the sequence (f_{n})_{n≥1} one can reduce the claim to the easy case, where ω and η have compact support. ⊓ ⊔

Theorem 1.57 (Hodge-de Rham Theorem). Let M be a complete Riemannian manifold without boundary. Then we obtain an orthogonal decomposition, the so called Hodge-de Rham decomposition

L^{2}Ω^{p}(M) = H_{(2)}^{p}(M) ⊕ clos(d^{p−1}(Ω_{c}^{p−1}(M))) ⊕ clos(δ^{p+1}(Ω_{c}^{p+1}(M))).

1.3.2 Analytic Definition of L^{2}-Betti Numbers

Suppose for a moment that M is a Riemannian manifold which is closed and that we require no group action. Then in view of Theorem 1.57 the space H^{p}(M) = {ω ∈ Ω^{p}(M) | Δ_{p}(ω) = 0} is isomorphic to the singular cohomology H_{sing}^{p}(M; ℂ). In particular H^{p}(M) is a finite dimensional ℂ-vector space and we can define the analytic p-th Betti number by its dimension. Of course the analytic p-th Betti number agrees with the classical p-th Betti number which is defined by the dimension of H_{sing}^{p}(M; ℂ). The analytic p-th Betti number can be interpreted in terms of the heat kernel e^{−tΔ_{p}}(x, y), namely by the following expression [220, 1.6.52 on page 56]

b_{p}(M) = lim_{t→∞} ∫_{M} tr_{ℂ}(e^{−tΔ_{p}}(x, x)) dvol .  (1.58)

Here (and elsewhere) e^{−tΔ_{p}}(x, y) denotes the heat kernel. This is a smooth section of the bundle hom(p_{1}^{∗} Alt^{p}(T M), p_{2}^{∗} Alt^{p}(T M)) over M × M for p_{k}: M × M → M the projection to the k-th factor. It is uniquely characterized by the property that for ω ∈ L^{2}Ω^{p}(M) and e^{−tΔ_{p}} the operator obtained from Δ_{p} by spectral calculus (see (1.64))

e^{−tΔ_{p}}(ω)(x) = ∫_{M} e^{−tΔ_{p}}(x, y)(ω_{y}) dvol_{y}

holds. The real number tr_{ℂ}(e^{−tΔ_{p}}(x, x)) is the trace of the endomorphism of finite dimensional vector spaces e^{−tΔ_{p}}(x, x): Alt^{p}(T_{x}M) → Alt^{p}(T_{x}M).

We want to generalize these results and notions to the L^{2}-setting. From now on M is a cocompact free proper G-manifold without boundary and with G-invariant Riemannian metric. Recall that an equivariant smooth triangulation K of M consists of a simplicial complex K with simplicial G-action such that for each open simplex σ and g ∈ G with gσ ∩ σ ≠ ∅ left multiplication with g induces the identity on σ, together with a G-homeomorphism f: |K| ⟶^{≅} M such that f restricted to each closed simplex is a smooth immersion of the simplex into M. This is the same as a lift of a smooth triangulation of G\M to M. Notice that K is a free cocompact G-CW -complex. In the sequel we will not distinguish between K and its realization |K|.

The key result, which is due to Dodziuk [145] and whose proof we will present in Section 1.4, is

Theorem 1.59 (L^{2}-Hodge-de Rham Theorem). Let M be a cocompact free proper G-manifold with G-invariant Riemannian metric and let K be an equivariant smooth triangulation of M. Suppose that M has no boundary. Let H_{(2)}^{p}(M) be the space of L^{2}-integrable harmonic smooth p-forms on M introduced in (1.55). Then integration defines an isomorphism of finitely generated Hilbert N (G)-modules

H_{(2)}^{p}(M) ⟶^{≅} H_{(2)}^{p}(K).

This allows us to define the analytic p-th L^{2}-Betti number by the von Neumann dimension of the finitely generated Hilbert N (G)-module H_{(2)}^{p}(M). Obviously this is the same as the cellular p-th L^{2}-Betti number. Moreover, this can be interpreted in terms of the heat kernel by the following expression [9, Proposition 4.16 on page 63] (or Theorem 3.136 (1))

b_{p}^{(2)}(M) = lim_{t→∞} ∫_{F} tr_{ℂ}(e^{−tΔ_{p}}(x, x)) dvol .  (1.60)

Here F is a fundamental domain for the G-action, i.e. an open subset F ⊂ M such that M is the union ⋃_{g∈G} g · clos(F) and gF ∩ F ≠ ∅ ⇒ g = 1 and the topological boundary of F is a set of measure zero. If one fixes a triangulation of G\M and fixes for each open simplex σ of dimension n = dim(M) a lift \overline{σ} ⊂ M, then the union of the lifts \overline{σ} for all n-simplices σ in G\M is a fundamental domain.

Recall that a manifold is called hyperbolic if it is equipped with a Riemannian metric whose sectional curvature is constant −1. This is equivalent to the statement that M comes with a Riemannian metric such that the universal covering M̃ with the induced Riemannian metric is isometrically diffeomorphic to the (real) hyperbolic space ℍ^{n} for n = dim(M) [218, Theorem 3.82 on page 131]. Recall that the Poincaré model for ℍ^{n} is the open unit disk {x ∈ ℝ^{n} | |x| < 1} with the Riemannian metric which is given at a point x ∈ ℝ^{n}, |x| < 1 by

T_{x}M × T_{x}M → ℝ, (v, w) ↦ \frac{4}{(1 − |x|^{2})^{2}} · ⟨v, w⟩_{Eucl},  (1.61)

where ⟨v, w⟩_{Eucl} is the standard Euclidean metric on T_{x}ℝ^{n} = ℝ^{n}.

Theorem 1.62. Let M be a hyperbolic closed Riemannian manifold of dimension n. Then

b_{p}^{(2)}(M̃) \{ \begin{matrix}= 0 & if 2p ≠ n \\ > 0 & if 2p = n \end{matrix} .

If n is even, then

(−1)^{n/2} · χ(M) > 0.

Proof. The statement about the Euler characteristic for M follows from the statement about the L^{2}-Betti numbers from Theorem 1.35 (2). Since the von Neumann dimension is faithful by Theorem 1.12 (1) it suffices to show, because of Theorem 1.59, that

H_{(2)}^{p}(ℍ^{n}) \{ \begin{matrix}= 0 & if 2p ≠ n \\ ≠ 0 & if 2p = n \end{matrix} .

This is done in [146] (even for all rotationally symmetric Riemannian manifolds). The idea of the proof is summarized as follows.

An element ω ∈ H_{(2)}^{p}(ℍ^{n}) is written (outside 0) in polar coordinates

ω = a(r, θ) ∧ dr + b(r, θ) ∧ dθ

for (p − 1)-forms a(r, θ) and b(r, θ) on S^{n−1} in the variable θ which depend on a parameter r ∈ (0, ∞). Then the conditions d^{p}ω = 0, δ^{p}(ω) = 0 and ∫_{ℍ^{n}} ω ∧ ∗ω < ∞ are explicitly rewritten in terms of a and b and the function f(r) for which the Riemannian metric with respect to the polar coordinates looks like ds^{2} = dr^{2} + f(r)^{2}dθ^{2}. The function f satisfies f(0) = 0, f′(0) = 1, f(r) > 0 for r > 0, and ∫_{1}^{∞} f(r)^{−1}dr < ∞. Then a further calculation shows that H_{(2)}^{p}(ℍ^{n}) ≠ 0 implies

∫_{1}^{∞} f^{n−2p−1}(r) dr < ∞.

Since the Hodge star operator yields an isomorphism H_{(2)}^{n−p}(ℍ^{n}) = H_{(2)}^{p}(ℍ^{n}), H_{(2)}^{p}(ℍ^{n}) ≠ 0 implies that both ∫_{1}^{∞} f^{n−2p−1}(r)dr and ∫_{1}^{∞} f^{−n+2p−1}(r)dr are finite. If 2p = n, these two integrals are equal and finite. However, if 2p ≠ n, then the exponents (n − 2p − 1) and (−n + 2p − 1) have different signs and one of the integrals has to diverge. This shows H_{(2)}^{p}(ℍ^{n}) = 0 for 2p ≠ n. One can show explicitly that H_{(2)}^{p}(ℍ^{n}) is an infinite dimensional vector space if 2p = n. ⊓ ⊔

We will extend Theorem 1.62 in Theorem 5.12 (1), where the computation of the L^{2}-Betti numbers of the universal covering M̃ of a closed Riemannian manifold M is given provided that M̃ is a symmetric space of non-compact type.

1.4 Comparison of Analytic and Cellular L^{2}-Betti Numbers

In this section we give the proof of the L^{2}-Hodge-de Rham Theorem 1.59.

1.4 Comparison of Analytic and Cellular L^{2}-Betti Numbers 55

1.4.1 Survey on Unbounded Operators and Spectral Families

We have to explain some facts about unbounded densely defined operators on Hilbert spaces and their domains, about spectral families and about the Spectral Theorem. For a general discussion we refer for instance to [434, Section VIII.1 - 3], [421, Chapter 5].

Let H be a Hilbert space and T: dom(T) → H be a (not necessarily bounded) linear operator defined on a dense linear subspace dom(T) which is called (initial) domain. We call T closed if its graph gr(T) := {(u, T (u)) | u ∈ dom(T)} ⊂ H × H is closed. We say that S: dom(S) → H is an extension of T and write T ⊂ S if dom(T) ⊂ dom(S) and S(u) = T (u) holds for all u ∈ dom(T). We write T = S if dom(T) = dom(S) and S(u) = T (u) holds for all u ∈ dom(T). We call T closable if and only if T has a closed extension. Since the intersection of an arbitrary family of closed sets is closed again, a closable unbounded densely defined operator T has a unique minimal closure, also called minimal closed extension , i.e. a closed operator T_{min} : dom(T_{min}) → H with T ⊂ T_{min} such that T_{min} ⊂ S holds for any closed extension S of T. Explicitly dom(T_{min}) consists of elements u ∈ H for which there exists a sequence (u_{n})_{n≥0} in dom(T) and an element v in H satisfying lim_{n→∞} u_{n} = u and lim_{n→∞} T (u_{n}) = v. Then v is uniquely determined by this property and we put T_{min}(u) = v. Equivalently, dom(T_{min}) is the Hilbert space completion of dom(T) with respect to the inner product

⟨u, v⟩_{gr} := ⟨u, v⟩_{H} + ⟨T (u), T (v)⟩_{H}.  (1.63)

If not stated otherwise we always use the minimal closed extension as the closed extension of a closable unbounded densely defined linear operator.

The adjoint of T is the operator T^{∗}: dom(T^{∗}) → H whose domain consists of elements v ∈ H for which there is an element u in H such that ⟨u′, u⟩ = ⟨T (u′), v⟩ holds for all u′ ∈ dom(T). Then u is uniquely determined by this property and we put T^{∗}(v) = u. Notice that T^{∗} may not have a dense domain in general. Its domain is dense if and only if T is closable. If T is closable, then T_{min}^{∗} = T^{∗} and T_{min} = (T^{∗})^{∗}. We call T symmetric if T ⊂ T^{∗} and selfadjoint if T = T^{∗}. Any selfadjoint operator is necessarily closed and symmetric. A bounded operator T: H → H is always closed and is selfadjoint if and only if it is symmetric. We call T essentially selfadjoint if T_{min} is selfadjoint.

A densely defined unbounded operator T: dom(f) ⊂ H → H for a Hilbert space H is called positive if T is selfadjoint and the real number ⟨T (v), v⟩ is ≥ 0 for all v ∈ dom(T). Notice that the Polar Decomposition T = US also exists for densely defined unbounded operators T: dom(T) ⊂ H_{1} → H_{2} [434, Theorem VIII.32 on page 297]. Here U and S are uniquely determined by the properties that U is a partial isometry, i.e. U^{∗}U is a projection, S is a positive selfadjoint operator and ker(U) = ker(T) = ker(S). The operator S is constructed by \sqrt{S^{∗}S} in the sense of functional calculus as explained below. We have U^{∗}US = S, U^{∗}T = S and U^{∗}UT = T.

Let H be a Hilbert space and T: dom(T) ⊂ H → H be a densely defined closed operator with domain dom(T). Define its resolvent set resolv(T) as the set of complex numbers λ ∈ ℂ for which there is a bounded operator S: H → H whose image is dom(T) and which satisfies (T −λ·id)∘S = id_{H} and S ∘(T −λ·id) = id_{dom(T)}. Define the spectrum spec(T) to be the complement of resolv(T) ⊂ ℂ. The spectrum of T is always a closed subset of ℂ. If T is selfadjoint, then spec(T) ⊂ ℝ. If T is positive, then spec(T) ⊂ [0, ∞). Notice the elementary facts that 0 ∉ spec(T) implies injectivity of T, but that there are everywhere defined bounded closed injective positive operators T with 0 ∈ spec(T).

We call a family {E_{λ} | λ ∈ ℝ} of orthogonal projections E_{λ}: H → H a spectral family if it satisfies for x ∈ H and λ, μ ∈ ℝ

lim_{λ→−∞} E_{λ}(x) = 0;
lim_{λ→∞} E_{λ}(x) = x;
lim_{λ→μ+} E_{λ}(x) = E_{μ}(x);
E_{λ}E_{μ} = E_{μ}E_{λ} = E_{min{λ,μ}}.

Denote by d⟨u, E_{λ}(u)⟩ the Borel measure on ℝ which assigns to the half open interval (λ, μ] the measure ⟨u, E_{μ}(u)⟩ − ⟨u, E_{λ}(u)⟩. If g: ℝ → ℝ is a Borel function, we obtain a selfadjoint operator on H

∫_{−∞}^{∞} g(λ) dE_{λ}  (1.64)

with dense domain

dom (∫_{−∞}^{∞} g(λ)dE_{λ}) := {u ∈ H | ∫_{−∞}^{∞} g(λ)^{2} d⟨u, E_{λ}(u)⟩ < ∞} .

It is uniquely determined by the property that for all u ∈ dom (∫_{−∞}^{∞} g(λ)dE_{λ})

⟨u , (∫_{−∞}^{∞} g(λ)dE_{λ}) (u)⟩ = ∫_{−∞}^{∞} g(λ)d⟨u, E_{λ}(u)⟩.  (1.65)

If g is a complex valued Borel function on ℝ, one defines ∫_{−∞}^{∞} g(λ)dE_{λ} using the Polar Decomposition g = ℜ(g) + i · ℑ(g).

Any selfadjoint operator T: dom(T) → H determines a spectral family {E_{λ}^{T} | λ ∈ ℝ}. It is uniquely determined by the property

T = ∫_{−∞}^{∞} λ dE_{λ}^{T} .  (1.66)

We often abbreviate for a Borel function g: ℝ → ℝ

g(T) := ∫_{−∞}^{∞} g(λ) dE_{λ}^{T} .  (1.67)

We have E_{λ}^{T} = χ_{(−∞,λ]}(T) = ∫_{−∞}^{∞} χ_{(−∞,λ]}dE_{λ}^{T} for χ_{(−∞,λ]} the characteristic function of (−∞, λ].

Definition 1.68 (Spectral family). We call the spectral family associated to a selfadjoint operator T by (1.66) the spectral family associated to T. The spectral family associated to an essentially selfadjoint operator T is the spectral family associated to T_{min}.

Notation 1.69. Given two unbounded operators S, T: H_{1} → H_{2} with domains dom(S) and dom(T) and complex numbers λ, μ, define the unbounded operator λ · S + μ · T: H_{1} → H_{2} with domain dom(λ · S + μ · T) = dom(S) ∩ dom(T) by (λ · S + μ · T)(x) = λ · S(x) + μ · T (x) for x ∈ dom(S) ∩ dom(T).

Given two unbounded operators S: dom(S) ⊂ H_{1} → H_{2} and T: dom(T) ⊂ H_{2} → H_{3}, define the unbounded operator T ∘ S with domain dom(T ∘ S) = S^{−1}(dom(T)) by T ∘ S(x) := T (S(x)) for x ∈ S^{−1}(dom(T)).

Given a selfadjoint operator T the functional calculus g ↦ g(T) is an essential homomorphism of ℂ-algebras in the following sense. Given Borel functions g_{1} and g_{2} and complex numbers λ_{1} and λ_{2}, the minimal closure of λ_{1} · g_{1}(T) + λ_{2} · g_{2}(T) is (λ_{1} · g_{1} + λ_{2} · g_{2})(T) and the minimal closure of g_{2}(T) ∘ g_{1}(T) is (g_{1} · g_{2})(T), where (λ_{1} · g_{1} + λ_{2} · g_{2}) and (g_{1} · g_{2}) are the obvious Borel functions given by pointwise addition and multiplication.

Let E and F be Hermitian vector bundles over a complete Riemannian manifold without boundary. Let D: C_{c}^{∞}(E) → C_{c}^{∞}(F) be an elliptic differential operator where C_{c}^{∞}(E) is the space of smooth sections with compact support. Our main examples are d^{p}: Ω_{c}^{p}(M) → Ω_{c}^{p+1}(M), δ^{p}: Ω_{c}^{p}(M) → Ω_{c}^{p−1}(M) and Δ_{p}: Ω_{c}^{p}(M) → Ω_{c}^{p}(M). Notice that there is a formally adjoint operator D^{t}: C_{c}^{∞}(F) → C_{c}^{∞}(E) which is uniquely determined by the property that ⟨D(u), v⟩_{L^{2}} = ⟨u, D^{t}(v)⟩_{L^{2}} holds for all u, v ∈ C_{c}^{∞}(E). It is again an elliptic differential operator. For instance, δ^{p+1} is the formal adjoint of d^{p}. The minimal closure D_{min} of D: C_{c}^{∞}(E) → L^{2}C^{∞}(F) has been defined above where L^{2}C^{∞}(F) is the Hilbert space completion of C_{c}^{∞}(F). The maximal closure D_{max} of D is defined by the adjoint of (D^{t})_{min}. Indeed, for any closure \overline{D} of D: C_{c}^{∞}(E) → L^{2}C^{∞}(F) we have D_{min} ⊂ \overline{D} ⊂ D_{max}. One can also describe dom(D_{max}) as the space of u ∈ L^{2}C^{∞}(E) for which the distribution D(u) actually lies in L^{2}C^{∞}(F).

Lemma 1.70. (1) Let M be a complete Riemannian manifold without boundary. Then the Laplacian Δ_{p}: Ω_{c}^{p}(M) → L^{2}Ω^{p}(M) is essentially selfadjoint. The minimal and maximal closures of d^{p}: Ω_{c}^{p}(M) → L^{2}Ω^{p+1}(M), δ^{p}: Ω_{c}^{p}(M) → L^{2}Ω_{c}^{p−1}(M) and Δ_{p}: Ω_{c}^{p}(M) → L^{2}Ω_{c}^{p}(M) agree;

(2) Let M be a cocompact G-manifold without boundary and with G-invariant Riemannian metric and let E and F be G-vector bundles with G-invariant Hermitian metrics. Let D: C_{c}^{∞}(E) → C_{c}^{∞}(F) be a G-equivariant elliptic operator. Then the minimal closure and the maximal closure of D: C_{c}^{∞}(E) → L^{2}C^{∞}(F) agree.

Proof. (1) The Laplacian is essentially selfadjoint by [109, Section 3],[216]. The claims about the equality of the minimal and maximal domains follow then from [73, Lemma 3.8 on page 113]

(2) is proved in [9, Proposition 3.1 on page 53]. ⊓ ⊔ Notice that Lemma 1.70 (1) allows us to talk about the spectral family

associated to the Laplace operator Δ_{p}: Ω_{c}^{p}(M) → L^{2}Ω^{p}(M) on a complete Riemannian manifold without boundary.

1.4.2 L^{2}-Hodge-de Rham Theorem

In this subsection we explain parts of the proof of the L^{2}-Hodge-de Rham-Theorem which we have already stated in Theorem 1.59. For more details we refer to the original proof in [145] (see also [147]).

Definition 1.71 (L^{2}-de Rham cohomology). Let M be a complete Riemannian manifold without boundary. Put

Z^{p}(M) := ker ((d^{p}: Ω_{c}^{p}(M) → L^{2}Ω^{p+1}(M))_{min});
B^{p}(M) := im ((d^{p−1} : Ω_{c}^{p−1}(M) → L^{2}Ω^{p}(M))_{min}) .

Define the unreduced L^{2}-de Rham cohomology by

H_{(2),unr}^{p}(M) := Z^{p}(M)/B^{p}(M)

and the (reduced) L^{2}-de Rham cohomology by the Hilbert space

H_{(2)}^{p}(M) := Z^{p}(M)/ clos(B^{p}(M)).

Notice that this definition makes sense since Z^{p}(M) ⊂ L^{2}Ω^{p}(M) is closed and B^{p}(M) lies in Z^{p}(M). Because of Lemma 1.70 (1) we get the same if we use the minimal closures instead of the maximal closures. The difference of the unreduced and reduced L^{2}-cohomology is measured by the Novikov-Shubin invariants which we will introduce in Chapter 2.

Lemma 1.72. Let M be a complete Riemannian manifold without boundary. The inclusion of H_{(2)}^{p}(M) into L^{2}Ω^{p}(M) induces an isometric isomorphism

H_{(2)}^{p}(M) ⟶^{≅} H_{(2)}^{p}(M).

Proof. The following inequalities can easily be derived from Lemma 1.56

clos(B^{p}(M)) = clos(d^{p−1}(Ω_{c}^{p−1}(M)));
Z^{p}(M) = clos(δ^{p+1}(Ω_{c}^{p+1}(M)))^{⊥}.

Now apply Theorem 1.57. ⊓ ⊔

Let M be a cocompact free proper G-manifold with G-invariant Riemannian metric. Suppose that M has no boundary. Let H^{k}Ω^{p}(M) for a nonnegative integer k be the k-th Sobolev space of p-forms on M, i.e. the Hilbert space completion of Ω_{c}^{p}(M) with respect to the inner product or norm

⟨ω, η⟩_{k} = ⟨(1 + Δ_{p})^{k/2}ω, (1 + Δ_{p})^{k/2}η⟩_{L^{2}} = ⟨ω, (1 + Δ_{p})^{k}η⟩_{L^{2}},
||ω||_{k} = \sqrt{||(1 + Δ_{p})^{k/2}ω||_{L^{2}}} = \sqrt{⟨ω, (1 + Δ_{p})^{k}ω⟩_{L^{2}}}

using the L^{2}-inner product (1.52) or L^{2}-norm of (1.53). In particular we get H^{0}Ω^{p}(M) = L^{2}Ω^{p}(M). One can identify H^{2k}Ω^{p}(M) with {ω ∈ L^{2}Ω^{p}(M) | (1 + Δ_{p})^{k}ω ∈ L^{2}Ω^{p}(M)}, where (1 + Δ_{p})^{k}ω is to be understood in the sense of distributions by Lemma 1.70 (2). The operator (1 + Δ_{p})^{k/2} : Ω_{c}^{p}(M) → Ω_{c}^{p}(M) induces for k ≤ l a G-equivariant isometric isomorphism

(1 + Δ_{p})^{k/2} : H^{l}Ω^{p}(M) ⟶^{≅} H^{l−k}Ω^{p}(M)  (1.73)

by the following argument. Obviously (1 + Δ_{p})^{k/2} is well-defined and isometric. In particular its image is closed. It remains to show that it is surjective. For this purpose it suffices to show that the orthogonal complement of the image of (1 + Δ_{p})^{k}: H^{2k}Ω^{p}(M) → L^{2}Ω^{p}(M) is trivial. Since this is the subspace of those elements ω ∈ L^{2}Ω^{p}(M) for which ⟨ω, (1 + Δ_{p})^{k}(η)⟩_{L^{2}} = 0 holds for all η ∈ Ω_{c}^{p}(M), ω lies in the kernel of ((1 + Δ_{p})^{k})_{max}. From Lemma 1.70 (2) we get ((1 + Δ_{p})^{k})_{max} = ((1 + Δ_{p})^{k})_{min}. Hence there is a sequence (ω_{n})_{n≥0} of elements in Ω_{c}^{p}(M) which converges in L^{2}Ω^{p}(M) to ω and for which (1 + Δ_{p})^{k}(ω_{n}) converges in L^{2}Ω^{p}(M) to zero. Since ||ω_{n}||_{L^{2}} ≤ ||(1 + Δ_{p})^{k}(ω_{n})||_{L^{2}} holds, we get ω = 0.

The definition of H^{k}Ω^{p}(ℝ^{n}) corresponds to the usual definition of Sobolev space in the literature (see for instance [473, §7 in Chapter I]). In [145], only Sobolev spaces H^{2k}Ω^{p}(M) are considered and denoted by H^{k}.

Notice that using a fundamental domain of the G-action one obtains a G-equivariant isometric isomorphism from L^{2}Ω^{p}(M) to the tensor product of Hilbert spaces l^{2}(G) ⊗ L^{2}Ω^{p}(G\M) where G acts on l^{2}(G) by left multiplication and on L^{2}Ω^{p}(G\M) trivially [9, 4.1 on page 57 and page 65]. This implies that H^{k}Ω^{p}(M) is a Hilbert N (G)-module in the sense of Definition 1.5 for all k, p ≥ 0. The exterior differential induces a G-equivariant bounded operator, denoted in the same way, d^{p}: H^{k+1}Ω^{p}(M) → H^{k}Ω^{p+1}(M) for all k, p ≥ 0. (To get bounded operators as differentials is the main reasons why we have introduced the Sobolev spaces.) Thus we obtain for l ≥ n with n = dim(M) a Hilbert N (G)-cochain complex H^{l−∗}Ω^{∗}(M) in the sense of Definition 1.15

… → 0 → H^{l}Ω^{0}(M) ⟶^{d^{0}} H^{l−1}Ω^{1}(M) ⟶^{d^{1}} H^{l−2}Ω^{2}(M) ⟶^{d^{2}} …

⟶^{d^{n−1}} H^{l−n}Ω^{n}(M) → 0 → … (1.74)

We have introduced its (reduced) L^{2}-cohomology H_{(2)}^{p}(H^{l−∗}Ω^{∗}(M)) in Definition 1.16. The next lemma will show that it is independent of l ≥ dim(M).

Lemma 1.75. Let l ≥ dim(M). Then the obvious inclusion of H_{(2)}^{p}(M) with the L^{2}-norm into H^{l−p}Ω^{p}(M) is a G-equivariant isometric embedding and induces a G-equivariant isometric isomorphism

H_{(2)}^{p}(M) ⟶^{≅} H_{(2)}^{p}(H^{l−∗}Ω^{∗}(M)).

Proof. Consider ω ∈ H_{(2)}^{p}(M). Obviously ||ω||_{k} = ||ω||_{0} < ∞ for all k ≥ 0. This implies that dω, δdω, δω and dδω are square-integrable. We conclude from Lemma 1.56

⟨ω, ω⟩_{0} = ⟨(1 + Δ_{p})ω, ω⟩_{0} = ⟨ω, ω⟩_{0} + ⟨d^{p}ω, d^{p}ω⟩_{0} + ⟨δ^{p}ω, δ^{p}ω⟩_{0}.

This implies dω = δω = 0. For η ∈ H^{l−p+1}Ω^{p−1}(M) we get from Lemma 1.56

⟨ω, d^{p−1}η⟩_{l−p} = ⟨ω, (1 + Δ_{p})^{l−p}d^{p−1}η⟩_{0} = ⟨δ^{p}ω, (1 + Δ_{p−1})^{l−p}η⟩_{0} = 0.

This shows that H_{(2)}^{p}(M) lies in ker(d^{p}) and is orthogonal to clos(im(d^{p−1})) in H^{l−p}Ω^{p}(M). It remains to show that the orthogonal complement of H_{(2)}^{p}(M) in ker(d^{p}) is contained in clos(im(d^{p−1})).

Given μ ∈ ker(d^{p}), we can decompose it orthogonally as μ = ω + η for ω ∈ H_{(2)}^{p}(M) and η in the orthogonal complement of H_{(2)}^{p}(M) in ker(d^{p}) ⊂ H^{l−p}Ω^{p}(M). Put η′ = (1 + Δ_{p})^{(l−p)/2}(η). Then d_{min}^{p}η′ = 0 and for every ν ∈ H_{(2)}^{p}(M) we get

⟨η′, ν⟩_{0} = ⟨(1 + Δ_{p})^{(l−p)/2}η, ν⟩_{0}
= ⟨(1 + Δ_{p})^{(l−p)/2}η, (1 + Δ_{p})^{(l−p)/2}ν⟩_{0}
= ⟨η, ν⟩_{l−p}
= 0.

From Theorem 1.57, we obtain a sequence of elements η_{n} ∈ Ω_{c}^{p−1}(M) such that η′ = lim_{n→∞} d^{p−1}η_{n} with respect to the L^{2}-norm. Now we derive from 1.73 in H^{l−p}Ω^{p}(M)

η = ((1 + Δ_{p})^{(l−p)/2})^{−1} η′ = lim_{n→∞} d^{p−1} ((1 + Δ_{p−1})^{(l−p)/2})^{−1} η_{n},

where we think of η_{n} as an element in H^{1}Ω^{p−1}(M). Hence η belongs to clos(im(d^{p−1})) in H^{l−p}Ω^{p}(M). ⊓ ⊔

Fix an equivariant smooth triangulation K of M. In particular K is a free cocompact G-CW -complex and we have introduced its cellular L^{2}-cochain complex C_{(2)}^{∗}(K) in Definition 1.29. Next we give a different model for it in terms of the cellular cochain complex C^{∗}(K) = hom_{ℤ}(C_{∗}(K), ℂ). Fix for any simplex in K some (arbitrary) orientation. Then we obtain a basis for C_{p}(K) for each p ≥ 0 and an element in C^{p}(K) is the same as a function f from the set S_{p}(K) of p-dimensional simplices of K to ℂ. We call f squaresummable if ∑_{σ∈S_{n}(K)} |f(σ)|^{2} < ∞. Let l^{2}C^{p}(K) ⊂ C^{p}(K) be the subspace of square-summable cochains. One easily checks that we obtain a subcomplex l^{2}C^{∗}(K) ⊂ C^{∗}(K), the cochain complex of square-summable cochains. It becomes a finite Hilbert N (G)-chain complex with respect to the inner product ⟨f, g⟩_{L^{2}} := ∑_{σ∈S_{n}(K)} f(σ)\overline{g(σ)}.

Lemma 1.76. There is a natural isometric G-chain isomorphism

f^{∗}: C_{(2)}^{∗}(K) ⟶^{≅} l^{2}C^{∗}(K).

Proof. The map f^{p} sends u ∈ hom_{ℤG}(C_{p}(K), l^{2}(G)) to the element f^{p}(u) ∈ map(S_{p}(K), ℂ) which assigns to a simplex σ the coefficient λ_{e} of the unit element e ∈ G in u(σ) = ∑_{g∈G} λ_{g} · g. ⊓ ⊔

If ω ∈ H^{k}Ω^{p}(M) for k > dim(M)/2 + 1, then ω is a C^{1}-form by the Sobolev inequality. In particular we can integrate ω over any oriented psimplex of K and thus obtain an element in C^{p}(K). It turns out that this element actually lies in l^{2}C^{p}(K). By this construction and the isomorphism appearing in Lemma 1.76 we obtain for large enough l > 0 a cochain map of Hilbert N (G)-cochain complexes, the L^{2}-de Rham cochain map

A^{∗}: H^{l−∗}Ω^{∗}(M) → C_{(2)}^{∗}(K).  (1.77)

Notice that C_{(2)}^{∗}(K), l^{2}C^{p}(K), f^{∗} and A^{∗} are independent of the choices of orientations of the simplices. A change of the orientation of a simplex σ affects ∫_{σ} ω and the basis for C_{p}(K) in a way which cancels out.

Next we define a right inverse of A^{∗} as follows. The construction is due to Whitney [522, VII.11 on page 226]. Let {U_{σ}}_{σ∈S_{0}(K)} be the open covering given by the open stars of the 0-simplices. Recall that the closed star st(σ) of a simplex σ consists of all simplices τ which are faces of some simplex τ′ which has σ as face. The open star is the interior of the closed star. Obviously gU_{σ} = U_{gσ} for σ ∈ S_{0}(K) and g ∈ G. Choose a G-invariant subordinate smooth partition {e_{σ}}_{σ∈S_{0}(K)} of unity, i.e. smooth function e_{σ}: M → [0, 1] with support in U_{σ} such that e_{gσ} ∘ l_{g} = e_{σ} for l_{g}: M → M left multiplication with g holds for all σ ∈ S_{0}(K) and g ∈ G and that the (locally finite) sum ∑_{σ∈S_{0}(K)} e_{σ} is constant 1. Given a p-simplex τ with vertices σ_{0}, σ_{1}, …, σ_{p}, let c_{τ} ∈ l^{2}C^{p}(M) be the characteristic function associated to τ and define a smooth p-form W (c_{τ}) with support in the star of τ by

W (c_{τ}) := p! ∑_{i=0}^{p}(−1)^{i}e_{σ_{i}}d^{0}e_{σ_{0}} ∧ … ∧ d^{0}e_{σ_{i−1}} ∧ d^{0}e_{σ_{i+1}} ∧ … ∧ d^{0}e_{σ_{p}}

if p > 0 and by W (c_{τ}) = e_{σ_{0}} if p = 0. Using the isomorphism appearing in Lemma 1.76 we obtain a well-defined cochain map of Hilbert N (G)-cochain complexes, the so called Whitney map.

W^{∗}: C_{(2)}^{∗}(K) → H^{l−∗}Ω^{∗}(M).  (1.78)

Again W^{∗} does not depend on the choice of orientations of the simplices. It satisfies

A^{∗} ∘ W^{∗} = id .  (1.79)

In particular we see that the map induced by A^{∗} on L^{2}-cohomology

H_{(2)}^{∗}(A^{∗}): H_{(2)}^{∗}(H^{l−∗}Ω^{∗}(M)) → H_{(2)}^{∗}(C_{(2)}^{∗}(K))  (1.80)

is surjective. Next we explain why it is injective. This is not done by constructing a homotopy between W^{∗} ∘ A^{∗} and id as it can be done for G\K and G\M since the construction on the quotients is not local and cannot be lifted to K and M. Instead one modifies the definition of W, namely one uses the partition given by barycentric coordinate functions e_{σ} for a vertex σ instead of the smooth partition before. Then the image of this new Whitney map consists of L^{2}-integrable forms and yields a bounded operator W^{p}: C_{(2)}^{p}(K) → L^{2}Ω^{p}(M) (see Lemma 2.79), but the image does not lie in H^{l}Ω^{p}(M) for all l ≥ 0 anymore, since the barycentric coordinate functions are not smooth and hence W (c_{τ}) is not smooth for a p-simplex τ. (We will see in the proof of Lemma 2.79 that W (c_{τ}) is at least continuous.) The advantage of this modified Whitney map lies in the following result [145, Lemma 3.9 on page 164].

Lemma 1.81. Let ω ∈ H^{k}Ω^{p}(M) be a fixed element and k > dim(M)/2+1. For every ϵ > 0 we can find an equivariant smooth triangulation K(ϵ) which is a subdivision of K such that

||ω − W_{K(ϵ)}^{p} ∘ A_{K(ϵ)}^{p}(ω)||_{L^{2}} < ϵ

holds for the L^{2}-de Rham map A_{K(ϵ)}^{p} and the modified Whitney map W_{K(ϵ)}^{p} associated to K(ϵ). ⊓ ⊔

This Lemma 1.81 is the key ingredient in the proof of injectivity of the map (1.80) as we explain next. Because of Lemma 1.72 and Lemma 1.75 it suffices to show for an element ω ∈ H_{(2)}^{p}(M) with H_{(2)}^{∗}(A^{∗})([ω]) = 0 that its class [ω] ∈ H_{(2)}^{p}(M) vanishes.

Fix ϵ > 0. Lemma 1.81 implies for an appropriate subdivision K(ϵ)

||ω − W_{K(ϵ)}^{p} ∘ A_{K(ϵ)}^{p}(ω)||_{L^{2}} < ϵ/2.

There is a cochain map l^{2}C^{∗}(K) → l^{2}C^{∗}(K(ϵ)) which induces an isomorphism on L^{2}-cohomology and is compatible with the de Rham cochain maps for K and K(ϵ). We conclude H_{(2)}^{∗}(A_{K(ϵ)}^{∗})([ω]) = 0. Hence we can find a cochain u ∈ l^{2}C^{p−1}(K(ϵ)) with

1.5 L^{2}-Betti Numbers of Manifolds with Boundary 63

||A_{K(ϵ)}^{p}(ω) − c_{K(ϵ)}^{p−1}(u)||_{L^{2}} < \frac{ϵ}{2 · ||W_{K(ϵ)}^{p}||}.

This implies

||ω − W_{K(ϵ)}^{p} ∘ c_{K(ϵ)}^{p−1}(u)||_{L^{2}}

≤ ||ω − W_{K(ϵ)}^{p} ∘ A_{K(ϵ)}^{p}(ω)||_{L^{2}} + ||W_{K(ϵ)}^{p} ∘ A_{K(ϵ)}^{p}(ω) − W_{K(ϵ)}^{p} ∘ c_{K(ϵ)}^{p−1}(u)||_{L^{2}}
≤ ||ω − W_{K(ϵ)}^{p} ∘ A_{K(ϵ)}^{p}(ω)||_{L^{2}} + ||W_{K(ϵ)}^{p}|| · ||A_{K(ϵ)}^{p}(ω) − c_{K(ϵ)}^{p−1}(u)||_{L^{2}}
≤ ϵ/2 + ||W_{K(ϵ)}^{p}|| · \frac{ϵ}{2 · ||W_{K(ϵ)}^{p}||}
= ϵ.

Now one checks that the image of W_{K(ϵ)}^{p} is contained in the domain of (d^{p}: Ω_{c}^{p}(M) → L^{2}Ω^{p+1}(M))_{max} and that d_{max}^{p−1}∘W_{K(ϵ)}^{p−1} = W_{K(ϵ)}^{p}∘c_{K(ϵ)}^{p−1}. Hence for ϵ > 0 there is v in the domain of (d^{p}: Ω_{c}^{p}(M) → L^{2}Ω^{p+1}(M))_{max} such that ||ω − d_{max}^{p−1}(v)||_{L^{2}} < ϵ holds. Hence the class represented by ω in the reduced L^{2}-de Rham cohomology H_{(2)}^{p}(M) is trivial. This shows that the map appearing in (1.80) is bijective. Now the map appearing in Theorem 1.59 is bijective since it is the composition of the bijective maps appearing in Lemma 1.75 and (1.80). This finishes the proof of Theorem 1.59. ⊓ ⊔

1.5 L^{2}-Betti Numbers of Manifolds with Boundary

We briefly discuss the case of a manifold with boundary. A useful discussion what smooth means for a function or a section of a bundle on a manifold with boundary is given in [467]. Lemma 1.56 says that for a complete Riemannian manifold with boundary and for smooth forms ω ∈ Ω^{p}(M) and η ∈ Ω^{p+1}(M) such that ω, d^{p}ω, η and δ^{p+1}η are square-integrable, we get

⟨d^{p}ω, η⟩_{L^{2}} − ⟨ω, δ^{p+1}η⟩_{L^{2}} = ∫_{∂M}(ω ∧ ∗^{p+1}η)|_{∂M}.  (1.82)

The classical de Rham isomorphism is an isomorphism for any manifold regardless whether it has a boundary or not (see Theorem 1.47).

Suppose that the boundary ∂M of the complete Riemannian manifold M is the disjoint union of ∂_{0}M and ∂_{1}M where we allow that ∂_{0}M, ∂_{1}M or both are empty. A form ω ∈ Ω^{p}(M) satisfies Dirichlet boundary conditions on ∂_{0}M if ω|_{∂_{0}M} = 0 holds. It satisfies Neumann boundary conditions on ∂_{1}M if ∗^{p}ω|_{∂_{1}M} = 0 holds. Define

Ω_{2}^{p}(M, ∂_{0}M) := {ω ∈ Ω_{c}^{p}(M) | ω|_{∂_{0}M} = 0, (δ^{p}ω)|_{∂_{0}M} = 0,
(∗^{p}ω)|_{∂_{1}M} = 0, (∗^{p+1}d^{p}ω)|_{∂_{1}M} = 0}; (1.83)
Ω_{1}^{p}(M, ∂_{0}M) := {ω ∈ Ω_{c}^{p}(M) | ω|_{∂_{0}M} = 0, (∗^{p}ω)|_{∂_{1}M} = 0}; (1.84)
Ω_{d}^{p}(M, ∂_{0}M) := {ω ∈ Ω_{c}^{p}(M) | ω|_{∂_{0}M} = 0};  (1.85)
Ω_{δ}^{p}(M, ∂_{0}M) := {ω ∈ Ω_{c}^{p}(M) | (∗^{p}ω)|_{∂_{1}M} = 0}.  (1.86)

Define the space of L^{2}-integrable harmonic smooth p-forms satisfying boundary conditions

H_{(2)}^{p}(M, ∂_{0}M) := {ω ∈ Ω_{2}^{p}(M, ∂_{0}M) | Δ_{p}(ω) = 0, ∫_{M} ω ∧ ∗ω < ∞}. (1.87)

We have the following version of the Hodge-de Rham Theorem 1.57.

Theorem 1.88. (Hodge-de Rham Theorem for manifolds with boundary). Let M be a complete Riemannian manifold whose boundary ∂M is the disjoint union ∂_{0}M and ∂_{1}M. Then we have the so called Hodge-de Rham decomposition

L^{2}Ω^{p}(M) = H_{(2)}^{p}(M, ∂_{0}M) ⊕ clos(d^{p−1}(Ω_{d}^{p−1}(M, ∂_{0}M))) ⊕
clos(δ^{p+1}(Ω_{δ}^{p+1}(M, ∂_{0}M))).

We have the following version of the L^{2}-Hodge-de Rham Theorem 1.59.

Theorem 1.89. (L^{2}-Hodge-de Rham Theorem for manifolds with boundary.) Let M be a cocompact free proper G-manifold with G-invariant Riemannian metric whose boundary ∂M is the disjoint union of G-spaces ∂_{0}M and ∂_{1}M. Let (K; ∂_{0}K, ∂_{1}K) be an equivariant smooth triangulation of the triad (M; ∂_{0}M, ∂_{1}M). Then integration defines an isomorphism of finitely generated Hilbert N (G)-modules

H_{(2)}^{p}(M, ∂_{0}M) ⟶^{≅} H_{(2)}^{p}(K, ∂_{0}K).

The detailed proof of Theorem 1.88 and of Theorem 1.89 in the more general context of Riemannian manifolds with bounded geometry can be found in [459, Theorem 5.10, Theorem 8.2 and Corollary 8.15] (see also [322, Theorem 5.13]). Theorem 1.88 remains true if one replaces Ω_{d}^{p−1}(M, ∂_{0}M) by Ω_{1}^{p−1}(M, ∂_{0}M) and Ω_{δ}^{p+1}(M, ∂_{0}M) by Ω_{1}^{p+1}(M, ∂_{0}M).

The expression (1.60) in terms of the heat kernel on M for the analytic L^{2}-Betti numbers which are defined by

b_{p}^{(2)}(M, ∂_{0}M) := dim_{N (G)}(H_{(2)}^{p}(M, ∂_{0}M))  (1.90)

carries over to the situation studied in Theorem 1.89 if one uses Ω_{2}(M, ∂_{0}M) as initial domain for Δ_{p}.

There is also the following version of Lemma 1.72 proved in [459, Theorem 6.2]. Define the (reduced) L^{2}-cohomology H_{(2)}^{p}(M, ∂_{0}M) as in Definition 1.71 but now using d^{p}: Ω_{d}^{p}(M, ∂_{0}M) → L^{2}Ω^{p+1}(M) instead of d^{p}: Ω_{c}^{p}(M) → L^{2}Ω^{p+1}(M).

Lemma 1.91. Let M be a complete Riemannian manifold whose boundary ∂M is the disjoint union of ∂_{0}M and ∂_{1}M. Then the inclusion of H_{(2)}^{p}(M, ∂_{0}M) into L^{2}Ω_{p}(M) induces an isomorphism

H_{(2)}^{p}(M, ∂_{0}M) ⟶^{≅} H_{(2)}^{p}(M, ∂_{0}M).

Notice that we do not require here (nor later in the context of Novikov-Shubin invariants) that the metric is a product near the boundary. However, this will be crucial when we deal with torsion and L^{2}-torsion in Remark 3.162.

1.6 Miscellaneous

The following result is sometimes useful since it gives a criterion for the reduced L^{2}-cohomology to be non-trivial.

Lemma 1.92. Let M be a complete Riemannian manifold without boundary. Denote by i: H_{c}^{p}(M; ℂ) → H^{p}(M; ℂ) the natural map from the cohomology with compact support to the cohomology of M. Then im(i) injects into the reduced L^{2}-cohomology H_{(2)}^{p}(M).

Proof. Let k: H_{c}^{p}(M; ℂ) → H_{(2)}^{p}(M) be the canonical map. Choose any linear map s: im(i) → H_{c}^{p}(M; ℂ) with i ∘ s = id. We want to show that k ∘ s is injective. It suffices to prove for a smooth closed p-form ω with compact support satisfying k([ω]) = 0 that i([ω]) = 0 holds.

Since k([ω]) = 0, there is a sequence (η_{n})_{n≥0} in Ω_{c}^{p−1}(M) such that ω = lim_{n→∞} d^{p−1}(η_{n}) with respect to the L^{2}-norm holds. Suppose i([ω]) ≠ 0. Let p: \overline{M} → M be the orientation covering of M. This is a double covering with orientable total space and the induced map p^{∗}: H^{p}(M; ℂ) → H^{p}(\overline{M}; ℂ) is injective. Hence p^{∗}ω represents a non-zero element in H^{p}(\overline{M}; ℂ). Put m = dim(M). By Poincaré duality [57, (5.4) on page 44 and Remark I.5.7 on page 46] there is a smooth (m − p)-form μ with compact support on \overline{M} such that d^{m−p}μ = 0 and ∫\overline{M} p^{∗}ω∧μ is different from zero. This yields the contradiction

∫\overline{M} p^{∗}ω ∧ μ = lim_{n→∞} ∫\overline{M} d^{p−1}p^{∗}η_{n} ∧ μ = lim_{n→∞} ∫\overline{M} d^{m−1}(p^{∗}η_{n} ∧ μ) = 0.⊓ ⊔

Cellular L^{2}-Betti numbers and L^{2}-(co-)homology can also be defined for a cocompact proper G-CW -complex X. Recall that the isotropy group of each point of a free G-CW -complex is trivial, whereas it is finite for a proper G-CW -complex. The point is that the cellular L^{2}-chain complex C_{∗}^{(2)}(X) = l^{2}(G) ⊗_{ℤG} C_{∗}(X) is still defined as Hilbert N (G)-chain complex for a proper G-CW -complex. After a choice of a characteristic map (Q_{i}^{n}, q_{i}^{n}): G/H_{i} × (D^{n}, S^{n−1}) → (X_{n}, X_{n−1}) for each element of the set I_{n} of equivariant ndimensional cells we obtain explicit isomorphisms

C_{n}(X) ≅ ⨁_{i∈I_{n}} ℤ[G/H_{i}];
C_{n}^{(2)}(X) ≅ ⨁_{i∈I_{n}} l^{2}(G/H_{i}),

and we put on C_{n}^{(2)}(X) the induced Hilbert N (G)-module structure. This is independent of the choice of characteristic maps since for a different choice the two identifications of C_{n}(X) differ by a direct sum of automorphisms of the shape ϵ_{i} · l_{g_{i}}: ℤ[G/H_{i}] → ℤ[G/H_{i}] for some ϵ_{i} ∈ {±1} and g_{i} ∈ G.

Theorem 1.35 (1), (3), (4), (8), (9) and (10) remain true if one replaces free by proper. The basic observation is that C_{∗}^{(2)}(X) can be written as l^{2}(G) ⊗_{ℂG} (ℂ ⊗_{ℤ} C_{∗}(X)) and ℂ ⊗_{ℤ} C_{n}(X) is a finitely generated projective ℂ[G]-module for all n. Theorem 1.35 (2) becomes

∑_{p≥0}(−1)^{p} · b_{p}^{(2)}(X) = ∑_{p≥0}(−1)^{p} · ∑_{i∈I_{p}} |H_{i}|^{−1}  (1.93)

since dim_{N (G)}(l^{2}(G/H_{i})) = |H_{i}|^{−1}. The space H_{(2)}^{p}(M) of harmonic L^{2}-forms of a cocompact proper G-manifold M with G-invariant Riemannian metric inherits the structure of a Hilbert N (G)-module and Theorem 1.59 remains true without the assumption that M is free.

We will give a combinatorial approach to the L^{2}-Betti numbers in Section 3.7 which is useful for concrete calculations.

We will explain a proportionality principle for L^{2}-Betti numbers in Theorem 3.183, Theorem 7.34 and Corollary 7.37.

We will compute the values of L^{2}-Betti numbers of the universal coverings of compact 3-manifolds in Theorem 4.1 and of locally symmetric spaces in Theorem 5.12 and (5.15).

A more algebraic treatment of Hilbert modules, dimension functions and L^{2}-Betti numbers and a definition of L^{2}-Betti numbers for arbitrary topological spaces with an arbitrary G-action will be given in Chapter 6. This will apply in particular to EG for any (discrete) group G.

We will discuss the behaviour of L^{2}-Betti numbers of groups under quasiisometry and measure equivalence in Section 7.5.

Lemma 1.94. Let B be the set of real numbers r, for which there exists a CW -complex X of finite type and an integer p ≥ 0 with b_{p}^{(2)}(X̃) = r. Then B is countable and contains {r | r ∈ ℚ, r ≥ 0}.

Proof. We get {r | r ∈ ℚ, r ≥ 0} ⊂ B from Example 1.38. Let B′ be the set of real numbers r, for which there exists a finitely presented group G, positive integers m and n and a matrix A ∈ M(m, n, ℤG) satisfying

r = dim_{N (G)} (ker (r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n})) ,

where r_{A}^{(2)} is given by right multiplication with A. The fundamental group π of a CW -complex of finite type X is finitely presented and

b_{p}^{(2)}(X̃) = dim_{N (π)}(ker(c_{p}^{(2)})) + dim_{N (π)}(ker(c_{p+1}^{(2)})) − n_{p+1}

follows from Additivity (see Theorem 1.12 (2)), where n_{p+1} is the number of (p + 1)-cells of X. Hence it suffices to show that B′ is countable. Since there are only countably many finite presentations and ℤG is countable for any finitely presented and hence countable group G, Lemma 1.94 follows. ⊓ ⊔

In Chapter 10 we deal with the question how large the set B is. At the time of writing no counterexample to the statement B = {q ∈ ℚ | q ≥ 0} is known to the author.

We will deal with the Singer Conjecture 11.1 that the L^{2}-Betti numbers of the universal covering of an aspherical closed manifold vanish outside the middle dimension in Chapter 11.

We will relate the L^{2}-Betti numbers of the universal covering of an aspherical closed manifold to its simplicial and its minimal volume (see Conjecture 14.1 and Subsection 14.2.6).

Question 1.95. (Vanishing of L^{2}-Betti numbers of the base and the total space of a fibration). Let F → E ⟶^{p} B be a fibration of connected finite CW -complexes. Suppose that all L^{2}-Betti numbers of B̃ are trivial. Does this imply that all the L^{2}-Betti numbers of Ẽ are trivial?

The answer is yes if B = S^{1} by Theorem 1.39. The answer is also known to be affirmative if one of the following conditions is satisfied [61]: i.) B is an S^{1}-CW -complex such that the inclusion of one (and hence all) orbits induces a map on the fundamental groups with infinite image. ii.) The map π_{1}(p): π_{1}(E) → π_{1}(B) is bijective and π_{1}(B) is virtually poly-cyclic. iii.) The map π_{1}(p): π_{1}(E) → π_{1}(B) is bijective and π_{1}(B) operates trivially on H_{∗}(F). If B is aspherical, more information will be given in Theorem 7.4.

Using the center-valued trace (see Theorem 9.5) one can define a centervalued von Neumann dimension and thus elements b_{p}^{u}(X; N (G)), which take values in the center Z(N (G)) of the von Neumann algebra N (G). Essentially we get for any conjugacy class (g), which contains only finitely many elements, a number. The value at (1) is the L^{2}-Betti number. Lott [321] defines analytically delocalized L^{2}-Betti numbers for the universal covering M̃ of a closed Riemannian manifold M, which assigns to any conjugacy class a number. The invariant is presently only defined under certain technical assumptions. For example, the p-th delocalized Betti number is well-defined if G is virtually abelian, or if G is virtually nilpotent or Gromov hyperbolic and there is a gap away from zero in the spectrum of Δ_{p} [321, Proposition 6].

Elek [171] defines an invariant b_{E}^{p}(X̃) for a finite CW -complex X replacing C_{(2)}^{∗}(X̃) by hom_{ℤ}(C_{∗}(X̃); 𝔽_{2}) for 𝔽_{2} the field of two elements and replacing the von Neumann dimension by the topological entropy of linear subshifts, provided that G is amenable. It has properties similar to the L^{2}-Betti numbers.

Finally we give some further references. Survey articles on L^{2}-cohomology and L^{2}-Betti numbers are [164], [168], [237, section 8], [332], [338], [341], [360] and [411]. More information about Morse inequalities in the L^{2}-context can be found for instance in [182], [363], [364], [401], [400], [475].

Exercises

1.1. Let H ⊂ G be a subgroup. Show that the Hilbert space l^{2}(G/H) with the obvious G-action is a Hilbert N (G)-module if and only if H is finite, and that in this case its von Neumann dimension is |H|^{−1}.

1.2. Show that Lemma 1.9 (5) remains true if one only requires that the rows are weakly exact instead of exact, provided that all Hilbert N (G)-modules have finite dimension.

1.3. Show by an example that Theorem 1.12 (4) becomes false if one drops the condition dim_{N (G)}(V) < ∞.

1.4. Consider the following diagram of Hilbert N (G)-modules of finite dimension with weakly exact rows

U_{1} ⟶^{u_{1}} U_{2} ⟶^{u_{2}} U_{3} ⟶^{u_{3}} U_{4} ⟶^{u_{4}} U_{5}
f_{1}↓  ^{f_{2}}↓  ^{f_{3}}↓  ^{f_{4}}↓  ^{f_{5}}↓
V_{1} ⟶^{v_{1}} V_{2} ⟶^{v_{2}} V_{3} ⟶^{v_{3}} V_{4} ⟶^{v_{4}} V_{5}

Suppose that f_{1} has dense image, f_{2} and f_{4} are weak isomorphisms and f_{5} is injective. Show that then f_{3} is a weak isomorphism.

1.5. The group von Neumann algebra N (G) carries an involution of rings given by taking adjoints of bounded G-equivariant operators l^{2}(G) → l^{2}(G). Show that the ring homomorphism i: N (H) → N (G) given by induction (see Definition 1.23) is compatible with these involutions.

1.6. Find as many different proofs as possible for the fact that all the L^{2}- Betti numbers of the universal covering of the n-dimensional torus T^{n} for n ≥ 1 vanish.

1.7. Let X be an aspherical CW -complex of finite type. Suppose that its fundamental group contains a normal infinite cyclic subgroup. Show that then all L^{2}-Betti numbers of X̃ vanish.

1.8. Show that a symmetric densely defined linear operator T: dom(T) → H is selfadjoint if and only if the range of T ± i · id is H and that it is essentially selfadjoint if and only if the range of T ± i · id in H is dense.

1.9. Let f: H → H be a linear map of finite dimensional Hilbert spaces which is symmetric. Show that the projection E_{λ}^{f} of the associated spectral family is the sum ⨁_{μ≤λ} pr_{μ}, where pr_{μ} is the projection onto the eigenspace of μ ∈ ℝ. Let u: H → ℂ^{n} be a unitary operator such that ufu^{−1} : ℂ^{n} → ℂ^{n} is given by a diagonal matrix D with real entries. Given a Borel function g: ℝ → ℂ, let g(D) be the diagonal matrix obtained from D by applying g to the diagonal entries. Show that ∫_{−∞}^{∞} g(λ)dE_{λ} in the sense of (1.64) is u^{−1}g(D)u.

1.10. Let M be a complete Riemannian manifold without boundary. Show that we obtain a Hilbert cochain complex (in the sense of Definition 1.15 for G = 1) by

… ⟶^{d_{min}^{p−2}} dom(d_{min}^{p−1}) ⟶^{d_{min}^{p−1}} dom(d_{min}^{p}) ⟶^{d_{min}^{p}} …

if we equip dom(d_{min}^{p}) with the Hilbert structure given by (1.63). Show that its unreduced and reduced L^{2}-cohomology in the sense of Definition 1.16 agrees with the notions of Definition 1.71.

1.11. Show by giving an example that Lemma 1.56 does not hold without the hypothesis that M is complete.

1.12. Show for any free ℤ-CW -complex X of finite type that b_{p}^{(2)}(X) ≤ b_{p}(ℤ\X) holds.

1.13. Let M be an orientable closed 4-manifold with π_{1}(M) = ℤ/2 ∗ ℤ/2. Show

b_{2}^{(2)}(M̃) = χ(M) = 2 + b_{2}(M).

1.14. Fix an integer n ≥ 1. Show that for a sequence r_{1}^{(2)}, r_{2}^{(2)}, …, r_{n}^{(2)} of non-negative rational numbers and a sequence r_{1}, r_{2}, …, r_{n} of non-negative integers there is a finite (n + 1)-dimensional CW -complex X with infinite fundamental group satisfying b_{p}^{(2)}(X̃) = r_{p}^{(2)} and b_{p}(X) = r_{p} for p = 1, …, n.

2. Novikov-Shubin Invariants

Introduction

In this chapter we introduce and study Novikov-Shubin invariants for Hilbert chain complexes and for regular coverings of CW -complexes of finite type or of compact manifolds.

We will associate to the Laplacian Δ_{p} of a cocompact free proper Gmanifold M with G-invariant Riemannian metric its spectral density function F_{p}^{Δ}(M): [0, ∞) → [0, ∞) which assigns to λ the von Neumann dimension of the image of E_{λ}, where {E_{λ} | λ ≥ 0} is the spectral family of Δ_{p}. We rediscover the p-th L^{2}-Betti number b_{p}^{(2)}(M) by F_{p}^{Δ}(M)(0). If G is finite F_{p}^{Δ}(M) measures how many eigenvalues ≤ λ, counted with multiplicity, Δ_{p} has, and this is the right intuition also for infinite G. F_{p}^{Δ}(M) contains a lot of information about the spectrum of Δ_{p}.

The spectral density function is an example of a so called density function F: [0, ∞) → [0, ∞], i.e. F is monotone non-decreasing and right-continuous. Two density functions F and G are called dilatationally equivalent if there are C > 0 and ϵ > 0 such that G(C^{−1} · λ) ≤ F (λ) ≤ G(C · λ) holds for all λ ∈ [0, ϵ]. The spectral density function itself depends on the Riemannian structure. However, it turns out that its dilatational equivalence class only depends on the G-homotopy type of M. Hence it becomes a very interesting object because it is defined in terms of analytic data but is also a topological invariant. The Novikov-Shubin invariant of a density function is defined by

α(F) = lim inf_{λ→0^{+}} \frac{ln(F (λ) − F (0))}{ln(λ)} ∈ [0, ∞],

provided that F (λ) > F (0) holds for all λ > 0. Otherwise, one puts formally α(F) = ∞^{+}. It measures how fast F (λ) approaches F (0) for λ → 0^{+} and takes the value ∞^{+} if and only if there is a gap in the spectrum at zero. It only depends on the dilatational equivalence class of F. Define the Novikov-Shubin invariant α_{p}^{Δ}(M) of M by the one of F_{p}^{Δ}(M). Novikov-Shubin invariants were originally defined in [400],[401]. In some sense Novikov-Shubin invariants measure how “thin” the spectrum of Δ_{p} at zero is. We will also introduce α_{p}(M) which is defined in terms of the differential d^{p−1}. It measures the difference between the unreduced and the reduced L^{2}-de Rham cohomology, the unreduced and the reduced one agree in dimension p if and only if α_{p}(M) = ∞^{+}. We have α_{p}^{Δ}(M) = 1/2 · min{α_{p}(M), α_{p+1}(M)}.

Via the Laplace transform the dilatational equivalence class of the spectral density function and the Novikov-Shubin invariants are invariants of the asymptotic behaviour of the heat kernel for large times. Such invariants reflect the global geometry but are in general very hard to study and to compute. Notice that index theory yields invariants of the asymptotic behaviour of the heat kernel for small times and hence local invariants in contrast to invariants of the large time asymptotics which cannot be expressed locally.

We will introduce and study spectral density functions and Novikov-Shubin invariants for Hilbert chain complexes and possibly unbounded operators in Section 2.1. In particular we will show that the spectral density function is a homotopy invariant and satisfies a subadditivity-relation for short exact sequences. In Section 2.2 we apply this to the cellular L^{2}-chain complex and thus obtain the cellular spectral density function and Novikov-Shubin invariants of a free G-CW -complex of finite type. We prove their main properties such as homotopy invariance, Poincaré duality, product formula and express the first Novikov-Shubin invariant in terms of the growth rate of the fundamental group. We give an explicit formula in the case G = ℤ. In Section 2.3 we introduce their analytic counterparts and show in Section 2.4 that the analytic and the cellular spectral density functions are dilatationally equivalent and hence the analytic and cellular Novikov-Shubin invariants agree [167], [240]. In Section 2.5 we discuss the conjecture that the Novikov-Shubin invariants are either rational positive numbers, ∞ or ∞^{+}. We briefly treat the case of a manifold with boundary in Section 2.6.

To get a quick overview about Novikov-Shubin invariants one should read through Sections 2.2, 2.3 and 2.5. The material of this chapter is rather independent of the following chapters which can be read without studying this chapter beforehand.

2.1 Spectral Density Functions

In this section we introduce and study the spectral density function associated to morphisms of Hilbert modules and to Hilbert chain complexes.

2.1.1 Spectral Density Functions of Morphisms

We next deal with the spectral density function of a map of Hilbert modules. We treat the more general case of an unbounded densely defined operator since this setting will be needed when we deal with the analytic case. If one only wants to investigate the cellular version it suffices to treat bounded operators.

Definition 2.1 (Spectral density function). Let U and V be Hilbert N (G)- modules. Let f: dom(f) ⊂ U → V be a G-equivariant closed densely defined operator. For λ ≥ 0 we define L(f, λ) as the set of Hilbert N (G)-submodules L ⊂ U such that L ⊂ dom(f) and ||f(x)|| ≤ λ · ||x|| holds for all x ∈ L. Define the spectral density function of f

F (f): [0, ∞) → [0, ∞], λ ↦ sup{dim_{N (G)}(L) | L ∈ L(f, λ)}.

Define the L^{2}-Betti number of f by

b^{(2)}(f) := dim_{N (G)}(ker(f)) = F (f)(0).

We call f Fredholm if there is ϵ > 0 such that F (f)(ϵ) < ∞.

We will conclude from Lemma 2.3 that the definition of Fredholm for a morphism of Hilbert N (G)-modules of Definition 1.20 and Definition 2.1 agree.

Notice that ker(f) ⊂ U is closed and hence a Hilbert N (G)-module and its von Neumann dimension is defined. The meaning of the spectral density function becomes more evident if one expresses it in terms of the spectrum of f^{∗}f, as we explain next.

Lemma 2.2. Let f: dom(f) ⊂ H_{1} → H_{2} be a closed densely defined operator of Hilbert spaces H_{1} and H_{2}. Define

dom(f^{∗}f) := {x ∈ H_{1} | x ∈ dom(f), f(x) ∈ dom(f^{∗})}

and thus an operator f^{∗}f: dom(f^{∗}f) ⊂ H_{1} → H_{1}. Then

(1) The subspace dom(f^{∗}f) is dense and f^{∗}f is a selfadjoint operator. Moreover, dom(f^{∗}f) is a core for dom(f), i.e. dom(f^{∗}f) ⊂ dom(f) and for any x ∈ dom(f) there is a sequence x_{n} ∈ dom(f^{∗}f) such that lim_{n→∞} x_{n} = x and lim_{n→∞} f(x_{n}) = f(x) holds;

(2) Let λ ≥ 0 and x ∈ dom(f). Then

||f(x)|| > λ · ||x||  if E_{λ^{2}}^{f^{∗}f}(x) = 0, x ≠ 0;
||f(x)|| ≤ λ · ||x||  if E_{λ^{2}}^{f^{∗}f}(x) = x,

where {E_{λ}^{f^{∗}f} | λ ∈ ℝ} denotes the spectral family of the selfadjoint operator f^{∗}f.

Proof. (1) see [287, Theorem V.3.24 on page 275].

(2) Suppose that E_{λ^{2}}^{f^{∗}f}(x) = 0 and x ≠ 0. Choose a sequence x_{n} ∈ dom(f^{∗}f) such that lim_{n→∞} x_{n} = x and lim_{n→∞} f(x_{n}) = f(x) holds. Since x ≠ 0, and 0 = E_{λ^{2}}^{f^{∗}f}(x) = lim_{μ→λ^{2}+} E_{μ}^{f^{∗}f}(x) holds, we can find ϵ > 0 with the property

⟨E_{λ^{2}+ϵ}^{f^{∗}f}(x), x⟩ < \frac{1}{2}⟨x, x⟩.

We get from the definition of the spectral family

||f(x_{n})||^{2} = ⟨f^{∗}f(x_{n}), x_{n}⟩
= ∫_{0}^{∞} μ d⟨E_{μ}^{f^{∗}f}(x_{n}), x_{n}⟩
≥ ∫_{(λ^{2},∞)} μ d⟨E_{μ}^{f^{∗}f}(x_{n}), x_{n}⟩
= ∫_{(λ^{2},λ^{2}+ϵ]} μ d⟨E_{μ}^{f^{∗}f}(x_{n}), x_{n}⟩ + ∫_{(λ^{2}+ϵ,∞)} μ d⟨E_{μ}^{f^{∗}f}(x_{n}), x_{n}⟩
≥ λ^{2} · (⟨E_{λ^{2}+ϵ}^{f^{∗}f}(x_{n}), x_{n}⟩ − ⟨E_{λ^{2}}^{f^{∗}f}(x_{n}), x_{n}⟩)
+(λ^{2} + ϵ) · (⟨x_{n}, x_{n}⟩ − ⟨E_{λ^{2}+ϵ}^{f^{∗}f}(x_{n}), x_{n}⟩)
= (λ^{2} + ϵ) · ⟨x_{n}, x_{n}⟩ − ϵ · ⟨E_{λ^{2}+ϵ}^{f^{∗}f}(x_{n}), x_{n}⟩ − λ^{2} · ⟨E_{λ^{2}}^{f^{∗}f}(x_{n}), x_{n}⟩.

Taking the limit n → ∞ yields

||f(x)||^{2} ≥ (λ^{2} + ϵ) · ⟨x, x⟩ − ϵ · ⟨E_{λ^{2}+ϵ}^{f^{∗}f}(x), x⟩ − λ^{2} · ⟨E_{λ^{2}}^{f^{∗}f}(x), x⟩
≥ (λ^{2} + ϵ) · ||x||^{2} − \frac{ϵ}{2} · ||x||^{2} − 0
= (λ^{2} + \frac{ϵ}{2}) · ||x||^{2}
> λ^{2} · ||x||^{2}

and hence the desired inequality ||f(x)|| > λ·||x||. The easier analogous proof of ||f(x)|| ≤ λ · ||x|| for E_{λ^{2}}^{f^{∗}f}(x) = x is left to the reader. ⊓ ⊔

Lemma 2.3. Let U and V be Hilbert N (G)-modules. Let f: dom(f) ⊂ U → V be a G-equivariant closed densely defined operator. Then for λ ∈ ℝ the spectral projection E_{λ^{2}}^{f^{∗}f} is G-equivariant and

F (f)(λ) = dim_{N (G)}(im(E_{λ^{2}}^{f^{∗}f})).

Proof. As f^{∗}f commutes with the unitary G-action, the same is true for its spectral projections. Since im(E_{λ^{2}}^{f^{∗}f}) ∈ L(f, λ) for λ ≥ 0 holds by Lemma 2.2 (2), we have dim_{N (G)}(im(E_{λ^{2}}^{f^{∗}f})) ≤ F (f)(λ). From Lemma 2.2 (2) and Theorem 1.12 (2) we conclude that E_{λ^{2}}^{f^{∗}f}|_{L}: L → im(E_{λ^{2}}^{f^{∗}f}) is injective and hence dim_{N (G)}(L) ≤ dim_{N (G)}(im(E_{λ^{2}}^{f^{∗}f})) for L ∈ L(f, λ). ⊓ ⊔

Lemma 2.4. Let U and V be Hilbert N (G)-modules. Let f: dom(f) ⊂ U → V be a G-equivariant closed densely defined operator. Let f = u|f| be its polar decomposition into a partial isometry u and a positive operator |f| with dom(f) = dom(|f|). Suppose that f is Fredholm and b^{(2)}(f^{∗}) is finite. Then |f| and f^{∗} are Fredholm,

b^{(2)}(f) = b^{(2)}(|f|)

and

F (f)(λ) − b^{(2)}(f) = F (|f|)(λ) − b^{(2)}(|f|) = F (f^{∗})(λ) − b^{(2)}(f^{∗}).

Proof. We can assume without loss of generality that f and f^{∗} are injective, otherwise pass to the induced operator ker(f)^{⊥} → clos(im(f)). Obviously F (u|f|)(λ) = F (|f|)(λ) = F (|f|u^{−1})(λ) and f^{∗} = |f|u^{−1} where dom(u|f|) = dom(|f|) and dom(|f|u^{−1}) = u(dom(f)). ⊓ ⊔

The intuitive meaning of the spectral density function becomes clear from the next example, namely, F (f)(λ) counts with multiplicity the eigenvalues μ of |f| satisfying μ ≤ λ.

Example 2.5. Let G be finite and f: U → V be a map of finitely generated Hilbert N (G)-modules, i.e. of finite dimensional unitary G-representations. Then F (f) is the right-continuous step function whose value at λ is the sum of the complex dimensions of the eigenspaces of f^{∗}f for eigenvalues μ ≤ λ^{2} divided by the order of G, or, equivalently, the sum of the complex dimensions of the eigenspaces of |f| for eigenvalues μ ≤ λ divided by the order of G.

Example 2.6. Let G = ℤ^{n}. In the sequel we use the notation and the identification N (ℤ^{n}) = L^{∞}(T^{n}) of Example 1.4. For f ∈ L^{∞}(T^{n}) the spectral density function F (M_{f}) of M_{f}: L^{2}(T^{n}) → L^{2}(T^{n}) sends λ to the volume of the set {z ∈ T^{n} | |f(z)| ≤ λ} (see Example 1.11).

Definition 2.7. We say that a function F: [0, ∞) → [0, ∞] is a density function if F is monotone non-decreasing and right-continuous. If F and G are two density functions, we write F ⪯ G if there are C > 0 and ϵ > 0 such that F (λ) ≤ G(C · λ) holds for all λ ∈ [0, ϵ]. We say that F and G are dilatationally equivalent (in signs F ≃ G) if F ⪯ G and G ⪯ F. We say that F is Fredholm if there exists λ > 0 such that F (λ) < ∞.

Of course, the spectral density function F (f) is a density function, and f is Fredholm if and only if the density function F (f) is Fredholm. Recall that for a function h: (0, ∞) → [0, ∞) its limit inferior for λ → 0+ is defined by

lim inf_{λ→0+} h(λ) := sup {inf{h(λ) | 0 < λ ≤ μ} | 0 < μ}  ∈ [0, ∞].

Definition 2.8 (Novikov-Shubin invariants). Let F be a Fredholm density function. The L^{2}-Betti number of F is

b^{(2)}(F) := F (0).

Its Novikov-Shubin invariant is

α(F) := lim inf_{λ→0+} \frac{ln(F (λ) − F (0))}{ln(λ)} ∈ [0, ∞],

provided that F (λ) > b^{(2)}(F) holds for all λ > 0. Otherwise, we put α(F) := ∞^{+}. If f is a Fredholm morphism of Hilbert N (G)-modules, we write

α(f) = α(F (f)).

Here ∞^{+} is a new formal symbol which should not be confused with ∞. We have α(F) = ∞^{+} if and only if there is an ϵ > 0 such that F (ϵ) = b^{(2)}(F). Notice that b^{(2)}(f) = b^{(2)}(F (f)) in the notation of Definition 2.1 and Definition 2.8. The Novikov-Shubin invariant of a spectral density function F measures how fast F (λ) approaches F (0) for λ → 0+. It is independent of the actual value F (0) and depends only on the germ of F (λ) at zero. The Novikov-Shubin invariant of a Fredholm map f: U → V of Hilbert N (G)- modules measures how “thick” the spectrum of |f| (or f^{∗}f) is at zero. It is ∞^{+} if and only if the spectrum of |f| (or f^{∗}f) has a gap at zero. In particular α(f) = ∞^{+} always holds if G is finite. This shows that the Novikov-Shubin invariants are only interesting for infinite G.

Example 2.9. In this example we show that any possible value can occur as Novikov-Shubin invariant of a map of Hilbert N (G)-modules and in particular of a spectral density function. Define spectral density functions F_{t} for t ∈ [0, ∞] ∐{∞^{+}} by F_{t}(0) = 0 and for λ > 0 by

F_{0}(λ) = \{ \begin{matrix}0 & λ = 0; \\ \frac{−1}{ln(λ)} & 0 < λ < e^{−1}; \\ 1 & e^{−1} < λ; \end{matrix}
F_{t}(λ) = λ^{t}  t ∈ (0, ∞);
F_{∞}(λ) = exp(−λ^{−1});
F_{∞^{+}}(λ) = 0.

Then one easily checks for t ∈ [0, ∞] ∐{∞^{+}}

α(F_{t}) = t.

Using Example 2.6 it is not hard to construct for t ∈ [0, ∞] ∐{∞^{+}} maps f_{t}: l^{2}(ℤ) → l^{2}(ℤ) of Hilbert N (ℤ)-modules such that F (f_{t})(λ) = F_{t}(λ) for small λ and hence α(f_{t}) = t holds.

Notation 2.10. Define an ordering on [0, ∞] ∐{∞^{+}} by the standard ordering on ℝ along with r < ∞ < ∞^{+} for all r ∈ ℝ. For all α, β ∈ [0, ∞] ∐{∞^{+}} we define

\frac{1}{α} ≤ \frac{1}{β} ⇔ α ≥ β.

Given α, β ∈ [0, ∞] ∐{∞^{+}}, we give meaning to γ in the expression

\frac{1}{α} + \frac{1}{β} = \frac{1}{γ}

as follows: If α, β ∈ (0, ∞), let γ be the real number for which this arithmetic expression of real numbers is true. If α = 0 or β = 0, put γ = 0. If α ∈ (0, ∞) and β ∈ {∞, ∞^{+}}, put γ to be α. If β ∈ (0, ∞) and α ∈ {∞, ∞^{+}}, put γ to be β. If α and β belong to {∞, ∞^{+}} and are not both ∞^{+}, put γ = ∞. If both α and β are ∞^{+}, put γ = ∞^{+}. For r, α ∈ [0, ∞] ∐{∞^{+}} we define

r · α ∈ [0, ∞] ∐{∞^{+}}

as follows. Given r ∈ [0, ∞) and α ∈ [0, ∞), we define r · α ∈ [0, ∞) to be the ordinary product of real numbers. For r ∈ (0, ∞) we put r · ∞ = ∞ · r = ∞. Put 0 · ∞ = ∞ · 0 = 0. Define ∞^{+} · α = α · ∞^{+} = ∞^{+} for α ∈ [0, ∞] ∐{∞^{+}}.

For example,

\frac{1}{∞} + \frac{1}{π} = \frac{1}{π} , \frac{1}{∞^{+}} + \frac{1}{π} = \frac{1}{π} , \frac{1}{∞} + \frac{1}{∞^{+}} = \frac{1}{∞}, \frac{1}{∞^{+}} + \frac{1}{∞^{+}} = \frac{1}{∞^{+}} ,

\frac{1}{α} ≤ \frac{1}{∞} + \frac{1}{4} + \frac{1}{2} ⇔ α ≥ 4/3, \frac{1}{α} ≤ \frac{1}{∞} + \frac{1}{∞^{+}} + \frac{1}{∞} ⇔ α ≥ ∞ and 5 · ∞ = ∞.

Lemma 2.11. Let F and F′ be density functions and f: U → V be a morphism of N (G)-Hilbert modules. Assume that F′ is Fredholm. Then

(1) If F ⪯ F′, then F is Fredholm and b^{(2)}(F) ≤ b^{(2)}(F′);

(2) If F ⪯ F′ and b^{(2)}(F) = b^{(2)}(F′), then α(F) ≥ α(F′);

(3) If F ≃ F′, then b^{(2)}(F) = b^{(2)}(F′) and α(F) = α(F′);

(4) α(F (λ^{r})) = r · α(F (λ)) for r ∈ (0, ∞);

(5) α(F) = α(F − b^{(2)}(F));

(6) b^{(2)}(f) = dim_{N (G)}(ker(f^{∗}f)) = dim_{N (G)}(ker(f));

(7) If f is zero and dim_{N (G)}(U) < ∞, then f is Fredholm and α(f) = ∞^{+};

(8) Suppose that dim_{N (G)}(U) = dim_{N (G)}(V) < ∞ or that U = V and f is selfadjoint. Then f: U → V is an isomorphism if and only if f is Fredholm, b^{(2)}(f) = 0 and α(f) = ∞^{+};

(9) Assume that i: V → V′ is injective with closed image and p: U′ → U is surjective with dim_{N (G)}(ker(p)) < ∞. Then f is Fredholm if and only if i ∘ f ∘ p is Fredholm, and in this case α(i ∘ f ∘ p) = α(f);

(10) If F and F′ are Fredholm then α(F + F′) = min {α(F), α(F′)};

(11) F (f^{∗}f)(λ^{2}) = F (f)(λ), b^{(2)}(f^{∗}f) = b^{(2)}(f) and α(f^{∗}f) = 1/2 · α(f);

(12) If F (0) = F′(0), then α(max{F, F′}) = min{α(F), α(F′)}. ⊓ ⊔

Proof. (1) to (7) are easy and left to the reader.

(8) The map f^{∗}f of Hilbert N (G)-modules is invertible if and only if there is ϵ > 0 such that E_{λ}^{f^{∗}f} is zero for λ < ϵ. If such ϵ > 0 exists, the inverse is given by ∫_{[ϵ,||f^{∗}f||]} λ^{−1}dE_{λ}^{f^{∗}f} (see (1.64)). Provided that f is selfadjoint, f^{∗}f = f^{2} and f^{∗}f is invertible if and only if f is invertible. Provided that dim_{N (G)}(U) = dim_{N (G)}(V) < ∞, Theorem 1.12 (1) and (2) imply that f is invertible if and only if f^{∗}f is invertible.

(9) The Inverse Mapping Theorem implies F (i ∘ f ∘ p)(λ) ≃ F (f)(λ) − dim_{N (G)}(ker(p)).

(10) Since b^{(2)}(F + F′) = b^{(2)}(F) + b^{(2)}(F′), we may assume without loss of generality by assertion (5) that b^{(2)}(F) = b^{(2)}(F′) = b^{(2)}(F + F′) = 0. As F, F′ ≤ F + F′, assertion (2) implies that α(F + F′) ≤ min{α(F), α(F′)}. To verify the reverse inequality, we may assume without loss of generality that α(F) ≤ α(F′). The cases α(F) = 0 and α(F) = ∞^{+} are trivial, and so we assume that 0 < α(F) ≤ ∞. Consider any real number α satisfying 0 < α < α(F). Then there exists a constant K > 0 such that for small positive λ we have F (λ), F′(λ) ≤ Kλ^{α}, and so F (λ) + F′(λ) ≤ 2K · λ^{α}, implying that α ≤ α(F + F′). The assertion follows.

The elementary proof of the other assertions is left to the reader. ⊓ ⊔ In the rest of this Subsection 2.1.1 we state and prove some basic proper-

ties of spectral density functions and Novikov-Shubin invariants for compositions and exact sequences which will be needed for the proofs of the main results in Subsection 2.1.2.

Lemma 2.12. Let f: U → V be a map of Hilbert N (G)-modules which is Fredholm and a weak isomorphism. Let L ⊂ V be a Hilbert N (G)-submodule. Then f restricts to a weak isomorphism from f^{−1}(L) to L and

dim_{N (G)}(L) = dim_{N (G)}(f^{−1}(L)).

Proof. From the Polar Decomposition of f, we can assume without loss of generality that U = V and f is positive. Obviously the restriction of f to f^{−1}(L) is injective. It remains to show that f(f^{−1}(L)) is dense in L because then f induces a weak isomorphism from f^{−1}(L) to L and the claim about the dimension follows from Theorem 1.12 (2).

Fix an orthogonal decomposition L = clos(f(f^{−1}(L))) ⊕ M, where M is an Hilbert N (G)-submodule of L. It remains to prove dim_{N (G)}(M) = 0 because then M = 0 follows from Theorem 1.12 (1). As f(f^{−1}(M)) ⊂ M and f(f^{−1}(M)) ⊂ f(f^{−1}(L)), we get f(f^{−1}(M)) = 0 and therefore M ∩ im(f) = 0. For λ > 0 restriction defines a map E_{λ}^{f}|_{M}: M → E_{λ}^{f}(U). If m ∈ ker(E_{λ}^{f}|_{M}) then the Spectral Theorem shows that m ∈ im(f), a preimage is given by (∫_{λ}^{||f||} μ^{−1}dE_{μ}^{f}) (m). Thus ker(E_{λ}^{f}|_{M}) = 0. Theorem 1.12 (2) shows

dim_{N (G)}(M) ≤ dim_{N (G)}(E_{λ}^{f}(U)). As f is injective and Fredholm and E_{λ} is right-continuous in λ, Theorem 1.12 (4) implies lim_{λ→0+} dim_{N (G)}(E_{λ}^{f}(U)) = dim_{N (G)}(E_{0}^{f}(U)) = 0. Thus dim_{N (G)}(M) = 0. ⊓ ⊔

Lemma 2.13. Let f: U → V and g: V → W be morphisms of Hilbert N (G)-modules. Then

(1) F (f)(λ) ≤ F (gf)(||g|| · λ);

(2) F (g)(λ) ≤ F (gf)(||f|| · λ) if f is Fredholm and has dense image; (3) F (gf)(λ) ≤ F (g)(λ^{1−r}) + F (f)(λ^{r}) for all r ∈ (0, 1).

Proof. (1) Consider L ∈ L(f, λ). For all x ∈ L, |gf(x)| ≤ ||g|| · |f(x)| ≤ ||g|| · λ · |x|. This implies that L ∈ L(gf, ||g|| · λ), and the claim follows.

(2) Consider L ∈ L(g, λ). For all x ∈ f^{−1}(L), we have |gf(x)| ≤ λ · |f(x)| ≤ λ · ||f|| · |x|, implying f^{−1}(L) ∈ L(gf, ||f|| · λ). Hence it remains to show dim_{N (G)}(L) ≤ dim_{N (G)}(f^{−1}(L)). Let p: U → U/ ker f be the projection and let \overline{f}: U/ ker(f) → V be the map induced by f. Clearly \overline{f} is also Fredholm. Since p is surjective and \overline{f} is a weak isomorphism, Theorem 1.12 (2) and Lemma 2.12 imply that dim_{N (G)} (f^{−1}(L)) ≥ dim_{N (G)} (p(f^{−1}(L))) =

dim_{N (G)} (\overline{f}^{−1}(L)) = dim_{N (G)}(L).

(3) Consider L ∈ L(gf, λ). Let L_{0} be the kernel of E_{λ^{2r}}^{f^{∗}f}|_{L}. We have a weakly exact sequence 0 → L_{0} → L → clos(E_{λ^{2r}}^{f^{∗}f}(L)) → 0. From Lemma 2.2 (2) we get that |f(x)| > λ^{r} · |x| for all nonzero x ∈ L_{0}. In particular, f|_{L_{0}} : L_{0} → clos(f(L_{0})) is a weak isomorphism, and so Theorem 1.12 (2) implies that dim_{N (G)}(L_{0}) = dim_{N (G)} (clos(f(L_{0}))). For x ∈ L_{0} we have

|gf(x)| ≤ λ · |x| ≤ \frac{λ}{λ^{r}} · |f(x)| = λ^{1−r} · |f(x)|.

Hence clos(f(L_{0})) ∈ L(g, λ^{1−r}). This shows that dim_{N (G)}(L_{0}) ≤ F (g)(λ^{1−r}). From Theorem 1.12 and Lemma 2.3 we conclude

dim_{N (G)} (clos(E_{λ^{2r}}^{f^{∗}f}(L))) ≤ dim_{N (G)} (im(E_{λ^{2r}}^{f^{∗}f})) = F (f)(λ^{r});
dim_{N (G)}(L) = dim_{N (G)}(L_{0}) + dim_{N (G)} (clos(E_{λ^{r}}^{f^{∗}f}(L))) .

This implies that dim_{N (G)}(L) ≤ F (g)(λ^{1−r}) + F (f)(λ^{r}). ⊓ ⊔

Lemma 2.14. Let f: U → V and g: V → W be morphisms of Hilbert N (G)-modules.

(1) If f and g are Fredholm, then the composition gf is Fredholm. If f and g are Fredholm and ker(g) ⊂ clos(im(f)) then

\frac{1}{α(gf)} ≤ \frac{1}{α(f)} + \frac{1}{α(g)};

(2) If gf is Fredholm, then f is Fredholm. If gf is Fredholm and ker(g) ∩ im(f) = 0, then

α(f) ≥ α(gf).

If gf is Fredholm and f has dense image, then g is Fredholm and

α(g) ≥ α(gf).

Proof. We can assume without loss of generality that f is injective by Lemma 2.11 (9), otherwise replace f by the induced map U/ ker(f) → V.

(1) We conclude from Lemma 2.13 (3) that gf is Fredholm. Now assume ker(g) ⊂ clos(im(f)). As f: U → clos(im(f)) is a weak isomorphism, Lemma 2.12 implies that b^{(2)}(gf) = b^{(2)}(g) = b^{(2)}(f) + b^{(2)}(g). From Lemma 2.13 (3) we conclude for 0 < r < 1

F (gf, λ) − b^{(2)}(gf) ≤ F (f, λ^{r}) − b^{(2)}(f) + F (g, λ^{1−r}) − b^{(2)}(g).

Assertions (2), (4), (5) and (10) of Lemma 2.11 show

α(gf) ≥ min {r · α(f), (1 − r) · α(g)} .

We only need to consider the case α(f), α(g) ∈ (0, ∞), the other cases being now obvious. Taking inverses gives

\frac{1}{α(gf)} ≤ max \{\frac{1}{r · α(f)}, \frac{1}{(1 − r) · α(g)}\} .

Since \frac{1}{r·α(f)} (resp. \frac{1}{(1−r)·α(g)}) is a strictly monotone decreasing (resp. increasing) function in r, the maximum on the right side, viewed as a function of r, obtains its minimum precisely when the two functions of r have the same value. One easily checks that this is the case if and only if r = \frac{α(g)}{α(f)+α(g)}, and the claim follows.

(2) This follows from Lemma 2.13 (1) and (2) and Lemma 2.11 (2), (5) and (9) since ker(g) ∩ im(f) = 0 ⇒ b^{(2)}(gf) = b^{(2)}(f) and b^{(2)}(gf) ≤ b^{(2)}(g) if f is injective. ⊓ ⊔

Lemma 2.15. Let u: U → U′ and v: V → V′ be morphisms of Hilbert N (G)-modules and let

0 ⟶ U_{1} ⟶ U_{0} ⟶ U_{2} ⟶ 0
f_{1}↓  ^{f_{0}}↓  ^{f_{2}}↓
0 ⟶ V_{1} ⟶ V_{0} ⟶ V_{2} ⟶ 0

be a commutative diagram of maps of Hilbert N (G)-modules whose rows are exact. Then

(1) u ⊕ v is Fredholm if and only if both u and v are Fredholm. In this case,

α(u ⊕ v) = min {α(u), α(v)};

(2) If f_{1} and f_{2} are Fredholm, then f_{0} is Fredholm. If f_{1} and f_{2} are Fredholm and f_{1} has dense image or f_{2} is injective, then

\frac{1}{α(f_{0})} ≤ \frac{1}{α(f_{1})} + \frac{1}{α(f_{2})};

(3) If f_{0} is Fredholm, then f_{1} is Fredholm. If f_{0} is Fredholm and f_{2} is injective, then α(f_{1}) ≥ α(f_{0}). If f_{0} is Fredholm and f_{1} has dense image, then f_{2} is Fredholm and α(f_{2}) ≥ α(f_{0}).

Proof. (1) This follows from Lemma 2.11 (10) using F (u ⊕ v, λ) = F (u, λ) + F (v, λ).

(2) Since an exact sequence of Hilbert N (G)-modules always splits by the Open Mapping Theorem we can assume without loss of generality by Lemma 2.11 (9) that U_{0} = U_{1} ⊕ U_{2} and V_{0} = V_{1} ⊕ V_{2} and f_{0} has the shape

f_{0} = ( \begin{matrix}f_{1} & f_{3} \\ 0 & f_{2} \end{matrix} )

for a morphism f_{3} : U_{2} → V_{1}. We have f_{0} = gf, where g = ( \begin{matrix}1 & f_{3} \\ 0 & f_{2} \end{matrix} ) and

f = ( \begin{matrix}f_{1} & 0 \\ 0 & 1 \end{matrix} ). Since f_{1} is Fredholm, f is Fredholm and α(f_{1}) = α(f) by

assertion (1). Since we can write

g = ( \begin{matrix}1 & 0 \\ 0 & f_{2} \end{matrix} ) · ( \begin{matrix}1 & f_{3} \\ 0 & 1 \end{matrix} )

we conclude from Lemma 2.11 (9) and assertion (1) that g is Fredholm and α(g) = α(f_{2}). If f_{2} is injective then g is injective, and if f_{1} has dense image then f has dense image. In both cases we have ker(g) ⊂ clos(im(f)). Now apply Lemma 2.14 (1).

(3) follows analogously using Lemma 2.14 (2). ⊓ ⊔

2.1.2 Spectral Density Functions of Hilbert Chain Complexes

Definition 2.16 (Spectral density function). Let C_{∗} be a Hilbert N (G)- chain complex. Define its p-th spectral density function to be the spectral density function (see Definition 2.1) of its p-th differential c_{p} restricted to im(c_{p+1})^{⊥}

F_{p}(C_{∗}) := F (c_{p}|_{im(c_{p+1})^{⊥}}: im(c_{p+1})^{⊥} → C_{p−1}) .

Suppose that C_{∗} is Fredholm, i.e. for p ∈ ℤ there exists λ_{p} > 0 with F_{p}(λ_{p}) < ∞ (see Definition 1.20 and Lemma 2.3). Define its p-th Novikov-Shubin invariant be the Novikov-Shubin invariant (see Definition 2.8) of F_{p}(C_{∗}),

α_{p}(C_{∗}) := α(F_{p}(C_{∗})).

Recall that Δ_{p}: C_{p} → C_{p} is the Laplacian of C_{∗} (see (1.17)). Put

F_{p}^{Δ}(C_{∗}) := F (Δ_{p});
α_{p}^{Δ}(C_{∗}) := α(Δ_{p}).

Define for a Hilbert N (G)-cochain complex C^{∗} which is Fredholm

F_{p}(C^{∗}) := F (c^{p}|_{im(c^{p−1})^{⊥}}: im(c^{p−1})^{⊥} → C^{p+1});
α_{p}(C^{∗}) := α(F_{p}(C^{∗})).

The dual (C_{∗})^{∗} of a Hilbert chain complex C_{∗} is the Hilbert cochain complex whose p-th cochain module is C_{p} and whose p-th codifferential c^{p} := c_{p+1}^{∗}: C_{p} → C_{p+1} is the adjoint of c_{p+1}.

Lemma 2.17. Let C_{∗} and D_{∗} be Hilbert chain complexes. Then

(1) If C_{∗} is Fredholm, then Δ_{p} is Fredholm and

α_{p}^{Δ}(C_{∗}) = 1/2 · min{α_{p}(C_{∗}), α_{p+1}(C_{∗})};

(2) Suppose that C_{∗} is Fredholm. Then (C_{∗})^{∗} is Fredholm and

F_{p+1}(C_{∗}) = F_{p}((C_{∗})^{∗});
b_{p}^{(2)}(C_{∗}) = b_{p}^{(2)}((C_{∗})^{∗});
α_{p+1}(C_{∗}) = α_{p}((C_{∗})^{∗});

(3) C_{∗} ⊕ D_{∗} is Fredholm if and only if both C_{∗} and D_{∗} are Fredholm. In this case

F_{p}(C_{∗} ⊕ D_{∗}) = F_{p}(C_{∗}) + F_{p}(D_{∗});
α_{p}(C_{∗} ⊕ D_{∗}) = min{α_{p}(C_{∗}), α_{p}(D_{∗})}.

Proof. (1) Lemma 1.18 and Lemma 2.15 (1) imply

F (Δ_{p}) − b^{(2)}(Δ_{p}) = F (c_{p}^{∗}c_{p}: ker(c_{p})^{⊥} → ker(c_{p})^{⊥})

+F (c_{p+1}c_{p+1}^{∗}: ker(c_{p+1}^{∗})^{⊥} → ker(c_{p+1}^{∗})^{⊥});

α(Δ_{p}) = min{α(c_{p}^{∗}c_{p}: ker(c_{p})^{⊥} → ker(c_{p})^{⊥}),

α(c_{p+1}c_{p+1}^{∗}: ker(c_{p+1}^{∗})^{⊥} → ker(c_{p+1}^{∗})^{⊥})}.

We conclude from Lemma 2.11 (5), (9) and (11)

1/2 · α_{p}(C_{∗}) = α(c_{p}^{∗}c_{p}: ker(c_{p})^{⊥} → ker(c_{p})^{⊥});
1/2 · α_{p+1}(C_{∗}) = α(c_{p+1}c_{p+1}^{∗}: ker(c_{p+1}^{∗})^{⊥} → ker(c_{p+1}^{∗})^{⊥}).

(2) This follows from Lemma 1.18 and Lemma 2.4.

(3) This follows from Lemma 2.15 (1). ⊓ ⊔ We recall that a Hilbert N (G)-chain complex C_{∗} is said to be contractible

if there exists a chain contraction γ_{∗}, i.e. a collection of morphisms γ_{p}: C_{p} → C_{p+1} for p ∈ ℤ such that γ_{p−1}c_{p} + c_{p+1}γ_{p} = id for all p.

Lemma 2.18. The following assertions are equivalent for a Hilbert N (G)- chain complex C_{∗}:

(1) C_{∗} is contractible;

(2) Δ_{p} is invertible for all p;

(3) C_{∗} is Fredholm, b_{p}^{(2)}(C_{∗}) = 0 and α_{p}(C_{∗}) = ∞^{+} for all p.

Proof. (1) ⇒ (3) We can construct morphisms \overline{c_{p}}: C_{p}/ clos(im(c_{p+1})) → C_{p−1} and \overline{γ_{p−1}} : C_{p−1} → C_{p}/ clos(im(c_{p+1})), using c_{p} and γ_{p−1}, such that \overline{γ_{p−1}} ∘ \overline{c_{p}} = id. Hence \overline{c_{p}} induces an invertible operator onto its image.

Lemma 2.11 (8) and (9) imply that \overline{c_{p}} is Fredholm and hence C_{∗} is Fredholm at p, b_{p}^{(2)}(C_{∗}) = b^{(2)}(\overline{c_{p}}) = 0 and α_{p}(C_{∗}) = α(\overline{c_{p}}) = ∞^{+}.

(3) ⇒ (2) From Lemma 1.18 and Lemma 2.17 (1) we conclude that Δ_{p} is Fredholm, b^{(2)}(Δ_{p}) = 0 and α(Δ_{p}) = ∞^{+} for all p. Now apply Lemma 2.11 (8).

(2) ⇒ (1) Suppose that Δ_{p} is invertible for all p. Then γ_{∗} with γ_{p} := Δ_{p+1}^{−1} ∘ c_{p+1}^{∗} is a chain contraction of C_{∗}. ⊓ ⊔

Next we reprove the homotopy invariance of the Novikov-Shubin invariants [240, Proposition 4.1].

Theorem 2.19. If f_{∗}: C_{∗} → D_{∗} is a chain homotopy equivalence of Hilbert N (G)-chain complexes, then for all p ∈ ℤ we have

F_{p}(C_{∗}) ≃ F_{p}(D_{∗}).

In particular C_{∗} is Fredholm at p if and only if D_{∗} is Fredholm at p. In this case

α_{p}(C_{∗}) = α_{p}(D_{∗}).

Proof. There are exact sequences of chain complexes 0 → C_{∗} → cyl_{∗}(f_{∗}) → cone_{∗}(f_{∗}) → 0 and 0 → D_{∗} → cyl_{∗}(f_{∗}) → cone_{∗}(C_{∗}) → 0 with cone_{∗}(f_{∗}) and cone_{∗}(C_{∗}) being contractible. We obtain chain isomorphisms C_{∗} ⊕ cone_{∗}(f_{∗}) ⟶^{≅} cyl_{∗}(f_{∗}) and D_{∗} ⊕ cone_{∗}(C_{∗}) ⟶^{≅} cyl_{∗}(f_{∗}) by the following general construction for an exact sequence 0 → C_{∗} ⟶^{j_{∗}} D_{∗} ⟶^{q_{∗}} E_{∗} → 0 with contractible E_{∗}: Choose a chain contraction ϵ_{∗} for E_{∗} and for each p a morphism t_{p}: E_{p} → D_{p} such that q_{p} ∘ t_{p} = id . Put

s_{p} = d_{p+1} ∘ t_{p+1} ∘ ϵ_{p} + t_{p} ∘ ϵ_{p−1} ∘ e_{p}.

This defines a chain map s_{∗}: E_{∗} → D_{∗} such that q_{∗} ∘ s_{∗} = id. Define a chain map u_{∗}: D_{∗} → C_{∗} by requiring that for x ∈ D_{p} its image u_{p}(x) is the unique y ∈ C_{p} such that x = s_{p}q_{p}(x) + j_{p}(y). Then j_{∗} + s_{∗} is a chain isomorphism C_{∗} ⊕E_{∗} → D_{∗} with inverse u_{∗} ⊕q_{∗}. Since C_{∗} ⊕cone_{∗}(f_{∗}) and D_{∗} ⊕cone_{∗}(C_{∗}) are isomorphic and cone_{∗}(f_{∗}) and cone_{∗}(C_{∗}) are contractible, Lemma 2.11 (9), Lemma 2.17 (3) and Lemma 2.18 imply F_{p}(C_{∗}) ≃ F_{p}(D_{∗}). Now the other statements follow from Lemma 2.11 (3). ⊓ ⊔

The next result is taken from [322, Theorem 2.3 on page 27].

Theorem 2.20. Let 0 → C_{∗} ⟶^{j_{∗}} D_{∗} ⟶^{q_{∗}} E_{∗} → 0 be an exact sequence of Hilbert N (G)-chain complexes. Suppose that two of the chain complexes are Fredholm. Then all three chain complexes are Fredholm and we have

\frac{1}{α_{p}(D_{∗})} ≤ \frac{1}{α_{p}(C_{∗})} + \frac{1}{α_{p}(E_{∗})} + \frac{1}{α(δ_{p})};
\frac{1}{α_{p}(E_{∗})} ≤ \frac{1}{α_{p−1}(C_{∗})} + \frac{1}{α_{p}(D_{∗})} + \frac{1}{α(H_{p−1}(j_{∗}))};
\frac{1}{α_{p}(C_{∗})} ≤ \frac{1}{α_{p}(D_{∗})} + \frac{1}{α_{p+1}(E_{∗})} + \frac{1}{α(H_{p}(q_{∗}))},

where δ_{p}: H_{p}^{(2)}(E_{∗}) → H_{p−1}^{(2)}(C_{∗}) is the boundary operator in the long weakly exact homology sequence (see Theorem 1.21).

Proof. We first treat the case where C_{∗} and E_{∗} are Fredholm. Then D_{∗} is Fredholm by Lemma 2.15 (2). We now show that

\frac{1}{α_{p}(D_{∗})} ≤ \frac{1}{α_{p}(C_{∗})} + \frac{1}{α_{p}(E_{∗})} + \frac{1}{α(δ_{p})}.  (2.21)

The given exact sequence 0 → C_{∗} ⟶^{j_{∗}} D_{∗} ⟶^{q_{∗}} E_{∗} → 0 induces the following commutative diagram with exact rows, where \overline{q_{p}}, \overline{d_{p}} and \overline{e_{p}} are the canonical homomorphisms induced by q_{p}, d_{p} and e_{p}, and i is the inclusion

0 ⟶ ker \overline{q_{p}} ⟶^{i} D_{p}/ clos(im(d_{p+1})) ⟶^{\overline{q_{p}}} E_{p}/ ker(e_{p}) ⟶ 0
∂_{p}↓  \overline{d_{p}}↓  \overline{e_{p}}↓
0 ⟶ C_{p−1} ⟶_{j_{p−1}}  D_{p−1}  ⟶_{q_{p−1}} E_{p−1} ⟶ 0

(2.22)

We define ∂_{p} in the above diagram as follows. Let x ∈ ker(e_{p}q_{p}) represent [x] ∈ ker(\overline{q_{p}}). Then d_{p}(x) = j_{p−1}(y) for a unique y ∈ C_{p−1}. We put ∂_{p}([x]) = y. (In fact, y ∈ ker(c_{p−1}).)

Next we construct a sequence which we will show to be weakly exact

C_{p} ⟶^{\overline{j_{p}}} ker(\overline{q_{p}}) \underrightarrow{q̂_{p}} H_{p}^{(2)}(E_{∗}) → 0  (2.23)

The map \overline{j_{p}} is induced by j_{p} in the obvious way. To define q̂_{p}, consider an x ∈ D_{p} whose class [x] ∈ D_{p}/ clos(im(d_{p+1})) lies in ker(\overline{q_{p}}). Then q_{p}(x) is in the kernel of e_{p} and determines a class [q_{p}(x)] in H_{p}^{(2)}(E_{∗}). Define q̂_{p}([x]) to be [q_{p}(x)]. One easily checks that q̂_{p} ∘ \overline{j_{p}} is zero and q̂_{p} is surjective. We will show that ker(q̂_{p}) is contained in clos(im(\overline{j_{p}})). Consider [x] ∈ ker(q̂_{p}) with representative x ∈ D_{p}. Since [q_{p}(x)] ∈ H_{p}^{(2)}(E_{∗}) is zero, there is a sequence (y_{n})_{n≥1} in E_{p+1} such that in E_{p}

lim_{n→∞} (q_{p}(x) − e_{p+1}(y_{n})) = 0.

As q_{p+1} is surjective, there is a sequence {u_{n}}_{n≥1} in D_{p+1} such that y_{n} = q_{p+1}(u_{n}). Thus

lim_{n→∞} q_{p} (x − d_{p+1}(u_{n})) = 0.

We write x − d_{p+1}(u_{n}) = j_{p}(w_{n}) + r_{n}, where w_{n} ∈ C_{p} and r_{n} ∈ im(j_{p})^{⊥}. Then we obtain lim_{n→∞} q_{p}(r_{n}) = 0. As the restriction of q_{p} to im(j_{p})^{⊥} is an isomorphism, we conclude lim_{n→∞} r_{n} = 0. Thus

x = lim_{n→∞} (j_{p}(w_{n}) + d_{p+1}(u_{n})) ,

and hence in D_{p}/ clos(im(d_{p+1}))

[x] = lim_{n→∞} \overline{j_{p}}(w_{n}).

This finishes the proof of weak exactness of (2.23).

Next, we construct a commutative diagram with exact rows

0 ⟶ ker(q̂_{p}) ⟶^{l_{1}} ker(\overline{q_{p}}) \underrightarrow{q̂_{p}} H_{p}^{(2)}(E_{∗}) ⟶ 0
\overline{∂_{p}}↓  ^{∂_{p}}↓  ^{δ_{p}}↓
0 ⟶ clos(im(c_{p})) ⟶_{l_{2}} ker(c_{p−1}) ⟶_{pr} H_{p−1}^{(2)}(C_{∗}) ⟶ 0

(2.24)

The maps l_{1} and l_{2} are the canonical inclusions and the map pr is the canonical projection. The map \overline{∂_{p}} is induced by ∂_{p}. One easily verifies that the diagram commutes and has exact rows.

Let ˜ \overline{j_{p}}: C_{p} → ker(q̂_{p}) be the morphism induced by \overline{j_{p}} whose image is dense by the weak exactness of (2.23). One easily checks that \overline{∂_{p}} ∘ ˜ \overline{j_{p}} = c_{p}. As c_{p} is Fredholm by assumption, Lemma 2.14 (2) implies that \overline{∂_{p}} is Fredholm and

α(\overline{∂_{p}}) ≥ α_{p}(C_{∗}).  (2.25)

As E_{∗} is Fredholm and hence H_{p}^{(2)}(E_{∗}) is finite dimensional, δ_{p} is Fredholm. As \overline{∂_{p}} has dense image, Lemma 2.15 (2) applied to (2.24) shows that ∂_{p} is Fredholm and

\frac{1}{α(∂_{p})} ≤ \frac{1}{α(\overline{∂_{p}})} + \frac{1}{α(δ_{p})}.  (2.26)

From Lemma 2.11 (9) \overline{e_{p}} is Fredholm and

α(\overline{e_{p}}) = α_{p}(E).  (2.27)

As \overline{e_{p}} is injective, Lemma 2.15 (2) applied to (2.22) shows

\frac{1}{α_{p}(D_{∗})} ≤ \frac{1}{α(∂_{p})} + \frac{1}{α(\overline{e_{p}})}.  (2.28)

Now (2.21) follows from (2.25), (2.26), (2.27) and (2.28).

Next we show under the assumption that D_{∗} and C_{∗} are Fredholm that E_{∗} is Fredholm and that

\frac{1}{α_{p}(E_{∗})} ≤ \frac{1}{α_{p−1}(C_{∗})} + \frac{1}{α_{p}(D_{∗})} + \frac{1}{α(H_{p−1}^{(2)}(j_{∗}))}.  (2.29)

There are an exact sequence 0 → D_{∗} → cyl_{∗}(q_{∗}) → cone_{∗}(q_{∗}) → 0 and chain homotopy equivalences E_{∗} → cyl_{∗}(q_{∗}) and ΣC_{∗} → cone_{∗}(q_{∗}). We conclude from Theorem 2.19 and (2.21) that E_{∗} is Fredholm and (2.29) is true since the connecting map from H_{p}^{(2)}(cone_{∗}(q_{∗})) → H_{p−1}^{(2)}(D_{∗}) agrees under the identification H_{p}^{(2)}(cone_{∗}(q_{∗})) = H_{p}^{(2)}(ΣC_{∗}) = H_{p−1}^{(2)}(C_{∗}) with the map H_{p−1}^{(2)}(j_{∗}): H_{p−1}^{(2)}(C_{∗}) → H_{p−1}^{(2)}(D_{∗}).

Analogously one shows under the assumption that D_{∗} and E_{∗} are Fredholm that C_{∗} is Fredholm and

\frac{1}{α_{p}(C_{∗})} ≤ \frac{1}{α_{p}(D_{∗})} + \frac{1}{α_{p+1}(E_{∗})} + \frac{1}{α(H_{p}^{(2)}(q_{∗}))}.  (2.30)

This finishes the proof of Theorem 2.20. ⊓ ⊔

2.1.3 Product Formula for Novikov-Shubin Invariants

In this subsection we deal with the Novikov-Shubin invariants of a tensor product of Hilbert chain complexes. We will only consider positive chain complexes C_{∗}, i.e. C_{n} = 0 for n < 0.

Lemma 2.31. Let G and H be discrete groups. Let f: U → U and g: V → V be positive maps of Hilbert N (G)-modules and N (H)-modules. Then f ⊗ id + id ⊗g: U ⊗ V → U ⊗ V is a positive map of Hilbert N (G × H)-modules and

F (f ⊗ id + id ⊗g) ≃ F (f) · F (g),

where F · G is defined in terms of Notation 2.10.

Proof. Notice for the sequel that dim_{N (G×H)}(U ⊗ V) = dim_{N (G)}(U) · dim_{N (H)}(V) (see Theorem 1.12 (5)). For x ∈ im(E_{λ/2}^{f}) and y ∈ im(E_{λ/2}^{g}) we compute using Lemma 2.2 (2)

||(f ⊗ id + id ⊗g)(x ⊗ y)|| ≤ ||(f ⊗ id)(x ⊗ y)|| + ||(id ⊗g)(x ⊗ y)||
= ||f(x)|| · ||y|| + ||x|| · ||g(y)||
≤ λ/2 · ||x|| · ||y|| + λ/2 · ||x|| · ||y||
= λ · ||x ⊗ y||.

This shows im(E_{λ/2}^{f}) ⊗ im(E_{λ/2}^{g}) ∈ L(f ⊗ id + id ⊗g, λ). Lemma 2.3 implies

F (f)(λ/2) · F (g)(λ/2) ≤ F (f ⊗ id + id ⊗g)(λ).  (2.32)

Since 0 ≤ f ⊗ id ≤ f ⊗ id + id ⊗g and f ⊗ id and f ⊗ id + id ⊗g commute, we have E_{λ}^{f⊗id + id ⊗g} ≤ E_{λ}^{f⊗id}. Analogously we get E_{λ}^{f⊗id + id ⊗g} ≤ E_{λ}^{id ⊗g}. This implies

im(E_{λ}^{f⊗id + id ⊗g}) ⊂ im(E_{λ}^{f⊗id}) ∩ im(E_{λ}^{id ⊗g}) = im(E_{λ}^{f}) ⊗ im(E_{λ}^{g}).

Hence we get from Theorem 1.12 (5)

F (f)(λ) · F (g)(λ) ≥ F (f ⊗ id + id ⊗g)(λ).  (2.33)

Now Lemma 2.31 follows from (2.32) and (2.33). ⊓ ⊔

Notation 2.34. Given a spectral density function F: [0, ∞) → [0, ∞], define

F^{⊥}: [0, ∞) → [0, ∞]

by F^{⊥}(λ) = F (λ) − F (0).

Define δ_{r} for r ∈ [0, ∞] by δ_{0} := ∞^{+} and by δ_{r} := 1 for r ≠ 0.

Notice that α(F) = α(F^{⊥}) by definition. Recall Notation 2.10.

Lemma 2.35. Let G and H be groups. Let C_{∗} be a Hilbert N (G)-chain complex and D_{∗} be a Hilbert N (H)-chain complex. Suppose that both C_{∗} and D_{∗} are Fredholm and positive. Then the tensor product of Hilbert chain complexes C_{∗} ⊗ D_{∗} is a Fredholm Hilbert N (G × H)-chain complex and

(1) We have

F_{n}^{Δ}(C_{∗} ⊗ D_{∗}) ≃ ∑_{i=0}^{n} F_{i}^{Δ}(C_{∗}) · F_{n−i}^{Δ}(D_{∗})

and

α_{n}^{Δ}(C_{∗} ⊗ D_{∗}) = min_{i=0,1,...,n}{α (F_{i}^{Δ}(C_{∗})^{⊥} · F_{n−i}^{Δ}(D_{∗})^{⊥}) ,
δ_{b_{i}^{(2)}(C_{∗})} · α_{n−i}^{Δ}(D_{∗}), α_{i}^{Δ}(C_{∗}) · δ_{b_{n−i}^{(2)}(D_{∗})}};

(2) We have

F_{n}(C_{∗} ⊗ D_{∗})^{⊥} ⪰ ∑_{i=0}^{n} F_{i}(C_{∗})^{⊥} · F_{n−i}(D_{∗})^{⊥} + b_{i}^{(2)}(C_{∗}) · F_{n−i}(D_{∗})^{⊥}
+F_{i}(C_{∗})^{⊥} · b_{n−i}^{(2)}(D_{∗});  (2.36)
F_{n}(C_{∗} ⊗ D_{∗})^{⊥} ⪰ ∑_{i=0}^{n} F_{i+1}(C_{∗})^{⊥} · F_{n−i}(D_{∗})^{⊥} + b_{i}^{(2)}(C_{∗}) · F_{n−i}(D_{∗})^{⊥}
+F_{i+1}(C_{∗})^{⊥} · b_{n−1−i}^{(2)}(D_{∗});  (2.37)
F_{n}(C_{∗} ⊗ D_{∗})^{⊥} ⪯ ∑_{i=0}^{n} F_{i+1}(C_{∗})^{⊥} · F_{n−i}(D_{∗})^{⊥} + F_{i}(C_{∗})^{⊥} · F_{n+1−i}(D_{∗})^{⊥}
+F_{i}(C_{∗})^{⊥} · F_{n−i}(D_{∗})^{⊥} + b_{i}^{(2)}(C_{∗}) · F_{n−i}(D_{∗})^{⊥}
+F_{i}(C_{∗})^{⊥} · b_{n−i}^{(2)}(D_{∗})  (2.38)

and

α_{n}(C_{∗} ⊗ D_{∗}) = min_{i=0,1,...,n}{α (F_{i+1}(C_{∗})^{⊥} · F_{n−i}(D_{∗})^{⊥}) ,
α (F_{i}(C_{∗})^{⊥} · F_{n−i}(D_{∗})^{⊥}) , δ_{b_{i}^{(2)}(C_{∗})} · α_{n−i}(D_{∗}),
α_{i}(C_{∗}) · δ_{b_{n−i}^{(2)}(D_{∗})}}.  (2.39)

Proof. (1) The Laplace operator Δ_{n}^{C_{∗}⊗D_{∗}}: (C_{∗} ⊗ D_{∗})_{n} → (C_{∗} ⊗ D_{∗})_{n} is the orthogonal sum

⨁_{i=0}^{n} Δ_{i}^{C_{∗}} ⊗ id + id ⊗Δ_{n−i}^{D_{∗}}: ⨁_{i=0}^{n} C_{i} ⊗ D_{n−i} → ⨁_{i=0}^{n} C_{i} ⊗ D_{n−i}.

Now apply Lemma 2.11 (10) and Lemma 2.31.

(2) Let e_{n}: (C_{∗} ⊗D_{∗})_{n} → (C_{∗} ⊗D_{∗})_{n−1} be the n-th differential of the N (G× H)-chain complex C_{∗} ⊗ D_{∗}. Consider the maps of Hilbert modules

e_{n⊥} := e_{n}|_{ker(e_{n+1}^{∗})} : ker(e_{n+1}^{∗}) → (C_{∗} ⊗ D_{∗})_{n−1};
c_{i⊥} := c_{i}|_{ker(c_{i+1}^{∗})} : ker(c_{i+1}^{∗}) → C_{i−1};
d_{n−i⊥} := d_{n−i}|_{ker(d_{n−i+1}^{∗})} : ker(d_{n−i+1}^{∗}) → D_{n−1−i}.

The following diagram commutes

ker(e_{n+1}^{∗})  ⟶^{(e_{n⊥})^{∗}e_{n⊥}}  ker(e_{n+1}^{∗})
↑  ↑
⨁_{i=0}^{n} ker(c_{i+1}^{∗}) ⊗ ker(d_{n−i+1}^{∗}) ⟶_{u} ⨁_{i=0}^{n} ker(c_{i+1}^{∗}) ⊗ ker(d_{n−i+1}^{∗})

where u is ⨁_{i=0}^{n}(c_{i⊥})^{∗}c_{i⊥} ⊗ id + id ⊗(d_{n−i⊥})^{∗}d_{n−i⊥} and the vertical maps are the canonical inclusions. Since (e_{n⊥})^{∗}e_{n⊥} is selfadjoint, it splits as the orthogonal sum of ⨁_{i=0}^{n}(c_{i⊥})^{∗}c_{i⊥} ⊗ id + id ⊗(d_{n−i⊥})^{∗}d_{n−i⊥} and some endomorphism of the orthogonal complement of ⨁_{i=0}^{n} ker(c_{i+1}^{∗}) ⊗ ker(d_{n−i+1}^{∗}) in ker(e_{n+1}^{∗}). Now Lemma 2.11 (11), Lemma 2.15 (1) and Lemma 2.31 imply

F_{n}(C_{∗} ⊗ D_{∗})(λ) = F_{n}(e_{n⊥})(λ)
= F_{n}((e_{n⊥})^{∗}e_{n⊥})(\sqrt{λ})
≥ ∑_{i=0}^{n} F ((c_{i⊥})^{∗}c_{i⊥} ⊗ id + id ⊗(d_{n−i⊥})^{∗}d_{n−i⊥}) (\sqrt{λ})

≃ ∑_{i=0}^{n} F ((c_{i⊥})^{∗}c_{i⊥})(\sqrt{λ}) · F ((d_{n−i⊥})^{∗}d_{n−i⊥})(\sqrt{λ})

= ∑_{i=0}^{n} F (c_{i⊥})(λ) · F (d_{n−i⊥})(λ)

= ∑_{i=0}^{n} F_{i}(C_{∗})(λ) · F_{n−i}(D_{∗})(λ).

Now equation (2.36) follows since b_{i}^{(2)}(C_{∗}) = F_{i}(C_{∗})(0) and analogously for D_{∗}.

Analogously we prove (2.37). We define maps of Hilbert modules

e_{n⊥}^{∗} := e_{n}^{∗}|_{ker(e_{n−1})} : ker(e_{n−1}) → (C ⊗ D)_{n};
c_{i+1⊥}^{∗} := c_{i+1}^{∗}|_{ker(c_{i})} : ker(c_{i}) → C_{i+1};
d_{n−i⊥}^{∗} := d_{n−i}^{∗}|_{ker(d_{n−i−1})} : ker(d_{n−i−1}) → D_{n−i}.

We obtain a commutative diagram with inclusions as vertical arrows

ker(e_{n−1})  ⟶^{(e_{n⊥}^{∗})^{∗}e_{n⊥}^{∗}}  ker(e_{n−1})
↑  ↑
⨁_{i=0}^{n} ker(c_{i}) ⊗ ker(d_{n−i−1}) ⟶_{v} ⨁_{i=0}^{n} ker(c_{i}) ⊗ ker(d_{n−i−1})

where v is given by ⨁_{i=0}^{n}(c_{i+1⊥}^{∗})^{∗}c_{i+1⊥}^{∗} ⊗ id + id ⊗(d_{n−i⊥}^{∗})^{∗}d_{n−i⊥}^{∗}. We conclude (2.37), where we use additionally the conclusion of Lemma 1.18

b^{(2)}(c_{i+1⊥}^{∗}) = b_{i}^{(2)}(C_{∗});
b^{(2)}(d_{n−i⊥}^{∗}) = b_{n−1−i}^{(2)}(D_{∗});
b^{(2)}(e_{n⊥}^{∗}) = b_{n−1}^{(2)}(E_{∗})

and the conclusion of Lemma 2.4

F (c_{i+1⊥}^{∗})^{⊥} = F (c_{i+1⊥})^{⊥};
F (d_{n−i⊥}^{∗})^{⊥} = F (d_{n−i⊥})^{⊥};
F (e_{n⊥}^{∗})^{⊥} = F (e_{n⊥})^{⊥}.

Next we prove (2.38). We have the orthogonal decomposition

C_{i} = ker(c_{i}) ⊕ ker(c_{i})^{⊥};
D_{n−i} = ker(d_{n−i}) ⊕ ker(d_{n−i})^{⊥};
C_{i} ⊗ D_{n−i} = ker(c_{i}) ⊗ ker(d_{n−i}) ⊕ ker(c_{i}) ⊗ ker(d_{n−i})^{⊥} ⊕
ker(c_{i})^{⊥} ⊗ ker(d_{n−i}) ⊕ ker(c_{i})^{⊥} ⊗ ker(d_{n−i})^{⊥}.

The summand ker(c_{i}) ⊗ ker(d_{n−i}) lies in the kernel of the n-th-differential e_{n} of C_{∗} ⊗ D_{∗}. Hence ker(e_{n})^{⊥} lies in the direct sum of the other three summands. Define maps by restricting the obvious maps

c_{i+1⊥}^{∗}: ker(c_{i}) → C_{i+1};
d_{n−i⊥}: ker(d_{n−i})^{⊥} → D_{n−1−i};
d_{n+1−i⊥}^{∗}: ker(d_{n−i}) → D_{n+1−i};
c_{i⊥}: ker(c_{i})^{⊥} → C_{i−1}.

Consider the following three maps

(c_{i+1⊥}^{∗})^{∗}c_{i+1⊥}^{∗} ⊗ id + id ⊗(d_{n−i⊥})^{∗}d_{n−i⊥}: ker(c_{i}) ⊗ ker(d_{n−i})^{⊥} →

ker(c_{i}) ⊗ ker(d_{n−i})^{⊥};

(c_{i⊥})^{∗}c_{i⊥} ⊗ id + id ⊗(d_{n+1−i⊥}^{∗})^{∗}d_{n+1−i⊥}^{∗}: ker(c_{i})^{⊥} ⊗ ker(d_{n−i}) →

ker(c_{i})^{⊥} ⊗ ker(d_{n−i});

(c_{i⊥})^{∗}c_{i⊥} ⊗ id + id ⊗(d_{n−i⊥})^{∗}d_{n−i⊥}: ker(c_{i})^{⊥} ⊗ ker(d_{n−i})^{⊥} →

ker(c_{i})^{⊥} ⊗ ker(d_{n−i})^{⊥}.

The orthogonal sum of these three selfadjoint maps is the orthogonal sum of the map

(e_{n⊥})^{∗}e_{n⊥}: ker(e_{n})^{⊥} → ker(e_{n})^{⊥}

and of an endomorphism of the orthogonal complement of ker(e_{n})^{⊥} in (ker(c_{i}) ⊗ ker(d_{n−i})^{⊥}) ⊕ (ker(c_{i})^{⊥} ⊗ ker(d_{n−i})) ⊕ (ker(c_{i})^{⊥} ⊗ ker(d_{n−i})^{⊥}). Now Lemma 2.4, Lemma 2.11 (11), Lemma 2.15 (1) and Lemma 2.31 imply (2.38) by the following calculation:

F_{n}(C_{∗} ⊗ D_{∗})^{⊥}(λ) = F (e_{n⊥})(λ)
= F ((e_{n⊥})^{∗}e_{n⊥})(\sqrt{λ})
≤ ∑_{i=0}^{n} F ((c_{i+1⊥}^{∗})^{∗}c_{i+1⊥}^{∗} ⊗ id + id ⊗(d_{n−i⊥})^{∗}d_{n−i⊥})(\sqrt{λ})

+F ((c_{i⊥})^{∗}c_{i⊥} ⊗ id + id ⊗(d_{n+1−i⊥}^{∗})^{∗}d_{n+1−i⊥}^{∗})(\sqrt{λ})

+F ((c_{i⊥})^{∗}c_{i⊥} ⊗ id + id ⊗(d_{n−i⊥})^{∗}d_{n−i⊥})(\sqrt{λ})
≃ ∑_{i=0}^{n} F (c_{i+1⊥}^{∗})^{∗}c_{i+1⊥}^{∗})(\sqrt{λ}) · F ((d_{n−i⊥})^{∗}d_{n−i⊥})(\sqrt{λ})
+F ((c_{i⊥})^{∗}c_{i⊥})(\sqrt{λ}) · F ((d_{n+1−i⊥}^{∗})^{∗}d_{n+1−i⊥}^{∗})(\sqrt{λ})
+F ((c_{i⊥})^{∗}c_{i⊥})(\sqrt{λ}) · F ((d_{n−i⊥})^{∗}d_{n−i⊥})(\sqrt{λ})
= ∑_{i=0}^{n} F (c_{i+1⊥}^{∗})(λ) · F (d_{n−i⊥})(λ)

+F (c_{i⊥})(λ) · F (d_{n+1−i⊥}^{∗})(λ) + F (c_{i⊥})(λ) · F (d_{n−i⊥})(λ)

= ∑_{i=0}^{n}(F_{i+1}(C_{∗})^{⊥}(λ) + b_{i}^{(2)}(C_{∗})) · F_{n−i}(D_{∗})^{⊥}(λ)
+F_{i}(C_{∗})^{⊥}(λ) · (F_{n+1−i}(D_{∗})^{⊥}(λ) + b_{n−i}^{(2)}(D_{∗}))
+F_{i}(C_{∗})^{⊥}(λ) · F_{n−i}(D_{∗})^{⊥}(λ).

Finally (2.39) follows from (2.36), (2.37), and (2.38) using Lemma 2.11 (2) and (10). ⊓ ⊔

Example 2.40. Let F and G be spectral density functions with F (0) = G(0) = 0. It is not hard to check that then

α(F) + α(G) ≤ α(F · G)

is true with respect to Notation 2.10. However, the other inequality is not true in general as the following example shows. Define (continuous) density functions F, G: [0, ∞) → [0, ∞] by

F (λ) = 2^{2^{2n+1}}λ^{3}  λ ∈ [2^{−2^{2n+1}}, 2^{−2^{2n}}];
G(λ) = 2^{−2^{2n+1}}  λ ∈ [2^{−2^{2n+1}}, 2^{−2^{2n}}];
F (λ) = 2^{−2^{2(n+1)}}  λ ∈ [2^{−2^{2(n+1)}}, 2^{−2^{2n+1}}];
G(λ) = 2^{2^{2(n+1)}}λ^{3}  λ ∈ [2^{−2^{2(n+1)}}, 2^{−2^{2n+1}}];
F (0) = 0;
G(0) = 0;
F (λ) = 1/2  λ ≥ 1/2;
G(λ) = 1/4  λ ≥ 1/2.

One easily checks F (λ) ≤ λ and G(λ) ≤ λ for λ ≥ 0. Since

F (2^{−2^{2(n+1)}}) = 2^{−2^{2(n+1)}};
lim_{n→∞} 2^{−2^{2(n+1)}} = 0;
G(2^{−2^{2n+1}}) = 2^{−2^{2n+1}};
lim_{n→∞} 2^{−2^{2n+1}} = 0,

we get α(F) = α(G) = 1 and α(F) + α(G) = 2. On the other hand F (λ) · G(λ) = λ^{3} and hence α(F · G) = 3.

Definition 2.41. A Fredholm spectral density function has the limit property if either F (λ) = F (0) for some λ > 0 or if F (λ) > F (0) for λ > 0 and the limit

lim_{λ→0+} \frac{ln(F (λ) − F (0))}{ln(λ)}

exists. A Fredholm Hilbert chain complex C_{∗} has the limit property if F_{p}(C_{∗}) has the limit property for all p ∈ ℤ. A G-CW -complex X of finite type has the limit property if its cellular L^{2}-chain complex has the limit property.

Remark 2.42. To the author’s knowledge there is no example of a G-CW - complex X of finite type which does not have the limit property. See also the discussion in [240, page 381].

Lemma 2.43. Let F, G: [0, ∞) → [0, ∞] be density functions which are Fredholm and have the limit property. Then F · G is a density function which is Fredholm and has the limit property and

α(F · G) = min{α(F) + α(G), δ_{b^{(2)}(F)} · α(G), α(F) · δ_{b^{(2)}(G)}}.

Proof. Since F (λ) = F^{⊥}(λ) + b^{(2)}(F), it suffices to treat the case F (0) = G(0) = 0 because of Lemma 2.11 (10). Since ln(F (λ) · G(λ)) = ln(F (λ)) + ln(G(λ)) holds and lim_{λ→0+} is compatible with + (in contrast to lim inf_{λ→0+}), Lemma 2.43 follows. ⊓ ⊔

We conclude from Lemma 2.35 and Lemma 2.43

Corollary 2.44. Let G and H be groups. Let C_{∗} be a Hilbert N (G)-chain complex and D_{∗} be a Hilbert N (H)-chain complex. Suppose that both C_{∗} and D_{∗} are positive, Fredholm and have the limit property. Then the tensor product of Hilbert chain complexes C_{∗} ⊗ D_{∗} is a Fredholm Hilbert N (G × H)- chain complex which has the limit property and we get

α_{n}^{Δ}(C_{∗} ⊗ D_{∗}) = min_{i=0,1,...,n}{α_{i}^{Δ}(C_{∗}) + α_{n−i}^{Δ}(D_{∗}),
δ_{b_{i}^{(2)}(C_{∗})} · α_{n−i}^{Δ}(D_{∗}), α_{i}^{Δ}(C_{∗}) · δ_{b_{n−i}^{(2)}(D_{∗}}}.

and

α_{n}(C_{∗} ⊗ D_{∗}) = min_{i=0,1,...,n}{α_{i+1}(C_{∗}) + α_{n−i}(D_{∗}), α_{i}(C_{∗}) + α_{n−i}(D_{∗}),
δ_{b_{i}^{(2)}(C_{∗})} · α_{n−i}(D_{∗}), α_{i}(C_{∗}) · δ_{b_{n−i}^{(2)}(D_{∗})}}.

2.1.4 The Laplacian in Dimension Zero

In this subsection G is a finitely generated group and S denotes a finite set of generators. We want to study the Novikov-Shubin invariant of the following operator

c_{S}: ⨁_{s∈S} l^{2}(G) \underrightarrow{⨁_{s∈S} r_{s−1}} l^{2}(G),

where r_{s−1} is right multiplication with s − 1 ∈ ℤ[G]. This is motivated by the following result.

Lemma 2.45. Let G be a finitely generated group and let X be a connected free G-CW -complex of finite type. Then for any finite set S of generators of G we have

α_{1}(X) = α(c_{S}).

Proof. The Cayley graph of G is the following connected one-dimensional free G-CW -complex. Its 0-skeleton is G. For each element s ∈ S we attach a free equivariant G-cell G × D^{1} by the attaching map G × S^{0} → G which sends (g, −1) to g and (g, 1) to gs. Since X and the Cayley graph C are connected we an choose a G-map f: X_{1} → C, where X_{1} is 1-skeleton of X. Theorem 2.55 (1) implies that α_{1}(X) = α_{1}(X_{1}) agrees with α_{1}(C) = α(c_{S}). ⊓ ⊔

If S is a finite set of generators for the group G, let b_{S}(n) be the number of elements in G which can be written as a word in n letters of S ∪ S^{−1} ∪ {1}. The group G has polynomial growth of degree not greater than d if there is C with b_{S}(n) ≤ Cn^{d} for all n ≥ 1. This property is a property of G and independent of the choice of the finite set S of generators. We say that G has polynomial growth if it has polynomial growth of degree not greater than d for some d > 0. A finitely generated group G is nilpotent if G possesses a finite lower central series

G = G_{1} ⊃ G_{2} ⊃ … ⊃ G_{s} = {1} G_{k+1} = [G, G_{k}].

If (P) is a property of groups, a group G is called virtually (P) if G contains a subgroup H ⊂ G of finite index such that H has property (P). Hence the notions of virtually finitely generated abelian, virtually free and virtually nilpotent are clear. In particular a group is virtually trivial if and only if it is finite. Let \overline{G} be virtually nilpotent. Let G ⊂ \overline{G} be a subgroup of finite index which is nilpotent and has the lower central series G = G_{1} ⊃ G_{2} ⊃ … ⊃ G_{s} = {1}. Let d_{i} be the rank of the finitely generated abelian group G_{i}/G_{i+1} and let d be the integer ∑_{i≥1} id_{i}. Then for any finite set S of generators of \overline{G} there is a constant C > 0 such that C^{−1}n^{d} ≤ b_{S}(n) ≤ Cn^{d} holds for any n ≥ 1 and in particular \overline{G} has polynomial growth precisely of degree d [22, page 607 and Theorem 2 on page 608]. A famous result of Gromov [231] says that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. The notion of an amenable group will be reviewed in Subsection 6.4.1.

Lemma 2.46. (1) α(c_{S}) = ∞^{+} if and only if G is non-amenable or finite;

(2) α(c_{S}) < ∞ if and only if G is infinite virtually nilpotent. If G is virtually nilpotent, α(c_{S}) is precisely the degree of the growth rate of G;

(3) α(c_{S}) = ∞ if and only if G is amenable and not virtually nilpotent.

Proof. Because of Lemma 2.45 α(c_{S}) is independent of the choice of the finite set S of generators. If G is finite, then obviously α(c_{S}) = ∞^{+}. Hence we can assume in the sequel that G is infinite and that S is symmetric, i.e. s ∈ S implies s^{−1} ∈ S. Define

P: l^{2}(G) \underrightarrow{∑_{s∈S} \frac{1}{|S|} ·r_{s}} l^{2}(G).

Then id −P = \frac{1}{2·|S|}c_{S} ∘c_{S}^{∗}. As G is infinite, dim_{N (G)}(l^{2}(G)/ clos(im(c_{S}))) = 0. This has been shown in the proof of Theorem 1.35 (8). Hence the kernel of c_{S}^{∗} is trivial. The spectrum of the selfadjoint operator P is contained in [-1,1] and we conclude from Lemma 2.3 and Lemma 2.4

tr_{N (G)}(χ_{[1−λ,1]}(P)) = F (c_{S})(\sqrt{2|S|λ}) − b^{(2)}(c_{S}),  (2.47)

where χ_{[1−λ,1]} is the characteristic function of [1 − λ, 1].

(1) From (2.47) α(c_{S}) has the value ∞^{+} if and only if the spectrum of the operator P does not contain 1. Since this operator is convolution with a probability distribution whose support contains S, namely

G → [0, 1], γ ↦ \{ \begin{matrix}|S|^{−1}, & γ ∈ S \\ 0, & γ ∉ S \end{matrix} ,

the spectrum of P contains 1 if and only if G is amenable by a result of Kesten [290, page 150], [524, Theorem 3.2 on page 7].

(2) The recurrency probability of the natural random walk on G is defined by

p(n) = tr_{N (G)}(P^{n}).

We will use the following result due to Varopoulos [502], which is also explained in [524, Theorem 6.5 and Theorem 6.6 on page 24]. (The assumption below that n even is needed since the period of the natural random walk on the Cayley graph with respect to symmetric set S of generators is 1 or 2.)

Theorem 2.48. The finitely generated group G has polynomial growth precisely of degree 2a if and only if there is a constant C > 0 such that C^{−1}n^{−a} ≤ p(n) ≤ Cn^{−a} holds for all even positive integers n. If the finitely generated group G does not have polynomial growth, then there is for each a > 0 a constant C(a) > 0 such that p(n) ≤ C(a)n^{−a} holds for all n ≥ 1.

In the sequel let n be an even positive integer. Notice that then P^{n} is positive. We have

(1 − λ)^{n}χ_{[1−λ,1]}(P) ≤ P^{n} ≤ (1 − λ)^{n}χ_{[0,1−λ]}(P) + χ_{[1−λ,1]}(P).

This implies because of ||χ_{[0,1−λ]}(P)|| ≤ 1

(1 − λ)^{n} tr_{N (G)}(χ_{[1−λ,1]}(P)) ≤ p(n) ≤ (1 − λ)^{n} + tr_{N (G)}(χ_{[1−λ,1]}(P)).

In the sequel we consider λ ∈ (0, 1/4) only. We conclude from (2.47)

\frac{ln(F (c_{S})(\sqrt{2|S|λ}) − b^{(2)}(c_{S}))}{ln(λ)} ≥ \frac{ln(p(n))}{ln(λ)} − n · \frac{ln(1 − λ)}{ln(λ)}; (2.49)
\frac{ln(F (c_{S})(\sqrt{2|S|λ}) − b^{(2)}(c_{S}))}{ln(λ)} ≤ \frac{ln(p(n) − (1 − λ)^{n})}{ln(λ)}.  (2.50)

Next we show for a > 0

α(c_{s}) ≥ 2a  if p(n) ≤ Cn^{−a} for n ≥ 1, n even;  (2.51)
α(c_{s}) ≤ 2a  if p(n) ≥ Cn^{−a} for n ≥ 1, n even,  (2.52)

where C > 0 is some constant independent of n. Suppose p(n) ≤ Cn^{−a} for all even positive integers n. Let n be the largest even integer which satisfies n ≤ λ^{−1}. Then n ≥ 2 since λ ∈ (0, 1/4). We estimate using (2.49)

\frac{ln(F (c_{S})(\sqrt{2|S|λ}) − b^{(2)}(c_{S}))}{ln(λ)} ≥ \frac{ln(Cn^{−a})}{ln(λ)} − n · \frac{ln(1 − λ)}{ln(λ)}
= a + \frac{ln(C) − a ln(λ · n)}{ln(λ)} − n · \frac{ln(1 − λ)}{ln(λ)}
≥ a + \frac{ln(C) − a ln(λ · n)}{ln(λ)} − \frac{ln(1 − λ)}{λ ln(λ)}.

Since | ln(C)−a ln(λ·n)| is bounded by | ln(C)|+a ln(2) and l’Hospital’s rule implies lim_{λ→0+} \frac{ln(1−λ)}{λ ln(λ)} = 0, we conclude (2.51) from Lemma 2.11 (4).

Suppose p(n) ≥ Cn^{−a} for all even positive integers n. Fix ϵ > 0. Put μ = 3^{−a−1}. Let [λ^{−(1+ϵ)}] be the largest integer which is less or equal to λ^{−(1+ϵ)}. Then we get for all λ ∈ (0, 1) and k ∈ {1, 2}

2μ ≤ (([λ^{−(1+ϵ)}] + k)λ^{1+ϵ})^{−a}  (2.53)

since 1 ≤ ([λ^{−(1+ϵ)}] + k)λ^{1+ϵ} ≤ 3 holds for λ ∈ (0, 1) and k ∈ {1, 2}. From l’Hospital’s rule we get

lim_{λ→0+} \frac{(ln(Cμ) + a(1 + ϵ) ln(λ)) · λ^{1+ϵ}}{ln(1 − λ)} = 0.

Hence we can choose λ_{0} ∈ (0, 1) such that for 0 < λ < λ_{0}

\frac{(ln(Cμ) + a(1 + ϵ) ln(λ)) · λ^{1+ϵ}}{ln(1 − λ)} ≤ \frac{1}{2}.

Put n = [λ^{−(1+ϵ)}] + k, where we choose k ∈ {1, 2} such that n is even. Since (1 − λ) ≤ 1 and λ^{−(1+ϵ)}/2 ≤ [λ^{−(1+ϵ)}] we get for 0 < λ < λ_{0}

(1 − λ)^{n} ≤ (1 − λ)^{[λ^{−(1+ϵ)}]} ≤ Cμ · λ^{a(1+ϵ)}.

Equation (2.53) implies 0 < Cμ ≤ C(([λ^{−(1+ϵ)}] + k)λ^{1+ϵ})^{−a} − Cμ. We conclude from (2.50) for λ ∈ (0, λ_{0})

\frac{ln(F (c_{S})(\sqrt{2|S|λ}) − b^{(2)}(c_{S}))}{ln(λ)}

≤ \frac{ln(C · n^{−a} − Cμ · λ^{a(1+ϵ)})}{ln(λ)}
= a(1 + ϵ) + \frac{ln(C(([λ^{−(1+ϵ)}] + k)λ^{1+ϵ})^{−a} − Cμ)}{ln(λ)}
≤ a(1 + ϵ) + \frac{ln(Cμ)}{ln(λ)}.

Lemma 2.11 (4) implies α(c_{S}) ≤ 2a(1 + ϵ). Since this is true for all ϵ > 0, (2.52) follows. Now assertion (2) follows from (2.51), (2.52) and Theorem 2.48.

(3) This follows from (1) and (2). This finishes the proof of Lemma 2.46. ⊓ ⊔

2.2 Cellular Novikov-Shubin Invariants

In this section we apply the invariants of Subsection 2.1.2 to the cellular L^{2}-chain complex and thus define Novikov-Shubin invariants for free G-CW - complexes of finite type. We will describe and prove their main properties.

Definition 2.54 (Novikov-Shubin invariants). Let X be a free G-CW - complex of finite type. Define its cellular p-th spectral density function and its cellular p-th Novikov-Shubin invariant by the corresponding notions (see Definition 2.16) of the cellular L^{2}-chain complex C_{∗}^{(2)}(X) of X (see Definition 1.29).

F_{p}(X; N (G)) := F_{p}(C_{∗}^{(2)}(X));
α_{p}(X; N (G)) := α_{p}(C_{∗}^{(2)}(X));
α_{p}^{Δ}(X; N (G)) := α_{p}^{Δ}(C_{∗}^{(2)}(X)).

If the group G and its action are clear from the context, we omit N (G) in the notation above.

If M is a cocompact free proper G-manifold, define its p-th cellular spectral density function and its cellular p-th Novikov-Shubin invariant by the corresponding notion for some equivariant smooth triangulation.

Since the Novikov-Shubin invariant will turn out to be homotopy invariant, the definition of the p-th Novikov-Shubin invariant for a cocompact free proper G-manifold is independent of the choice of equivariant smooth triangulation. This is also true for the dilatational equivalence class of the spectral density function. All these definitions extend in the obvious way to pairs.

Theorem 2.55 (Novikov-Shubin invariants).

(1) Homotopy invariance Let f: X → Y be a G-map of free G-CW -complexes of finite type. Suppose that the map induced on homology with complex coefficients H_{p}(f; ℂ): H_{p}(X; ℂ) → H_{p}(Y; ℂ) is an isomorphism for p ≤ d − 1. Then

F_{p}(X) ≃ F_{p}(Y)  for p ≤ d;
α_{p}(X) = α_{p}(Y)  for p ≤ d.

In particular, if f is a weak homotopy equivalence, we get for all p ≥ 0

F_{p}(X) ≃ F_{p}(Y);
α_{p}(X) = α_{p}(Y);

(2) Poincaré duality Let M be a cocompact free proper G-manifold of dimension n which is orientable. Then

F_{p}(M) ≃ F_{n+1−p}(M, ∂M);
α_{p}(M) = α_{n+1−p}(M, ∂M);

(3) Product formula Let X be a free G-CW -complex of finite type and let Y be a free H-CW - complex of finite type. Suppose that both X and Y have the limit property (see Definition 2.41). Then X × Y has the limit property and α_{p}(X × Y) is the minimum in [0, ∞] ∐{∞^{+}} of the union of the following four sets

{α_{i+1}(X) + α_{p−i}(Y) | i = 0, 1, …, (p − 1)};
{α_{i}(X) + α_{p−i}(Y) | i = 1, …, (p − 1)};
{α_{p−i}(Y) | i = 0, 1, …, (p − 1), b_{i}^{(2)}(X) ≠ 0};
{α_{i}(X) | i = 1, 2, …, p, b_{p−i}^{(2)}(Y) ≠ 0};

(4) Connected sums Let M_{1}, M_{2}, …, M_{r} be compact connected m-dimensional manifolds, with m ≥ 3. Let M be their connected sum M_{1}# … #M_{r}. Then

α_{p}(M̃) = min {α_{p}(M̃_{j}) | 1 ≤ j ≤ r} for 2 ≤ p ≤ m − 1.

If π_{1}(M_{i}) is trivial for all i except for i = i_{0}, then α_{1}(M̃) = α_{1}(M̃_{i_{0}}). If π_{1}(M_{i}) is trivial for all i except for i ∈ {i_{0}, i_{1}}, i_{0} ≠ i_{1} and π_{1}(M_{i_{0}}) = π_{1}(M_{i_{1}}) = ℤ/2, then α_{1}(M̃) = 1. In all other cases α_{1}(M̃) = ∞^{+};

(5) First Novikov-Shubin invariant Let X be a connected free G-CW -complex of finite type. Then G is finitely generated and

(a) α_{1}(X) is finite if and only if G is infinite and virtually nilpotent. In this case α_{1}(X) is the growth rate of G;

(b) α_{1}(X) is ∞^{+} if and only if G is finite or non-amenable;

(c) α_{1}(X) is ∞ if and only if G is amenable and not virtually nilpotent;

(6) Restriction Let X be a free G-CW -complex of finite type and let H ⊂ G be a subgroup of finite index [G : H]. Let res_{G}^{H} X be the H-space obtained from X by restricting the G-action to an H-action. Then this is a free H-CW - complex of finite type and we get for p ≥ 0

F_{p}(X; N (G)) = \frac{1}{[G : H]} · F_{p}(res_{G}^{H} X; N (H));
α_{p}(X; N (G)) = α_{p}(res_{G}^{H} X; N (H));

(7) Induction Let H be a subgroup of G and let X be a free H-CW -complex of finite type. Then G ×_{H} X is a free G-CW -complex of finite type and

F_{p}(G ×_{H} X; N (G)) = F_{p}(X; N (H));
α_{p}(G ×_{H} X; N (G)) = α_{p}(X; N (H)).

Proof. (1) We can assume without loss of generality that f is cellular. We have the canonical exact sequence of finitely generated free ℂG-chain complexes 0 → C_{∗}(X) → cyl(C_{∗}(f)) → cone(C_{∗}(f)) → 0, where C_{∗}(f) is the ℂG-chain map induced by f on the cellular ℂG-chain complexes. Let C_{∗}, D_{∗} and E_{∗} be the d-dimensional finitely generated ℂG-chain complexes which are obtained from C_{∗}(X), cyl(C_{∗}(f)) and cone(C_{∗}(f)) by truncating everything in dimension d+1 and higher. We get an exact sequence of finitely generated free ℂG-chain complexes 0 → C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{p_{∗}} E_{∗} → 0. Since the canonical inclusion C_{∗}(Y) → cyl(C_{∗}(f)) is a ℂG-chain homotopy equivalence, it induces a chain homotopy equivalence of finitely generated Hilbert N (G)-chain complexes C_{∗}^{(2)}(Y) → l^{2}(G) ⊗_{ℂG} cyl(C_{∗}(f)). We conclude from Theorem 2.19 that F_{p}(l^{2}(G) ⊗_{ℂG} C_{∗}) ≃ F_{p}(X) and F_{p}(l^{2}(G) ⊗_{ℂG} D_{∗}) ≃ F_{p}(Y) holds for p ≤ d. Hence it remains to show

F_{p}(l^{2}(G) ⊗_{ℂG} C_{∗}) ≃ F_{p}(l^{2}(G) ⊗_{ℂG} D_{∗}) for p ≤ d.

The map H_{p}(i_{∗}): H_{p}(C_{∗}) → H_{p}(D_{∗}) is bijective for p ≤ d − 1 since H_{p}(f; ℂ) is bijective for p ≤ d−1. This implies that H_{p}(E_{∗}) = 0 for p ≤ d−1. Let P be the kernel of e_{d}: E_{d} → E_{d−1}. Since 0 → P → E_{d} → E_{d−1} → E_{d−2} → … → E_{0} → 0 is an exact ℂG-sequence and each E_{i} is finitely generated free, there is a finitely generated free ℂG-module F′ such that F := P ⊕ F′ is finitely generated free and P is a direct summand in E_{d}. For a ℂG-module W let d[W]_{∗} be the ℂG-chain complex concentrated in dimension d with W as d-th chain module. Let D′_{∗} be the preimage of d[F]_{∗} = d[P]_{∗}⊕d[F′]_{∗} ⊂ E_{∗}⊕d[F′]_{∗} under p_{∗} ⊕ id_{d[F′]_{∗}}: D_{∗} ⊕ d[F′]_{∗} → E_{∗} ⊕ d[F′]_{∗}. We obtain a commutative diagram of ℂG-chain complexes with exact rows

0 ⟶ C_{∗} ⟶^{i_{∗}} D_{∗} ⊕ d[F′]_{∗} ⟶^{p_{∗}⊕id_{d[F′]∗}} E_{∗} ⊕ d[F′]_{∗} ⟶ 0
id↑  ^{j_{∗}}↑  ^{k_{∗}}↑
0 ⟶ C_{∗} ⟶^{i′_{∗}}  D′_{∗}  ⟶^{p′_{∗}}  d[F]_{∗} ⟶ 0

where j_{∗} and k_{∗} are the inclusions. Since id_{C_{∗}} and k_{∗} are ℂG-homotopy equivalences, j_{∗}: D′_{∗} → D_{∗} ⊕ d[F′]_{∗} is a ℂG-chain homotopy equivalence. Theorem 2.19 and Lemma 2.11 (9) imply for p ≤ d

F_{p}(l^{2}(G) ⊗_{ℂG} D′_{∗}) ≃ F_{p}(l^{2}(G) ⊗_{ℂG} (D_{∗} ⊕ d[F′]_{∗})) ≃ F_{p}(l^{2}(G) ⊗_{ℂG} D_{∗}).

Hence it remains to show

F_{p}(l^{2}(G) ⊗_{ℂG} C_{∗}) ≃ F_{p}(l^{2}(G) ⊗_{ℂG} D′_{∗}) for p ≤ d.

There is a ℂG-chain complex C′_{∗} such that C′_{p} = C_{p} and c′_{p} = c_{p} for p ≠ d, C′_{d} = C_{d}⊕F, c′_{d} = c_{d}⊕u: C_{d}⊕F → C_{d−1} for some ℂG-map u: F → C_{d−1} and a chain isomorphism g_{∗}: C′_{∗} ⟶^{≅} D′_{∗} such that g_{∗} composed with the obvious inclusion l_{∗}: C_{∗} → C′_{∗} is i′_{∗}. Hence F_{p}(l^{2}(G) ⊗_{ℂG} C′_{∗}) ≃ F_{p}(l^{2}(G) ⊗_{ℂG} D′_{∗}) for p ≤ d and H_{d−1}(l_{∗}): H_{d−1}(C_{∗}) → H_{d−1}(C′_{∗}) is bijective. It remains to show

F (id_{l^{2}(G)} ⊗_{ℂG}(c_{d} ⊕ u): l^{2}(G) ⊗_{ℂG} (C_{d} ⊕ F) → l^{2}(G) ⊗_{ℂG} C_{d−1})

≃ F (id_{l^{2}(G)} ⊗_{ℂG}c_{d}: l^{2}(G) ⊗_{ℂG} C_{d} → l^{2}(G) ⊗_{ℂG} C_{d−1}) . (2.56)

Since H_{d−1}(l_{∗}) is bijective, im(c_{d}) = im(c′_{d}). This shows im(u) ⊂ im(c_{d}). As F is finitely generated free, we can find a ℂG-map v: F → C_{d} with u = c_{d} ∘v. The map id_{C_{d}} ⊕v: C_{d} ⊕ F → C_{d} is split surjective. Hence the map

id_{l^{2}(G)} ⊗_{ℂG}(id_{C_{d}} ⊕v): l^{2}(G) ⊗_{ℂG} (C_{d} ⊕ F) → l^{2}(G) ⊗_{ℂG} C_{d}

is a surjective map of finitely generated Hilbert N (G)-modules whose composition with id_{l^{2}(G)} ⊗_{ℂG}c_{d}: l^{2}(G)⊗_{ℂG}C_{d} → l^{2}(G)⊗_{ℂG}C_{d−1} is id_{l^{2}(G)} ⊗_{ℂG}(c_{d}⊕ u): l^{2}(G) ⊗_{ℂG} (C_{d} ⊕ F) → l^{2}(G) ⊗_{ℂG} C_{d−1}. Now (2.56) and thus assertion (1) follow from Lemma 2.11 (9).

(2) is proved analogously to Theorem 1.35 (3) using Lemma 2.17 (2) and Theorem 2.19.

(3) This follows from Corollary 2.44 since the cellular chain complex of X ×Y is the tensor product of the one of X and the one of Y.

(4) This is proved analogously to Theorem 1.35 (6) for 2 ≤ p ≤ m − 1. The claim for α_{1} follows from assertion (5) and assertion (6) since a free product G_{1} ∗ G_{2} of non-trivial groups is amenable if and only if G_{1} = G_{2} = ℤ/2, the group ℤ/2 ∗ ℤ/2 contains ℤ as a subgroup of finite index and α_{1}(S̃^{1}) = 1 by Lemma 2.58.

(5) Since X is connected and X → G\X is a regular covering, there is a short exact sequence 1 → π_{1}(X) → π_{1}(G\X) → G → 1. Since G\X has finite 1skeleton by assumption, π_{1}(G\X) and hence G are finitely generated. Now we can apply Lemma 2.45 and Lemma 2.46.

(6) This follows from Theorem 1.12 (6).

(7) This follows from Lemma 1.24 (2), since for each morphism f: U → V of Hilbert N (H)-modules i_{∗}E_{λ}^{f^{∗}f} = E_{λ}^{(i_{∗}f)^{∗}i_{∗}f} and hence

F_{N (H)}(f) = F_{N (G)}(i_{∗}f)  (2.57)

holds. ⊓ ⊔

A more conceptual proof for Theorem 2.55 (1) can be given after we have developed some theory in Section 6.7. We will see that F_{p}(X) ≃ F_{p}(Y) for p ≤ d holds if H_{p}^{G}(f; N (G)): H_{p}^{G}(X; N (G)) → H_{p}^{G}(Y; N (G)) is bijective for p ≤ d − 1. This will only use that the Novikov-Shubin invariants of homotopy equivalent N (G)-Hilbert chain complexes agree (see Theorem 2.19). Here H_{p}^{G}(X; N (G)) is the homology of the chain complex of modules over the ring N (G) given by N (G) ⊗_{ℂG} C_{∗}(X), where only the ring structure of N (G) enters. The bijectivity of H_{p}^{G}(f; N (G)) for p ≤ d − 1 follows from the universal coefficients spectral sequence [518, Theorem 5.6.4 on page 143] which converges to H_{p+q}^{G}(X; N (G)) and has as E^{2}-term E_{p,q}^{2} = Tor_{p}^{ℂG}(N (G), H_{q}(X; ℂ)) since by assumption H_{p}(f; ℂ): H_{p}(X; ℂ) → H_{p}(Y; ℂ) is bijective for p ≤ d − 1.

We will see in Example 3.110 that the condition “orientable” in Theorem 2.55 (2) is necessary.

Lemma 2.58. Let C_{∗} be a free ℂ[ℤ]-chain complex of finite type. Since ℂ[ℤ] is a principal ideal domain [15, Proposition V.5.8 on page 151 and Corollary V.8.7 on page 162], we can write

H_{p}(C_{∗}) = ℂ[ℤ]^{n_{p}} ⊕ (⨁_{i_{p}=1}^{s_{p}} ℂ[ℤ]/((z − a_{p,i_{p}})^{r_{p,ip}}))

for a_{p,i_{p}} ∈ ℂ and n_{p}, s_{p}, r_{p,i_{p}} ∈ ℤ with n_{p}, s_{p} ≥ 0 and r_{p,i_{p}} ≥ 1, where z is a fixed generator of ℤ.

Then l^{2}(ℤ) ⊗_{ℂ[ℤ]} C_{∗} has the limit property and

b_{p}^{(2)}(l^{2}(ℤ) ⊗_{ℂ[ℤ]} C_{∗}) = n_{p}.

If s_{p} ≥ 1 and {i_{p} = 1, 2 …, s_{p}, |a_{p,i_{p}}| = 1} ≠ ∅, then

α_{p+1}(l^{2}(ℤ) ⊗_{ℂ[ℤ]} C_{∗}) = min\{\frac{1}{r_{p,i_{p}}} | i_{p} = 1, 2 …, s_{p}, |a_{p,i_{p}}| = 1\},

otherwise

α_{p+1}(l^{2}(ℤ) ⊗_{ℂ[ℤ]} C_{∗}) = ∞^{+}.

Proof. The statement about the L^{2}-Betti numbers has already been proved in Lemma 1.34.

Let P (n_{p})_{∗} be the chain complex concentrated in dimension 0 whose 0-th module is ℂ[ℤ]^{n_{p}}. Let Q(a_{p,i_{p}}, r_{p,i_{p}})_{∗} be the ℂ[ℤ]-chain complex concentrated in dimensions 0 and 1 whose first differential M_{(z−a_{p,ip})^{r_{p,ip}}} : ℂ[ℤ] → ℂ[ℤ] is multiplication with (z − a_{p,i_{p}})^{r_{p,ip}}. Notice that its homology is trivial except in dimension 0 where it is given by ℂ[ℤ]/((z − a_{i})^{r_{p,ip}}). One easily constructs a ℂ[ℤ]-chain map

f_{∗}: ⨁_{p≥0} Σ^{p} (P (n_{p})_{∗} ⊕ ⨁_{i_{p}=1}^{s_{p}} Q(a_{p,i_{p}}, r_{p,i_{p}})_{∗}) → C_{∗}

which induces an isomorphism on homology and is therefore a ℂ[ℤ]-chain equivalence. Because of Lemma 2.17 (3) and Theorem 2.19 it suffices to show for a ∈ ℂ, r ∈ ℤ, r ≥ 1

α_{1}(Q(a, r)_{∗}) = \{ \begin{matrix}\frac{1}{r} & if |a| = 1 \\ ∞^{+} & if |a| ≠ 1 \end{matrix} .

Because of Example 2.6 we get

F_{1}(Q(a, r)_{∗})(λ) = vol{z ∈ S^{1} | |(z − a)^{r}| ≤ λ}.

If |a| ≠ 1, then F_{1}(Q(a, r)_{∗})(λ) = 0 for 0 ≤ λ < |1 − |a||^{r} and hence α_{1}(Q(a, r)_{∗}) = ∞^{+}. If |a| = 1 we conclude α_{1}(Q(a, r)_{∗}) = \frac{1}{r} from

vol{z ∈ S^{1} | |(z − a)^{r}| ≤ λ} = vol{cos(ϕ) + i sin(ϕ) | |2 − 2 cos(ϕ)|^{r/2} ≤ λ};
lim_{ϕ→0} \frac{2 − 2 cos(ϕ)}{ϕ^{2}} = 1.

This finishes the proof of Lemma 2.58. ⊓ ⊔

Example 2.59. For G = ℤ we conclude from Lemma 2.58 that α_{1}(S̃^{1}) = 1 and α_{p}(S̃^{1}) = ∞^{+} for p ≥ 2 and that S̃^{1} has the limit property. We get for the n-torus T^{n} from Theorem 2.55 (3) that T̃^{n} has the limit property and

α_{p}(T̃^{n}) = \{ \begin{matrix}n & 1 ≤ p ≤ n \\ ∞^{+} & otherwise \end{matrix} .

Example 2.60. Let t_{1} and t_{2} be the generators of ℤ^{2} and let f: l^{2}(ℤ^{2}) → l^{2}(ℤ^{2}) be given by right multiplication with (t_{1} − 1)(t_{2} − 1). Then we get from Example 2.6 for small λ > 0

F (f)(λ) = vol{(z_{1}, z_{2}) ∈ T^{2} | |z_{1} − 1| · |z_{2} − 1| ≤ λ}
= vol{(u_{1}, u_{2}) ∈ [−π, π] × [−π, π] |

|2 − 2 cos(u_{1})|^{1/2} · |2 − 2 cos(u_{1})|^{1/2} ≤ λ}

≃ vol{(u_{1}, u_{2}) ∈ [−π, π] × [−π, π] | |u_{1}||u_{2}| ≤ λ}
= 4 · (∫_{λ/π}^{π} \frac{λ}{u}du + \frac{λ}{π} π)
= 4 · (λ ln(π) − λ ln(λ/π) + λ)
= 4λ · (− ln(λ) + 2 ln(π) + 1)
≃ − ln(λ) · λ.

This shows that α(f) = 1 and that F (f) is not dilatationally equivalent to λ.

Theorem 2.61 (Novikov-Shubin invariants and S^{1}-actions). Let X be a connected S^{1}-CW -complex of finite type. Suppose that for one orbit S^{1}/H (and hence for all orbits) the inclusion into X induces a map on π_{1} with infinite image. (In particular the S^{1}-action has no fixed points.) Let X̃ be the universal covering of X with the canonical π_{1}(X)-action. Then we get for all p ≥ 0

b_{p}^{(2)}(X̃) = 0;
α_{p}(X̃) ≥ 1.

Proof. We show for any S^{1}-CW -complex Y of finite type together with a S^{1}-map f: Y → X that b_{p}^{(2)}(f^{∗}X̃; N (π)) = 0 and α_{p}(f^{∗}X̃; N (π)) ≥ 1 for all p ≥ 0, where f^{∗}X̃ is the pullback of the universal covering of X and π = π_{1}(X). The statement about the L^{2}-Betti numbers has already been proved in Theorem 1.40. The proof of the statement about the Novikov-Shubin invariants is analogous using Theorem 2.20, Theorem 2.55 (7) and the conclusion from Lemma 2.58 that

α_{p}(S̃^{1}) = \{ \begin{matrix}1 & if p = 1; \\ ∞^{+} & otherwise. \end{matrix}  ⊓ ⊔

The notion of elementary amenable group will be explained in Definition 6.34. The next result is taken from [260, Corollary 2 on page 240]. Notice that a group H, which has a finite-dimensional model for BH, is torsionfree.

Lemma 2.62. Let 1 → H → G → Q → 1 be an extension of groups such that H is elementary amenable and BG has a finite-dimensional model. Then G contains a normal torsionfree abelian subgroup.

Theorem 2.63. (Novikov Shubin invariants and aspherical CW - complexes). Let X be an aspherical finite CW -complex. Suppose that its fundamental group contains an elementary amenable infinite normal subgroup H. Then

b_{p}^{(2)}(X̃) = 0 for p ≥ 0;
α_{p}(X̃) ≥ 1 for p ≥ 1.

Proof. The claim for the L^{2}-Betti numbers is a special case of Theorem 1.44. Since X is a finite-dimensional model for Bπ_{1}(X), we can assume without loss of generality by Lemma 2.62 that H is torsionfree abelian. In the case H = ℤ^{n}, the claim follows from [343, Theorem 3.9 (6) on page 174]. In the general case, one has to notice that any finitely generated subgroup of H is isomorphic to ℤ^{n} for some n ≤ dim(X). Now the claim follows from [343, Theorem 3.7 on page 172]. More details can also be found in [515, section 4.5]. ⊓ ⊔

Theorem 2.63 still makes sense and still is true (by the same proof) if one replaces finite by finite-dimensional and uses the extension of the definition of Novikov-Shubin invariant to arbitrary spaces in [343].

2.3 Analytic Novikov-Shubin Invariants

In this section we introduce the analytic version of spectral density functions and Novikov-Shubin invariants.

Definition 2.64 (Analytic spectral density function). Let M be a cocompact free proper G-manifold without boundary and with G-invariant Riemannian metric. Let d_{min}^{p} and (Δ_{p})_{min} be the minimal closures of the densely defined operators d^{p}: Ω_{c}^{p}(M) → L^{2}Ω^{p+1}(M) and Δ_{p}: Ω_{c}^{p}(M) → L^{2}Ω^{p}(M). Let d_{min}^{p⊥}: dom(d_{min}^{p}) ∩ im(d_{min}^{p−1})^{⊥} → im(δ_{min}^{p+2})^{⊥} be the operator induced by d_{min}^{p}. Define the analytic p-th spectral density function of M by

F_{p}(M) := F (d_{min}^{p⊥}): [0, ∞) → [0, ∞)

and define

F_{p}^{Δ}(M) := F ((Δ_{p})_{min}): [0, ∞) → [0, ∞).

Define the analytic p-th Novikov-Shubin invariant of M by

α_{p}(M) := α(F_{p−1}(M)).

Put

α_{p}^{Δ}(M) := α(F_{p}^{Δ}(M)).

Notice that here we define α_{p}(M) to be α(F_{p−1}(M)) and not to be α(F_{p}(M)) as in Definition 2.16 and Definition 2.54 because here we are dealing with cochain complexes whereas in the cellular context we use chain complexes and we want to show later that both definitions agree (see Lemma 2.17 (2)).

Notation 2.65. Denote by (δ^{p+1}d^{p})_{min}^{⊥} and (d^{p−1}δ^{p})_{min}^{⊥} the minimal closure of the operators

δ^{p+1}d^{p}: Ω_{c}^{p}(M) ∩ im(d_{min}^{p−1})^{⊥} → im(d_{min}^{p−1})^{⊥};
d^{p−1}δ^{p}: Ω_{c}^{p}(M) ∩ im(δ_{min}^{p+1})^{⊥} → im(δ_{min}^{p+1})^{⊥}.

Let d_{min}^{p⊥}: dom(d_{min}^{p}) ∩ im(d_{min}^{p−1})^{⊥} → im(δ_{min}^{p+2})^{⊥} and δ_{min}^{p⊥}: dom(δ_{min}^{p}) ∩ im(δ_{min}^{p+1})^{⊥} → im(d_{min}^{p−2})^{⊥} be the closed densely defined operators induced by d_{min}^{p} and δ_{min}^{p}. Denote by E_{λ}^{Δ_{p}}(x, y) the smooth Schwartz kernel for the projection E_{λ}^{Δ_{p}} appearing in the spectral family of the selfadjoint operator (Δ_{p})_{min}.

The existence of the smooth Schwartz kernel E_{λ}^{Δ_{p}}(x, y) is for instance proved in [9, Proposition 2.4]. It is a smooth section of the vector bundle hom(p_{1}^{∗} Alt^{p}(T M), p_{2}^{∗} Alt^{p}(T M)) over M × M for p_{k}: M × M → M the projection to the k-th factor. It is uniquely characterized by the property that for ω ∈ L^{2}Ω^{p}(M)

E_{λ}^{Δ_{p}}(ω)(x) = ∫_{M} E_{λ}^{Δ_{p}}(x, y)(ω_{y}) dvol_{y}

holds.

Lemma 2.66. Let M be a cocompact free proper G-manifold without boundary and with G-invariant Riemannian metric. Then (1) The values of the functions F_{p}(M)(λ) and F_{p}^{Δ}(M)(λ) are finite for all

λ ∈ ℝ. In particular F_{p}(M) and F_{p}^{Δ}(M) are Fredholm;

(2) We have

F_{p}^{Δ}(M)(λ^{2}) − b_{p}^{(2)}(M) = (F_{p−1}(M)(λ) − b_{p−1}^{(2)}(M))
+(F_{p}(M)(λ) − b_{p}^{(2)}(M));
α_{p}^{Δ}(M) = 1/2 · min{α_{p}(M), α_{p+1}(M)};

(3) We have

F_{p}^{Δ}(M)(λ) = ∫_{F} tr(E_{λ}^{Δ^{p}}(x, x)) dvol_{x},

where F is a fundamental domain of the G-action;

(4) If F_{p}(H^{l−∗}Ω^{∗}(M))(λ) is the p-th spectral density function of the Hilbert N (G)-cochain complex H^{l−∗}Ω^{∗}(M) (see (1.74)), then

F_{p}(M) ≃ F_{p}(H^{l−∗}Ω^{∗}(M)).

Proof. We get from [9, Proposition 4.16 on page 63]

tr_{N (G)}(E_{λ}^{Δ_{p}}) = ∫_{F} tr(E_{λ}^{Δ_{p}}(x, x)) dvol_{x} .

This and F_{p}^{Δ}(M)(λ)) = tr_{N (G)}(E_{λ}^{Δ_{p}}) imply that F_{p}^{Δ}(M)(λ)) is finite for all λ ∈ ℝ and satisfies

F_{p}^{Δ}(M)(λ) = ∫_{F} tr(E_{λ}^{Δ_{p}}(x, x)) dvol_{x} .  (2.67)

One easily checks the following equalities of densely defined operators using Lemma 1.70 (1)

(Δ_{p})_{min}|_{im(d_{min}^{p−1})^{⊥}} = (δ^{p+1}d^{p})_{min}^{⊥};
(Δ_{p})_{min}|_{im(δ_{min}^{p+1})^{⊥}} = (d^{p−1}δ^{p})_{min}^{⊥};
δ_{min}^{p⊥} = (d_{min}^{(p−1)⊥})^{∗};
(δ^{p+1}d^{p})_{min}^{⊥} = (d_{min}^{p⊥})^{∗}d_{min}^{p⊥};
(d^{p−1}δ^{p})_{min}^{⊥} = (δ_{min}^{p⊥})^{∗}δ_{min}^{p⊥}.

We conclude from Theorem 1.57

ker((Δ_{p})_{min}) = ker(d_{min}^{p⊥}) = ker((δ^{p+1}d^{p})_{min}^{⊥}) = ker((d^{p−1}δ^{p})_{min}^{⊥})

= ker(δ_{min}^{p⊥}) = H_{(2)}^{p}(M).

We conclude from Theorem 1.57, Lemma 2.3 and Lemma 2.4

F_{p}^{Δ}(M)(λ) + dim_{N (G)}(H_{(2)}^{p}(M)) = F ((δ^{p+1}d^{p})_{min}^{⊥})(λ) + F ((d^{p−1}δ^{p})_{min}^{⊥})(λ);
F ((δ^{p+1}d^{p})_{min}^{⊥})(λ^{2}) = F_{p}(M)(λ);
F ((d^{p−1}δ^{p})_{min}^{⊥})(λ^{2}) − b_{p}^{(2)}(M) = F_{p−1}(M)(λ) − b_{p−1}^{(2)}(M).

This together with (2.67) proves assertions (1), (2) and (3). We will prove (4) in the next Section 2.4. ⊓ ⊔

The following result is due to Efremov [167]. We will prove it in the next Section 2.4.

Theorem 2.68. (Analytic and combinatorial Novikov-Shubin invariants). Let M be a cocompact free proper G-manifold without boundary and with G-invariant Riemannian metric. Then the cellular and the analytic spectral density functions (see Definition 2.54 and Definition 2.64) are dilatationally equivalent and the cellular and analytic Novikov-Shubin invariants agree in each dimension p.

Example 2.69. The Novikov-Shubin invariants of the universal covering M̃ of a closed hyperbolic manifold of dimension n have been computed [316, Proposition 46 on page 499] using the analytic approach, namely

α_{p}(M̃) = \{ \begin{matrix}1 & if n is odd and 2p = n ± 1 \\ ∞^{+} & otherwise \end{matrix} .

Moreover, the spectral density function has the limit property.

Example 2.70. In the notation of Example 1.36 we get

α_{p}(F̃_{g}^{d}) = \{ \begin{matrix}2 & if g = 1, d = 0, p = 1, 2 \\ 1 & if g = 0, d = 2, p = 1 \\ ∞^{+} & otherwise \end{matrix} .

This follows from Theorem 2.55 (1) and (5), Example 2.59, Example 2.69 and the facts that F_{g}^{d} is homotopy equivalent to ⋁_{i=1}^{2g+d−1} S^{1} for d ≥ 1, F_{g}^{0} is hyperbolic for g ≥ 2, F_{1}^{0} = T^{2}, π_{1}(F_{0}^{0}) = 1 and a free group of rank ≥ 2 is not amenable.

2.4 Comparison of Analytic and Cellular Novikov-Shubin Invariants

In this section we give the proofs of Lemma 2.66 (4) and of Theorem 2.68. They will follow from Lemma 2.71, Lemma 2.72 and Lemma 2.80.

Lemma 2.71. For any equivariant smooth triangulation K we have

F_{p}(K) ⪯ F_{p}(H^{l−∗}Ω^{∗}(M)).

Proof. We have constructed cochain maps of Hilbert N (G)-modules in (1.77) and (1.78)

A^{∗}: H^{l−∗}Ω^{∗}(M) → C_{(2)}^{∗}(K);
W^{∗}: C_{(2)}^{∗}(K) → H^{l−∗}Ω^{∗}(M)

such that A^{∗} ∘ W^{∗} = id holds (see (1.79)). Now apply Lemma 2.17 (3). ⊓ ⊔

Lemma 2.72. We have

F_{p}(H^{l−∗}Ω^{∗}(M))) ⪯ F_{p}(M).

Proof. The following diagram commutes and the vertical maps are given by the isometric isomorphisms of Hilbert N (G)-modules of (1.73)

H^{l−p}Ω^{p}(M) ⟶^{d_{H^{l−p}}^{p}} H^{l−p−1}Ω^{p+1}(M)
(1+Δ_{p})^{(l−p−1)/2}↓  ^{(1+Δ_{p})^{(l−p−1)/2}}↓
H^{1}Ω^{p}(M) ⟶^{d_{H^{1}}^{p}} L^{2}Ω^{p+1}(M)

where d_{H^{l−p}}^{p}: H^{l−p}Ω^{p}(M) → H^{l−p−1}Ω^{p+1}(M) is the bounded operator which is induced by d^{p}: Ω_{c}^{p}(M) → Ω_{c}^{p+1}(M). Hence it remains to show

F (d_{H^{1}}^{p⊥}) ⪯ F (d_{min}^{p⊥}),  (2.73)

where d_{H^{1}}^{p⊥}: im(d_{H^{2}}^{p−1})^{⊥} → L^{2}Ω^{p+1}(M) is the operator which is induced by d_{H^{1}}^{p}: H^{1}Ω^{p}(M) → L^{2}Ω^{p+1}(M).

For ω ∈ Ω_{c}^{p−1}(M) and η ∈ im(d_{H^{2}}^{p−1})^{⊥} we conclude using the isomorphism (1.73) and partial integration (see Lemma 1.56)

⟨d^{p−1}(ω), η⟩_{L^{2}} = ⟨(1 + Δ_{p}) ∘ d_{H^{2}}^{p−1} ∘ (1 + Δ_{p})^{−1}(ω), η⟩_{L^{2}}
= ⟨(1 + Δ_{p})^{1/2} ∘ d_{H^{2}}^{p−1} ∘ (1 + Δ_{p})^{−1}(ω), (1 + Δ_{p})^{1/2}(η)⟩_{L^{2}}
= ⟨d_{H^{2}}^{p−1}((1 + Δ_{p})^{−1}(ω)), η⟩_{H^{1}}
= 0.  (2.74)

One easily checks that for ω ∈ Ω_{c}^{p}(M) and hence for all ω ∈ H^{1}Ω^{p}(M)

||ω||_{1}^{2} = ||ω||_{L^{2}}^{2} + ||d^{p}ω||_{L^{2}}^{2} + ||δ^{p}ω||_{L^{2}}^{2}  (2.75)

holds. Now (2.74) and (2.75) imply that we obtain a well-defined injective morphism of Hilbert N (G)-modules induced by the inclusion H^{1}Ω^{p}(M) → L^{2}Ω^{p}(M)

j: im(d_{H^{2}}^{p−1})^{⊥} → im(d_{min}^{p−1})^{⊥}.

Consider 0 ≤ λ < 1 and L ∈ L(d_{H^{1}}^{p⊥}, λ). For ω ∈ L we have j(w) ∈ im(d_{min}^{p−1})^{⊥} and hence δ^{p}(ω) = 0 and we get from (2.75)

||d^{p}(ω)||_{L^{2}}^{2} ≤ λ^{2} · ||ω||_{1}^{2} = λ^{2} · ||ω||_{L^{2}}^{2} + λ^{2} · ||d^{p}(ω)||_{L^{2}}^{2}.

This implies

||d^{p}(ω)||_{L^{2}} ≤ \frac{λ}{\sqrt{1 − λ^{2}}} · ||ω||_{L^{2}}^{2}.

We conclude from Lemma 2.2 (2) that the composition

E_{λ^{2}/(1−λ^{2})}^{(d_{min}^{p⊥})^{∗}d_{min}^{p⊥}} ∘ j: im(d_{H^{2}}^{p−1})^{⊥} → im(E_{λ^{2}/(1−λ^{2})}^{(d_{min}^{p⊥})^{∗}d_{min}^{p⊥}})

is injective on L. This implies using Lemma 2.3

dim_{N (G)}(L) ≤ dim_{N (G)}(im(E_{λ^{2}/(1−λ^{2})}^{(d_{min}^{p⊥})^{∗}d_{min}^{p⊥}})) = F_{p}(M)(λ/\sqrt{1 − λ^{2}})

and hence

F_{p}(H^{l−∗}Ω^{∗}(M))(λ) ≤ F_{p}(M)(λ/\sqrt{1 − λ^{2}}).

This finishes the proof of Lemma 2.72. ⊓ ⊔ The mesh of a triangulation is defined by

mesh(K) := sup{d(p, q) | p, q vertices of a 1-simplex},

where d(p, q) is the metric on M induced by the Riemannian metric. The fullness of a triangulation is defined by

full(K) := inf \{ \frac{vol(σ)}{dim(M) mesh(K)}| σ ∈ S_{dim(M)}(K)\} .

The next result is taken from [145, page 165]. Its proof is based on [150, Proposition 2.4 on page 8]. Notice that from now on W^{∗} is given in terms of barycentric coordinate functions.

Lemma 2.76. Let M be a cocompact free proper G-manifold without boundary and with G-invariant Riemannian metric. Fix θ > 0, k > dim(M)/2 + 1 and an equivariant smooth triangulation K. Then there is a constant C > 0 such that for any equivariant barycentric subdivision K′ of M with fullness full(K′) ≥ θ and any p-form ω ∈ H^{k}Ω^{p}(M)

||ω − W_{K′}^{p} ∘ A_{K′}^{p}(ω)||_{L^{2}} ≤ C · mesh(K′) · ||ω||_{k}

holds.

Fix ϵ > 0. Lemma 2.2 (2) implies for ω ∈ im(E_{ϵ^{2}}^{(d_{min}^{p⊥})^{∗}d_{min}^{p⊥}})

||ω||_{k}^{2} = ⟨ω, (1 + Δ_{p})^{k}(ω)⟩_{L^{2}}
= ⟨ω, (1 + (d_{min}^{p⊥})^{∗}d_{min}^{p⊥})^{k}(ω)⟩_{L^{2}}
≤ (1 + ϵ^{2k}) · ||ω||_{L^{2}}^{2}.

Hence the inclusion of im(E_{ϵ^{2}}^{(d_{min}^{p⊥})^{∗}d_{min}^{p⊥}}) into L^{2}Ω^{p}(M) induces a bounded G-equivariant operator

i_{ϵ}: im(E_{ϵ^{2}}^{(d_{min}^{p⊥})^{∗}d_{min}^{p⊥}}) → H^{l−p}Ω^{p}(M).  (2.77)

Lemma 2.78. Fix k > dim(M)/2 + 1 and ϵ > 0. Then there is an equivariant smooth triangulation K and a constant C_{1} > 0 such that for all ω ∈ im(E_{ϵ^{2}}^{(d_{min}^{p⊥})^{∗}d_{min}^{p⊥}})

||ω||_{L^{2}} ≤ C_{1} · || pr ∘A^{p} ∘ i_{ϵ}(ω)||_{L^{2}}

holds, where pr: l^{2}C^{p}(K) → l^{2}C^{p}(K) is the orthogonal projection onto im (c^{p−1} : l^{2}C^{p−1}(K) → l^{2}C^{p}(K))^{⊥}.

Proof. Given θ > 0 and an equivariant smooth triangulation of M, we can find an equivariant subdivision whose fullness is bounded from below by θ and whose mesh is arbitrary small [522]. Hence we can find by Lemma 2.76 an equivariant smooth triangulation K and a constant 0 < C_{0} < 1 such that for any p-form η ∈ H^{k}Ω^{p}(M)

||η − W^{p} ∘ A^{p}(η)||_{L^{2}} ≤ C_{0} · ||η||_{L^{2}}

holds. Let pr′: L^{2}Ω^{p}(M) → L^{2}Ω^{p}(M) be the orthogonal projection onto im(d_{min}^{p−1})^{⊥}. Recall that W^{p} sends im(c^{p−1}) to im(d_{min}^{p−1}) and hence pr′ ∘W^{p} ∘

pr = pr′ ∘W^{p}. Now we estimate for ω ∈ im(E_{ϵ^{2}}^{(d_{min}^{p⊥})^{∗}d_{min}^{p⊥}}) using pr′(ω) = ω

||ω||_{L^{2}} ≤ || pr′ ∘W^{p} ∘ A^{p} ∘ i_{ϵ}(ω)||_{L^{2}} + ||ω − pr′ ∘W^{p} ∘ A^{p} ∘ i_{ϵ}(ω)||_{L^{2}}
= || pr′ ∘W^{p} ∘ pr ∘A^{p} ∘ i_{ϵ}(ω)||_{L^{2}} + ||ω − pr′ ∘W^{p} ∘ A^{p} ∘ i_{ϵ}(ω)||_{L^{2}}
≤ ||W^{p} ∘ pr ∘A^{p} ∘ i_{ϵ}(ω)||_{L^{2}} + || pr′(ω − W^{p} ∘ A^{p} ∘ i_{ϵ}(ω))||_{L^{2}}
≤ ||W^{p}|| · || pr ∘A^{p} ∘ i_{ϵ}(ω)||_{L^{2}} + ||ω − W^{p} ∘ A^{p} ∘ i_{ϵ}(ω)||_{L^{2}}
≤ ||W^{p}|| · || pr ∘A^{p} ∘ i_{ϵ}(ω)||_{L^{2}} + C_{0} · ||ω||_{L^{2}}.

If we put C_{1} = max\{\frac{||W^{p}||}{1−C_{0}} | p = 0, 1, …, dim(M)\}, the claim follows. ⊓ ⊔

Lemma 2.79. Given an equivariant smooth triangulation K, there is a constant C_{2} such that for all u ∈ l^{2}C^{p}(M)

||u||_{L^{2}} ≤ C_{2} · ||W_{K}^{p}(u)||_{L^{2}}.

Proof. For an element u ∈ l^{2}C^{p}(M) the element W^{p}(u) ∈ L^{2}Ω^{p}(M) is smooth outside the (dim(M) − 1)-skeleton. The p-form W^{p}(u), which is a priori defined in the sense of distributions, restricted to the interior of a dim(M)-simplex σ is smooth and has a unique smooth extension to σ itself which we denote in the sequel by W (u)|_{σ}. If τ is a p-dimensional face of both the dim(M)-simplices σ_{0} and σ_{1}, then i_{0}^{∗}W (u)|_{σ_{0}} = i_{1}^{∗}W (u)|_{σ_{1}} for the inclusions i_{k}: τ → σ_{k} for k = 0, 1. Hence we can define for any p-simplex τ a smooth p-form W (u)|_{τ} by i^{∗}W (u)|_{σ} for any dim(M)-simplex σ which contains τ as face and i: τ → σ the inclusion. In particular W (c_{τ}) which is a priori given in a distributional sense is a continuous (not necessarily smooth) p-form. Since G acts cocompactly, there is a constant K > 0 such that ||W (c_{τ})||_{L^{2}} ≤ K holds for all p-simplices τ and hence W (u) ∈ L^{2}Ω^{p}(M) for all u ∈ l^{2}C^{p}(M). If τ and σ are p-simplices, then we get for the characteristic functions c_{τ} , c_{σ} ∈ l^{2}C^{p}(K) of σ and τ

W (c_{τ})|_{σ} = 0  if τ ≠ σ;
∫_{σ} W (c_{σ}) = 1.

Recall that G acts freely and cocompactly. Hence there are numbers D > 0 and S > 0 such that for any p-simplex σ

∫_{σ} ||W (c_{σ})||_{x}^{2} dvol_{x}^{σ} ≥ 2 · D;
|{τ | τ ∈ st(σ}| ≤ S;
|{τ | σ ∈ st(τ}| ≤ S,

where τ runs through all simplices.

Recall that the support of W (c_{τ}) lies in the star st(τ) of τ. For any psimplex σ we can choose an open neighborhood U(σ) of the interior int(σ) which is obtained by thickening the interior of σ into the dim(M)-simplices having σ as faces such that

||W (c_{τ})||_{x}^{2} ≤ \frac{D}{2 · (4S − 2)S vol(σ)}  if τ ≠ σ, x ∈ U(σ);
g · U(σ) = U(g · σ);
U(σ) ∩ U(τ) = ∅ for σ ≠ τ;
W (c_{τ})(x) = 0 if x ∈ U(σ) and σ ∉ st(τ).

holds. There is a number δ > 0 such that possibly by shrinking U(σ) to something which is up to small error a product neighbourhood int(σ)×(−\frac{δ}{2D} , \frac{δ}{2D}) we can additionally achieve that for all g ∈ G and p-simplices σ and τ

∫_{U(σ)} ||W^{p}(c_{τ}))||_{x}^{2} dvol_{x} \{ \begin{matrix}≥ δ & τ = σ \\ ≤ \frac{δ}{(4S−2)S} & τ ≠ σ \end{matrix} .

2.4 Comparison of Analytic and Cellular Novikov-Shubin Invariants 111

holds. Then we get for u = ∑_{σ} u_{σ} · c_{σ} in l^{2}C^{p}(M) using the inequality || ∑_{i=1}^{r} a_{i}||^{2} ≤ (2r − 1) · ∑_{i=1}^{r} ||a_{i}||^{2}.

||u||_{L^{2}}^{2} = ∑_{σ} |u_{σ}|^{2}
≤ \frac{1}{δ} · ∑_{σ} |u_{σ}|^{2} · ∫_{U(σ)} ||W (c_{σ})||_{x}^{2} dvol_{x}
≤ \frac{1}{δ} · ∑_{σ} ∫_{U(σ)} ||u_{σ} · W (c_{σ})||_{x}^{2} dvol_{x}
≤ \frac{1}{δ} · ∑_{σ} ∫_{U(σ)} (|| ∑_{τ} u_{τ} · W (c_{τ})||_{x}^{2} + || ∑_{τ≠σ} u_{τ} · W (c_{τ})||_{x}^{2}) dvol_{x}
≤ \frac{1}{δ} · ∫_{∐_{σ} U(σ)} || ∑_{τ} u_{τ} · W (c_{τ})||_{x}^{2} dvol_{x} +

\frac{1}{δ} · ∑_{σ} ∫_{U(σ)} || ∑_{τ≠σ,σ∈st(τ)} u_{τ} · W (c_{τ})||_{x}^{2} dvol_{x}

≤ \frac{1}{δ} · ∫_{M} || ∑_{τ} u_{τ} · W (c_{τ})||_{x}^{2} dvol_{x} +

\frac{1}{δ} · ∑_{σ} ∫_{U(σ)}(2S − 1) · ∑_{τ≠σ,σ∈st(τ)} ||u_{τ} · W (c_{τ})||_{x}^{2} dvol_{x}

≤ \frac{1}{δ} · ||W (u)||_{L^{2}} + \frac{2S − 1}{δ} · ∑_{σ} ∑_{τ≠σ,σ∈st(τ)} |u_{τ} |^{2} · ∫_{U(σ)} ||W (c_{τ})||_{x}^{2} dvol_{x}
≤ \frac{1}{δ} · ||W (u)||_{L^{2}} + \frac{2S − 1}{δ} · ∑_{σ} ∑_{τ≠σ,σ∈st(τ)} |u_{τ} |^{2} · \frac{δ}{(4S − 2)S}
≤ \frac{1}{δ} · ||W (u)||_{L^{2}} + \frac{2S − 1}{δ} · S · ∑_{τ} |u_{τ} |^{2} · \frac{δ}{(4S − 2)S}
≤ \frac{1}{δ} · ||W (u)||_{L^{2}} + \frac{1}{2} · ∑_{τ} |u_{τ} |^{2}
≤ \frac{1}{δ} · ||W (u)||_{L^{2}} + \frac{1}{2} · ||u||_{L^{2}}^{2}.

This implies

||u||_{L^{2}} ≤ \frac{2}{δ} · ||W (u)||_{L^{2}}

and hence Lemma 2.79 is proved. ⊓ ⊔ Lemma 2.80. For any equivariant smooth triangulation K we have

F_{p}(M) ⪯ F_{p}(K).

Proof. Because of Theorem 2.55 (1) it suffices to show the claim for one equivariant smooth triangulation. Let K be the equivariant smooth triangulation and C_{1} > 0 be the constant appearing in Lemma 2.78. Recall that pr: l^{2}C^{p}(K) → l^{2}C^{p}(K) is the orthogonal projection onto im(c^{p−1})^{⊥} and pr′: L^{2}Ω^{p}(M) → L^{2}Ω^{p}(M) is the orthogonal projection onto im(d_{min}^{p−1})^{⊥}. Fix ϵ > 0. Next we show that there is a constant C_{3} > 0 such that for all λ ≤ ϵ and ω ∈ im(E_{λ^{2}}^{(d_{min}^{p⊥})^{∗}d_{min}^{p⊥}})

||c^{p} ∘ pr ∘A^{p} ∘ i_{ϵ}(ω)||_{L^{2}} ≤ C_{3} · λ · || pr ∘A^{p} ∘ i_{ϵ}(ω)||_{L^{2}}  (2.81)

holds. We estimate using Lemma 2.2 (2), Lemma 2.78 and Lemma 2.79

||c^{p} ∘ pr ∘A^{p} ∘ i_{ϵ}(ω)||_{L^{2}}

= ||c^{p} ∘ A^{p} ∘ i_{ϵ}(ω)||_{L^{2}}
≤ C_{2} · ||W^{p+1} ∘ c^{p} ∘ A^{p} ∘ i_{ϵ}(ω)||_{L^{2}}
= C_{2} · ||W^{p+1} ∘ A^{p+1} ∘ d^{p} ∘ i_{ϵ}(ω)||_{L^{2}}
= C_{2} · ||W^{p+1} ∘ A^{p+1} ∘ i_{ϵ} ∘ d_{min}^{p⊥}(ω)||_{L^{2}}
≤ C_{2} · ||W^{p+1} ∘ A^{p+1} ∘ i_{ϵ}|| · ||d_{min}^{p⊥}(ω)||_{L^{2}}
≤ C_{2} · ||W^{p+1} ∘ A^{p+1} ∘ i_{ϵ}|| · λ · ||ω||_{L^{2}}
≤ C_{2} · ||W^{p+1} ∘ A^{p+1} ∘ i_{ϵ}|| · λ · C_{1} · || pr ∘A^{p} ∘ i_{ϵ}(ω)||_{L^{2}}.

If we put C_{3} := C_{1} · C_{2} · ||W^{p+1} ∘ A^{p+1} ∘ i_{ϵ}||, then (2.81) follows.

We conclude from (2.81) that clos(pr ∘A^{p}∘i_{ϵ}(im(E_{λ^{2}}^{(d_{min}^{p⊥})^{∗}d_{min}^{p⊥}}))) belongs to L(c^{p}, C_{3}λ). Since pr ∘A^{p} ∘i_{ϵ} is injective by Lemma 2.78 we get from Theorem 1.12 (2) and Lemma 2.2 (2) for λ ≤ ϵ

F_{p}(M)(λ) = dim_{N (G)} (im(E_{λ^{2}}^{(d_{min}^{p⊥})^{∗}d_{min}^{p⊥}}))
= dim_{N (G)} (clos(pr ∘A^{p} ∘ i_{ϵ}(im(E_{λ^{2}}^{(d_{min}^{p⊥})^{∗}d_{min}^{p⊥}}))))
≤ F_{p}(K)(C_{3}λ).

This finishes the proof of Lemma 2.80. ⊓ ⊔

2.5 On the Positivity and Rationality of the Novikov-Shubin Invariants

The following conjecture is taken from [322, Conjecture 7.1 on page 56].

Conjecture 2.82. (Positivity and rationality of Novikov-Shubin invariants). Let G be a group. Then for any free G-CW -complex X of finite type its Novikov-Shubin invariants α_{p}(X) are positive rational numbers unless they are ∞ or ∞^{+}.

This conjecture is equivalent to the statement that for any matrix A ∈ M(m, n, ℤG) the Novikov-Shubin invariant of the induced morphism of finitely generated Hilbert N (G)-modules l^{2}(G)^{m} → l^{2}(G)^{n} is a positive rational number, ∞ or ∞^{+}. It is also equivalent to the version where X is any cocompact free proper G-manifold without boundary and with G-invariant Riemannian metric. The proof of these equivalent formulations is analogous to the proof of Lemma 10.5.

Here is some evidence for Conjecture 2.82. Unfortunately, all the evidence comes from computations, no convincing conceptual reason is known. Conjecture 2.82 has been proved for G = ℤ in Lemma 2.58. Conjecture 2.82 is true for virtually abelian G by [316, Proposition 39 on page 494]. (The author of [316] informs us that his proof of this statement is correct when G = ℤ but has a gap when G = ℤ^{k} for k > 1. The nature of the gap is described in [321, page 16]. The proof in this case can be completed by the same basic method used in [316]. Moreover, the value ∞ does not occur for G = ℤ^{k}.) D. Voiculescu informs us that Conjecture 2.82 is also true for a free group G. Details of the proof will appear in the Ph. D.thesis of Roman Sauer [456]. If Conjecture 2.82 is true for the free G-CW -complex X of finite type and for the cocompact free H-CW -complex Y of finite type and both X and Y have the limit property (see Definition 2.41), then Conjecture 2.82 holds for the G × H-CW -complex X × Y by Theorem 2.55 (3). In all examples, where Novikov-Shubin invariants can be computed explicitly, the result confirms Conjecture 2.82. For any finitely generated group G it is true for α_{1}(X) by Theorem 2.55 (5). For 3-manifolds we refer to Theorem 4.2, for Heisenberg groups to Theorem 2.85 and for symmetric spaces to Theorem 5.12 (2).

Here is further evidence if one replaces positive rational number by positive number in Conjecture 2.82. If X is a finite aspherical CW -complex such that its fundamental group contains an elementary amenable infinite normal subgroup, then α_{p}(X̃) ≥ 1 holds for p ≥ 1 (see Theorem 2.63). We get α_{p}(X̃) ≥ 1 for all p ≥ 1 if X is a connected S^{1}-CW -complex of finite type such that for one orbit S^{1}/H the inclusion into X induces a map on π_{1} with infinite image by Theorem 2.61. Let p: E → B be a fibration such that B is a connected finite CW -complex and the fiber F has the homotopy type of a connected finite CW -complex. Suppose that the inclusion of F into E induces an injection on the fundamental groups and that b_{p}^{(2)}(F̃) = 0 for all p ≥ 0. Then α_{p}(Ẽ) > 0 for all p ≥ 1 if α_{p}(F̃) > 0 for all p ≥ 1 (see Theorem 3.100 and Remark 3.184). A similar statement for pushouts follows from Theorem 3.96 (2) and Remark 3.184.

Conjecture 2.82 is related to the question whether a cocompact free G-CW -complex is of determinant class (see Theorem 3.14 (4)) and to Theorem 3.28. The question about determinant class arises in the construction of L^{2}- torsion in Subsection 3.3.1 and has a positive answer for the groups appearing in the class G as explained in Chapter 13. Some evidence for Conjecture 2.82 comes from Theorem 13.7, where an estimate of the shape F_{p}(λ) ≤ \frac{C}{− ln(λ)} is proved for small λ.

We do not know of an example of an aspherical closed manifold M such that α_{p}(M̃) < 1 holds for some p ≥ 0.

2.6 Novikov-Shubin Invariants of Manifolds with Boundary

In this section we briefly discuss the case of a manifold with boundary. Let M be a cocompact free proper G-manifold with G-invariant Riemannian metric. Suppose that the boundary ∂M of the complete Riemannian manifold M is the disjoint union ∂_{0}M and ∂_{1}M where we allow that ∂_{0}M, ∂_{1}M or both are empty. Denote by d_{min}^{p}: L^{2}Ω^{p}(M) → L^{2}Ω^{p+1}(M) the minimal closure of the operator d^{p}: Ω_{c}^{p}(M, ∂_{0}M) → L^{2}Ω^{p+1}(M), where Ω_{c}^{p}(M, ∂_{0}M) is the subspace of Ω_{c}^{p}(M) consisting of those p-forms ω with compact support whose restriction to ∂_{0}M vanishes. Then im(d_{min}^{p−1}) is contained in the domain of d_{min}^{p} and we obtain a closed densely defined operator d_{min}^{p⊥}: im(d_{min}^{p−1})^{⊥} → L^{2}Ω^{p}(M). Define the p-th analytic spectral density function and the p-th Novikov-Shubin invariant of (M, ∂_{0}M) by

F_{p}(M, ∂_{0}M) := F (d_{min}^{p⊥});
α_{p}(M, ∂_{0}M) := α(F_{p−1}(∂_{0}M)).

This is consistent with [240, Definition 3.1 on page 387]. Let (K, ∂_{0}K, ∂_{1}K) → (M, ∂_{0}M, ∂_{1}M) be any equivariant smooth triangulation. Then we have the following version of Theorem 2.68

F_{p}(M, ∂_{0}M) ≃ F_{p}(K, ∂_{0}K);  (2.83)
α_{p}(M, ∂_{0}M) = α_{p}(K, ∂_{0}K).  (2.84)

Analogously the notions and results for the Laplacian carry over to (M, ∂_{0}M) if we use as initial domain the space Ω_{2}^{p}(M, ∂_{0}M) (see (1.83)).

We briefly explain the idea of the proof of F_{p}(M, ∂_{0}M) ≃ F_{p}(K, ∂_{0}K). Let g_{0} and g_{1} be two G-invariant Riemannian metrics on M. Since M is cocompact the L^{2}-norms on Ω_{c}^{p}(M) with respect to g_{0} and g_{1} are equivalent. This implies that we obtain a commutative square with invertible bounded G-equivariant operators induced by the identity on Ω_{c}^{p}(M) as vertical arrows

L_{g_{0}}^{2}Ω^{p}(M) ⟶^{d_{min}^{p}} L_{g_{0}}^{2}Ω^{p+1}(M)
≅↓  ≅↓
L_{g_{1}}^{2}Ω^{p}(M) ⟶^{d_{min}^{p}} L_{g_{1}}^{2}Ω^{p+1}(M)

This implies that the spectral density functions of (M, ∂_{0}M) with respect to g_{0} and g_{1} are dilatationally equivalent. Hence we can assume without loss of generality that the G-invariant Riemannian metric on M is a product near the boundary and hence induces a G-invariant Riemannian metric on M ∪_{∂_{1}M} M. Let τ: M ∪_{∂_{1}M} M → M ∪_{∂_{1}M} M be the isometric involution given by flipping the two copies of M inside M ∪_{∂_{1}M} M. It induces an isometric involution on all spaces L^{2}Ω^{p}(M) and the operators d_{min}^{p} commute with this involution. Hence the whole picture decomposes orthogonally into a +-part where this involution is the identity and a −-part where this involution is − id. So we can define (d_{min}^{p})^{±}: L^{2}Ω^{p}(M ∪_{∂_{1}M} M)^{±} → L^{2}Ω^{p+1}(M ∪_{∂_{1}M} M)^{±} and F_{p}(M ∪_{∂_{1}M} M)^{±} and analogously F_{p}(K ∪_{∂_{1}K} K)^{±}. One checks that (d_{min}^{p})^{+} : L^{2}Ω^{p}(M ∪_{∂_{1}M} M)^{+} → L^{2}Ω^{p+1}(M ∪_{∂_{1}M} M)^{+} is the same as d_{min}^{p}: L^{2}Ω^{p}(M) → L^{2}Ω^{p+1}(M) for the pair (M, ∂_{0}M) and for − instead of + one gets it for the pair (M, ∂_{1}M). In particular

F_{p}(M ∪_{∂_{1}M} M)^{+} = F_{p}(M, ∂_{0}M);
F_{p}(M ∪_{∂_{1}M} M)^{−} = F_{p}(M, ∂_{1}M).

Similarly one gets

F_{p}(K ∪_{∂_{1}K} K)^{+} = F_{p}(K, ∂_{0}K);
F_{p}(K ∪_{∂_{1}K} K)^{−} = F_{p}(K, ∂_{1}K).

Put ∂_{1}M = ∂M. Then M ∪_{∂M} M has no boundary. The proof of Theorem 2.68 that F_{p}(M ∪_{∂M} M) = F_{p}(K ∪_{∂K} K) can be easily modified to show

F_{p}(M ∪_{∂M} M)^{+} = F_{p}(K ∪_{∂K} K)^{+};
F_{p}(M ∪_{∂M} M)^{−} = F_{p}(K ∪_{∂K} K)^{−},

because everything is compatible with the involution and ||ω||_{k}^{2} = ||ω^{+}||_{k}^{2} + ||ω^{−}||_{k}^{2} holds for all ω ∈ H^{k}Ω^{p}(M). Hence the claim is true in the case ∂_{1}M = ∂M. Now repeating this doubling trick allows to conclude the general case by inspecting M ∪_{∂_{1}M} M.

2.7 Miscellaneous

Let M be a closed Riemannian manifold. The analytic Laplace operator Δ_{p}: L^{2}Ω^{p}(M̃) → L^{2}Ω^{p}(M̃) on the universal covering of M has zero in its spectrum if and only if b_{p}^{(2)}(M̃) ≠ 0 or α_{p}^{Δ}(M̃) ≠ ∞^{+}. In Chapter 12 we will deal with the conjecture that for an aspherical closed Riemannian manifold M there is at least one p ≥ 0 such that Δ_{p} has zero in the spectrum. We conclude from Theorem 1.35 (8), Theorem 2.55 (5b) and Theorem 2.68 the result of Brooks [68] that for a closed Riemannian manifold M the Laplacian Δ_{0} : L^{2}Ω^{0}(M̃) → L^{2}Ω^{0}(M̃) acting on functions on the universal covering has zero in its spectrum if and only if π_{1}(M) is amenable. Moreover, the result of Brooks extends to the case where M is compact and has a boundary if one uses Neumann boundary conditions on M̃.

We have interpreted the L^{2}-Betti numbers in terms of the heat kernel (see (1.60)). We will do this also for the Novikov-Shubin invariants α_{p}^{Δ}(M) in Theorem 3.136 (3) and (4).

The notions of spectral density function and Novikov-Shubin invariants make also sense for proper G-CW -complexes of finite type because in this setting the cellular L^{2}-chain complex is still defined as a Hilbert N (G)-chain complex as explained in Section 1.6. Theorem 2.55 (1), (2), (3), (5), (6) and (7) remain true word by word for proper G-CW -complexes of finite type. The L^{2}-de Rham complex is also defined for a cocompact proper G-manifold with G-invariant Riemannian metric. Hence the analytic versions of spectral density function and Novikov-Shubin invariants are still well-defined in this context and Theorem 2.68 remains true if one drops the condition free.

We will explain a proportionality principle for Novikov-Shubin invariants in Theorem 3.183.

A combinatorial approach to the Novikov-Shubin invariants which is useful for concrete calculations will be given in Section 3.7.

We will give further computations of Novikov-Shubin invariants for universal coverings of compact 3-manifolds in Theorem 4.2 and for universal coverings of closed locally symmetric spaces in Theorem 5.12 (2).

We will discuss the behaviour of L^{2}-Betti numbers and Novikov-Shubin invariants of groups under quasi-isometry and measure equivalence in Section 7.5).

Finally we mention the following computation due to Rumin [450, Corollary 7.15 on page 449]. Notice that the invariants computed there corresponds in our notation to α_{p}^{Δ}(M̃) and one can easily deduce the values of α_{p}(M̃) from the relation 2 · α_{p}^{Δ}(M̃) = min{α_{p}(M̃), α_{p+1}(M̃)}.

Theorem 2.85. Let M be a closed Riemannian manifold whose universal covering M̃ is the Heisenberg group H^{2n+1}. Then M is a non-trivial S^{1}- bundle over a torus and

α_{p}(M̃) = \{ \begin{matrix}n + 1 & if p = n + 1; \\ 2 · (n + 1) & if 1 ≤ p ≤ dim(M), p ≠ n + 1; \\ ∞^{+} & otherwise. \end{matrix}

In [343] the notion of Novikov-Shubin invariants for free G-CW -complexes of finite type is extended to arbitrary G-spaces. In particular α_{p}(G) can be defined by α_{p}(EG) for any group G.

Exercises

2.1. Let f: U → V and g: V → W be morphisms of Hilbert N (G)-modules. Show: i.) if f and g are weak isomorphisms, then g ∘ f is a weak isomorphism and ii.) if two of the morphisms f, g and g ∘ f are weak isomorphisms and Fredholm, then all three are weak isomorphisms and Fredholm. Find a counterexample to assertion ii.) if one replaces “weak isomorphisms and Fredholm” by “weak isomorphisms”.

2.2. Let G be a group which contains an element of infinite order. Then for any integer n ≥ 1 there is an element u ∈ ℂG such that α(r_{u}) = 1/n and b^{(2)}(r_{u}) = 0 holds for the map r_{u}: l^{2}(G) → l^{2}(G) given by right multiplication with u. Moreover, if we additionally assume that G is finitely presented we can find for any sequence α_{3}, α_{4}, … of elements α_{p} ∈ {1/n | n ∈ ℤ, n ≥ 1} ∐{∞^{+}} a connected CW -complex X of finite type with π_{1}(X) = G such that b_{p}^{(2)}(X̃) = 0 and α_{p}(X̃) = α_{p} holds for p ≥ 3.

2.3. Let G be locally finite, i.e. any finitely generated subgroup is finite. Let A ∈ M(m, n, ℂG) be any matrix. It induces a map of Hilbert N (G)- modules r_{A}: ⨁_{i=1}^{m} l^{2}(G) → ⨁_{i=1}^{n} l^{2}(G). Then α(r_{A}) = ∞^{+}. If X is any free G-CW -complex of finite type, then α_{p}(X) = ∞^{+} for all p ≥ 1.

2.4. Show that there is no connected CW -complex X of finite type with α_{1}(X̃) = 1 and α_{2}(X̃) ≠ ∞^{+}.

2.5. Show for two connected closed 4-dimensional manifolds M and N with isomorphic fundamental groups that α_{p}(M̃) = α_{p}(Ñ) holds for all p ≥ 1. Show the analogous statement if M and N are connected compact 3-manifolds possibly with boundary.

2.6. Let S^{1} → E → B be a principal S^{1}-bundle of CW -complexes of finite type such that B is simply connected. Show that either α_{p}(Ẽ) = 1 or α_{p}(Ẽ) = ∞^{+} for p ≥ 1 holds.

2.7. Show that l^{2}(ℤ^{n}) viewed as ℂ[ℤ^{n}]-module is flat if and only if n ≤ 1.

2.8. Let M be a closed hyperbolic manifold. Compute the L^{2}-Betti numbers and the Novikov-Shubin invariants of the universal covering of T^{n} × M.

2.9. Let M and N be two compact manifolds (possibly with boundary) whose dimension is less or equal to 2. Compute the L^{2}-Betti numbers and Novikov-Shubin invariants of { \atop M × N}.

2.10. Let X and Z be connected CW -complexes of finite type. Suppose that π_{1}(Z) is finite and X̃ has the limit property. Show

α_{p}({ \atop X × Z}) = min\{α_{i}(X̃) | i = 1, …, p, H_{p−i}(Z̃; ℂ) ≠ 0\};
α_{p}({ \atop X × T^{n}}) = \{ \begin{matrix}min\{n + α_{i}(X̃) | i = p − n, …, p\} & if b_{i}^{(2)}(X̃) = 0 for \\ & p − n ≤ i ≤ p − 1; \\ n & otherwise. \end{matrix}

2.11. Let F → E → B be a fibration of connected CW -complexes such that the inclusion induces an injection π_{1}(F) → π_{1}(E). Suppose that b_{p}^{(2)}(F̃) = 0 and α_{p}(F̃) > 0 holds for all p ≥ 0. Show that then the same is true for Ẽ.

2.12. Let A be the set of real numbers r, for which there exists a CW - complex X of finite type and an integer p ≥ 0 with α_{p}(X̃) = r. Then A is countable and contains {r | r ∈ ℚ, r ≥ 0}.

3. L^{2}-Torsion

Introduction

In this chapter we introduce and study L^{2}-torsion for Hilbert chain complexes and for regular coverings of finite CW -complexes or of compact manifolds.

There are various notions of torsion invariants, such as Reidemeister torsion, Whitehead torsion and analytic Ray-Singer torsion, which have been intensively studied since the twenties and have remained in the focus of attention. They will be reviewed in Section 3.1. L^{2}-torsion is the L^{2}-analog of Reidemeister torsion as L^{2}-Betti numbers are the L^{2}-analogs of the classical Betti numbers. The situation for the classical Reidemeister torsion is best if the homology vanishes, and the same is true in the L^{2}-context. Therefore we will consider in this introduction only the case of the universal covering X̃ of a finite CW -complex X for which all L^{2}-Betti numbers are trivial. In many interesting geometric situations this assumption will be satisfied.

For the construction of Reidemeister torsion the notion of a determinant is crucial and we will define and investigate its L^{2}-version, the Fuglede-Kadison determinant, in Section 3.2. In the classical context of Reidemeister torsion the vanishing of the homology implies that the chain complexes under consideration are contractible. In the L^{2}-context the vanishing of the L^{2}-Betti numbers does not imply that the Hilbert chain complex under consideration is contractible, the Novikov-Shubin invariants of Chapter 2 measure the difference. Notice that the zero-in-the-spectrum Conjecture 12.1 says in the aspherical case that the cellular Hilbert N (π_{1}(X))-chain complex of X̃ is never contractible so that it is too restrictive to demand contractibility. This forces us to deal with Fuglede-Kadison determinants for weak automorphisms and with the problem whether X̃ is of determinant class, which means that the Fuglede-Kadison determinant of each differential is different from zero and hence the L^{2}-torsion is defined as a real number. The universal covering X̃ is of determinant class, provided that all its Novikov-Shubin invariants are positive or that π_{1}(X) belongs to the class G, which will be investigated in Subsection 13.1.3. There is the Conjecture 3.94 that X̃ is always of determinant class. We will deal with it in Chapter 13.

In Section 3.3 we introduce L^{2}-torsion for finite Hilbert N (G)-chain complexes and define ρ^{(2)}(X̃) in terms of these invariants applied to the cellular L^{2}-chain complex in Section 3.4. It turns out that the L^{2}-torsion ρ^{(2)}(X̃) has the same formal properties as the classical Euler characteristic χ(X) if one ensures that relevant maps induce injections on the fundamental groups. For instance, ρ^{(2)}(X̃) depends only on the simple homotopy type of X and, provided that a certain map in K-theory is always trivial, it depends on the homotopy type of X only. There are sum formulas, product formulas and fibration formulas and Poincaré duality holds. For a self map f: X → X inducing an isomorphism on π_{1}(X) one can compute ρ^{(2)}(T̃_{f}) of the mapping torus T_{f} in terms of the map induced on the L^{2}-homology of X̃ by f. If X is an aspherical finite CW -complex such that π_{1}(X) contains an elementary amenable infinite normal subgroup and is of det ≥ 1-class, then ρ^{(2)}(X̃) vanishes. There are explicit formulas in terms of H_{p}(X̃; ℂ) if π_{1}(X) = ℤ.

In Section 3.5 we introduce the analytic version of ρ^{(2)}(M̃) for a closed Riemannian manifold M in terms of the Laplace operator on differential forms on M̃ following Lott and Mathai. The analytic and topological version agree by a deep result of Burghelea, Friedlander, Kappeler and McDonald. The L^{2}-torsion ρ^{(2)}(M̃) of an odd-dimensional closed hyperbolic manifold is up to a dimension constant, which is computable and different from zero, the volume of M. Manifolds with boundary are briefly discussed in Section 3.6.

In Section 3.7 we give a combinatorial approach for the computation of L^{2}-Betti numbers and L^{2}-torsion. Namely, we give an algorithm to produce monotone decreasing sequences of rational numbers which converge to the L^{2}-Betti numbers and the Fuglede-Kadison determinants respectively. It is easier to compute than for instance the analytic versions which are defined in terms of the heat kernel (and meromorphic extensions). In practice this yields numerical upper bounds for the L^{2}-Betti numbers and the Fuglede-Kadison determinants. The speed of convergence is ≈ n^{−α}, where α is the relevant Novikov-Shubin invariant.

In order to get a quick overview one should read through Sections 3.1, (only if the reader is not familiar with the classical concept of torsion invariants) 3.4 and 3.5 and skip the very technical Sections 3.2 and 3.3.

3.1 Survey on Torsion Invariants

In this section we give a brief review of torsion invariants in order to motivate the definition of L^{2}-torsion.

3.1.1 Whitehead Groups

Let R be an associative ring with unit. Denote by GL(n, R) the group of invertible (n, n)-matrices with entries in R. Define the group GL(R) by the colimit of the system indexed by the natural numbers … ⊂ GL(n, R) ⊂ GL(n + 1, R) ⊂ … where the inclusion GL(n, R) to GL(n + 1, R) is given by stabilization

A ↦ ( \begin{matrix}A & 0 \\ 0 & 1 \end{matrix} ) .

Define K_{1}(R) by the abelianization GL(R)/[GL(R), GL(R)] of GL(R). Define the Whitehead group Wh(G) of a group G to be the cokernel of the map G × {±1} → K_{1}(ℤG) which sends (g, ±1) to the class of the invertible (1, 1)- matrix (±g). This will be the group where Whitehead torsion will take its values in.

The Whitehead group Wh(G) is known to be trivial if G is the free abelian group ℤ^{n} of rank n [25] or the free group ∗_{i=1}^{n}ℤ of rank n [481]. There is the conjecture that it vanishes for any torsionfree group. This has been proved by Farrell and Jones [190], [191], [193], [194], [196] for a large class of groups. This class contains any subgroup G ⊂ G′ where G′ is a discrete cocompact subgroup of a Lie group with finitely many path components and any group G which is the fundamental group of a non-positively curved closed Riemannian manifold or of a complete pinched negatively curved Riemannian manifold. The Whitehead group satisfies Wh(G ∗ H) = Wh(G) ⊕ Wh(H) [481].

If G is finite, then Wh(G) is very well understood (see [406]). Namely, Wh(G) is finitely generated, its rank as abelian group is the number of conjugacy classes of unordered pairs {g, g^{−1}} in G minus the number of conjugacy classes of cyclic subgroups and its torsion subgroup is isomorphic to the kernel SK_{1}(G) of the change of coefficient homomorphism K_{1}(ℤG) → K_{1}(ℚG). For a finite cyclic group G the Whitehead group Wh(G) is torsionfree. For instance the Whitehead group Wh(ℤ/p) of a cyclic group of order p for an odd prime p is the free abelian group of rank (p − 3)/2 and Wh(ℤ/2) = 0. The Whitehead group of the symmetric group S_{n} is trivial. The Whitehead group of ℤ^{2} × ℤ/4 is not finitely generated as abelian group.

3.1.2 Whitehead Torsion

Let R be an associative ring with unit. Let C_{∗} be a based free finite R-chain complex, where based free means that each chain module C_{p} is equipped with an (ordered) basis. Suppose that C_{∗} is acyclic. Choose a chain contraction γ_{∗}: C_{∗} → C_{∗+1}. In the sequel we write :

C_{odd} := ⨁_{n∈ℤ} C_{2n+1};
C_{ev} := ⨁_{n∈ℤ} C_{2n}.

Then we obtain an isomorphism (c + γ)_{odd} : C_{odd} → C_{ev}. Since we have a basis for the source and target of this isomorphism, it determines an invertible matrix and hence an element

ρ(C_{∗}) ∈ K_{1}(R).  (3.1)

Let f: X → Y be a G-homotopy equivalence of finite free G-CW - complexes. It induces a chain homotopy equivalence C_{∗}(f): C_{∗}(X) → C_{∗}(Y) of the cellular ℤG-chain complexes. Let cone_{∗}(C_{∗}(f)) be its mapping cone which is a contractible finite free ℤG-chain complex. The G-CW -complex-structure determines a cellular ℤG-basis which is not quite unique, one may permute the basis elements or multiply a basis element with an element of the form ±g ∈ ℤG. Hence we can define the Whitehead torsion

τ(f) ∈ Wh(G)  (3.2)

by the image of the element ρ(cone_{∗}(C_{∗}(f))) defined in (3.1) under the canonical projection K_{1}(ℤG) → Wh(G). The Whitehead torsion τ(f) depends only on the G-homotopy class of f and satisfies τ(g ∘ f) = τ(f) + τ(g) for G-homotopy equivalences f: X → Y and g: Y → Z of finite free G-CW - complexes [116, (22.4)]. There are sum and product formulas for Whitehead torsion [116, (23.1) and (23.2)].

Given a homotopy equivalence f: X → Y of connected finite CW - complexes, we can pick a lift f̃: X̃ → Ỹ to the universal coverings and obtain an element τ(f̃) ∈ Wh(π_{1}(Y)). The vanishing of this element has a specific meaning, namely it is zero if and only if f is a simple homotopy equivalence, i.e. up to homotopy f can be written as a finite sequence of combinatorial moves, so called elementary collapses and elementary expansions [116, (22.2)]. If f is a homeomorphism, then it is a simple homotopy equivalence, or equivalently τ(f̃) = 0, by a result of Chapman [99], [100]. A map between finite polyhedra is a simple homotopy equivalence if and only if there are regular neighbourhoods i_{X}: X → N_{X} and i_{Y} : Y → N_{Y} in high dimensional Euclidean space and a homeomorphism g: N_{X} → N_{Y} such that g ∘ i_{X} and i_{Y} ∘ f are homotopic [445, Chapter 3]. If Y is not connected, we define Wh(π_{1}(Y)) := ⨁_{C∈π_{0}(Y)} Wh(π_{1}(C)) and τ(f) by {τ(f|_{f^{−1}(C)} : f^{−1}(C) → C) | C ∈ π_{0}(Y)}.

The main importance of Whitehead torsion lies in the s-Cobordism Theorem we will explain next. A (n + 1)-dimensional h-cobordism is a compact (n + 1)-dimensional manifold W whose boundary is the disjoint union ∂W = ∂_{0}W ∐ ∂_{1}W such that the inclusions i_{k}: ∂_{k}W → W for k = 0, 1 are homotopy equivalences. Given a closed n-dimensional manifold M, an h-cobordism over M is a (n + 1)-dimensional h-cobordism W together with a diffeomorphism f: M → ∂_{0}W. Two such h-cobordisms (W, f) and (W′, f′) over M are diffeomorphic if there is a diffeomorphism F: W → W′ with F ∘ f = f′. The next result is due to Barden, Mazur, Stallings (for its proof see for instance [288], [337, Section 1]), its topological version was proved by Kirby and Siebenmann [292, Essay II].

Theorem 3.3 (S-Cobordism Theorem). Let M be a closed smooth ndimensional manifold. Suppose n ≥ 5. Then the map from the set of diffeomorphism classes of h-cobordisms (W, f) over M to the Whitehead group Wh(π_{1}(M)) which sends the class of (W, f) to the image of the Whitehead torsion τ(ĩ_{0} : { \atop ∂_{0}W} → W̃) under the inverse of the isomorphism Wh(π_{1}(M)) → Wh(π_{1}(W)) induced by i_{0} ∘ f, is a bijection. In particular an h-cobordism (W, f) over M is trivial, i.e. diffeomorphic to the trivial hcobordism (M × [0, 1], i_{0}), if and only if τ(ĩ: { \atop ∂_{0}W} → W̃) = 0.

Notice that the Whitehead group of the trivial group vanishes. Hence any h-cobordism over M is trivial if M is simply connected. This implies the Poincaré Conjecture in dimensions ≥ 5 which says that a closed ndimensional manifold is homeomorphic to S^{n} if it is homotopy equivalent to S^{n}. The s-Cobordism Theorem is known to be false in dimension n = 4 [151] but it is still true for “good” fundamental groups in the topological category by results of Freedman [202], [203]. This implies that the Poincaré Conjecture is true also in dimension 4. Counterexamples to the s-Cobordism Theorem in dimension n = 3 are constructed by Cappell and Shaneson [90]. The Poincaré Conjecture in dimension 3 is open at the time of writing.

The s-Cobordism Theorem is one key ingredient in surgery theory which is designed to classify manifolds up to diffeomorphism. For instance, in order to show that two closed manifolds M and N are diffeomorphic, the strategy is to construct a cobordism W′ (with appropriate bundle data) between M and N, then to modify W′ to an h-cobordism W over M via surgery on the interior of W′ such that the Whitehead torsion is trivial, and finally to apply the s-Cobordism Theorem to conclude that W is diffeomorphic to M ×[0, 1] and in particular its two ends M and N are diffeomorphic. More information about Whitehead torsion can be found for instance in [116], [375]. Generalizations like bounded, controlled, equivariant or stratified versions can be found for instance in [4], [270], [326], [424], [425], [426], [483], [519].

3.1.3 Reidemeister Torsion

Let X be a finite free G-CW -complex. Let V be an orthogonal finite dimensional G-representation. Suppose that H_{p}^{G}(X; V) := H_{p}(V ⊗_{ℤG} C_{∗}(X)) vanishes for all p ≥ 0. After a choice of a cellular ℤG-basis, we obtain an isomorphism ⨁_{i=1}^{l_{p}} V ⟶^{≅} V ⊗_{ℤG} C_{p}(X). Now choose any orthonormal basis of V and equip V ⊗_{ℤG} C_{∗}(X) with the induced basis. With these choices we obtain a well-defined element ρ(V ⊗_{ℤG} C_{∗}(X)) ∈ K_{1}(ℝ) (see (3.1)). The determinant induces an isomorphism det_{ℝ}: K_{1}(ℝ) ⟶^{≅} ℝ^{inv}. The Reidemeister torsion of X with coefficients in V is defined to be the real number

ρ(X; V) := ln (|det_{ℝ}(ρ(V ⊗_{ℤG} C_{∗}(X)))|) .  (3.4)

It is independent of the choices of a cellular basis for X and an orthonormal basis for V. If f: X′ → X is a G-homeomorphism, then ρ(X′; V) = ρ(X; V).

Let V and W be two orthogonal ℤ/n-representations for some n ≥ 3 such that ℤ/n acts freely on the unit spheres SV and SW. If U is any orthogonal representation with trivial fixed point set, then one computes H_{p}^{ℤ/n}(SV; U) = H_{p}^{ℤ/n}(SW; U) = 0 for all p ≥ 0 and hence ρ(SV; U) and ρ(SW; U) are defined. Suppose that SV and SW are ℤ/n-homeomorphic. Since then ρ(SV; U) = ρ(SW; U) holds for any such U, one can compute using Franz’ Lemma [201] that V and W are isomorphic as orthogonal ℤ/n-representations. Reidemeister torsion was the first invariant in algebraic topology which is not a homotopy invariant. Namely, for suitable choices of V and W, the associated lens spaces L(V) := (ℤ/n)\SV and L(W) are homotopy equivalent but not homeomorphic [436], [116, chapter V]. The difference of the diffeomorphism type is detected by ρ(SV; U) for suitable U. On the other hand Reidemeister torsion can be used to prove rigidity. Namely, one can show using Reidemeister torsion that L(V) and L(W) are homeomorphic if and only if they are isometrically diffeomorphic with respect to the Riemannian metric induced by the orthogonal structure on V and W. Lens spaces with this Riemannian metric have constant positive sectional curvature. A closed Riemannian manifold with constant positive sectional curvature and cyclic fundamental group is isometrically diffeomorphic to a lens space after possibly rescaling the Riemannian metric with a constant [525]. The result above for free representations is generalized by De Rham’s Theorem [133] (see also [323, Proposition 3.2 on page 478], [327, page 317], [443, section 4]) as follows. It says for a finite group G and two orthogonal G-representations V and W whose unit spheres SV and SW are G-diffeomorphic that V and W are isomorphic as orthogonal G-representations. This remains true if one replaces G-diffeomorphic by G-homeomorphic provided that G has odd order (see [265], [353]), but not for any finite group G (see [89], [91], [244] and [245]).

The Alexander polynomial of a knot can be interpreted as a kind of Reidemeister torsion of the canonical infinite cyclic covering of the knot complement (see [374], [497]). Reidemeister torsion appears naturally in surgery theory [352]. Counterexamples to the (polyhedral) Hauptvermutung that two homeomorphic simplicial complexes are already PL-homeomorphic are given by Milnor [373] (see also [431]) and detected by Reidemeister torsion. Seiberg-Witten invariants for 3-manifolds are essentially given by torsion invariants [498].

Definition (3.4) can be extended to the case where H_{p}^{G}(X; V) is not trivial, provided a Hilbert space structure is specified on each H_{p}^{G}(X; V). Namely, there is up to chain homotopy precisely one chain map i_{∗}: H_{∗}^{G}(X; V) → V ⊗_{ℤG} C_{∗}(X) which induces the identity on homology where we consider H_{∗}^{G}(X; V) as a chain complex with trivial differentials. Its mapping cone cone_{∗}(i_{∗}) is an acyclic ℝ-chain complex whose chain modules are Hilbert spaces. As above choose for each chain module an orthonormal basis and define using (3.4)

ρ(X; V) := ln (|det_{ℝ} (ρ(cone_{∗}(i_{∗})))|) .  (3.5)

The real number ρ(X; V) is independent of the choice of i_{∗} and orthonormal basis (see also [375, page 365]). There is one preferred Hilbert structure on H_{∗}^{G}(X; V) which is induced by the one on V ⊗_{ℤG} C_{∗}(X). However, with this choice ρ(X; V) is not invariant under barycentric subdivision if X is a free cocompact simplicial G-complex with non-trivial H_{∗}^{G}(X; V) so that we do not get a useful invariant for manifolds (see [345, Example 5.1 on page 240]).

The situation is better in the presence of a G-invariant Riemannian metric on the cocompact free proper G-manifold M. Let f: X → M be any equivariant smooth triangulation of M. Then we can use a variant of the Hodgede Rham isomorphism H^{p}(M; V) ⟶^{≅} H_{G}^{p}(X; f^{∗}V) (see Theorem 1.57), the natural isomorphism H_{G}^{p}(X; f^{∗}V)^{∗} ⟶^{≅} H_{p}^{G}(X; f^{∗}V) and the Hilbert space structure on H^{p}(M; V) coming from the Riemannian metric to put a preferred Hilbert space structure on H_{p}^{G}(X; f^{∗}V). With respect to this preferred Hilbert space structure on H_{p}^{G}(X; f^{∗}V) we define the topological Reidemeister torsion

ρ_{top}(M; V) := ρ(X; f^{∗}V),  (3.6)

where ρ(X; f^{∗}V) was defined in (3.5). The topological Reidemeister torsion is independent of the choice of (X, f) and is invariant under isometric Gdiffeomorphisms. If H^{p}(M; V) is trivial for all p ≥ 0, then we are back in the situation of (3.4) and ρ_{top}(M; V) depends only on the G-diffeomorphism type of M but not any more on the G-invariant Riemannian metric.

Ray-Singer [432] defined the analytic counterpart of topological Reidemeister torsion using a regularization of the zeta-function as follows. The first observation is that one can compute ρ(X; V), which was introduced in (3.5) with respect to the Hilbert structure on H_{∗}^{G}(X; V) induced by the one on V ⊗_{ℤG} C_{∗}(X), in terms of the cellular Laplace operator Δ_{p}: V ⊗_{ℤG} C_{p}(X) → V ⊗_{ℤG} C_{p}(X) by

ρ(C_{∗}) = −\frac{1}{2} · ∑_{p∈ℤ}(−1)^{p} · p · ln(det_{ℝ}(Δ_{p}^{⊥})),  (3.7)

where Δ_{p}^{⊥}: ker(Δ_{p})^{⊥} → ker(Δ_{p})^{⊥} is the positive automorphism of finite dimensional Hilbert spaces induced by Δ_{p} (cf. Lemma 3.30). Let M be a cocompact free proper G-manifold with G-invariant Riemannian metric. Let Δ_{p}: Ω^{p}(M; V) → Ω^{p}(M; V) be the Laplace operator acting on smooth p-forms on M with coefficients in the orthogonal (finite dimensional) Grepresentation V. The Laplacian above on M is an essentially selfadjoint operator with discrete spectrum since M is cocompact. One wants to use the expression (3.7) also for the analytic Laplace operator and has to take into account that it is defined on infinite-dimensional spaces and hence ln(det_{ℝ}(Δ_{p}^{⊥})) does not make sense a priori. This is done as follows. The zeta-function is defined by

ζ_{p}(s) := ∑_{λ>0} λ^{−s},  (3.8)

where λ runs through the positive eigenvalues of Δ_{p} listed with multiplicity. Since the eigenvalues grow fast enough, the zeta-function is holomorphic for ℜ(s) > dim(M)/2. Moreover it has a meromorphic extension to ℂ with no pole in 0 [468]. So its derivative for s = 0 is defined. The analytic Reidemeister torsion or Ray-Singer torsion of M is defined by [432, Definition 1.6 on page 149]

ρ_{an}(M; V) := \frac{1}{2} · ∑_{p≥0}(−1)^{p} · p · \frac{d}{ds} ζ_{p}(s)|_{s=0} .  (3.9)

The basic idea is that \frac{d}{ds}ζ_{p}(s)|_{s=0} is a generalization of the (logarithm of) the ordinary determinant det_{ℝ}. Namely, if f: V → V is a positive linear automorphism of the finite-dimensional real vector space V and the positive real numbers λ_{1}, λ_{2}, …, λ_{r} are the eigenvalues of f listed with multiplicity, then we get

\frac{d}{ds} ζ_{p}(s)|_{s=0} = \frac{d}{ds} ∑_{i=1}^{r} λ_{i}^{−s}|_{s=0}

= ∑_{i=1}^{r} (− ln(λ_{i}) · λ_{i}^{−s})|_{s=0}

= − ln (∏_{i=1}^{r} λ_{i})
= − ln (det_{ℝ}(f)) .

Ray and Singer conjectured that the analytic and topological Reidemeister torsion agree. This conjecture was proved independently by Cheeger [103] and Müller [390]. Manifolds with boundary and manifolds with symmetries, sum (= glueing) formulas and fibration formulas are treated in [74], [82], [126], [127], [323], [327], [348], [503], [504], [505]. Non-orthogonal coefficient systems are studied in [49], [50], [80], [393]. Further references are [42], [44], [43], [45], [46], [47], [48], [62], [83], [136], [181], [198], [206], [207], [221], [295], [296], [317], [387], [423], [433], [506].

3.2 Fuglede-Kadison Determinant

In this section we extend the notion of the Fuglede-Kadison determinant for invertible morphisms of finite dimensional Hilbert N (G)-modules [208] to arbitrary morphisms and study its main properties. It is not enough to consider only invertible morphisms because then we could construct L^{2}-torsion only for finite free G-CW -complexes whose L^{2}-Betti numbers are all trivial and whose Novikov-Shubin invariants are all ∞^{+}. But this condition is in view of the zero-in-the-spectrum Conjecture 12.1 much too restrictive.

Let f: U → V be a morphism of finite dimensional Hilbert N (G)- modules, i.e. dim_{N (G)}(U), dim_{N (G)}(V) < ∞. Recall that {E_{λ}^{f^{∗}f} | λ ∈ ℝ} is the spectral family of the positive operator f^{∗}f and that the spectral density function F = F (f): [0, ∞) → [0, ∞) sends λ to dim_{N (G)}(im(E_{λ^{2}}^{f^{∗}f})) (see Lemma 2.3). Recall that F is a monotone non-decreasing right-continuous function. Denote by dF the measure on the Borel σ-algebra on ℝ which is uniquely determined by its values on the half open intervals (a, b] for a < b

dF ((a, b]) = F (b) − F (a).  (3.10)

Notice that the measure of the one point set {a} is lim_{x→0+} F (a) − F (a − x) and is zero if and only if F is left-continuous in a. We will use here and in the sequel the convention that ∫_{a}^{b}, ∫_{a+}^{b}, ∫_{a}^{∞} and ∫_{a+}^{∞} respectively means integration over the interval [a, b], (a, b], [a, ∞) and (a, ∞) respectively.

Definition 3.11 (Fuglede-Kadison determinant). Let f: U → V be a morphism of finite dimensional Hilbert N (G)-modules with spectral density function F = F (f). Define its (generalized) Fuglede-Kadison determinant

det_{N (G)}(f) ∈ [0, ∞)

by det_{N (G)}(f) := exp (∫_{0+}^{∞} ln(λ) dF ) if ∫_{0+}^{∞} ln(λ) dF > −∞ and by

det_{N (G)}(f) := 0 if ∫_{0+}^{∞} ln(λ) dF = −∞. We call f of determinant class if and only if ∫_{0+}^{∞} ln(λ) dF > −∞. Often we omit N (G) from the notation.

Notice that for ∫_{0+}^{∞} ln(λ) dF there is only a problem of convergence at zero where ln(λ) goes to −∞ but not at ∞ because we have F (λ) = F (||f||_{∞}) for all λ ≥ ||f||_{∞} and hence

∫_{0+}^{∞} ln(λ) dF = ∫_{0+}^{a} ln(λ) dF for a ≥ ||f||_{∞}.

The notion of determinant class and the first investigations of its basic properties are due to Burghelea-Friedlander-Kappeler-McDonald [84, Definition 4.1 on page 800, Definition 5.7 on page 817].

If 0: U → V is the zero homomorphism for finite-dimensional Hilbert N (G)-modules U and V, then det(0: U → V) = 1 by definition.

Example 3.12. Let G be finite and f: U → V be a morphism of finite dimensional Hilbert N (G)-modules, i.e. a linear G-equivariant map of finitedimensional unitary G-representations. Let λ_{1}, λ_{2}, …, λ_{r} be the positive eigenvalues of the positive map f^{∗}f. Then we conclude from Example 2.5

det(f) = (∏_{i=1}^{r} λ_{i})^{\frac{1}{2·|G|}} .

If f is an isomorphism det(f) is the |G|-th root of the classical determinant of the positive automorphism of complex vector spaces |f|: U → U.

Example 3.13. Let G = ℤ^{n}. In the sequel we use the notation and the identification N (ℤ^{n}) = L^{∞}(T^{n}) of Example 1.4. We conclude from Example 2.6 for f ∈ L^{∞}(T^{n})

det (M_{f}: L^{2}(T^{n}) → L^{2}(T^{n})) = exp (∫_{T^{n}} ln(|f(z)|) · χ_{{u∈S^{1}|f(u)≠0}} dvol_{z})

using the convention exp(−∞) = 0.

The next result says that the main important properties of the classical determinant of endomorphisms of finite-dimensional complex vector spaces carry over to the (generalized) Fuglede-Kadison determinant. But the proof in our context is of course more complicated than the one in the classical case.

Theorem 3.14 (Kadison-Fuglede determinant). (1) Let f: U → V and g: V → W be morphisms of finite dimensional Hilbert N (G)-modules such that f has dense image and g is injective. Then

det(g ∘ f) = det(f) · det(g);

(2) Let f_{1} : U_{1} → V_{1}, f_{2} : U_{2} → V_{2} and f_{3} : U_{2} → V_{1} be morphisms of finite dimensional Hilbert N (G)-modules such that f_{1} has dense image and f_{2} is injective. Then

det ( \begin{matrix}f_{1} & f_{3} \\ 0 & f_{2} \end{matrix} ) = det(f_{1}) · det(f_{2});

(3) Let f: U → V be a morphism of finite dimensional Hilbert N (G)-

modules. Then

det(f) = det(f^{∗});

(4) If the Novikov-Shubin invariant of the morphism f: U → V of finite dimensional Hilbert N (G)-modules satisfies α(f) > 0, then f is of determinant class;

(5) Let H ⊂ G be a subgroup of finite index [G : H]. Let res f: res U → res V be the morphism of finite dimensional Hilbert N (H)-modules obtained from f by restriction. Then

det_{N (H)}(res f) = det_{N (G)}(f)^{[G:H]};

(6) Let i: H → G be an injective group homomorphism and let f: U → V be a morphism of finite dimensional Hilbert N (H)-modules. Then

det_{N (G)}(i_{∗}f) = det_{N (H)}(f).

Before we can give the proof of Theorem 3.14, we need

Lemma 3.15. Let f: U → V be a morphism of finite dimensional Hilbert N (G)-modules and let F: [0, ∞) → [0, ∞) be a density function. Then

(1) We have for 0 < ϵ < a

∫_{ϵ+}^{a} ln(λ) dF = − ∫_{ϵ}^{a} \frac{1}{λ} · (F (λ) − F (0)) dλ
+ ln(a) · (F (a) − F (0)) − ln(ϵ) · (F (ϵ) − F (0));
∫_{0+}^{a} ln(λ) dF = lim_{ϵ→0+} ∫_{ϵ+}^{a} ln(λ) dF;
∫_{0+}^{a} \frac{1}{λ} · (F (λ) − F (0)) dλ = lim_{ϵ→0+} ∫_{ϵ}^{a} \frac{1}{λ} · (F (λ) − F (0)) dλ.

We have ∫_{0+}^{a} ln(λ) dF > −∞ if and only if ∫_{0+}^{a} \frac{1}{λ} ·(F (λ)−F (0)) dλ < ∞, and in this case

lim_{λ→0+} ln(λ) · (F (λ) − F (0)) = 0;
∫_{0+}^{a} ln(λ) dF = − ∫_{0+}^{a} \frac{1}{λ} · (F (λ) − F (0)) dλ
+ ln(a) · (F (a) − F (0));

(2) If f is invertible, we get

det(f) = exp (\frac{1}{2} · tr(ln(f^{∗}f)));

(3) If f^{⊥}: ker(f)^{⊥} → clos(im(f)) is the weak isomorphism induced by f we

get

det(f) = det(f^{⊥});

(4) det(f) = det(f^{∗}) = \sqrt{det(f^{∗}f)} = \sqrt{det(ff^{∗})};

(5) If f: U → U is an injective positive operator, then

lim_{ϵ→0+} det(f + ϵ · id_{U}) = det(f);

(6) If f ≤ g for injective positive morphisms f, g: U → U, then

det(f) ≤ det(g);

(7) If f and g are morphisms of finite dimensional Hilbert N (G)-modules,

then

det(f ⊕ g) = det(f) · det(g).

Proof. (1) The first equation follows from partial integration for a continuously differentiable function g on (0, ∞) for ϵ < a (see for instance [330, page 95])

∫_{ϵ+}^{a} g(λ) dF = − ∫_{ϵ}^{a} g′(λ) · F (λ) · dλ + g(a) · F (a) − g(ϵ) · F (ϵ) (3.16)

and the second and third from Levi’s Theorem of monotone convergence. We conclude from the first three equations that

− ∫_{0+}^{1} \frac{1}{λ} · (F (λ) − F (0)) dλ ≤ ∫_{0+}^{1} ln(λ) dF,

and that it remains to show

∫_{0+}^{1} ln(λ) dF > −∞ ⇒ lim_{λ→0+} ln(λ) · (F (λ) − F (0)) = 0.  (3.17)

Suppose that lim_{λ→0+} ln(λ) · (F (λ) − F (0)) = 0 is not true. Then we can find a number C < 0 and a monotone decreasing sequence 1 > λ_{1} > λ_{2} > λ_{3} > … of positive real numbers converging to zero such that ln(λ_{i})·(F (λ_{i})−F (0)) ≤ C holds for all i ≥ 0. Since lim_{i→∞} F (λ_{i}) = F (0) holds we can assume without loss of generality 2·(F (λ_{i+1})−F (0)) ≤ F (λ_{i})−F (0), otherwise pass to an appropriate subsequence of (λ_{i})_{i}. We have for each natural number n and each λ ∈ (0, 1) that ln(λ) ≤ ∑_{i=1}^{n} ln(λ_{i}) · χ_{(λ_{i+1},λ_{i}]}(λ) holds, where χ_{(λ_{i+1},λ_{i}]}(λ) is the characteristic function of (λ_{i+1}, λ_{i}]. This implies for all n ≥ 1

∫_{0+}^{1} ln(λ) dF ≤ ∫_{0+}^{1} ∑_{i=1}^{n} ln(λ_{i}) · χ_{(λ_{i+1},λ_{i}]}(λ) dF

= ∑_{i=1}^{n} ln(λ_{i}) · (F (λ_{i}) − F (λ_{i+1}))

≤ ∑_{i=1}^{n} ln(λ_{i}) · \frac{F (λ_{i}) − F (0)}{2}

≤ n · \frac{C}{2} .

We conclude ∫_{0+}^{1} ln(λ) dF = −∞. Hence (3.17) and therefore assertion (1) are proved.

(2) We conclude from (1.65) or the more general fact that the trace is linear and ultra-weakly continuous

tr(ln(f^{∗}f)) = tr (∫_{0+}^{∞} ln(λ) dE_{λ}^{f^{∗}f})
= ∫_{0+}^{∞} ln(λ) d (tr(E_{λ}^{f^{∗}f}))
= ∫_{0+}^{∞} ln(λ^{2}) d (tr(E_{λ^{2}}^{f^{∗}f}))
= 2 · ∫_{0+}^{∞} ln(λ) dF (f).

(3) If F^{⊥} is the spectral density function of f^{⊥}, then F (λ) = F^{⊥}(λ) + F (0).

(4) This follows from assertion (1) and the conclusion from Lemma 2.4

F (f)(λ) − F (f)(0) = F (f^{∗})(λ) − F (f^{∗})(0) = F (f^{∗}f)(λ^{2}) − F (f^{∗}f)(0) = F (ff^{∗})(λ^{2}) − F (ff^{∗})(0).

(5) We have F (f + ϵ · id_{U})(λ) = F (f)(λ − ϵ). Since F (f)(0) = 0, we get

∫_{0+}^{∞} ln(λ) dF (f + ϵ · id_{U}) = ∫_{(−ϵ)+}^{∞} ln(λ + ϵ) dF (f)
= ∫_{(−ϵ)+}^{0} ln(λ + ϵ) dF (f) + ∫_{0+}^{∞} ln(λ + ϵ) dF (f)
= ln(ϵ) · F (f)(0) + ∫_{0+}^{∞} ln(λ + ϵ) dF (f)
= ∫_{0+}^{∞} ln(λ + ϵ) dF (f).

We conclude from Levi’s Theorem of monotone convergence

lim_{ϵ→0+} ∫_{0+}^{∞} ln(λ + ϵ) dF (f) = ∫_{0+}^{∞} ln(λ) dF (f).

This shows

lim_{ϵ→0+} det(f + ϵ · id_{U}) = det(f).

(6) For u ∈ U we get

||f^{1/2}(u)|| = ⟨f^{1/2}(u), f^{1/2}(u)⟩^{1/2}
= ⟨f(u), u⟩^{1/2}
≤ ⟨g(u), u⟩^{1/2}
= ⟨g^{1/2}(u), g^{1/2}(u)⟩^{1/2}
= ||g^{1/2}(u)||.

This implies for the spectral density functions

F (g^{1/2})(λ) ≤ F (f^{1/2})(λ).

Next we give the proof of the claim under the additional hypothesis that f and g are invertible. Then we can choose ϵ > 0 such that F (f)(λ) = F (g)(λ) = 0 for λ ≤ ϵ. Fix a ≥ ||g|| ≥ ||f||. Since F (f^{1/2})(λ) = F (g^{1/2})(λ) = dim_{N (G)}(U) for λ ≥ \sqrt{a}, we conclude from assertion (1)

∫_{0}^{∞} ln(λ) dF (g^{1/2}) ≥ ∫_{0}^{∞} ln(λ) dF (f^{1/2}).

This implies det(f^{1/2}) ≤ det(g^{1/2}). Now the claim for invertible f and g follows from assertion (4). The general case follows now from assertion (5) since f + ϵ · id_{U} and g + ϵ · id_{U} are invertible and f + ϵ · id_{U} ≤ g + ϵ · id_{U} holds for all ϵ > 0.

(7) Obviously F (f ⊕ g) = F (f) + F (g). Now apply assertion (1). ⊓ ⊔ For the proof of Theorem 3.14 we will need the next lemma where we will

use holomorphic calculus [282, Theorem 3.3.5 on page 206].

Lemma 3.18. Let f: D → ℂ be a holomorphic function defined on a domain D in ℂ whose boundary is a smooth closed curve γ: S^{1} → ℂ. Let X(t) for 0 ≤ t ≤ 1 be a differentiable family of morphisms X(t): U → U for a finite dimensional Hilbert N (G)-module U, where differentiable is to be understood with respect to the operator norm. Suppose that the spectrum spec(X(t)) := {z ∈ ℂ | z − X(t) is not invertible} of each X(t) lies in the interior of D. We define (motivated by the Cauchy integral formula)

f(X(t)) := \frac{1}{2πi} ∫_{γ} f(z) · (z − X(t))^{−1} dz.

Then f(X(t)) is differentiable with respect to t and

tr (\frac{d}{dt}f(X(t))) = tr(f′(X(t)) ∘ X′(t)),

where f′(z) = \frac{d}{dz} f(z) and X′(t) = \frac{d}{dt}X(t).

Proof. From

\frac{f(X(t + h)) − f(X(t))}{h}

= \frac{1}{2πi} ∫_{γ} f(z) · \frac{(z − X(t + h))^{−1} − (z − X(t))^{−1}}{h} dz
= \frac{1}{2πi} ∫_{γ} f(z) · (z − X(t + h))^{−1} ∘ \frac{X(t + h) − X(t)}{h} ∘ (z − X(t))^{−1} dz

we conclude

\frac{d}{dt}f(X(t)) = \frac{1}{2πi} ∫_{γ} f(z) · (z − X(t))^{−1} ∘ X′(t) ∘ (z − X(t))^{−1} dz.

Since

\frac{d}{dz} (f(z) · (z − X(t))^{−1}) = f′(z) · (z − X(t))^{−1} − f(z) · (z − X(t))^{−2}

and γ is closed, partial integration gives

f′(X(t)) = \frac{1}{2πi} ∫_{γ} f(z) · (z − X(t))^{−2} dz.

Since

tr (f(z) · (z − X(t))^{−1} ∘ X′(t) ∘ (z − X(t))^{−1})
= tr (f(z) · (z − X(t))^{−2} ∘ X′(t))

holds and tr commutes with integration, Lemma 3.18 follows. ⊓ ⊔ Now we are ready to give the proof of Theorem 3.14.

Proof. (1) Next we show for positive invertible morphisms f, g: U → U

tr(ln(gf^{2}g)) = 2 · (tr(ln(g)) + tr(ln(f))).  (3.19)

Consider the families g(t · f^{2} + (1 − t) · id)g and t · f^{2} + (1 − t) · id. There are real numbers 0 < a < b such that the spectrum of each member of these families lies in [a, b]. Choose a domain D in the half plane of complex numbers with positive real part which is bounded by a smooth curve and contains [a, b]. Let ln be a holomorphic extension of the logarithm (0, ∞) → ℝ to a holomorphic function on ℂ with the negative real numbers removed. Notice that D lies in the domain of ln. Since the definition of ln(h) of 1.64 and the one by holomorphic calculus in Lemma 3.18 for invertible h agree, Lemma 3.18 implies

\frac{d}{dt} tr (ln(g(t · f^{2} + (1 − t) · id)g))
= tr (\frac{d}{dt} ln(g(t · f^{2} + (1 − t) · id)g))
= tr ((g(t · f^{2} + (1 − t) · id)g)^{−1} ∘ (g(f^{2} − id)g))
= tr ((t · f^{2} + (1 − t) · id)^{−1} ∘ (f^{2} − id))
= \frac{d}{dt} tr (ln(t · f^{2} + (1 − t) · id)) .

Now (3.19) follows since ln(h^{2}) = 2·ln(h) for invertible positive h and ln(id) = 0.

Next we show for injective positive morphisms f, g: U → U

det(gf^{2}g) = det(f)^{2} · det(g)^{2}.  (3.20)

Notice that (3.20) holds under the additional assumption that f and g are invertible because of Lemma 3.15 (2), the equation ln(h^{2}) = 2 · ln(h) for invertible positive h and (3.19). The general case is reduced to this special case as follows.

Choose a constant C such that for all 0 < ϵ ≤ 1

gf^{2}g ≤ g(f + ϵ · id_{U})^{2}g ≤ gf^{2}g + Cϵ · id_{U}

holds. We conclude from Lemma 3.15 (4), (5) and (6)

det(gf^{2}g) = lim_{ϵ→0+} det(g(f + ϵ · id_{U})^{2}g);
det(f) = lim_{ϵ→0+} det(f + ϵ · id_{U});
det(gf^{2}g) = det(fg^{2}f).

Since g + ϵ and f + ϵ for ϵ > 0 are invertible, we get

det(gf^{2}g) = lim_{ϵ→0+} det(g(f + ϵ · id_{U})^{2}g)
= lim_{ϵ→0+} det((f + ϵ · id_{U})g^{2}(f + ϵ · id_{U}))
= lim_{ϵ→0+} lim_{δ→0+} det((f + ϵ · id_{U})(g + δ · id_{U})^{2}(f + ϵ · id_{U}))
= lim_{ϵ→0+} lim_{δ→0+} det(f + ϵ · id_{U})^{2} · det(g + δ · id_{U})^{2}
= det(f)^{2} · det(g)^{2}.

Hence (3.20) holds for all injective positive morphisms f and g.

Given morphisms f: U → V and g: V → W of finite dimensional Hilbert N (G)-modules such that f has dense image and g is injective, it remains to show

det(g ∘ f) = det(f) · det(g).

We can assume without loss of generality that both f and g are injective. Otherwise replace f by the injective map f^{⊥}: ker(f)^{⊥} → V induced by f and use the conclusion from Lemma 3.15 (7) that det(f^{⊥}) = det(f) and det(g ∘ f^{⊥}) = det(g ∘ f) holds.

If u is a unitary and v some morphism, then u^{−1}vu and v have the same spectral density function and hence

det(u^{−1}vu) = det(v).  (3.21)

In order to prove assertion (1) we use the polar decomposition f = au and g = vb where u and v are unitary isomorphisms and a and b are positive. Notice that g is injective, f has dense image and hence f^{∗} is injective. Therefore both a and b are injective. We compute using Lemma 3.15 (4), (3.20) and (3.21)

det(gf) = det(vbau) = \sqrt{det((vbau)^{∗}(vbau))} = \sqrt{det(u^{−1}ab^{2}au)}

= \sqrt{det(ab^{2}a)} = \sqrt{det(a)^{2} · det(b)^{2}} = \sqrt{det(ff^{∗}) · det(g^{∗}g)} = det(f)·det(g).

(2) The claim is already proved in Lemma 3.15 (7) if f_{3} is trivial. Because of the equation

( \begin{matrix}f_{1} & f_{3} \\ 0 & f_{2} \end{matrix} ) = ( \begin{matrix}1 & 0 \\ 0 & f_{2} \end{matrix} ) · ( \begin{matrix}1 & f_{3} \\ 0 & 1 \end{matrix} ) · ( \begin{matrix}f_{1} & 0 \\ 0 & 1 \end{matrix} )

assertion (1) implies that it suffices to prove

det ( \begin{matrix}1 & f_{3} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) = 1.

Since this matrix can be written as a commutator,

( \begin{matrix}1 & f_{3} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & −f_{3} & 1 \end{matrix} ) · ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) · ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & −f_{3} & 1 \end{matrix} )^{−1} · ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )^{−1}

the claim follows from assertion (1).

(3) has already been proved in Lemma 3.15 (4).

(4) Because of Lemma 3.15 (1) it suffices to show:

lim_{ϵ→0+} ∫_{ϵ}^{a} \frac{1}{λ} · (F (λ) − F (0)) · dλ < ∞.

Since α(f) is assumed to be positive, there is 0 < δ and 0 < α < α(f) such that

F (λ) − F (0) ≤ λ^{α}

holds for 0 ≤ λ ≤ δ. Now assertion (4) follows from

lim_{ϵ→0+} ∫_{ϵ}^{a} λ^{α−1} dλ = lim_{ϵ→0+} \frac{1}{α} · (a^{α} − ϵ^{α}) = \frac{1}{α} · a^{α}.

(5) This follows from Theorem 1.12 (6).

(6) This follows from (2.57). This finishes the proof of Theorem 3.14. ⊓ ⊔

Example 3.22. Consider a non-trivial element p ∈ ℂ[ℤ]. We want to compute the Fuglede-Kadison determinant of the morphism R_{p}: l^{2}(ℤ) → l^{2}(ℤ) given by multiplication with p ∈ ℂ[ℤ]. We can write

p(z) = C · z^{n} · ∏_{k=1}^{l}(z − a_{k})

for complex numbers C, a_{0}, a_{1}, …, a_{l} and integers n, l with C ≠ 0 and l ≥ 0. We want to show

ln(det(R_{p})) = ln(|C|) + ∑_{1≤k≤l,|a_{k}|>1} ln(|a_{k}|).  (3.23)

We get from Theorem 3.14 (1)

det(R_{p}) = det(R_{C}) · det(R_{z})^{n} · ∏_{k=1}^{l} det(R_{(z−a_{i})}).

Hence it remains to show

det(R_{(z−a)}) = \{ \begin{matrix}|a| & for |a| ≥ 1 \\ 1 & for |a| ≤ 1 \end{matrix}  (3.24)

Because of Example 3.13 it suffices to show

∫_{S^{1}} ln((z − a)\overline{(z − a)}) dvol = \{ \begin{matrix}2 · ln(|a|) & for |a| ≥ 1 \\ 0 & for |a| ≤ 1 \end{matrix}  (3.25)

for a ∈ ℂ, where we equip S^{1} with the obvious measure satisfying vol(S^{1}) = 1. Because ∫_{S^{1}} ln((z −a)\overline{(z − a)}) dvol = ∫_{S^{1}} ln((z −|a|)(z^{−1} −|a|)) dvol we may suppose a ∈ ℝ^{≥0} in the sequel.

We compute for a ≠ 1 and the path γ: [0, 1] → S^{1}, t ↦ exp(2πit) using the Residue Theorem

∫_{S^{1}} \frac{d}{da} ln((z − a)(z^{−1} − a)) dvol

= ∫_{S^{1}} \frac{1}{a − z} + \frac{1}{a − z^{−1}} dvol
= 2 · ∫_{S^{1}} \frac{1}{a − z} dvol
= 2 · ∫_{S^{1}} \frac{1}{(a − z) · 2πiz} · 2πiz · dvol
= \frac{2}{2πi} · ∫_{γ} \frac{1}{(a − z) · z} dz
= \{ \begin{matrix}\frac{2}{a} & for a > 1 \\ 0 & for a < 1 \end{matrix} .

This implies for a ∈ ℝ^{≥0}, a ≠ 1

\frac{d}{da} ∫_{S^{1}} ln((z − a)(z^{−1} − a)) dvol = \{ \begin{matrix}\frac{2}{a} & for a > 1 \\ 0 & for a < 1 \end{matrix} .

We conclude for an appropriate number C

∫_{S^{1}} ln((z − a)(z^{−1} − a)) dvol = 2 · ln(a) + C for a > 1 ∫_{S^{1}} ln((z − a)(z^{−1} − a)) dvol = 0 for a < 1 .

We get from Levi’s Theorem of monotone convergence

∫_{S^{1}} ln((z − 1)(z^{−1} − 1)) dvol = C.

We get from Lebesgue’s Theorem of majorized convergence

∫_{S^{1}} ln((z − 1)(z^{−1} − 1)) dvol = 0.

This proves (3.25) and hence (3.24) and (3.23).

Example 3.26. The following examples show that the conditions appearing in Theorem 3.14 (1) and (2) are necessary. We will use the same notation as in Example 3.13. Let u ∈ L^{∞}(T^{n}) be a given function such that {z ∈ T^{n} | u(z) = 0} has measure zero. Put

\begin{aligned}g: & L^{2}(T^{n})^{2} & → & L^{2}(T^{n})^{2}, & (a, b) & ↦ & (a, 0); \\ f: & L^{2}(T^{n})^{2} & → & L^{2}(T^{n})^{2}, & (a, b) & ↦ & (u · a + b, b); \\ f_{1} : & L^{2}(T^{n}) & → & L^{2}(T^{n}), & a & ↦ & u · a; \\ f_{2} : & L^{2}(T^{n}) & → & L^{2}(T^{n}), & a & ↦ & 0; \\ f_{3} : & L^{2}(T^{n}) & → & L^{2}(T^{n}), & a & ↦ & a. \end{aligned}

Then one easily checks using Lemma 3.15, Theorem 3.14 (2) and Example 3.13.

det(g ∘ f) = det ( \begin{matrix}u & 1 \\ 0 & 0 \end{matrix} ) = (det ( \begin{matrix}u & 1 \\ 0 & 0 \end{matrix} ) ( \begin{matrix}u & 1 \\ 0 & 0 \end{matrix} )^{∗})^{1/2}

= (det ( \begin{matrix}uu^{∗} + 1 & 0 \\ 0 & 0 \end{matrix} ))^{1/2} = exp (\frac{1}{2} · ∫_{T^{n}} ln(1 + |u(z)|^{2}) dvol_{z});

det(f) = det ( \begin{matrix}u & 1 \\ 0 & 1 \end{matrix} ) = det(u) = exp (∫_{T^{n}} ln(|u(z)|) dvol_{z});

det ( \begin{matrix}f_{1} & f_{3} \\ 0 & f_{2} \end{matrix} ) = det ( \begin{matrix}u & 1 \\ 0 & 0 \end{matrix} ) = exp (\frac{1}{2} · ∫_{T^{n}} ln(1 + |u(z)|^{2}) dvol_{z});

det(g) = 1;
det(f_{1}) = exp (∫_{T^{n}} ln(|u(z)|) dvol_{z});
det(f_{2}) = 1.

If we put for instance n = 1 and u(exp(2πit)) = exp(\frac{−1}{t}) for t ∈ (0, 1], then we get ∫_{T^{n}} ln(|u(z)|) dvol_{z} = −∞ and ∫_{T^{n}} ln(1 + |u(z)|^{2}) dvol_{z} ≥ 0 and hence

det(g ∘ f) ≥ 1;
det(f) = 0;
det(g) = 1;
det ( \begin{matrix}f_{1} & f_{3} \\ 0 & f_{2} \end{matrix} ) ≥ 1;
det(f_{1}) = 0;
det(f_{2}) = 1.

Notice that f has dense image and g is not injective and det(g ∘ f) ≠ det(f) · det(g). Moreover, f^{∗} is injective and g^{∗} has not dense image and det(f^{∗}∘g^{∗}) ≠ det(f^{∗}) · det(g^{∗}) because of Lemma 3.15 (4). A similar statement holds for

( \begin{matrix}f_{1} & f_{3} \\ 0 & f_{2} \end{matrix} ).

We have shown in Theorem 3.14 (4) for a morphisms f: U → V of finite dimensional Hilbert N (G)-modules that α(f) > 0 implies det(f) > 0. The converse is not true in general as the following example shows.

Example 3.27. We give an example of a morphism of finitely generated Hilbert N (ℤ)-modules such that f is of determinant class but its Novikov Shubin invariant is zero (cf Lemma 3.14 (4)). Fix ϵ > 0. Define a monotone non-decreasing continuous function F: [0, ∞) → [0, ∞) by F (0) = 0, F (λ) = | ln(λ/(1 + ϵ))|^{−1−ϵ} for 0 < λ ≤ 1 and F (λ) = | ln(1/(1 + ϵ))|^{−1−ϵ} for 1 ≤ λ. Since lim_{λ→0+} \frac{λ^{α}}{F (λ)} = 0 holds for any α > 0, we get α(F) = 0. We conclude from Levi’s Theorem of monotone convergence

∫_{0+}^{1} \frac{F (λ) − F (0)}{λ} dλ = ∫_{0+}^{1} \frac{1}{λ · | ln(λ/(1 + ϵ))|^{1+ϵ}} dλ
= lim_{δ→0+} ∫_{δ}^{1} \frac{1}{λ · | ln(λ/(1 + ϵ))|^{1+ϵ}} dλ
= lim_{δ→0+} −\frac{1}{ϵ · | ln(1/(1 + ϵ)|} + \frac{1}{ϵ · ln(δ/(1 + ϵ))}
= −\frac{1}{ϵ · ln(1 + ϵ)}
< ∞.

Define f: S^{1} → ℝ by sending exp(2πit) to F (t) for 0 ≤ t < 1. Then F is the spectral density function of the morphism M_{f}: L^{2}(S^{1}) → L^{2}(S^{1}) of finitely generated Hilbert N (ℤ)-modules (see Example 2.6). Lemma 3.15 (1) implies

det(f) > 0;
α(f) = 0.

3.3 L^{2}-Torsion of Hilbert Chain Complexes 139

Hence det(f) > 0 does not imply α(f) > 0. But we can get from det(f) > 0 the following related conclusion for the asymptotic behaviour of the spectral density function of f at zero which of course is weaker than the condition α(f) > 0

Theorem 3.28. Let f: U → V be a morphism of finite dimensional Hilbert N (G)-modules. Let F be its spectral density function. Put K = max{1, ||f||_{∞}}. Suppose that det(f) > 0. Then we get for all λ ∈ (0, 1]

F (λ) − F (0) ≤ (ln(K) · (dim(U) − dim(ker(f))) − ln(det(f))) · \frac{1}{− ln(λ)}.

Proof. We conclude from Lemma 3.15 (1) that ∫_{0+}^{K} \frac{F (ξ)−F (0)}{ξ}dξ < ∞ and

ln(det(f)) = ln(K) · (dim(U) − dim(ker(f))) − ∫_{0+}^{K} \frac{F (ξ) − F (0)}{ξ}dξ.

We estimate for λ ∈ (0, 1]

(F (λ) − F (0)) · (ln(K) − ln(λ))

= ∫_{λ}^{K} \frac{F (λ) − F (0)}{ξ}dξ
≤ ∫_{λ}^{K} \frac{F (ξ) − F (0)}{ξ}dξ
≤ ∫_{0+}^{K} \frac{F (ξ) − F (0)}{ξ}dξ
= ln(K) · (dim(U) − dim(ker(f))) − ln(det(f)).

This implies

F (λ) − F (0) ≤ \frac{ln(K) · (dim(U) − dim(ker(f))) − ln(det(f))}{ln(K) − ln(λ)}
≤ (ln(K) · (dim(U) − dim(ker(f))) − ln(det(f))) · \frac{1}{− ln(λ)}.⊓ ⊔

The obvious version of Theorem 3.28 for analytic spectral density functions is stated in Theorem 13.7.

3.3 L^{2}-Torsion of Hilbert Chain Complexes

In this section we introduce and study the L^{2}-torsion of finite Hilbert N (G)- chain complexes.

3.3.1 Basic Definitions and Properties of L^{2}-Torsion

Definition 3.29 (L^{2}-torsion). We call a Hilbert N (G)-chain complex C_{∗} dim-finite if dim(C_{p}) < ∞ for all p and C_{p} = 0 for |p| ≥ N for some integer N. A dim-finite Hilbert N (G)-chain complex C_{∗} is of determinant class if the differential c_{p}: C_{p} → C_{p−1} is of determinant class in the sense of Definition 3.11 for each p ∈ ℤ. A Hilbert N (G)-chain complex C_{∗} is called weakly acyclic or equivalently L^{2}-acyclic if its L^{2}-homology is trivial. It is called det-L^{2}-acyclic if C_{∗} is of determinant class and weakly acyclic.

If C_{∗} is of determinant class, define its L^{2}-torsion by

ρ^{(2)}(C_{∗}) := − ∑_{p∈ℤ}(−1)^{p} · ln(det(c_{p})) ∈ ℝ .

Notice that we prefer to take the logarithm of the determinant in order to get later additive instead of multiplicative formulas and because this will fit better with the analytic version. The next lemma expresses the L^{2}-torsion in terms of the Laplace operator Δ_{p}: C_{p} → C_{p} (see (1.17)) which will motivate the analytic definition later.

Lemma 3.30. A dim-finite Hilbert N (G)-chain complex C_{∗} is of determinant class if and only if Δ_{p} is of determinant class for all p ∈ ℤ. In this case

ρ^{(2)}(C_{∗}) = − \frac{1}{2} · ∑_{p∈ℤ}(−1)^{p} · p · ln(det(Δ_{p})).

Proof. From Lemma 1.18 we obtain an orthogonal decomposition

C_{p} = ker(c_{p})^{⊥} ⊕ clos(im(c_{p+1})) ⊕ ker(Δ_{p});
Δ_{p} = ((c_{p}^{⊥})^{∗} ∘ c_{p}^{⊥}) ⊕ (c_{p+1}^{⊥} ∘ (c_{p+1}^{⊥})^{∗}) ⊕ 0,

where c_{j}^{⊥}: ker(c_{j})^{⊥} → clos(im(c_{j})) is the weak isomorphism induced by c_{j}. Lemma 3.15 (4) and (7) imply that C_{∗} is of determinant class if and only Δ_{p} is of determinant class for all p ∈ ℤ and that in this case

−\frac{1}{2} · ∑_{p∈ℤ}(−1)^{p} · p · ln(det(Δ_{p}))

= −\frac{1}{2} · ∑_{p∈ℤ}(−1)^{p} · p · ln (det (((c_{p}^{⊥})^{∗} ∘ c_{p}^{⊥}) ⊕ (c_{p+1}^{⊥} ∘ (c_{p+1}^{⊥})^{∗}) ⊕ 0))

= −\frac{1}{2} · ∑_{p∈ℤ}(−1)^{p} · p · (ln (det ((c_{p}^{⊥})^{∗} ∘ c_{p}^{⊥}))
+ ln (det (c_{p+1}^{⊥} ∘ (c_{p+1}^{⊥})^{∗})) + ln(det(0)))
= −\frac{1}{2} · ∑_{p∈ℤ}(−1)^{p} · p · (2 · ln (det (c_{p})) + 2 · ln (det (c_{p+1})))
= − ∑_{p∈ℤ}(−1)^{p} · ln (det (c_{p})) .  ⊓ ⊔

Definition 3.31. Let f_{∗}: C_{∗} → D_{∗} be a chain map of dim-finite Hilbert N (G)-chain complexes. We call it of determinant class if its mapping cone cone_{∗}(f_{∗}) is of determinant class. In this case we define the L^{2}-torsion of f_{∗} by

t^{(2)}(f_{∗}) := ρ^{(2)}(cone_{∗}(f_{∗})).

Before we can state the main properties of these invariants ρ^{(2)}(C_{∗}) and t^{(2)}(f_{∗}), we need some preparations. Let 0 → U ⟶^{i} V ⟶^{p} W → 0 be a weakly exact sequence of finite dimensional Hilbert N (G)-modules. We call it of determinant class if the 2-dimensional weakly acyclic chain complex which it defines with W in dimension 0 is of determinant class, and we define in this case

ρ^{(2)}(U, V, W) ∈ ℝ  (3.32)

by the L^{2}-torsion of this chain complex in the sense of Definition 3.29. If the sequence is of determinant class we will later see in Lemma 3.41 or directly from Theorem 3.14 that for any choice of map s: W → V for which p ∘ s is a weak isomorphism of determinant class also i ⊕ s: U ⊕ W → V is a weak isomorphism of determinant class and

ρ^{(2)}(U, V, W) = − ln(det(i)) + ln(det(p))
= − ln(det(i ⊕ s)) + ln(det(p ∘ s)).  (3.33)

If 0 → U ⟶^{i} V ⟶^{p} W → 0 is exact, then it is of determinant class. If i is isometric and p induces an isometric isomorphism ker(p)^{⊥} → W, then ρ^{(2)}(U, V, W) = 0. This applies to the canonical exact sequence 0 → U → U ⊕ W → W → 0.

If 0 → C_{∗} → D_{∗} → E_{∗} → 0 is a weakly exact sequence of dim-finite Hilbert N (G)-chain complexes, we call it of determinant class if each exact sequence 0 → C_{p} → D_{p} → E_{p} → 0 is of determinant class and define in this case

ρ^{(2)}(C_{∗}, D_{∗}, E_{∗}) = ∑_{p∈ℤ}(−1)^{p} · ρ^{(2)}(C_{p}, D_{p}, E_{p}).  (3.34)

Let LHS_{∗}(C_{∗}, D_{∗}, E_{∗}) be the weakly acyclic dim-finite Hilbert N (G)-chain complex given by the weakly exact long homology sequence associated to an exact sequence 0 → C_{∗} → D_{∗} → E_{∗} → 0 of dim-finite Hilbert N (G)-chain complexes (see Theorem 1.21), where we use the convention that H_{0}^{(2)}(E_{∗}) sits in dimension zero. A chain map f_{∗}: C_{∗} → D_{∗} is called a weak homology equivalence if H_{p}^{(2)}(f_{∗}) is a weak isomorphism for all p ∈ ℤ.

The next result reflects the main properties of the torsion invariants defined above. These properties are very similar to the one for the classical notions for finite based free acyclic chain complexes over a field. However, the proof, which will be given in Subsection 3.3.3 after some preliminaries in Subsection 3.3.2, is more complicated in our context. To understand the basic properties of L^{2}-torsion it suffices to study the next theorem, and the reader may skip its proof and pass directly to Section 3.4. The main ideas of its proof come from [345, Section 6]. A proof of Theorem 3.35 (1) can also be found in [82, Theorem 2.7 on page 40].

Theorem 3.35 (L^{2}-torsion of Hilbert chain complexes). (1) Let 0 → C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{q_{∗}} E_{∗} → 0 be an exact sequence of dim-finite Hilbert N (G)- chain complexes. Suppose that three of the Hilbert N (G)-chain complexes C_{∗}, D_{∗}, E_{∗} and LHS_{∗} are of determinant class. Then all four are of determinant class and

ρ^{(2)}(C_{∗}) − ρ^{(2)}(D_{∗}) + ρ^{(2)}(E_{∗})

= ρ^{(2)}(C_{∗}, D_{∗}, E_{∗}) − ρ^{(2)}(LHS_{∗}(C_{∗}, D_{∗}, E_{∗}));

(2) Let

0 ⟶ C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{p_{∗}} E_{∗} ⟶ 0
f_{∗}↓  ^{g_{∗}}↓  ^{h_{∗}}↓
0 ⟶ C′_{∗} ⟶^{i′_{∗}} D′_{∗} ⟶^{p′_{∗}} E′_{∗} ⟶ 0

be a commutative diagram of dim-finite Hilbert N (G)-chain complexes which are of determinant class. Suppose that two of the chain maps f_{∗}, g_{∗} and h_{∗} are weak homology equivalences of determinant class and that the rows are weakly exact and of determinant class. Then all three chain maps f_{∗}, g_{∗} and h_{∗} are weak homology equivalences of determinant class and

t^{(2)}(f_{∗}) − t^{(2)}(g_{∗}) + t^{(2)}(h_{∗}) = ρ^{(2)}(C′_{∗}, D′_{∗}, E′_{∗}) − ρ^{(2)}(C_{∗}, D_{∗}, E_{∗});

(3) Let f_{∗}, g_{∗}: C_{∗} → D_{∗} be weak homology equivalences of dim-finite N (G)- chain complexes such that f_{∗} or g_{∗} is of determinant class. Suppose that f_{∗} and g_{∗} are homotopic. Then both are of determinant class and

t^{(2)}(f_{∗}) = t^{(2)}(g_{∗});

(4) Let C_{∗}, D_{∗} and E_{∗} be dim-finite Hilbert N (G)-chain complexes and let f_{∗}: C_{∗} → D_{∗} and g_{∗}: D_{∗} → E_{∗} be chain maps. Suppose that two of the chain maps f_{∗}, g_{∗} and g_{∗} ∘f_{∗} are weak homology equivalences of determinant class. Then all three are weak homology equivalences of determinant class and

t^{(2)}(g ∘ f) = t^{(2)}(f) + t^{(2)}(g);

(5) Let C_{∗} and D_{∗} be dim-finite Hilbert N (G)-chain complexes of determinant class and f_{∗}: C_{∗} → D_{∗} be a weak homology equivalence. Then f_{∗} is of determinant class if and only if H_{p}^{(2)}(f_{∗}) is of determinant class for all p ∈ ℤ and in this case

t^{(2)}(f_{∗}) = ρ^{(2)}(D_{∗}) − ρ^{(2)}(C_{∗}) + ∑_{p∈ℤ}(−1)^{p} · ln (det (H_{p}^{(2)}(f_{∗})));

(6) Let f_{∗}: C_{∗} → C′_{∗} and g_{∗}: D_{∗} → D′_{∗} be chain maps of dim-finite Hilbert N (G)-chain complexes and N (H)-chain complexes. Denote by χ^{(2)}(C_{∗}) ∈ ℝ the L^{2}-Euler characteristic ∑_{p∈ℤ}(−1)^{p} · b_{p}^{(2)}(C_{∗}). Then

(a) If D_{∗} is det-L^{2}-acyclic, then the dim-finite Hilbert N (G × H)-chain complex C_{∗} ⊗ D_{∗} is det-L^{2}-acyclic and

ρ^{(2)}(C_{∗} ⊗ D_{∗}) = χ^{(2)}(C_{∗}) · ρ^{(2)}(D_{∗});

(b) If C_{∗} and D_{∗} are of determinant class, then the dim-finite Hilbert N (G × H)-chain complex C_{∗} ⊗ D_{∗} is of determinant class and

ρ^{(2)}(C_{∗} ⊗ D_{∗}) = χ^{(2)}(C_{∗}) · ρ^{(2)}(D_{∗}) + χ^{(2)}(D_{∗}) · ρ^{(2)}(C_{∗});

(c) If f_{∗} and g_{∗} are weak homology equivalences of determinant class, then the chain map f_{∗} ⊗ g_{∗} of Hilbert N (G × H)-chain complexes is a weak homology equivalence of determinant class and

t^{(2)}(f_{∗} ⊗ g_{∗}) = χ^{(2)}(C_{∗}) · t^{(2)}(g_{∗}) + χ^{(2)}(D_{∗}) · t^{(2)}(f_{∗});

(7) Let H ⊂ G be a subgroup of finite index [G : H] and let C_{∗} be a dimfinite Hilbert N (G)-chain complex. Then C_{∗} is det-L^{2}-acyclic if and only if the dim-finite Hilbert N (H)-chain complex res C_{∗} obtained from C_{∗} by restriction is det-L^{2}-acyclic, and in this case

ρ^{(2)}(res C_{∗}) = [G : H] · ρ^{(2)}(C_{∗});

(8) Let i: H → G be an inclusion of groups and let C_{∗} be a dim-finite Hilbert N (H)-chain complex. Then C_{∗} is det-L^{2}-acyclic if and only if the dimfinite Hilbert N (G)-chain complex i_{∗}C_{∗} obtained by induction with i (see Definition 1.23) is det-L^{2}-acyclic, and in this case

ρ^{(2)}(C_{∗}) = ρ^{(2)}(i_{∗}C_{∗}).

Example 3.36. We give an example that in Theorem 3.35 (3) the condition that f_{∗} and g_{∗} are weak homology equivalences is necessary. The same is true for Theorem 3.35 (4) and (5). Let C_{∗} and D_{∗} respectively be the Hilbert N (G)-chain complexes concentrated in dimensions 0 and 1 whose first differentials are id: N (G) → N (G) and 0: N (G) → N (G) respectively. For any morphism γ: N (G) → N (G) we obtain a chain map f(γ)_{∗}: C_{∗} → D_{∗} by putting f(γ)_{1} = γ and f(γ)_{0} = 0. The chain map f(γ)_{∗} is homotopic to 0 : C_{∗} → D_{∗}, a chain homotopy is given by γ itself. However, one easily computes that t^{(2)}(f(γ)_{∗}) is not independent of γ, namely

t^{(2)}(f(γ)_{∗}) = − \frac{1}{2} · ln (det (id +γ^{∗}γ)) .

3.3.2 L^{2}-Torsion and Chain Contractions

Before we give the proof of Theorem 3.35 in Subsection 3.3.3, we reformulate the definition of L^{2}-torsion in terms of weak chain contractions. This formulation will be useful for the proofs since it is more flexible. Moreover, it applies to more general situations and is closer to standard notions such as Whitehead torsion.

Lemma 3.37. (1) Let f: U → V and g: V → W be morphisms of finite dimensional Hilbert N (G)-modules. If two of the maps f, g and g ∘ f are weak isomorphisms (of determinant class), then also the third;

(2) Let

0 ⟶ U_{1} ⟶^{i} U_{0} ⟶^{p} U_{2} ⟶ 0
f_{1}↓  ^{f_{0}}↓  ^{f_{2}}↓
0 ⟶ V_{1} ⟶^{j} V_{0} ⟶^{q} V_{2} ⟶ 0

be a commutative diagram of maps of finite dimensional Hilbert N (G)- modules whose rows are weakly exact (and of determinant class). If two of the three maps f_{1}, f_{0} and f_{2} are weak isomorphisms (of determinant class), then also the third;

(3) Let f_{∗}: C_{∗} → D_{∗} be a chain map of Hilbert N (G)-chain complexes such that C_{p} and D_{p} have finite dimension for all p ∈ ℤ. Then f_{∗} is a weak homology equivalence if and only if cone(f_{∗}) is weakly acyclic.

Proof. (1) This follows from Lemma 1.13 and Theorem 3.14 (1).

(2) The given diagram induces the following commutative diagram with exact rows

0 ⟶ ker(p) ⟶ U_{0} ⟶ ker(p)^{⊥} ⟶ 0
f′_{1}↓  ^{f_{0}}↓  ^{f′_{2}}↓
0 ⟶ ker(q) ⟶ V_{0} ⟶ ker(q)^{⊥} ⟶ 0 .

The induced maps i: U_{1} → ker(p), p^{⊥}: ker(p)^{⊥} → U_{2}, j: V_{1} → ker(q), and q^{⊥}: ker(q)^{⊥} → V_{2} are weak isomorphisms (of determinant class) by assumption. Because of assertion (1) f′_{1} and f′_{2} respectively are weak isomorphisms (of determinant class) if and only if f_{1} and f_{2} respectively are weak isomorphisms (of determinant class.) We conclude from Theorem 1.21 and Theorem 3.14 (2) that already all three maps f′_{1}, f_{0} and f′_{2} are weak isomorphisms (of determinant class) if two of them are weak isomorphisms (of determinant class).

(3) follows from the long weakly exact homology equivalence (see Theorem 1.21) associated to the exact sequence 0 → D_{∗} → cone(f_{∗}) → ΣC_{∗} → 0 since the boundary map is H_{∗}^{(2)}(f_{∗}). ⊓ ⊔

A chain map f_{∗}: C_{∗} → D_{∗} of Hilbert N (G)-chain complexes is called weak chain isomorphism if f_{p}: C_{p} → D_{p} is a weak isomorphism for all p.

Definition 3.38. A weak chain contraction for a Hilbert N (G)-chain complex C_{∗} is a pair (γ_{∗}, u_{∗}) which consists of a weak chain isomorphism u_{∗}: C_{∗} → C_{∗} and a chain homotopy γ_{∗}: u_{∗} ≃ 0 satisfying γ_{∗} ∘ u_{∗} = u_{∗} ∘ γ_{∗}.

A chain contraction in the ordinary sense is just a weak chain contraction (γ_{∗}, u_{∗}) with u_{∗} = id.

Lemma 3.39. The following statements are equivalent for a dim-finite Hilbert N (G)-chain complex.

(1) C_{∗} is weakly acyclic (and of determinant class);

(2) Δ_{p}: C_{p} → C_{p} is a weak isomorphism (of determinant class) for all p ∈ ℤ;

(3) There is a weak chain contraction (γ_{∗}, u_{∗}) with γ_{∗} ∘ γ_{∗} = 0 (such that u_{∗} is of determinant class);

(4) There is a weak chain contraction (γ_{∗}, u_{∗}) (such that u_{∗} is of determinant class).

Proof. (1) ⇒ (2) This follows from Lemma 1.18 and Lemma 3.30 since a selfadjoint endomorphism of a Hilbert N (G)-module (such as Δ_{p}) is a weak isomorphism if and only if it is injective.

(2) ⇒ (3) Put γ_{p} = c_{p}^{∗}. Then (γ_{∗}, Δ_{∗}) is the desired weak chain contraction.

(3) ⇒ (4) is trivial.

(4) ⇒ (1) Since γ_{∗} is a chain homotopy between u_{∗} and 0_{∗}, we get H_{p}^{(2)}(u_{∗}) = 0 for p ∈ ℤ. Since u_{∗} is a weak isomorphism, H_{p}^{(2)}(u_{∗}) is a weak isomorphism for all p ∈ ℤ by Lemma 3.44 which we will prove later. Hence C_{∗} is weakly acyclic. We split

u_{p} = ( \begin{matrix}u_{p}^{0} & u_{p}^{1} \\ 0 & u_{p}^{⊥} \end{matrix} ) : C_{p} = ker(c_{p}) ⊕ ker(c_{p})^{⊥} → C_{p} = ker(c_{p}) ⊕ ker(c_{p})^{⊥};
c_{p} = ( \begin{matrix}0 & c_{p}^{⊥} \\ 0 & 0 \end{matrix} ) : C_{p} = ker(c_{p}) ⊕ ker(c_{p})^{⊥} → C_{p−1} = ker(c_{p−1}) ⊕ ker(c_{p−1})^{⊥};
γ_{p} = ( \begin{matrix}γ_{p}^{0} & γ_{p}^{1} \\ γ_{p}^{2} & γ_{p}^{3} \end{matrix} ) : C_{p} = ker(c_{p}) ⊕ ker(c_{p})^{⊥} → C_{p+1} = ker(c_{p+1}) ⊕ ker(c_{p+1})^{⊥}.

Since u_{p}^{⊥} is an endomorphism of a finite dimensional Hilbert N (G)-module with dense image, we conclude from Lemma 1.13 that u_{p}^{⊥} is a weak isomorphism. From c_{p+1} ∘ γ_{p} + γ_{p−1} ∘ c_{p} = u_{p} we conclude γ_{p−1}^{2} ∘ c_{p}^{⊥} = u_{p}^{⊥}. Since c_{p}^{⊥} and u_{p}^{⊥} are weak isomorphisms, γ_{p−1}^{2} is a weak isomorphism by Lemma 3.37 (1). If u_{p} is of determinant class, c_{p} is of determinant class by Theorem 3.14 (1) and Lemma 3.15 (3). Hence C_{∗} is of determinant class if u_{∗} is of determinant class for all p ∈ ℤ. ⊓ ⊔

The proof of the next lemma is a direct calculation.

Lemma 3.40. Let (γ_{∗}, u_{∗}) and (δ_{∗}, v_{∗}) be weak chain contractions for the dim-finite Hilbert N (G)-chain complex C_{∗}. Define Θ: C_{ev} → C_{ev} by

Θ := (v_{∗} ∘ u_{∗} + δ_{∗} ∘ γ_{∗}) = ( \begin{matrix}... & ⋮ & ⋮ & ⋮ & ... \\ … & vu & 0 & 0 & … \\ … & δγ & vu & 0 & … \\ … & 0 & δγ & vu & … \\ ... & ⋮ & ⋮ & ⋮ & ... \end{matrix} ) .

Then the composition

Θ′ : C_{odd} ⟶^{(uc+γ)_{odd}} C_{ev} ⟶^{Θ} C_{ev} ⟶^{(vc+δ)_{ev}} C_{odd}

is given by the lower triangle matrix

( \begin{matrix}... & ⋮ & ⋮ & ⋮ & ... \\ … & (v^{2}u^{2})_{2n−1} & 0 & 0 & … \\ … & ∗ & (v^{2}u^{2})_{2n+1} & 0 & … \\ … & ∗ & ∗ & (v^{2}u^{2})_{2n+3} & … \\ ... & ⋮ & ⋮ & ⋮ & ... \end{matrix} ) .

Lemma 3.41. Let C_{∗} be a weakly acyclic dim-finite Hilbert N (G)-chain complex of determinant class. Let (γ_{∗}, u_{∗}) and (δ_{∗}, v_{∗}) be weak chain contractions such that u_{p} and v_{p} are of determinant class for p ∈ ℤ.

Then the maps (uc + γ)_{odd} : C_{odd} → C_{ev}, (uc + γ)_{ev} : C_{ev} → C_{odd}, u_{odd} and u_{ev} are weak isomorphisms and of determinant class and we get

ln (det((uc + γ)_{odd})) − ln (det(u_{odd})) = − ln (det((vc + δ)_{ev})) + ln (det(v_{ev})) ;
ln (det(u_{odd})) = ln (det(u_{ev})) ;
ρ^{(2)}(C_{∗}) = ln (det((uc + γ)_{odd})) − ln (det(u_{odd})) .

Proof. Since u_{p} and v_{p} are weak isomorphisms and of determinant class, we conclude from Lemma 1.13, Theorem 3.14 (1) and (2), Lemma 3.37 (1) and (2) and Lemma 3.40 that the maps (uc + γ)_{odd}, (vc + δ)_{ev}, u_{odd}, v_{odd}, u_{ev} and v_{ev} are weak isomorphisms and of determinant class and that

2 · (ln (det(v_{odd})) + ln (det(u_{odd}))) = ln (det((vc + δ)_{ev})) + ln (det(v_{ev}))

+ ln (det(u_{ev})) + ln (det((uc + γ)_{odd})) .

We conclude ln (det(u_{odd})) = ln (det(u_{ev})) from Theorem 3.14 (1) applied to (uc + γ)_{odd} ∘ u_{odd} = u_{ev} ∘ (uc + γ)_{odd} and analogously ln (det(v_{odd})) = ln (det(v_{ev})). This proves the first two equations.

The first equation shows that ln (det((uc + γ)_{odd})) − ln (det(u_{odd})) is independent of the choice of the weak chain contraction (γ_{∗}, u_{∗}) with u_{p} a weak isomorphism of determinant class for all p ∈ ℤ. Hence it suffices to prove the third equation for the special choice (γ_{∗}, u_{∗}) = ((c_{∗})^{∗}, Δ_{∗}).

Let f_{p}: C_{p} → C_{p} be the p-fold composition (Δ_{p})^{p} = Δ_{p} ∘ … ∘ Δ_{p}. Then the following square commutes

C_{odd} ⟶^{(Δc+c^{∗})_{odd}} C_{ev}
f_{odd}↓  ^{f_{ev}}↓
C_{odd} ⟶^{(Δc^{∗}+c)_{odd}} C_{ev}

We conclude from Theorem 3.14, Lemma 3.15 (4) and Lemma 3.30

2 · ln (det ((Δc + c^{∗})_{odd}))

= ln (det ((Δc + c^{∗})_{odd}^{∗} ∘ (Δc + c^{∗})_{odd}))
= ln (det ((Δc + c^{∗})_{odd}^{∗} ∘ f_{ev} ∘ (Δc + c^{∗})_{odd})) − ln (det (f_{ev}))
= ln (det ((Δc + c^{∗})_{odd}^{∗} ∘ (Δc^{∗} + c)_{odd} ∘ f_{odd})) − ln (det (f_{ev}))
= ln (det ((Δc^{∗} + c)_{ev} ∘ (Δc^{∗} + c)_{odd})) + ln (det (f_{odd})) − ln (det (f_{ev}))

and

2 · (ln (det ((Δc + c^{∗})_{odd})) − ln (det (Δ_{odd})))

= ln (det (f_{odd})) − ln (det (f_{ev})) + ln (det ((Δc^{∗} + c)_{ev} ∘ (Δc^{∗} + c)_{odd}))
−2 · ln (det (Δ_{odd}))
= − ∑_{p} (−1)^{p} · p · ln (det (Δ_{p})) + ln (det ((Δc^{∗} + c)_{ev} ∘ (Δc^{∗} + c)_{odd}))
−2 · ln (det (Δ_{odd}))
= 2 · ρ^{(2)}(C_{∗}) + ln (det ((Δc^{∗} + c)_{ev})) + ln (det ((Δc^{∗} + c)_{odd}))
−2 · ln (det (Δ_{odd})) .

Hence it remains to show

ln (det ((Δc^{∗} + c)_{ev})) + ln (det ((Δc^{∗} + c)_{odd})) = 2 · ln (det (Δ_{odd})) .

The dual chain complex (C_{∗})^{∗} has the chain contraction (c_{∗}, Δ_{∗}). If we apply to it the first and second equation, which we have already proved, we obtain

ln (det ((Δc^{∗} + c)_{odd})) − ln (det (Δ_{odd})) = − ln (det ((Δc^{∗} + c)_{ev}))
+ ln (det (Δ_{ev})) ;
ln (det (Δ_{odd})) = ln (det (Δ_{ev})) .

This finishes the proof of Lemma 3.41. ⊓ ⊔ 3.3.3 Proofs of the Basic Properties of L^{2}-Torsion

This subsection is devoted to the proof of Theorem 3.35. We will need the following lemmas.

Lemma 3.42. Let 0 → C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{q_{∗}} E_{∗} → 0 be an exact sequence of dim-finite N (G)-chain complexes. Suppose that (ϵ_{∗}, w_{∗}) is a weak chain contraction for E_{∗} (such that w_{p} is of determinant class for p ∈ ℤ). Then there is a chain map s_{∗}: E_{∗} → D_{∗} such that q_{∗} ∘ s_{∗} = w_{∗}: E_{∗} → E_{∗} and that i_{p} ⊕ s_{p}: C_{p} ⊕ E_{p} → D_{p} is a weak isomorphism (of determinant class) for p ∈ ℤ.

Proof. For each p ∈ ℤ choose a morphism σ_{p}: E_{p} → D_{p} with q_{p} ∘ σ_{p} = id. Now define

s_{p} := d_{p+1} ∘ σ_{p+1} ∘ ϵ_{p} + σ_{p} ∘ ϵ_{p−1} ∘ e_{p}: E_{p} → D_{p}.

Then s_{∗}: D_{∗} → E_{∗} is a chain map with p_{∗} ∘ s_{∗} = w_{∗} and the following diagram commutes

0 ⟶ C_{∗} ⟶ C_{∗} ⊕ E_{∗} ⟶ E_{∗} ⟶ 0
id↓  ^{i_{∗}⊕s_{∗}}↓  ^{w_{∗}}↓
0 ⟶ C_{∗} ⟶^{i_{∗}} D_{∗}  ⟶^{q_{∗}} E_{∗} ⟶ 0

where the upper horizontal row is the canonical one. Now the claim follows from Lemma 3.37 (2). ⊓ ⊔

Lemma 3.43. Let C_{∗} be a Hilbert N (G)-chain complex with trivial differentials. Let u_{∗}, v_{∗}: C_{∗} → D_{∗} be chain maps to a dim-finite Hilbert N (G)-chain complex D_{∗} of determinant class with H_{∗}^{(2)}(u_{∗}) = H_{∗}^{(2)}(v_{∗}). Then there is a weak chain isomorphism g_{∗}: D_{∗} → D_{∗} such that each g_{p} : D_{p} → D_{p} is of determinant class, H_{p}^{(2)}(g_{∗}) = id and there is a chain homotopy g_{∗}∘u_{∗} ≃ g_{∗}∘v_{∗}.

Proof. Since d_{p}^{⊥}: ker(d_{p})^{⊥} → clos(im(d_{p})) is a weak isomorphism, we can

choose an isomorphism ψ_{p−1} : clos(im(d_{p})) ⟶^{≅} ker(d_{p})^{⊥} by the Polar Decomposition Theorem. Define g_{p}: D_{p} → D_{p} by the orthogonal sum of

d_{p+1}|_{ker(d_{p+1})^{⊥}} ∘ ψ_{p} : clos(im(d_{p+1})) → clos(im(d_{p+1}));
id: ker(d_{p}) ∩ clos(im(d_{p+1}))^{⊥} → ker(d_{p}) ∩ clos(im(d_{p+1}))^{⊥};
ψ_{p−1} ∘ d_{p}^{⊥}: ker(d_{p})^{⊥} → ker(d_{p})^{⊥}.

Since D_{∗} is of determinant class, g_{p} is a weak isomorphism of determinant class by Lemma 3.15 (3) and (7) and Lemma 3.37 (1). Define γ_{p}: C_{p} → D_{p+1} by the composition of u_{p}−v_{p}: C_{p} → clos(im(d_{p+1})) and ψ_{p}: clos(im(d_{p+1})) → ker(d_{p+1})^{⊥} ⊂ D_{p+1}. Now one easily checks using Lemma 1.18 that we obtain a chain map g_{∗}: D_{∗} → D_{∗} with H_{∗}^{(2)}(g_{∗}) = id and γ_{∗} defines a chain homotopy between g_{∗} ∘ u_{∗} and g_{∗} ∘ v_{∗}. ⊓ ⊔ Lemma 3.44. Let C_{∗} and D_{∗} be dim-finite Hilbert N (G)-chain complexes. If f_{∗}: C_{∗} → D_{∗} is a weak chain isomorphism (such that f_{p} is of determinant class for all p ∈ ℤ), then H_{p}^{(2)}(f_{∗}) is a weak isomorphism (of determinant class) for all p ∈ ℤ. If f_{∗}: C_{∗} → D_{∗} is a weak chain isomorphism (such that f_{p} is of determinant class for all p ∈ ℤ) and C_{∗} or D_{∗} is of determinant class, then both C_{∗} and D_{∗} are of determinant class and

ρ^{(2)}(D_{∗})−ρ^{(2)}(C_{∗}) = ∑_{p∈ℤ}(−1)^{p}·ln (det(f_{p}))−∑_{p∈ℤ}(−1)^{p}·ln (det (H_{p}^{(2)}(f_{∗}))) .

Proof. Fix n_{0} ∈ ℤ such that C_{p} = D_{p} = 0 for p < n_{0}. We use induction over n for which C_{p} = D_{p} = 0 for p > n. The induction beginning n ≤ n_{0} is trivial since then C_{∗} and D_{∗} are concentrated in dimension n_{0}. The induction step from n to n + 1 is done as follows.

Define C′_{∗} as the subchain complex of C_{∗} with C′_{p} = C_{p} for p ≥ n + 1, C_{n} = clos(im(c_{n+1})) and C′_{p} = 0 for p ≤ n − 1. Define C′′_{∗} as the quotient chain complex of C_{∗} with C′′_{p} = 0 for p ≥ n+1, C′′_{n} = im(c_{n+1})^{⊥} and C′′_{p} = C_{p} for p ≤ n − 1. There is an obvious exact sequence 0 → C′_{∗} → C_{∗} → C′′_{∗} → 0. Lemma 3.15 (3) implies that both C′_{∗} and C′′_{∗} are of determinant class if and only if C_{∗} is of determinant class and in this case

ρ^{(2)}(C_{∗}) = ρ^{(2)}(C′_{∗}) + ρ^{(2)}(C′′_{∗}).  (3.45)

We obtain a commutative diagram with exact rows

\begin{aligned}0 & ⟶ & C′_{n} & ⟶ & C_{n} & ⟶ & C′′_{n} & ⟶ & 0 \\ & ^{f′_{n}}↓ & ^{f_{n}}↓ & ^{f′′_{n}}↓ \\ 0 & ⟶ & D′_{n} & ⟶ & D_{n} & ⟶ & D′′_{n} & ⟶ & 0 \end{aligned}  (3.46)

The map f′_{n}: C′_{n} → D′_{n} has dense image as its composition with c′_{n+1} : C′_{n+1} → C′_{n} is the composition of the maps which both have dense image f_{n+1} : C_{n+1} → D_{n+1} and d_{n+1} : D_{n+1} → clos(im(d_{n+1})). Since the middle vertical arrow in diagram (3.46) is a weak isomorphism and the left vertical arrow in diagram (3.46) has dense image and hence is a weak isomorphism, all three vertical arrows in diagram (3.46) are weak isomorphisms by Lemma 3.37 (2). Theorem 3.14 (2) applied to diagram (3.46) shows that for all p ∈ ℤ the maps f′_{p} and f′′_{p} are weak isomorphisms of determinant class and

ln(det(f_{p})) = ln(det(f′_{p})) + ln(det(f′′_{p})),  (3.47)

provided that f_{p} is of determinant class for all p ∈ ℤ.

Next we prove Lemma 3.44 for f′_{∗}: C′_{∗} → D′_{∗}. Suppose that f′_{∗}: C′_{∗} → D′_{∗} is a weak chain isomorphism. Since C′_{p} = 0 for p ≥ n + 2, we obtain a commutative diagram with exact rows

0 ⟶ H_{n+1}^{(2)}(C′_{∗}) ⟶ C′_{n+1} ⟶^{pr} ker(c′_{n+1})^{⊥} ⟶ 0
H_{n+1}^{(2)}(f′_{∗})↓  ^{f′_{n+1}}↓  ^{f_{n+1}^{′⊥}}↓
0 ⟶ H_{n+1}^{(2)}(D′_{∗}) ⟶ D′_{n+1} ⟶^{pr} ker(d′_{n+1})^{⊥} ⟶ 0

(3.48)

and the commutative diagram

\begin{matrix}ker(c′_{n+1})^{⊥} & ⟶^{c_{n+1}^{′⊥}} & C′_{n} \\ ^{f_{n+1}^{′⊥}}↓ & ^{f′_{n}}↓ \\ ker(d′_{n+1})^{⊥} & ⟶^{d_{n+1}^{′⊥}} & D′_{n} \end{matrix}  (3.49)

Since the horizontal arrows and the right vertical arrow in diagram (3.49) are weak isomorphisms, f_{n+1}^{′⊥} is a weak isomorphism by Lemma 3.37 (1). Lemma 3.37 (2) applied to diagram (3.48) shows that H_{n+1}^{(2)}(f′_{∗}) is a weak isomorphism.

Now suppose that f′_{p}: C′_{p} → D′_{p} is a weak isomorphism of determinant class for all p ∈ ℤ. Theorem 3.14 (2) applied to diagram (3.48) shows that H_{n+1}^{(2)}(f′_{∗}) and f_{n+1}^{⊥} are of determinant class and satisfy

ln(det(f′_{n+1})) = ln (det (H_{n+1}^{(2)}(f′_{∗}))) + ln (det (f_{n+1}^{⊥})) . (3.50)

If C′_{∗} or D′_{∗} is of determinant class, we conclude from Lemma 3.15 (3) and Lemma 3.37 (1) applied to diagram (3.49) that both C′_{∗} and D′_{∗} are of determinant class. In this case Theorem 3.14 (1) and Lemma 3.15 (3) applied to (3.49) imply

(−1)^{n} · ln(det(f′_{n})) + ρ^{(2)}(C′_{∗}) = (−1)^{n} · ln(det(f_{n+1}^{′⊥})) + ρ^{(2)}(D′_{∗}). (3.51)

We conclude from (3.50) and (3.51) that H_{n+1}(f′_{∗}) is a weak isomorphism of determinant class and that both C′_{∗} and D′_{∗} are of determinant class and

ρ^{(2)}(D′_{∗}) − ρ^{(2)}(C′_{∗}) = ∑_{p∈ℤ}(−1)^{p} · ln (det(f′_{p}))
− ∑_{p∈ℤ}(−1)^{p+1} · ln (det (H_{p+1}^{(2)}(f′_{∗}))) . (3.52)

Notice that the induction hypothesis applies to f′′_{∗}: C′′_{∗} → D′′_{∗}. We conclude that C′′_{∗} and D′′_{∗} are of determinant class and H_{p}^{(2)}(f′′_{∗}) is a weak isomorphism of determinant class for all p ∈ ℤ and

ρ^{(2)}(D′′_{∗}) − ρ^{(2)}(C′′_{∗}) = ∑_{p∈ℤ}(−1)^{p} · ln (det(f′′_{p}))
− ∑_{p∈ℤ}(−1)^{p} · ln (det (H_{p}^{(2)}(f′′_{∗}))) . (3.53)

Now Lemma 3.44 follows from (3.45), (3.47), (3.52) and (3.53). ⊓ ⊔ The next lemma contains a kind of rotation principle.

Lemma 3.54. Let 0 → C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{q_{∗}} E_{∗} → 0 be an exact sequence of dim-finite Hilbert N (G)-chain complexes. Then there is an exact sequence of dim-finite Hilbert N (G)-chain complexes 0 → D_{∗} → Ẽ_{∗} → C̃_{∗} → 0 with the following properties

(1) C_{∗}, E_{∗} and LHS_{∗}(C_{∗}, D_{∗}, E_{∗}) (defined in Theorem 3.35 (1)) respectively are of determinant class if and only if C̃_{∗}, Ẽ_{∗} and LHS_{∗}(D_{∗}, Ẽ_{∗}, C̃_{∗}) respectively are of determinant class;

(2) There is a chain isomorphism

LHS_{∗}(C_{∗}, D_{∗}, E_{∗}) ⟶^{≅} Σ (LHS_{∗}(D_{∗}, Ẽ_{∗}, C̃_{∗}));

(3) If C_{∗}, D_{∗}, E_{∗} and LHS_{∗}(C_{∗}, D_{∗}, E_{∗}) are of determinant class, then D_{∗}, Ẽ_{∗}, C̃_{∗} and LHS_{∗}(D_{∗}, Ẽ_{∗}, C̃_{∗}) are of determinant class and

ρ^{(2)}(C_{∗}) − ρ^{(2)}(D_{∗}) + ρ^{(2)}(E_{∗}) −ρ^{(2)}(C_{∗}, D_{∗}, E_{∗}) + ρ^{(2)}(LHS_{∗}(C_{∗}, D_{∗}, E_{∗}))

= −ρ^{(2)}(D_{∗}) + ρ^{(2)}(Ẽ_{∗}) − ρ^{(2)}(C̃_{∗})

+ρ^{(2)}(D_{∗}, Ẽ_{∗}, C̃_{∗}) − ρ^{(2)}(LHS_{∗}(D_{∗}, Ẽ_{∗}, C̃_{∗})).

Proof. The desired exact sequence is the canonical exact sequence 0 → D_{∗} → cyl_{∗}(q_{∗}) → cone_{∗}(q_{∗}) → 0. It remains to show the various claims.

Notice for any dim-finite Hilbert N (G)-chain complex F_{∗}, that cone_{∗}(F_{∗}) = cone_{∗}(id: F_{∗} → F_{∗}) is contractible and satisfies

ρ^{(2)}(cone_{∗}(F_{∗})) = 0.  (3.55)

This follows from the fact that

( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) : F_{p−1} ⊕ F_{p} → F_{p} ⊕ F_{p+1}

is an explicit chain contraction for cone_{∗}(F_{∗}). We have the canonical short exact sequences 0 → ΣC_{∗} ⟶^{j_{∗}} cone_{∗}(q_{∗}) → cone_{∗}(E_{∗}) → 0 and 0 → E_{∗} ⟶^{k_{∗}} cyl_{∗}(q_{∗}) → cone(D_{∗}) → 0. From Lemma 3.42 we obtain chain isomorphisms u_{∗}: ΣC_{∗} ⊕ cone_{∗}(E_{∗}) ⟶^{≅} cone_{∗}(q_{∗}) and v_{∗}: E_{∗} ⊕ cone(D_{∗}) ⟶^{≅} cyl_{∗}(q_{∗}) such that under the obvious identifications H_{p}^{(2)}(ΣC_{∗} ⊕ cone_{∗}(E_{∗})) = H_{p}(ΣC_{∗}) and H_{p}^{(2)}(E_{∗} ⊕ cone(D_{∗})) = H_{p}^{(2)}(E_{∗})

H_{p}^{(2)}(u_{∗}) = H_{p}^{(2)}(j_{∗});  (3.56)
H_{p}^{(2)}(v_{∗}) = H_{p}^{(2)}(k_{∗}).  (3.57)

Moreover, we conclude from (3.33)

ρ(ΣC_{∗}, cone_{∗}(q_{∗}), cone_{∗}(E_{∗})) = − ∑_{p∈ℤ}(−1)^{p} · ln(det(u_{p})); (3.58)
ρ(E_{∗}, cyl_{∗}(q_{∗}), cone(D_{∗})) = − ∑_{p∈ℤ}(−1)^{p} · ln(det(v_{p})). (3.59)

One easily checks using Theorem 3.14 (2)

ρ(ΣC_{∗}, cone_{∗}(q_{∗}), cone_{∗}(E_{∗})) = −ρ(C_{∗}, D_{∗}, E_{∗});  (3.60)
ρ(E_{∗}, cyl_{∗}(q_{∗}), cone(D_{∗})) = 0;  (3.61)
ρ^{(2)}(D_{∗}, cyl_{∗}(q_{∗}), cone_{∗}(q_{∗})) = 0.  (3.62)

We conclude from (3.58), (3.59), (3.60) and (3.61)

∑_{p∈ℤ}(−1)^{p} · ln(det(u_{p})) = ρ(C_{∗}, D_{∗}, E_{∗});  (3.63)
∑_{p∈ℤ}(−1)^{p} · ln(det(v_{p})) = 0.  (3.64)

From Lemma 3.44 and equations (3.55), (3.56) and (3.57) we conclude that cone_{∗}(q_{∗}) and cyl_{∗}(q_{∗}) respectively are of determinant class if and only if C_{∗} and E_{∗} respectively are of determinant class and in this case

ρ^{(2)}(cone_{∗}(q_{∗})) − ρ^{(2)}(ΣC_{∗})
= ∑_{p∈ℤ}(−1)^{p} · ln (det(u_{p})) − ∑_{p∈ℤ}(−1)^{p} · ln (det (H_{p}^{(2)}(j_{∗}))); (3.65)
ρ^{(2)}(cyl_{∗}(q_{∗})) − ρ^{(2)}(E_{∗})
= ∑_{p∈ℤ}(−1)^{p} · ln (det(v_{p})) − ∑_{p∈ℤ}(−1)^{p} · ln (det (H_{p}^{(2)}(k_{∗}))) . (3.66)

Recall that j_{∗} and k_{∗} induce isomorphisms H_{p}^{(2)}(ΣC_{∗}) ⟶^{≅} H_{p}^{(2)}(cone_{∗}(q_{∗})) and H_{p}^{(2)}(E_{∗}) ⟶^{≅} H_{p}^{(2)}(cyl_{∗}(q_{∗})) and we have the obvious isomorphisms id: H_{p}^{(2)}(D_{∗}) ⟶^{≅} H_{p}^{(2)}(D_{∗}). They induce a chain isomorphism

LHS_{∗}(C_{∗}, D_{∗}, E_{∗}) ⟶^{≅} Σ (LHS_{∗}(D_{∗}, cyl_{∗}(q_{∗}), cone_{∗}(q_{∗})))

and Lemma 3.44 shows

−ρ^{(2)}(LHS_{∗}(D_{∗}, cyl_{∗}(q_{∗}), cone_{∗}(q_{∗}))) − ρ^{(2)}(LHS_{∗}(C_{∗}, D_{∗}, E_{∗}))
= − ∑_{p∈ℤ}(−1)^{p} · ln (det (H_{p}^{(2)}(j_{∗})))
+ ∑_{p∈ℤ}(−1)^{p} · ln (det (H_{p}^{(2)}(k_{∗}))) . (3.67)

Now Lemma 3.54 follows from (3.62), (3.63), (3.64), (3.65), (3.66) and (3.67).

⊓ ⊔

The next lemma is the decisive step in the proof of Theorem 3.35 (1). It proves additivity and all other properties are consequences.

Lemma 3.68. Let C_{∗}, D_{∗} and E_{∗} be dim-finite Hilbert N (G)-chain complexes. Let 0 → C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{q_{∗}} E_{∗} → 0 be a weakly exact sequence of determinant class. Suppose that two of the chain complexes C_{∗}, D_{∗} and E_{∗} are weakly acyclic and of determinant class. Then all three are weakly acyclic and of determinant class and

ρ^{(2)}(C_{∗}) − ρ^{(2)}(D_{∗}) + ρ^{(2)}(E_{∗}) = ρ^{(2)}(C_{∗}, D_{∗}, E_{∗}).

Proof. The given exact sequence induces weak isomorphisms \overline{i}_{∗}: C_{∗} → ker(q_{∗}) and \overline{q}_{∗}: ker(q_{∗})^{⊥} → E_{∗} such that \overline{i_{p}} and \overline{q_{p}} are of determinant class for all p ∈ ℤ and

ρ^{(2)}(C_{∗}, D_{∗}, E_{∗}) = ∑_{p∈ℤ}(−1)^{p} · (− ln(det(\overline{i}_{p})) + ln(det(\overline{q}_{p}))) . (3.69)

From Lemma 3.44 we conclude that ker(q_{∗}) and ker(q_{∗})^{⊥} respectively are weakly acyclic of determinant class if and only if C_{∗} and E_{∗} respectively are weakly acyclic of determinant class and in this case we get

ρ^{(2)}(ker(q_{∗})) − ρ^{(2)}(C_{∗}) = ∑_{p∈ℤ}(−1)^{p} · ln(det(\overline{i}_{p}));  (3.70)
ρ^{(2)}(E_{∗}) − ρ^{(2)}(ker(q_{∗})^{⊥}) = ∑_{p∈ℤ}(−1)^{p} · ln(det(\overline{q}_{p}))  (3.71)

respectively. Because of (3.69), (3.70) and (3.71) it remains to show the claim for 0 → ker(q_{∗}) → D_{∗} → ker(q_{∗})^{⊥} → 0. Because of Lemma 3.54 it suffices to show under the assumption that ker(q_{∗}) and ker(q_{∗})^{⊥} are weakly acyclic and of determinant class that D_{∗} is weakly acyclic and of determinant class and

ρ^{(2)}(ker(q_{∗})) − ρ^{(2)}(D_{∗}) + ρ^{(2)}(ker(q_{∗})^{⊥}) = 0  (3.72)

holds.

We can write the differential d_{p} of D_{∗} by

d_{p} = ( \begin{matrix}d′_{p} & \overline{d}_{p} \\ 0 & d′′_{p} \end{matrix} ) : D_{p} = ker(q_{p}) ⊕ ker(q_{p})^{⊥} → D_{p−1} = ker(q_{p−1}) ⊕ ker(q_{p−1})^{⊥},

where d′_{p} and d′′_{p} are the differentials of ker(q_{∗}) and ker(q_{∗})^{⊥}. By Lemma 3.39 we can choose weak chain contractions (γ′_{∗}, u′_{∗}) and (γ′′_{∗}, u′′_{∗}) for ker(q_{∗}) and ker(q_{∗})^{⊥} such that u′_{p} and u′′_{p} are weak isomorphism of determinant class for p ∈ ℤ. Define a chain homotopy γ_{∗}: u_{∗} ≃ 0 for D_{∗} by

γ_{p} := ( \begin{matrix}γ′_{p} & 0 \\ 0 & γ′′_{p} \end{matrix} ) : D_{p} → D_{p+1};
u_{p} := ( \begin{matrix}u′_{p} & \overline{d}_{p+1} ∘ γ′′_{p} + γ′_{p−1} ∘ \overline{d}_{p} \\ 0 & u′′_{p} \end{matrix} ) : D_{p} → D_{p}

with respect to the orthogonal decomposition D_{p} = ker(q_{p}) ⊕ ker(q_{p})^{⊥}. We conclude from Theorem 3.14 (2), Lemma 3.37 (2) and Lemma 3.41 that (γ_{∗}, u_{∗}) is a weak chain contraction such that u_{p} is of determinant class for all p ∈ ℤ and that (3.72) holds. This finishes the proof of Lemma 3.68. ⊓ ⊔

Now we are ready to give the proof of Theorem 3.35.

Proof. (1) Step 1: If E_{∗} is weakly acyclic and of determinant class, then LHS_{∗} is of determinant class and assertion (1) is true.

We get from Lemma 3.39 and Lemma 3.42 a chain map s_{∗}: E_{∗} → D_{∗} such that q_{p} ∘ s_{p}: E_{p} → E_{p} and i_{p} ⊕ s_{p}: C_{p} ⊕ E_{p} → D_{p} are weak isomorphisms of determinant class for p ∈ ℤ. Because of Lemma 3.44 the induced map H_{p}^{(2)}(i_{∗} ⊕ s_{∗}) is a weak isomorphism of determinant class for p ∈ ℤ. Hence the long weakly exact homology sequence LHS_{∗} is of determinant class since H_{p}^{(2)}(E_{∗}) = 0 for p ∈ ℤ by assumption and in particular H_{p}^{(2)}(i_{∗} ⊕ s_{∗}) = H_{p}^{(2)}(i_{∗}). Provided that C_{∗} or D_{∗} is of determinant class, we conclude from Lemma 3.15 (7) and Lemma 3.44 that both C_{∗} and D_{∗} are of determinant class and

ρ^{(2)}(D_{∗}) − ρ^{(2)}(C_{∗}) − ρ^{(2)}(E_{∗}) = ∑_{p∈ℤ}(−1)^{p} · ln(det(i_{p} ⊕ s_{p}))
+ρ^{(2)}(LHS_{∗}(C_{∗}, D_{∗}, E_{∗})). (3.73)

Since E_{∗} is weakly acyclic and of determinant class by assumption and q_{∗} ∘ s_{∗}: E_{∗} → E_{∗} is a weak chain isomorphism such that each q_{p} ∘ s_{p} is a weak isomorphism of determinant class, Lemma 3.44 applied to q_{∗} ∘ s_{∗} shows

∑_{p∈ℤ}(−1)^{p} · ln(det(q_{p} ∘ s_{p})) = 0.

Hence (3.33) shows

ρ^{(2)}(C_{∗}, D_{∗}, E_{∗}) = − ∑_{p∈ℤ}(−1)^{p} · ln(det(i_{p} ⊕ s_{p})).  (3.74)

Now Step 1 follows from (3.73) and (3.74). Step 2: If one of the chain complexes C_{∗}, D_{∗} and E_{∗} is weakly acyclic and of determinant class, then LHS_{∗}(C_{∗}, D_{∗}, E_{∗}) is of determinant class and assertion (1) is true.

This follows from Lemma 3.54 and Step 1. Step 3: Assertion (1) is true provided that the differentials of C_{∗} and E_{∗} are trivial.

Since i_{∗} and q_{∗} induce isomorphisms C_{∗} → ker(q_{∗}) and ker(q_{∗}) → E_{∗}, (3.33) and Lemma 3.44 shows that we can assume without loss of generality that D_{p} = C_{p} ⊕ E_{p} and i_{p} and q_{p} are the obvious inclusions and projections for all p ∈ ℤ. Obviously C_{∗} and E_{∗} are of determinant class and satisfy

ρ^{(2)}(C_{∗}) = ρ^{(2)}(E_{∗}) = 0.  (3.75)

If we write

d_{p} = ( \begin{matrix}0 & x_{p} \\ 0 & 0 \end{matrix} ) : D_{p} = C_{p} ⊕ E_{p} → D_{p−1} = C_{p−1} ⊕ E_{p−1},

then one easily checks that the associated long weakly exact homology sequence looks like

… ⟶^{x_{p+1}} C_{p} ⟶^{j} C_{p} ⊕ ker(x_{p}) ⟶^{pr} E_{p} ⟶^{x_{p}} C_{p−1} ⟶^{j} …,

where j is the canonical inclusion onto the first factor and pr is induced by the projection onto the second factor. Hence LHS_{∗}(C_{∗}, D_{∗}, E_{∗}) is of determinant class if and only if x_{p} is of determinant class for all p ∈ ℤ and in this case

ρ^{(2)}(LHS_{∗}(C_{∗}, D_{∗}, E_{∗})) = − ∑_{p∈ℤ}(−1)^{p} · ln(det(x_{p})).  (3.76)

Lemma 3.15 (3) implies that D_{∗} is of determinant class if and only if x_{p} is of determinant class for all p ∈ ℤ and in this case

ρ^{(2)}(D_{∗}) = − ∑_{p∈ℤ}(−1)^{p} · ln(det(x_{p})).  (3.77)

Now the claim follows from (3.75), (3.76) and (3.77). Step 4: Assertion (1) is true provided that C_{∗} is of determinant class and the differentials of E_{∗} are trivial.

In the sequel we write \overline{C}_{p} = ker(Δ_{p}) and \overline{\overline{C}}_{p} = ker(Δ_{p})^{⊥}. Denote by k_{∗}: \overline{\overline{C}}_{∗} → C_{∗} the canonical inclusion and by pr_{∗}: C_{∗} → \overline{C}_{∗} the canonical projection. We conclude from Lemma 1.18 that we have an orthogonal decomposition of Hilbert N (G)-chain complexes C_{∗} = \overline{C}_{∗} ⊕\overline{\overline{C}}_{∗}, the differentials of \overline{C}_{∗} are all trivial and \overline{\overline{C}}_{∗} is weakly acyclic. From Lemma 3.15 (7) we conclude that \overline{\overline{C}}_{∗} is of determinant class and

ρ^{(2)}(\overline{\overline{C}}_{∗}) = ρ^{(2)}(C_{∗}).  (3.78)

Notice that the chain map pr_{∗} ⊕i_{∗}: C_{∗} → \overline{C}_{∗} ⊕ D_{∗} is injective and has closed image since this is true for i_{∗}. Define a dim-finite Hilbert N (G)-chain complex D̂_{∗} by the orthogonal complement in \overline{C}_{∗} ⊕ D_{∗} of the image of pr_{∗} ⊕i_{∗}: C_{∗} → \overline{C}_{∗} ⊕D_{∗}. We obtain a commutative diagram of dim-finite Hilbert N (G)-chain complexes with exact rows and columns

0  0  0
↓  ↓  ↓
0 ⟶ \overline{\overline{C}}_{∗} ⟶^{id} \overline{\overline{C}}_{∗} ⟶ 0 ⟶ 0
k_{∗}↓  ^{i_{∗}∘k_{∗}}↓  ↓
0 ⟶ C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{q_{∗}} E_{∗} ⟶ 0
pr^{∗}↓  p̂r_{∗}↓  ^{id}↓
0 ⟶ \overline{C}_{∗} \underrightarrow{î_{∗}} D̂_{∗} \underrightarrow{q̂_{∗}} E_{∗} ⟶ 0
↓  ↓  ↓
0  0  0

Since \overline{\overline{C}}_{∗} is weakly acyclic and of determinant class, Step 2 applied to the middle column shows that the induced map H_{p}^{(2)}(p̂r_{∗}): H_{p}^{(2)}(D_{∗}) → H_{p}^{(2)}(D̂_{∗}) is a weak isomorphism of determinant class for all p ∈ ℤ. Moreover, D_{∗} is of determinant class if and only if D̂_{∗} is of determinant class, and in this case

ρ^{(2)}(\overline{\overline{C}}_{∗}) − ρ^{(2)}(D_{∗}) + ρ^{(2)}(D̂_{∗})
= ρ^{(2)}(\overline{\overline{C}}_{∗}, D_{∗}, D̂_{∗}) − ∑_{p∈ℤ}(−1)^{p} · ln (det (H_{p}^{(2)}(p̂r_{∗}))) . (3.79)

The map from the middle row to the lower row induces a weak chain isomorphism from LHS_{∗}(C_{∗}, D_{∗}, E_{∗}) to LHS_{∗}(\overline{C}_{∗}, D̂_{∗}, E_{∗}) which is in each dimension of determinant class. Lemma 3.44 implies that LHS_{∗}(C_{∗}, D_{∗}, E_{∗}) is of determinant class if and only if LHS_{∗}(\overline{C}_{∗}, D̂_{∗}, E_{∗}) is of determinant class, and in this case

ρ^{(2)}(LHS_{∗}(\overline{C}_{∗}, D̂_{∗}, E_{∗})) − ρ^{(2)}(LHS_{∗}(C_{∗}, D_{∗}, E_{∗}))
= − ∑_{p∈ℤ}(−1)^{p} · ln (det (H_{p}^{(2)}(p̂r_{∗}))) . (3.80)

Step 3 applied to the lower row shows that D̂_{∗} is of determinant class if and only if LHS_{∗}(\overline{C}_{∗}, D̂_{∗}, E_{∗}) is of determinant class, and in this case

− ρ^{(2)}(D̂_{∗}) = ρ^{(2)}(\overline{C}_{∗}, D̂_{∗}, E_{∗}) − ρ^{(2)}(LHS_{∗}(\overline{C}_{∗}, D̂_{∗}, E_{∗})). (3.81)

One easily checks using Lemma 3.68

ρ^{(2)}(\overline{C}_{∗}, D̂_{∗}, E_{∗}) = ρ^{(2)}(C_{∗}, D_{∗}, E_{∗});  (3.82)
ρ^{(2)}(\overline{\overline{C}}_{∗}, D_{∗}, D̂_{∗}) = 0.  (3.83)

Now the claim follows from (3.78), (3.79), (3.80), (3.81), (3.82) and (3.83).

Step 5: Assertion (1) is true provided that C_{∗} and E_{∗} are of determinant class.

Define D̂_{∗} as the kernel of the chain map q_{∗} ⊕ k_{∗}: D_{∗} ⊕ \overline{E}_{∗} → E_{∗}. Then we obtain a commutative diagram with exact rows and columns

0  0  0
↓  ↓  ↓
0 ⟶ C_{∗} ⟶^{î_{∗}∘pr_{∗}} D̂_{∗} ⟶ \overline{E}_{∗} ⟶ 0
id↓  k̂_{∗}↓  ^{k_{∗}}↓
0 ⟶ C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{q_{∗}} E_{∗} ⟶ 0
↓  ^{pr_{∗} ∘q_{∗}}↓  ^{pr}^{∗}↓
0 ⟶ 0 ⟶ \overline{\overline{E}}_{∗} ⟶^{id} \overline{\overline{E}}_{∗} ⟶ 0
↓  ↓  ↓
0  0  0

Now we proceed analogously to Step 4 by applying Step 1 to the middle column and Step 4 to the upper row. Step 6: Assertion (1) is true.

This follows from Lemma 3.54 and Step 5. This finishes the proof of assertion (1) of Theorem 3.35.

(2) follows from Lemma 3.37 (3) and Lemma 3.68 applied to the induced exact sequence 0 → cone_{∗}(f_{∗}) → cone_{∗}(g_{∗}) → cone_{∗}(h_{∗}) → 0.

(3) Let γ_{∗}: f_{∗} ≃ g_{∗} be a chain homotopy. Consider the isomorphism of dimfinite N (G)-chain complexes u_{∗}: cone_{∗}(f_{∗}) → cone_{∗}(g_{∗}) given by

u_{p} = ( \begin{matrix}id & 0 \\ γ_{p−1} & id \end{matrix} ) : C_{p−1} ⊕ D_{p} → C_{p−1} ⊕ D_{p}.

We get from Lemma 3.37 (3) and Lemma 3.44 that both cone_{∗}(f_{∗}) and cone_{∗}(g_{∗}) are weakly acyclic and of determinant class and

t^{(2)}(g_{∗}) − t^{(2)}(f_{∗}) = ρ^{(2)}(cone_{∗}(g_{∗})) − ρ^{(2)}(cone_{∗}(f_{∗}))
= ∑_{p∈ℤ}(−1)^{p} · ln(det(u_{p}))
= 0.

(4) Consider the chain map h_{∗}: Σ^{−1} cone_{∗}(g_{∗}) → cone_{∗}(f_{∗}) given by

h_{p} = ( \begin{matrix}0 & 0 \\ − id & 0 \end{matrix} ) : D_{p} ⊕ E_{p+1} → C_{p−1} ⊕ D_{p}.

There are obvious exact sequences 0 → cone_{∗}(f_{∗}) → cone_{∗}(h_{∗}) → cone_{∗}(g_{∗}) → 0 and 0 → cone_{∗}(g_{∗} ∘ f_{∗}) ⟶^{i_{∗}} cone_{∗}(h_{∗}) → cone_{∗}(D_{∗}) → 0, where i_{∗} is given by

i_{p} := ( \begin{matrix}f_{p−1} & 0 \\ 0 & 1 \\ 1 & 0 \\ 0 & 0 \end{matrix} ) : C_{p−1} ⊕ D_{p} → D_{p−1} ⊕ E_{p} ⊕ C_{p−1} ⊕ D_{p}

and the other maps are the canonical ones. Because of Lemma 3.68 f_{∗}, h_{∗}, g_{∗} and g_{∗} ∘ f_{∗} are weak homology equivalences of determinant class. One easily checks

ρ^{(2)}(cone_{∗}(f_{∗}), cone_{∗}(h_{∗}), cone_{∗}(g_{∗})) = 0;
ρ^{(2)}(cone_{∗}(g_{∗} ∘ f_{∗}), cone_{∗}(h_{∗}), cone_{∗}(D_{∗})) = 0.

Hence we get from Lemma 3.68

ρ^{(2)}(cone_{∗}(f_{∗})) − ρ^{(2)}(cone_{∗}(h_{∗})) + ρ^{(2)}(cone_{∗}(g_{∗})) = 0; (3.84)
ρ^{(2)}(cone_{∗}(g_{∗} ∘ f_{∗})) − ρ^{(2)}(cone_{∗}(h_{∗})) + ρ^{(2)}(cone_{∗}(D_{∗})) = 0. (3.85)

Now assertion (4) follows from (3.55), (3.84) and (3.85).

(5) Consider H_{∗}^{(2)}(C_{∗}) and H_{∗}^{(2)}(D_{∗}) as chain complexes with the trivial differential. We get from Lemma 1.18 chain maps i_{∗}: H_{∗}^{(2)}(C_{∗}) → C_{∗} and j_{∗}: H_{∗}^{(2)}(D_{∗}) → D_{∗} such that i_{p} and j_{p} are isometric inclusions for p ∈ ℤ and in particular both i_{∗} and j_{∗} are of determinant class, H_{∗}^{(2)}(i_{∗}) = id and H_{∗}^{(2)}(j_{∗}) = id. Moreover, we get from the definitions and Lemma 3.15 (3) and (7)

ρ^{(2)}(C_{∗}) = t^{(2)}(i_{∗});  (3.86)
ρ^{(2)}(D_{∗}) = t^{(2)}(j_{∗}).  (3.87)

We obtain from Lemma 3.43 applied to u_{∗} = f_{∗} ∘ i_{∗} and v_{∗} = j_{∗} ∘ H_{∗}^{(2)}(f_{∗}) a chain map g_{∗}: D_{∗} → D_{∗} such that H_{p}^{(2)}(g_{∗}) = id, each g_{p} is a weak isomorphism of determinant class and g_{∗} ∘ f_{∗} ∘ i_{∗} and g_{∗} ∘ j_{∗} ∘ H_{∗}^{(2)}(f_{∗}) are chain homotopic. Because of assertion (3) and (4) f_{∗} is of determinant class if and only if H_{∗}^{(2)}(f_{∗}) is of determinant class and in this case

t^{(2)}(f_{∗}) + t^{(2)}(i_{∗}) = t^{(2)}(H_{∗}^{(2)}(f_{∗})) + t^{(2)}(j_{∗}).  (3.88)

Since t^{(2)}(H_{∗}^{(2)}(f_{∗})) = ∑_{p∈ℤ}(−1)^{p} · ln (det (H_{p}^{(2)}(f_{∗}))) holds by assertion (2), assertion (5) follows from (3.86), (3.87) and (3.88). (6a) Fix n_{0} with C_{p} = 0 for p < n_{0}. We use induction over n with C_{p} = 0 for p > n. For the induction beginning n = n_{0} we have to show for a finite dimensional Hilbert N (G)-module U and a weak isomorphism of determinant class f: V → W for finite dimensional N (H)-modules V and W that id ⊗f: U ⊗ V → U ⊗ W is a weak isomorphism of determinant class and

ln(det_{N (G×H)}(id ⊗f)) = dim(U) · ln(det_{N (H)}(f)).

This follows from the equation of spectral density functions

F_{N (G×H)}(id ⊗f) = dim(U) · F_{N (H)}(f),

which is a consequence of the equality id ⊗E_{λ}^{f^{∗}f} = E_{λ}^{(id ⊗f)^{∗}(id ⊗f)}. Next we explain the induction step from n to n + 1. Notice that Theorem 1.12 (2) implies

χ^{(2)}(C_{∗}) = ∑_{p∈ℤ}(−1)^{p} · dim(C_{p}).  (3.89)

Let C_{∗}|n be obtained from C_{∗} by truncating in dimensions > n and let (n + 1)[C_{∗}] be the chain complex whose (n + 1)-th chain module is C_{n+1} and whose other chain modules are all trivial. Then the induction step follows from Lemma 3.68 applied to the short exact sequence 0 → C_{∗}|n ⊗ D_{∗} → C_{∗} ⊗ D_{∗} → (n + 1)[C_{∗}] ⊗ D_{∗} → 0 and the induction beginning applied to (n + 1)[C_{∗}] ⊗ D_{∗} and the induction hypothesis applied to C_{∗}|n ⊗ D_{∗}. This finishes the proof of assertion (6a).

(6b) Recall that we have introduced \overline{C}_{∗} and \overline{\overline{C}}_{∗} before (see (3.78)). We conclude from (3.78) and assertion (6a)

ρ^{(2)}(C_{∗} ⊗ D_{∗}) = ρ^{(2)}(\overline{C}_{∗} ⊗ \overline{D}_{∗}) + ρ^{(2)}(\overline{C}_{∗} ⊗ \overline{\overline{D}}_{∗}) + ρ^{(2)}(\overline{\overline{C}}_{∗} ⊗ \overline{D}_{∗})
+ρ^{(2)}(\overline{\overline{C}}_{∗} ⊗ \overline{\overline{D}}_{∗})
= 0 + χ^{(2)}(C_{∗}) · ρ^{(2)}(D_{∗}) + χ^{(2)}(D_{∗}) · ρ^{(2)}(C_{∗}) + 0.

(6c) Since f_{∗} ⊗ g_{∗} = f_{∗} ⊗ id ∘ id ⊗g_{∗} holds, it suffices because of assertion (4) to show

t^{(2)}(id ⊗g_{∗}) = χ^{(2)}(C_{∗}) · t^{(2)}(g_{∗}).

This is done as above by induction over the dimension of C_{∗}.

(7) This follows from Theorem 3.14 (5).

(8) This follows from Theorem 3.14 (6). This finishes the proof of Theorem 3.35. ⊓ ⊔

Remark 3.90. Theorem 3.35 remains true if one replaces “of determinant class” everywhere by “with positive Novikov-Shubin invariant”. In view of Theorem 3.14 (4) it remains to check that the property “with positive Novikov-Shubin invariant” is inherited as claimed in all the lemmas and theorems. Notice that the proof of inheritance of the property “of determinant class” is a formal consequence of the following two facts: i.) given two weak isomorphisms f: U → V and g: V → W of finite dimensional Hilbert N (G)- modules, both f and g are of determinant class if and only if g ∘ f is of determinant class (see Theorem 3.14 (1)) and ii.) given a commutative diagram of maps of finite dimensional Hilbert N (G)-modules whose rows are weakly exact and of determinant class and whose vertical arrows are weak isomorphisms

0 ⟶ U_{1} ⟶^{i} U_{0} ⟶^{p} U_{2} ⟶ 0
f_{1}↓  ^{f_{0}}↓  ^{f_{2}}↓
0 ⟶ V_{1} ⟶^{j} V_{0} ⟶^{q} V_{2} ⟶ 0

then f_{0} is of determinant class if and only if both f_{1} and f_{2} are of determinant class (see Theorem 3.14 (2)). The corresponding statements i.) and ii.) remain true if one replaces “of determinant class” everywhere by “with positive Novikov-Shubin invariant” because of Lemma 2.14 and Lemma 2.15.

3.4 Cellular L^{2}-Torsion

In this section we introduce and study cellular L^{2}-torsion. Essentially we apply the material of Section 3.3 to the cellular L^{2}-chain complex of a finite free G-CW -complex. There are two interesting cases, the case where the L^{2}- homology vanishes and the case where the underlying space is a cocompact free proper G-manifold with a G-invariant Riemannian metric. Cellular L^{2}- torsion has been introduced in [93] and [345]. The definition of determinant class is taken from [84, Definition 4.1 on page 800].

3.4.1 Cellular L^{2}-Torsion in the Weakly-Acyclic Case

Definition 3.91 (L^{2}-torsion). Let X be a finite free G-CW -complex. We call it det-L^{2}-acyclic if its cellular L^{2}-chain complex C_{∗}^{(2)}(X) is det-L^{2}- acyclic (see Definition 3.29). In this case we define its cellularL^{2}-torsion

ρ^{(2)}(X; N (G)) := ρ^{(2)}(C_{∗}^{(2)}(X))

by the L^{2}-torsion of C_{∗}^{(2)}(X) (see Definition 3.29). Often we omit N (G) from the notation. ⊓ ⊔

3.4 Cellular L^{2}-Torsion 161

Since for two equivariant smooth triangulations f: K → M and g: L → M of a cocompact free proper G-manifold M the Whitehead torsion of g^{−1} ∘ f: K → L in Wh(G) is trivial, K is det-L^{2}-acyclic if and only if L is det-L^{2}- acyclic and in this case ρ^{(2)}(K) = ρ^{(2)}(L) by Theorem 3.93 (1). Hence we can define M to be det-L^{2}-acyclic if K is det-L^{2}-acyclic for some (and hence each) equivariant smooth triangulation and put in this case ρ^{(2)}(M) = ρ^{(2)}(K). The notions above extend in the obvious way to pairs.

We obtain a homomorphism

Φ = Φ^{G}: Wh(G) → ℝ  (3.92)

by sending the class of an invertible (n, n) matrix A over ℤG to the logarithm of the Fuglede-Kadison determinant det(R_{A}) (see Definition 3.11) of the morphism R_{A}: l^{2}(G)^{n} → l^{2}(G)^{n} induced by A. This is well-defined by Theorem 3.14. The next theorem presents the basic properties of L^{2}-torsion.

Theorem 3.93 (Cellular L^{2}-torsion). (1) Homotopy invariance Let f: X → Y be a G-homotopy equivalence of finite free G-CW - complexes. Let τ(f) ∈ Wh(G) be its Whitehead torsion (see (3.2)). Suppose that X or Y is det-L^{2}-acyclic. Then both X and Y are det-L^{2}-acyclic and

ρ^{(2)}(Y) − ρ^{(2)}(X) = Φ^{G}(τ(f));

(2) Sum formula Consider the G-pushout of finite free G-CW -complexes such that j_{1} is an inclusion of G-CW -complexes, j_{2} is cellular and X inherits its G-CW - complex structure from X_{0}, X_{1} and X_{2}

X_{0} ⟶^{j_{1}} X_{1}
j_{2}↓  ^{i_{1}}↓
X_{2} ⟶^{i_{2}} X

Assume that three of the G-CW -complexes X_{0}, X_{1}, X_{2} and X are det-L^{2}-acyclic. Then all four G-CW -complexes X_{0}, X_{1}, X_{2} and X are det-L^{2}-acyclic and

ρ^{(2)}(X) = ρ^{(2)}(X_{1}) + ρ^{(2)}(X_{2}) − ρ^{(2)}(X_{0});

(3) Poincaré duality Let M be a cocompact free proper G-manifold without boundary of even dimension which is orientable and det-L^{2}-acyclic. Then

ρ^{(2)}(M) = 0;

(4) Product formula Let X be a finite free G-CW -complex and let Y be a finite free H-CW - complex. Suppose that X is det-L^{2}-acyclic. Then the finite free G × H-CW -complex X × Y is det-L^{2}-acyclic and

ρ^{(2)}(X × Y, N (G × H)) = χ(H\Y) · ρ^{(2)}(X, N (G));

(5) Restriction Let X be a finite free G-CW -complex and let H ⊂ G be a subgroup of finite index [G : H]. Let res_{G}^{H} X be the finite H-CW -complex obtained from X by restricting the G-action to an H-action. Then X is det-L^{2}- acyclic if and only if res_{G}^{H} X is det-L^{2}-acyclic, and in this case

ρ^{(2)}(X; N (G)) = [G : H] · ρ^{(2)}(res_{G}^{H} X; N (H));

(6) Induction Let H be a subgroup of G and let X be a finite free H-CW -complex. Then the finite free G-CW -complex G ×_{H} X is det-L^{2}-acyclic if and only if X is det-L^{2}-acyclic, and in this case

ρ^{(2)}(G ×_{H} X; N (G)) = ρ^{(2)}(X; N (H));

(7) Positive Novikov-Shubin invariants and determinant class

If X is a finite free G-CW -complex with b_{p}^{(2)}(X) = 0 and α_{p}(X) > 0 for all p ≥ 0, then X is det-L^{2}-acyclic.

Proof. (1) This follows from Theorem 3.35 (5) and Lemma 3.41.

(2) We obtain an exact sequence of Hilbert N (G)-chain complexes 0 → C_{∗}^{(2)}(X_{0}) → C_{∗}^{(2)}(X_{1}) ⊕ C_{∗}^{(2)}(X_{2}) → C_{∗}^{(2)}(X) → 0. Now apply Theorem 3.35 (1).

(3) There is a subgroup G_{0} ⊂ G of index 1 or 2 which acts orientation preserving on M. Since ρ^{(2)}(M; N (G_{0})) = [G : G_{0}] · ρ^{(2)}(M; N (G_{0})) by assertion (5) we can assume without loss of generality that G = G_{0}, i.e. G\M is orientable. Fix an equivariant smooth triangulation f: K → M of M. Put π = π_{1}(K) and n = dim(M). Let [G\K] be the image of the fundamental class of [G\M] under the isomorphism H_{n}(G\M) → H_{n}(G\K) induced by G\f^{−1}. The Poincaré ℤG-chain homotopy equivalence

∩[G\K]: C^{n−∗}(K) → C_{∗}(K)

has trivial Whitehead torsion with respect to the cellular basis [510, Theorem 2.1 on page 23]. It induces a chain homotopy equivalence of finite Hilbert N (G)-chain complexes f_{∗}: l^{2}(G)⊗_{ℤG} C^{n−∗}(K) → C_{∗}^{(2)}(K) with t^{(2)}(f_{∗}) = 0. We get from Theorem 3.35 (5) and Lemma 3.41 that l^{2}(G) ⊗_{ℤG} C^{n−∗}(K) is det-L^{2}-acyclic and ρ^{(2)}(l^{2}(G) ⊗_{ℤG} C^{n−∗}(K)) = ρ^{(2)}(C_{∗}^{(2)}(K)). We conclude ρ^{(2)}(M) = (−1)^{n+1} · ρ^{(2)}(M) from Theorem 3.35 (5). Since n is even by assumption, assertion (3) follows.

(4) This follows from Theorem 3.35 (6a).

(5) This follows from Theorem 3.35 (7).

(6) This follows from Theorem 3.35 (8).

(7) This follows from Theorem 3.14 (4). This finishes the proof of Theorem 3.93. ⊓ ⊔

Part (1) and (2) of the following conjecture is taken from [330, Conjecture 1.5]. We will later prove it for a large class of groups (see Lemma 13.6 and Theorem 13.3).

Conjecture 3.94 (Homotopy invariance of L^{2}-torsion). We have for any group G:

(1) The homomorphism

Φ = Φ^{G}: Wh(G) → ℝ

sending the class [A] of an invertible matrix A ∈ GL_{n}(ℤG) to ln(det(r_{A}^{(2)})) (which we have already defined in (3.92)) is trivial;

(2) If X and Y are det-L^{2}-acyclic finite G-CW -complexes, which are Ghomotopy equivalent, then their L^{2}-torsion agree

ρ^{(2)}(X; N (G)) = ρ^{(2)}(Y; N (G));

(3) Let A ∈ M_{n}(ℤG) be a (n, n)-matrix over ℤG. Then r_{A}^{(2)} : l^{2}(G)^{n} → l^{2}(G)^{n} is of determinant class (see Definition 3.11).

Conjecture 3.94 (1) is obviously true if Wh(G) vanishes. There is the conjecture that the Whitehead group Wh(G) vanishes if G is torsionfree.

In most applications X will occur as the universal covering of a finite CW -complex. Therefore we will discuss this special case here. Since we also want to deal with non-connected CW -complexes, we introduce

Notation 3.95. Let X be a (not necessarily connected) finite CW -complex. We say that X̃ is det-L^{2}-acyclic, if the universal covering C̃ of each connected component C of X is det-L^{2}-acyclic in the sense of Definition 3.91. In this case we write

ρ^{(2)}(X̃) := ∑_{C∈π_{0}(X)} ρ^{(2)}(C̃),

where ρ^{(2)}(C̃) is the L^{2}-torsion of the finite free π_{1}(C)-CW -complex C̃ of Definition 3.91.

The next theorem presents the basic properties of ρ^{(2)}(X̃). It is a consequence of Theorem 3.93. Notice the formal analogy between the behaviour of ρ^{(2)}(X̃) and the ordinary Euler characteristic χ(X).

Theorem 3.96. (Cellular L^{2}-torsion for universal coverings).

(1) Homotopy invariance Let f: X → Y be a homotopy equivalence of finite CW -complexes. Let τ(f) ∈ Wh(π_{1}(Y)) be its Whitehead torsion. Suppose that X̃ or Ỹ is det-L^{2}-acyclic. Then both X̃ and Ỹ are det-L^{2}-acyclic and

ρ^{(2)}(Ỹ) − ρ^{(2)}(X̃) = Φ^{π_{1}(Y)}(τ(f)),

where Φ^{π_{1}(Y)} : Wh(π_{1}(Y)) = ⨁_{C∈π_{0}(Y)} Wh(π_{1}(C)) → ℝ is the sum of the maps Φ^{π_{1}(C)} of (3.92);

(2) Sum formula Consider the pushout of finite CW -complexes such that j_{1} is an inclusion of CW -complexes, j_{2} is cellular and X inherits its CW -complex structure from X_{0}, X_{1} and X_{2}

X_{0} ⟶^{j_{1}} X_{1}
j_{2}↓  ^{i_{1}}↓
X_{2} ⟶^{i_{2}} X

Assume X̃_{0}, X̃_{1}, and X̃_{2} are det-L^{2}-acyclic and that for k = 0, 1, 2 the map π_{1}(i_{k}): π_{1}(X_{k}) → π_{1}(X) induced by the obvious map i_{k}: X_{k} → X is injective for all base points in X_{k}. Then X̃ is det-L^{2}-acyclic and we get

ρ^{(2)}(X̃) = ρ^{(2)}(X̃_{1}) + ρ^{(2)}(X̃_{2}) − ρ^{(2)}(X̃_{0});

(3) Poincaré duality

Let M be a closed manifold of even dimension such that M̃ is det-L^{2}- acyclic. Then

ρ^{(2)}(M̃) = 0;

(4) Product formula

Let X and Y be finite CW -complexes. Suppose that X̃ is det-L^{2}-acyclic. Then { \atop X × Y} is det-L^{2}-acyclic and

ρ^{(2)}({ \atop X × Y}) = χ(Y) · ρ^{(2)}(X̃);

(5) Multiplicativity Let X → Y be a finite covering of finite CW -complexes with d sheets. Then X̃ is det-L^{2}-acyclic if and only if Ỹ is det-L^{2}-acyclic and in this case

ρ^{(2)}(X̃) = d · ρ^{(2)}(Ỹ);

(6) Positive Novikov-Shubin invariants and determinant class

If X is a finite CW -complex with b_{p}^{(2)}(X̃) = 0 and α_{p}(X̃) > 0 for all p ≥ 0, then X̃ is det-L^{2}-acyclic.

Next we want to deal with the behaviour of the cellular L^{2}-torsion under a fibration p: E → B with fiber F. We begin with introducing simple structures. A simple structure ξ = [(X, f)] on a topological space Z is an equivalence class of pairs (X, f) consisting of a finite CW -complex X and a homotopy equivalence f: X → Z, where we call two such pairs (X, f) and (Y, g) equivalent if the Whitehead torsion τ(g^{−1} ∘ f) vanishes, i.e. g^{−1} ∘ f is a simple homotopy equivalence. If g: Z_{1} → Z_{2} is a homotopy equivalence of topological spaces and we have specified simple structures ξ_{i} = [(X_{i}, f_{i})] on Z_{i}, we can still define the Whitehead torsion

τ(g) ∈ Wh(π_{1}(Z_{2}))

by the image of τ(f_{2}^{−1} ∘ g ∘ f_{1}) under the isomorphism f_{2∗}: Wh(π_{1}(X_{2})) → Wh(π_{1}(Z_{2})). If for some representative (X_{1}, f_{1}) of ξ_{1} and a given G-covering \overline{Z_{1}} → Z_{1} the total space of the pullback \overline{X_{1}} → X_{1} with f_{1} is det-L^{2}-acyclic with respect to the action of the group of deck transformations, then this is true for all representatives by Theorem 3.93 (1), and we say that (\overline{Z_{1}}, ξ_{1}) is det-L^{2}-acyclic. In this case we can still define the cellular L^{2}-torsion of \overline{Z_{1}} with respect to ξ_{1} by

ρ^{(2)}(\overline{Z_{1}}, ξ_{1}) := ρ^{(2)}(\overline{X_{1}}).  (3.97)

This is independent of the choice of the representative (X_{1}, f_{1}) by Theorem 3.93 (1).

Let E ⟶^{p} B be a fibration such that the fiber F has the homotopy type of a finite CW -complex and B is a connected finite CW -complex. Recall that F is only determined up to homotopy. We can associate to p an element

θ(p) ∈ H^{1}(B; Wh(π_{1}(E)))  (3.98)

by specifying a homomorphism π_{1}(B, b) → Wh(π_{1}(E)) for a fixed base point b ∈ B as follows. The fiber transport [488, 15.12 on page 343] of an element w ∈ π_{1}(B, b) determines a homotopy class of selfhomotopy equivalences t_{w}: F_{b} → F_{b} of F_{b} := p^{−1}(b). If we choose a simple structure ξ(F_{b}) on F_{b}, we can take the Whitehead torsion τ(t_{w}) in Wh(π_{1}(F_{b})) and push it forward to Wh(π_{1}(E)) using the map induced by the inclusion k_{b}: F_{b} → E. It turns out that the class of θ(p) is independent of the base point b ∈ B and the simple structure on F_{b}. We call p simple if θ(p) = 0.

Suppose that p is simple. If we fix a base point b ∈ B and a simple structure ξ(F_{b}) on F_{b}, there is a preferred simple structure ξ_{b,ξ(F_{b})}(E) on E. If we choose another base point b′ and simple structure ξ(F_{b′}) on F_{b′}, we obtain another simple structure on ξ_{b′,ξ(F_{b′})}(E). If w is any path in B from b′ to b, the fiber transport yields a homotopy class of homotopy equivalences t_{w}: F_{b} → F_{b′} and we get

τ (id: (E, ξ_{b,ξ(F_{b})}(E)) → (E, ξ_{b′,ξ(F_{b′})}(E)))
= χ(B) · k_{b′∗} (τ (t_{w}: (F_{b}, ξ(F_{b})) → (F_{b′}, ξ(F_{b′})))) . (3.99)

Notice that the right side is independent of the choice of w because of the assumption θ(p) = 0.

Details of the construction of ξ_{b,ξ(F_{b})}(E) and the claims above can be found in [325, section 2] or in the more general context of equivariant CW - complexes in [326, section 15]. It is based on the observation that over a cell e in B there is an up to fiber homotopy unique (strong) fiber homotopy equivalence from the restriction of E to the cell e to the trivial fibration e × F_{b} → e, provided we have fixed a homotopy class of paths from b to some point in e. The point is that after a choice of b ∈ B and simple structure ξ(F_{b}) on F_{b} we obtain uniquely a simple structure ξ_{b,ξ(F_{b})}(E) on E and ρ^{(2)}(Ẽ, ξ_{b,ξ(F_{b})}(E)) is defined, and we can ask for the relation of ρ^{(2)}(Ẽ, ξ_{b,ξ(F_{b})}(E)) and ρ^{(2)}(F̃_{b}, ξ(F_{b})).

Theorem 3.100 (L^{2}-torsion and fibrations). Let p: E → B be a fibration with θ(p) = 0 such that B is a connected finite CW -complex and the fiber has the homotopy type of a finite CW -complex. Suppose that the inclusion of F_{b} into E induces an injection on the fundamental groups for all base points in F_{b}. Fix b ∈ B and a simple structure ξ(F_{b}) on F_{b}. Suppose that F̃_{b} is det-L^{2}-acyclic.

Then E is det-L^{2}-acyclic and

ρ^{(2)}(Ẽ, ξ_{b,ξ(F_{b})}(E)) = χ(B) · ρ^{(2)}(F̃_{b}, ξ(F_{b})).

Before we give the proof of Theorem 3.100, we discuss some interesting special cases.

Example 3.101. If we make the additional assumption that χ(B) = 0, then E has a preferred simple structure ξ(E) independent of the choices of b ∈ B and ξ(F_{b}) because of (3.99) and Theorem 3.100 says ρ(Ẽ, ξ(E)) = 0.

Remark 3.102. Suppose for one (and hence) all base points b ∈ B that the

composition Wh(π_{1}(F_{b})) ⟶^{k_{b}} Wh(π_{1}(E)) ⟶^{Φ^{π_{1}(E)}} ℝ is trivial (cf. Conjecture 3.94 (1)). Then ρ^{(2)}(Ẽ, ξ_{b,ξ(F_{b})}(E)) and χ(B)·ρ^{(2)}(F̃_{b}, ξ(F_{b})) are independent of the choice of b ∈ B and ξ(F_{b}) by Theorem 3.96 (1).

Let F → E ⟶^{p} B be a (locally trivial) fiber bundle of finite CW -complexes with connected base space B. Then θ(p) = 0 and ρ^{(2)}(F̃) and ρ^{(2)}(Ẽ) are independent of the choice of a finite CW -structure on F and E since the Whitehead torsion of a homeomorphism of finite CW -complexes is always trivial [99], [100] and Theorem 3.96 (1) holds. It turns out that ξ_{b,ξ(F_{b})}(E) is the simple structure on E given by any finite CW -structure if ξ(F_{b}) is the simple structure on F_{b} given by any finite CW -structure. Hence Theorem 3.100 yields

Corollary 3.103. Suppose that F → E ⟶^{p} B is a (locally trivial) fiber bundle of finite CW -complexes with connected B. Suppose that for one (and hence all) b ∈ B the inclusion of F_{b} into E induces an injection on the fundamental groups for all base points in F_{b} and F̃_{b} is det-L^{2}-acyclic. Then Ẽ is det-L^{2}-acyclic and

ρ^{(2)}(Ẽ) = χ(B) · ρ^{(2)}(F̃).

Next we give the proof of Theorem 3.100. Theorem 3.93 (6) implies that Ẽ|_{F_{b}} is det-L^{2}-acyclic if and only if F̃_{b} is det-L^{2}-acyclic and in this case ρ^{(2)}(Ẽ|_{F_{b}}) = ρ(F̃_{b}) since by assumption the inclusion of F into E induces an injection on the fundamental groups for all base points in F_{b}. Hence Theorem 3.100 is the special case \overline{E} = Ẽ of the following slightly stronger statement.

Lemma 3.104. Let p: E → B be a fibration with θ(p) = 0 such that B is a connected finite CW -complex and the fiber has the homotopy type of a finite CW -complex. Let q: \overline{E} → E be a G-covering and let \overline{F_{b}} → F_{b} be its restriction to F_{b} for some fixed b ∈ B. Suppose that \overline{F_{b}} is det-L^{2}-acyclic. Then \overline{E} is det-L^{2}-acyclic and

ρ^{(2)}(\overline{E}, ξ_{b,ξ(F_{b})}(E)) = χ(B) · ρ^{(2)}(\overline{F_{b}}, ξ(F_{b})).

Proof. We use induction over the number of cells of B. The induction beginning B = ∅ is trivial. We have to deal with the situation that B is obtained from B_{0} by attaching a cell, i.e. there is a pushout

S^{n−1} ⟶^{q} B_{0}
i↓  ^{j}↓
D^{n} ⟶^{Q} B

The pullback construction applied to p ∘ q: \overline{E} → B yields a G-pushout

\overline{E_{S}} ⟶^{\overline{q}} \overline{E_{0}}
\overline{i}↓  \overline{j}↓
\overline{E_{D}} ⟶^{\overline{Q}} \overline{E}

Let Y be a finite CW -complex and g: Y → F_{b} be a homotopy equivalence representing the given simple structure ξ(F_{b}) on F_{b}. The pullback of q: \overline{E} → E with g yields a G-covering \overline{Y} → Y. By inspecting the construction of the simple structure ξ_{b,ξ(F_{b})}(E) on E and applying the induction hypothesis to \overline{E_{0}}|_{C} → C for each component C of B_{0} taking (3.99) into account, one checks that there is a commutative square of finite G-CW -complexes

\overline{Y} × S^{n−1} ⟶^{j_{1}} \overline{X_{0}}
j_{2}↓  ^{i_{1}}↓
\overline{Y} × D^{n} ⟶^{i_{2}} \overline{X}

such that j_{1} is an inclusion of G-CW -complexes, j_{2} is cellular and \overline{X} inherits its G-CW -complex structure from \overline{Y} × S^{n−1}, \overline{Y} × D^{n} and \overline{X_{0}}, the finite G-CW -complexes \overline{Y} × S^{n−1}, \overline{Y} × D^{n} and \overline{X_{0}} are det-L^{2}-acyclic and

ρ^{(2)}(\overline{X_{0}}) = χ(B_{0}) · ρ^{(2)}(\overline{F_{b}}, ξ(F_{b}));
ρ^{(2)}(\overline{E}, ξ_{b,ξ(F_{b})}(E)) = ρ^{(2)}(\overline{X}).

Theorem 3.93 (2) and (4) imply that \overline{X} is det-L^{2}-acyclic and

ρ^{(2)}(\overline{E}, ξ_{b,ξ(F_{b})}(E)) = ρ^{(2)}(\overline{X})
= ρ^{(2)}(\overline{X_{0}}) + ρ^{(2)}(\overline{Y} × D^{n}) − ρ^{(2)}(\overline{Y} × S^{n−1})
= ρ^{(2)}(\overline{X_{0}}) + χ(D^{n}) · ρ^{(2)}(\overline{Y}) − χ(S^{n−1}) · ρ^{(2)}(\overline{Y})
= χ(B_{0}) · ρ^{(2)}(\overline{F_{b}}, ξ(F_{b})) + χ(D^{n}) · ρ^{(2)}(\overline{F_{b}}, ξ(F_{b}))

− χ(S^{n−1}) · ρ^{(2)}(\overline{F_{b}}, ξ(F_{b}))

= χ(B) · ρ^{(2)}(\overline{F_{b}}, ξ(F_{b})).  ⊓ ⊔

Theorem 3.105 (L^{2}-torsion and S^{1}-actions). Let X be a connected S^{1}- CW -complex of finite type. Suppose that for one orbit S^{1}/H (and hence for all orbits) the inclusion into X induces a map on π_{1} with infinite image. (In particular the S^{1}-action has no fixed points.) Let X̃ be the universal covering of X with the canonical π_{1}(X)-action. Then X̃ is det-L^{2}-acyclic and

α_{p}(X̃) ≥ 1  for all p;
ρ^{(2)}(X̃) = 0.

Proof. Theorem 2.61 shows that b_{p}^{(2)}(X̃) = 0 for p ≥ 0 and α_{p}(X̃) ≥ 1 for p ≥ 1. In particular X̃ is det-L^{2}-acyclic by Theorem 3.96 (6). The proof of ρ^{(2)}(X̃) = 0 is analogous to the one of Theorem 1.40 using Theorem 3.93 (2) and (4) and the conclusion ρ^{(2)}(S̃^{1}) = 0 from (3.24) appearing in Example 3.22. ⊓ ⊔

Next we deal with the mapping torus T_{f} of a selfmap f: X → X of a connected finite CW -complex. If p: T_{f} → S^{1} is the canonical projection,

let π_{1}(T_{f}) ⟶^{ϕ} G ⟶^{ψ} ℤ be a factorization of the epimorphism π_{1}(T_{f}) ⟶^{π_{1}(p)} π_{1}(S^{1}) = ℤ into epimorphisms ϕ and ψ. If i: X → T_{f} is the obvious inclusion,

let L ⊂ G be the image of the composition π_{1}(X) ⟶^{π_{1}(i)} π_{1}(T_{f}) ⟶^{ϕ} G. Let \overline{T_{f}} be the covering of T_{f} associated to ϕ which is a free G-CW -complex. Denote by \overline{X} → X the L-covering of X associated to the epimorphism ι := ϕ ∘ π_{1}(i): π_{1}(X) → L. There is an automorphism μ: L → L uniquely determined by the property that ι∘π_{1}(f) = μ∘ι. Then G is the semidirect product L⋊_{μ}ℤ. Let \overline{f}: \overline{X} → \overline{X} be a (μ : L → L)-equivariant lift of f. Then \overline{T_{f}} is the mapping telescope of \overline{f} infinite to both sides, i.e., the identification space

\overline{T_{f}} = ∐_{n∈ℤ} \overline{X} × [n, n + 1]/ ∼,

where the identification ∼ is given by (x, n + 1) ∼ (\overline{f}(x), n). The group of deck transformations G is the semidirect product L ⋊_{μ} ℤ and acts in the obvious way.

Let j: L → G be the inclusion. We obtain a ℤG-chain map, a Hilbert N (G)-chain map and a morphism of Hilbert N (G)-modules

\overline{C_{∗}(\overline{f})}_{∗}: ℤG ⊗_{ℤL} C_{∗}(\overline{X}) → ℤG ⊗_{ℤL} C_{∗}(\overline{X});
\overline{C_{∗}(\overline{f})}_{∗}^{(2)} : l^{2}(G) ⊗_{ℤL} C_{∗}(\overline{X}) → l^{2}(G) ⊗_{ℤL} C_{∗}(\overline{X});
\overline{H_{p}^{(2)}(\overline{f})}: j_{∗}H_{p}^{(2)}(\overline{X}) = \overline{ℂG ⊗_{ℂL} H_{p}^{(2)}(\overline{X})} → j_{∗}H_{p}^{(2)}(\overline{X}) = \overline{ℂG ⊗_{ℂL} H_{p}^{(2)}(\overline{X})}

by sending g ⊗ u to g ⊗ u − gt ⊗ C_{∗}(\overline{f})(u) or g ⊗ u − gt ⊗ H_{p}^{(2)}(\overline{f})(u) respectively. Then the cellular ℤG-chain complex C_{∗}(T̃_{f}) is the mapping cone cone_{∗}(\overline{C_{∗}(\overline{f})}_{∗}). Under the obvious identification of j_{∗}H_{p}^{(2)}(\overline{X}) with

H_{p}^{(2)}(l^{2}(G) ⊗_{ℤL} C_{∗}(\overline{X})) the map \overline{H_{p}^{(2)}(\overline{f})} becomes the endomorphism of

H_{p}^{(2)}(l^{2}(G) ⊗_{ℤL} C_{∗}(\overline{X})) induced by \overline{C_{∗}(\overline{f})}_{∗}^{(2)}.

Theorem 3.106 (L^{2}-torsion of mapping tori). Let f: X → X and ϕ, ψ be given as above. Suppose that the G-CW -complex \overline{T_{f}} is of determinant class. Then \overline{T_{f}} is det-L^{2}-acyclic, for any p ≥ 0 the endomorphism of finitely

generated Hilbert N (π_{1}(T_{f}))-modules \overline{H_{p}^{(2)}(\overline{f})}: j_{∗}H_{p}^{(2)}(\overline{X}) → j_{∗}H_{p}^{(2)}(\overline{X}) is a weak isomorphism of determinant class and

ρ^{(2)}(T̃_{f}) = ∑_{p≥0}(−1)^{p} · ln (det (\overline{H_{p}^{(2)}(\overline{f})})) .

Proof. We conclude from Theorem 1.39 that H_{p}^{(2)}(T̃_{f}) = H_{p}^{(2)}(\overline{C_{∗}(\overline{f})}_{∗}^{(2)}) vanishes for all p ≥ 0. Hence the long weakly exact homology sequences associated to 0 → l^{2}(G) ⊗_{ℤL} C_{∗}(\overline{X}) → cone_{∗}(\overline{f}_{∗}^{(2)}) → Σ l^{2}(G) ⊗_{ℤL} C_{∗}(\overline{X}) → 0 looks like

… → 0 → H_{p}^{(2)}(l^{2}(G) ⊗_{ℤL} C_{∗}(\overline{X}))

⟶^{H_{p}^{(2)}(\overline{C_{∗}(\overline{f})}_{∗}^{(2)})} H_{p}^{(2)}(l^{2}(G) ⊗_{ℤL} C_{∗}(\overline{X})) → 0 → …

Now Theorem 3.35 (1) implies that H_{p}^{(2)}(\overline{C_{∗}(\overline{f})}_{∗}^{(2)}) is a weak isomorphism of determinant class and

ρ^{(2)}(T̃_{f}) = ∑_{p≥0}(−1)^{p} · ln (det (H_{p}^{(2)}(\overline{C_{∗}(\overline{f})}_{∗}^{(2)}))) .

Since H_{p}^{(2)}(\overline{C_{∗}(\overline{f})}_{∗}^{(2)}) and \overline{H_{p}^{(2)}(\overline{f})} are conjugated by an isomorphism, the claim follows from Theorem 3.14 (1). ⊓ ⊔

The assumption in Theorem 3.106 that \overline{T_{f}} is of determinant class is for instance satisfied if ϕ: π_{1}(T_{f}) → G is bijective and π_{1}(X) belongs to the class G (see Definition 13.9, Lemma 13.6, Lemma 13.11 (4) and Theorem 13.3) because π_{1}(T_{f}) is the mapping torus group of the endomorphism of π_{1}(X) induced by f appearing in Lemma 13.11 (4).

Example 3.107. Let f: X → X be a selfmap of a connected finite CW - complex. Let \overline{T_{f}} → T_{f} be the canonical infinite cyclic covering of the mapping torus. It is the covering associated to the canonical epimorphism π_{1}(T_{f}) → ℤ or, equivalently, the pullback of the universal covering of S^{1} with respect to the canonical map T_{f} → S^{1}. Let H_{p}(f; ℂ): H_{p}(X; ℂ) → H_{p}(X; ℂ) be the endomorphism of a finite dimensional complex vector space induced by f. By the Jordan Normal Form Theorem it is conjugated by an isomorphism H_{p}(X; ℂ) → ℂ^{n} to an automorphism of ℂ^{n} which is a direct sum of automorphisms of the form

B(λ_{i_{p}}, n_{i_{p}}) = ( \begin{matrix}λ_{i_{p}} & 1 & 0 & … & 0 \\ 0 & λ_{i_{p}} & 1 & … & 0 \\ 0 & 0 & λ_{i_{p}} & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & λ_{i_{p}} \end{matrix} )

where the size of the block B(λ_{i_{p}}, n_{i_{p}}) is n_{i_{p}}, λ_{i_{p}} ∈ ℂ and i_{p} = 1, 2 …, r_{p}. One easily checks that then H_{p}(\overline{T_{f}}; ℂ) = ⨁_{i_{p}=1}^{r_{p}} ℂ[ℤ]/((t − λ_{i_{p}})^{n_{ip}}). We conclude from Lemma 2.58 that \overline{T_{f}} is of determinant class and

b_{p}^{(2)}(\overline{T_{f}}; N (G)) = 0;  (3.108)
α_{p+1}(\overline{T_{f}}; N (G)) = max \{ \frac{1}{n_{i_{p}}} | i_{p} ∈ \{1, 2 …, r_{p}\}, |λ_{i_{p}}| = 1\} , (3.109)

where the maximum over the empty set is defined to be ∞^{+}. In the notation of Theorem 3.106 we see that \overline{H_{p}^{(2)}(\overline{f})} is conjugate to a direct sum of automorphisms of the shape

( \begin{matrix}1 − λ_{i_{p}}t & −t & 0 & … & 0 \\ 0 & 1 − λ_{i_{p}}t & −t & … & 0 \\ 0 & 0 & 1 − λ_{i_{p}}t & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & 1 − λ_{i_{p}}t \end{matrix} ) : l^{2}(ℤ)^{n_{ip}} → l^{2}(ℤ)^{n_{ip}} .

We conclude from Theorem 3.14 (2), (3.24) appearing in Example 3.22 and Theorem 3.106

ρ^{(2)}(\overline{T_{f}}; N (G)) = ∑_{p≥0}(−1)^{p} · (  ∑_{1≤i_{p}≤r_{p},|λ_{ip} |>1} n_{i_{p}} · ln(|λ_{i_{p}}|)) .

Example 3.110. We construct a 4-dimensional non-orientable cocompact free proper ℤ-manifold M without boundary such that M is det-L^{2}-acyclic, α_{2}(M; N (ℤ)) ≠ α_{3}(M; N (ℤ)) and ρ^{(2)}(M; N (ℤ)) ≠ 0. This shows that the condition that M is orientable is necessary in the statements about Poincaré duality in Theorem 2.55 (2) and Theorem 3.93 (3).

Equip T^{3} with the orientation reversing free ℤ/2-action which sends (z_{1}, z_{2}, z_{3}) ∈ S^{1} × S^{1} × S^{1} to (\overline{z_{1}}, −z_{2}, −z_{3}), where we think of S^{1} ⊂ ℂ and \overline{z_{1}} is the complex conjugate. The map f: T^{2} ⟶^{≅} T^{2}, which sends (z_{2}, z_{3}) to

(z_{2}^{4}z_{3}^{7}, z_{2}z_{3}^{2}), is an automorphism, since the integral (2, 2)-matrix A = ( \begin{matrix}4 & 7 \\ 1 & 2 \end{matrix} )

has determinant 1. The automorphism id ×f: T^{3} → T^{3} is ℤ/2-equivariant and hence induces an automorphism g: (ℤ/2)\T^{3} ⟶^{≅} (ℤ/2)\T^{3} of the nonorientable closed 3-manifold (ℤ/2)\T^{3}. Let M be \overline{T_{g}} for the infinite cyclic covering \overline{T_{g}} → T_{g} associated to the canonical epimorphism π_{1}(T_{f}) → ℤ. Obviously M is a 4-dimensional non-orientable cocompact free proper ℤ-manifold without boundary. Let h: T^{2} → M be the composition of the map T^{2} → T^{3} sending (z_{2}, z_{3}) to (1, z_{2}, z_{3}) and the projection T^{3} → (ℤ/2)\T^{3}. Then the following diagram commutes and has isomorphisms as vertical maps

H_{p}(T^{2}; ℂ) ⟶^{H_{p}(f;ℂ)} H_{p}(T^{2}; ℂ)
H_{p}(h;ℂ)↓  ^{H_{p}(h;ℂ)}↓
H_{p}((ℤ/2)\T^{3}; ℂ) ⟶^{H_{p}(g;ℂ)} H_{p}((ℤ/2)\T^{3}; ℂ)

The vertical maps are isomorphisms by the following argument. The trivial ℂ[ℤ/2]-module ℂ is ℂ[ℤ/2]-projective and hence H_{p}((ℤ/2)\T^{3}; ℂ) = H_{p}(T^{3}; ℂ)⊗_{ℂ[ℤ/2]}ℂ holds. The map S^{1} to S^{1} sending z to \overline{z} or −z respectively induces − id or id respectively on H_{1}(S^{1}; ℂ) and id on H_{0}(S^{1}; ℂ). Now apply the Künneth formula. The endomorphism H_{p}(f; ℂ) is the identity on ℂ for p = 0, 2 and is the automorphism of ℂ^{2} given by the matrix A for p = 1. Since the complex eigenvalues of the matrix A are 3 − \sqrt{8} and 3 + \sqrt{8}, we conclude from Example 3.107

α_{p}(M; N (ℤ)) = 1  for p = 1, 3;
α_{p}(M; N (ℤ)) = ∞^{+}  otherwise;
ρ^{(2)}(M; N (ℤ)) = − ln(3 + \sqrt{8}).

3.4.2 Cellular L^{2}-Torsion in the Weakly-Acyclic and Aspherical Case

Theorem 3.111. Let M be an aspherical closed manifold with non-trivial S^{1}-action. Then the action has no fixed points and the inclusion of any orbit into M induces an injection on the fundamental groups. Moreover, M̃ is det-L^{2}-acyclic and

α_{p}(M̃) ≥ 1  for all p;
ρ^{(2)}(M̃) = 0.

Proof. This follows from Corollary 1.43 and Theorem 3.105. ⊓ ⊔

Definition 3.112 (det ≥ 1-class). A group G is of det ≥ 1-class if for any A ∈ M(m, n, ℤG) the Fuglede-Kadison determinant (see Definition 3.11) of the induced morphism r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n} satisfies

det(r_{A}^{(2)}) ≥ 1.

Schick [462] uses the phrase “has semi-integral determinant” instead of the phrase “of det ≥ 1-class” which we prefer. There is no group G known which is not of det ≥ 1-class. We will later present in Subsection 13.1.3 a class of groups for which it is known that they are of det ≥ 1-class. It includes amenable groups and countable residually finite groups. The assertion for the L^{2}-torsion in the theorem below is the main result of [515] (see also [516]). Its proof is interesting as it preshadows a more ring theoretic approach to L^{2}-Betti numbers, which we will present in Chapter 6, and localization techniques for non-commutative rings, which will play a role in Chapter 8 and Chapter 10.

Theorem 3.113 (L^{2}-torsion and aspherical CW -complexes). Let X be an aspherical finite CW -complex. Suppose that its fundamental group π_{1}(X) contains an elementary amenable infinite normal subgroup H and π_{1}(X) is of det ≥ 1-class. Then

b_{p}^{(2)}(X̃) = 0 for p ≥ 0;
α_{p}(X̃) ≥ 1 for p ≥ 1;
ρ^{(2)}(X̃) = 0.

The claims for the L^{2}-Betti numbers and Novikov-Shubin invariants have already been proved in Theorem 2.63. We conclude from Theorem 3.93 (7) that X̃ is det-L^{2}-acyclic. The proof for the claim about the L^{2}-torsion needs some preparation.

Lemma 3.114. Let G be a group which is of det ≥ 1-class. Let Y be a finite free G-CW -complex which is det-L^{2}-acyclic. Let S ⊂ ℤG be a multiplicatively closed subset such that for any s ∈ S the induced morphism r_{s}^{(2)} : l^{2}(G) → l^{2}(G) sending u to us is a weak isomorphism and its Fuglede-Kadison determinant satisfies det(r_{s}^{(2)}) = 1. Assume that (ℤG, S) satisfies the Ore condition (see Definition 8.14). Suppose that the Ore localization (see Definition 8.14) S^{−1}H_{p}(X) vanishes for all p ≥ 0. Then

ρ^{(2)}(Y; N (G)) = 0.

Proof. Let f: N (G)^{m} → N (G)^{n} be an N (G)-homomorphism of (left) N (G)- modules. Choose a (m, n)-matrix A ∈ M(m, n, N (G)) such that f sends x to xA. Define

ν(f): l^{2}(G)^{m} → l^{2}(G)^{n}, y ↦ (A^{∗}y^{t})^{t}

where y^{t} is obtained from y by transposing and applying elementwise the involution l^{2}(G) → l^{2}(G) which sends ∑_{g∈G} λ_{g} · g to ∑_{g∈G} \overline{λ_{g}} · g, the matrix A^{∗} is obtained from A by transposing and applying the involution ∗: N (G) → N (G) to each entry, and Ay^{t} is understood in the sense of matrices and plugging y_{j} into an element a: l^{2}(G) → l^{2}(G) in N (G). Notice that ν(g∘f) = ν(g) ∘ ν(f) and that ν is compatible with direct sums and, more generally, with block decompositions of matrices. Moreover ν(λ · f) = λ · ν(f) for λ ∈ ℂ and ν(f + g) = ν(f) + ν(g). (This construction will be analysed further and be extended to finitely generated projective N (G)-modules in Section 6.2).

Given a ℤG-homomorphism f: ℤG^{m} → ℤG^{n}, define det_{ℤG}(f) ∈ [0, ∞) by the Fuglede-Kadison determinant of ν(id_{N (G)} ⊗_{ℤG}f): l^{2}(G)^{m} → l^{2}(G)^{n}. We call f a weak isomorphism or of determinant class if ν(id_{N (G)} ⊗_{ℤG}f) has this property. A based free finite ℤG-chain complex C_{∗} is called det-L^{2}-acyclic, if ν(N (G) ⊗_{ℤG} C_{∗}) is det-L^{2}-acyclic, and in this case we define ρ^{(2)}(C_{∗}) by ρ^{(2)}(ν(N (G) ⊗_{ℤG} C_{∗})). The point is that these notions can be extended to S^{−1}ℤG-modules and S^{−1}ℤG-chain complexes as follows.

Let f: S^{−1}ℤG^{m} → S^{−1}ℤG^{n} be a S^{−1}ℤG-homomorphism. Then there exist elements s_{1}, s_{2} ∈ S such that r_{s_{1}} ∘ f ∘ r_{s_{2}} maps ℤG^{m} ⊂ S^{−1}ℤG^{m} to ℤG^{n} ⊂ S^{−1}ℤG^{n}, where r_{s_{1}} and r_{s_{2}} are given by right multiplication with s_{1} and s_{2}. (It is possible to choose s_{1} = 1 or s_{2} = 1.) This follows from the fact that any element in S^{−1}ℤG can be written as us^{−1} or t^{−1}v for u, v ∈ ℤG and s, t ∈ S. Fix such elements s_{1} and s_{2}. We say that f is a weak isomorphism resp. is of determinant class if the ℤG-homomorphism r_{s_{1}} ∘ f ∘ r_{s_{2}} : ℤG^{m} → ℤG^{n} has this property. If f is a weak isomorphism, we define

det_{S^{−1}ℤG}(f) := det_{ℤG}(r_{s_{1}} ∘ f ∘ r_{s_{2}})  ∈ [0, ∞).  (3.115)

We conclude from Theorem 3.14 (1) that this is independent of the choice of s_{1} and s_{2} and assertions (1) and (2) of Theorem 3.14 for the Fuglede-Kadison determinant carry over to both det_{ℤG} and det_{S^{−1}ℤG}. Moreover, we get

det_{S^{−1}ℤG}(f) ≥ 1  (3.116)

from the following argument. Choose s ∈ S such that r_{s} ∘ f = S^{−1}g holds for an appropriate ℤG-homomorphisms g: ℤG^{m} → ℤG^{n}. Since G is of det ≥ 1class by assumption, we conclude from the definition (3.115)

det_{S^{−1}ℤG}(f) = det_{ℤG}(g) ≥ 1.

We call a based free finite S^{−1}ℤG-chain complex D_{∗} det-L^{2}-acyclic if there is a weak chain contraction (δ_{∗}, v_{∗}) in the sense that δ_{∗} is a S^{−1}ℤG-chain homotopy δ_{∗}: v_{∗} ≃ 0 satisfying δ_{∗} ∘ v_{∗} = v_{∗} ∘ δ_{∗} and v_{p}: D_{p} → D_{p} is a weak S^{−1}ℤG-isomorphism of determinant class for all p ∈ ℤ (cf. Definition 3.38). In this case we define the L^{2}-torsion of D_{∗}

ρ^{(2)}(D_{∗}) := ln (det_{S^{−1}ℤG} (vd + δ)_{odd})) − ln (det_{S^{−1}ℤG} (v_{odd})) ∈ ℝ.

This is independent of the choice of (δ_{∗}, v_{∗}) by the same argument as in Subsection 3.3.2 (cf. Definition 3.38, Lemma 3.40 and Lemma 3.41). If E_{∗} is a det-L^{2}-acyclic based free finite ℤG-chain complex, then S^{−1}E_{∗} is a det-L^{2}- acyclic based free finite S^{−1}ℤG-chain complex and ρ^{(2)}(E_{∗}) = ρ^{(2)}(S^{−1}E_{∗}).

Now suppose that C_{∗} is a based free finite ℤG-chain complex such that S^{−1}H_{p}(C_{∗}) = 0 for p ∈ ℤ and C_{∗} is det-L^{2}-acyclic. Lemma 8.15 (3) implies that H_{p}(S^{−1}C_{∗}) = 0 for p ∈ ℤ. Hence we can find a chain contraction γ_{∗}: S^{−1}C_{∗} → S^{−1}C_{∗+1}. Both compositions of the maps (S^{−1}c + γ)_{odd} : S^{−1}C_{odd} → S^{−1}C_{ev} and (S^{−1}c + γ)_{ev} : S^{−1}C_{ev} → S^{−1}C_{odd} are given by upper triangular automorphisms of S^{−1}C_{odd} or S^{−1}C_{ev} with identity maps on the diagonal (cf. Lemma 3.40). Hence both maps are isomorphisms and satisfy

det_{S^{−1}ℤG} ((S^{−1}c + γ)_{odd}) · det_{S^{−1}ℤG} ((S^{−1}c + γ)_{ev}) = 1. (3.117)

Since det_{S^{−1}ℤG} ((S^{−1}c + γ)_{odd}) ≥ 1 and det_{S^{−1}ℤG} ((S^{−1}c + γ)_{ev}) ≥ 1 holds by (3.116), we get det_{S^{−1}ℤG} ((S^{−1}c + γ)_{odd}) = 1. Since γ_{∗} is a chain contraction (and not only a weak chain contraction), we conclude

ρ^{(2)}(C_{∗}) = ln (det_{S^{−1}ℤG} ((S^{−1}c + γ)_{odd})) = 0.

Now apply this to C_{∗} = C_{∗}(Y) with a cellular ℤG-basis and Lemma 3.114 follows. ⊓ ⊔

Remark 3.118. The assumption in Lemma 3.114 that Y is det-L^{2}-acyclic is not necessary. With a little effort and some additional knowledge it follows from the other assumptions by the following argument. Obviously it suffices to show that b_{p}^{(2)}(Y) = 0 for all p ∈ ℤ since G is of det ≥ 1-class by assumption. We will introduce the algebra U(G) of affiliated operators in Chapter 8. We only have to know that U(G) is the Ore localization of N (G) with respect to the set of non-zero-divisors of N (G) (see Theorem 8.22 (1)). Hence we get an embedding S^{−1}ℤG ⊂ U(G). Since S^{−1}C_{∗}(Y) is contractible, U(G)⊗_{ℤG}C_{∗}(Y) is contractible and hence H_{p}(U(G)⊗_{ℤG}C_{∗}(Y)) = 0 for p ≥ 0. We will later show for an appropriate notion of dimension that b_{p}^{(2)}(Y) = dim_{U(G)} (H_{p} (U(G) ⊗_{ℤG} C_{∗}(Y))) (see Theorem 8.31). This implies b_{p}^{(2)}(Y) = 0.

Lemma 3.119. Let G be a group and let A ⊂ G be a normal abelian infinite torsionfree subgroup. Put S = {x ∈ ℤA | x ≠ 0, det(r_{x}^{(2)}) = 1}, where r_{x}^{(2)} : l^{2}(G) → l^{2}(G) is given by right multiplication with x.

Then for each element x ∈ S the morphism r_{x}^{(2)} : l^{2}(G) → l^{2}(G) is a weak isomorphism. The set S is multiplicatively closed. The pair (ℤG, S) satisfies the Ore condition (see Definition 8.14). The trivial ℤG-module ℤ satisfies S^{−1}ℤ = 0.

Proof. Consider x ∈ ℤA. There is a finitely generated subgroup B ⊂ A with x ∈ ℤB. Since B is a finitely generated torsionfree abelian group, it is isomorphic to ℤ^{n} for some n ∈ ℤ. We conclude from Lemma 1.34 (1) that dim_{N (B)} (ker (r_{x}^{(2)} : l^{2}(B) → l^{2}(B))) ∈ {0, 1}. Since ker (r_{x}^{(2)}) is a proper

closed subspace of l^{2}(B) because of x ≠ 0, its von Neumann dimension cannot be dim_{N (B)}(l^{2}(B)) = 1. This implies that its dimension is zero. We conclude

from Lemma 1.24 (2) and (3) that dim_{N (G)} (ker (r_{x}^{(2)} : l^{2}(G) → l^{2}(G))) = 0.

Hence r_{x}^{(2)} : l^{2}(G) → l^{2}(G) is a weak isomorphism by Lemma 1.13. We conclude from Theorem 3.14 (1) that S is multiplicatively closed. Next we show that (ℤG, S) satisfies the right Ore condition. (Notice ℤG

is a ring with involution which leaves S invariant so that then also the left Ore condition is satisfied.) Since S contains no non-trivial zero-divisors, it suffices to show for (r, s) ∈ ℤG × S that there exists (r′, s′) ∈ ℤG × S satisfying rs′ = sr′. Let {g_{i} | i ∈ I} be a complete system of representatives for the cosets Ag ∈ A\G. We can write r = ∑_{i∈I} f_{i}g_{i} for f_{i} ∈ ℤA, where almost all f_{i} are zero. Since A is normal in G, we get g^{−1}sg ∈ ℤA for each g ∈ G. Obviously g^{−1}sg ≠ 0. We have

det (r_{g^{−1}sg}^{(2)}) = det (r_{g}^{(2)} ∘ r_{s}^{(2)} ∘ r_{g^{−1}}^{(2)})
= det (r_{g}^{(2)}) · det (r_{s}^{(2)}) · det (r_{g^{−1}}^{(2)})
= det(r_{s}^{(2)}) = 1.

This shows g^{−1}sg ∈ S for all g ∈ G. Since S is abelian, we can define

s′ = ∏_{{j∈I|f_{j}≠0}} g_{j}^{−1}sg_{j};
s′_{i} = ∏_{{j∈I|f_{j}≠0,j≠i}} g_{i}g_{j}^{−1}sg_{j}g_{i}^{−1};
r′ = ∑_{i∈I} s′_{i}f_{i}g_{i}.

Since S is multiplicatively closed, we have s′, s′_{i} ∈ S. Since ℤA is abelian, we conclude

rs′ = ∑_{i∈I} f_{i}g_{i}s′ = ∑_{{i∈I|f_{i}≠0}} f_{i}g_{i}s′g_{i}^{−1}g_{i} = ∑_{{i∈I|f_{i}≠0}} g_{i}s′g_{i}^{−1}f_{i}g_{i}

= ∑_{{i∈I|f_{i}≠0}} ss′_{i}f_{i}g_{i} = sr′.

Consider a subgroup ℤ ⊂ A. Fix a generator t ∈ ℤ. We conclude from Lemma 3.14 (6) and Example 3.13

det (r_{1−t}^{(2)} : l^{2}(G) → l^{2}(G)) = det (r_{1−t}^{(2)} : l^{2}(ℤ) → l^{2}(ℤ)) = 1.

Hence 1 − t ∈ S. If ℤ is the trivial ℤG-module, then (1 − t) acts by multiplication with zero on ℤ. Hence S^{−1}ℤ = 0. This finishes the proof of Lemma 3.119. ⊓ ⊔

Now we are ready to finish the proof of Theorem 3.113. By Lemma 2.62 we can assume without loss of generality that π_{1}(X) contains a normal abelian infinite torsionfree subgroup A ⊂ π_{1}(X). Notice that H_{p}(X̃) is zero for p ≠ 0 and is the trivial ℤ[π_{1}(X)]-module ℤ for p = 0. Now Theorem 3.113 follows from Lemma 3.114 and Lemma 3.119. ⊓ ⊔

We refer to Conjecture 11.3, which deals with the L^{2}-torsion of universal coverings of aspherical closed manifolds.

3.4.3 Topological L^{2}-Torsion for Riemannian Manifolds

In this subsection we introduce another variant of L^{2}-torsion where we assume that the underlying G-space is a cocompact free proper G-manifold without boundary and with G-invariant Riemannian metric. The G-invariant Riemannian metric together with the L^{2}-Hodge-de Rham Theorem 1.59 will allow us to drop the condition that all L^{2}-Betti numbers vanish.

Definition 3.120 (Topological L^{2}-torsion). Let M be a cocompact free proper G-manifold without boundary and with G-invariant Riemannian metric. Let f: K → M be an equivariant smooth triangulation. Assume that M is of determinant class. Hence ρ^{(2)}(K) := ρ^{(2)}(C_{∗}^{(2)}(K)) ∈ ℝ is defined (see Definition 3.29). Let

A_{K}^{p}: H_{(2)}^{p}(M) ⟶^{≅} H_{(2)}^{p}(K)

be the L^{2}-Hodge-de Rham isomorphism of Theorem 1.59, where we use on H_{(2)}^{p}(M) the Hilbert N (G)-structure coming from the Riemannian metric. Define the topological L^{2}-torsion

ρ_{top}^{(2)}(M) = ρ^{(2)}(K) − ∑_{p≥0}(−1)^{p} · ln (det (A_{K}^{p}: H_{(2)}^{p}(M) ⟶^{≅} H_{(2)}^{p}(K))) .

We have to check that this is independent of the choice of equivariant smooth triangulation. Let g: L → M be another choice. Then the composition of H_{(2)}^{p}(g^{−1} ∘ f): H_{(2)}^{p}(L) → H_{(2)}^{p}(K) with A_{L}^{p} is just A_{K}^{p}. Theorem 3.14 (1) implies

ln(det(A_{K}^{p})) = ln(det(A_{L}^{p})) + ln (det (H_{(2)}^{p}(g^{−1} ∘ f))) . (3.121)

We conclude from Theorem 3.35 (5)

ρ^{(2)}(L) − ρ^{(2)}(K) = − ∑_{p≥0}(−1)^{p} · ln (det (H_{p}^{(2)}(g^{−1} ∘ f))) . (3.122)

We obtain from Lemma 1.18 a commutative square of finitely generated Hilbert N (G)-modules with isometric isomorphisms as vertical arrows

H_{(2)}^{p}(L) ⟶^{H_{(2)}^{p}(g^{−1}∘f)} H_{(2)}^{p}(K)
≅↓  ≅↓
H_{p}^{(2)}(L) ⟶^{(H_{p}^{(2)}(g^{−1}∘f))^{∗}} H_{p}^{(2)}(K)

We conclude from Theorem 3.14 (1) and Lemma 3.15 (4)

ln (det (H_{p}^{(2)}(g^{−1} ∘ f))) = ln (det (H_{(2)}^{p}(g^{−1} ∘ f))) . (3.123)

We get from (3.121), (3.122) and (3.123)

ρ^{(2)}(K) − ∑_{p≥0} det (A_{K}^{p}: H_{(2)}^{p}(M) ⟶^{≅} H_{(2)}^{p}(K))

= ρ^{(2)}(L) − ∑_{p≥0} det (A_{L}^{p}: H_{(2)}^{p}(M) ⟶^{≅} H_{(2)}^{p}(L)) . (3.124)

Hence Definition 3.120 makes sense since ρ_{top}^{(2)}(M) is independent of the choice of equivariant smooth triangulation by (3.124). Notice that ρ^{(2)}(M) does depend on the choice of Riemannian metric. If M happens to be det-L^{2}- acyclic, then ρ^{(2)}(M) of Definition 3.91 and ρ_{top}^{(2)}(M) of Definition 3.120 agree.

There is an obvious analog of Theorem 3.93 for ρ_{top}^{(2)}(M). If f: M → N is a G-homotopy equivalence of cocompact free proper G manifolds without boundary and with G-invariant Riemannian metrics, then we get

ρ_{top}^{(2)}(N) − ρ_{top}^{(2)}(M) = ϕ^{G}(τ(f)) − ∑_{p≥0}(−1)^{p}·

ln (det (H_{(2)}^{p}(f): H_{(2)}^{p}(N) → H_{(2)}^{p}(M))) , (3.125)

where H_{(2)}^{p}(f) is obtained from H_{(2)}^{p}(f) by conjugating with the L^{2}-Hodge-de

Rham isomorphism. We still have Poincaré duality, i.e. ρ_{top}^{(2)}(M) = 0 if M is orientable and has even dimension. There is a product formula

ρ_{top}^{(2)}(M × N) = χ(G\M) · ρ_{top}^{(2)}(N) + χ(H\N) · ρ_{top}^{(2)}(M), (3.126)

where M and N respectively are Riemannian manifolds without boundary and with cocompact free proper actions by isometries of the group G and H respectively. Restriction and induction also carry over in the obvious way. Given a closed Riemannian manifold M, ρ_{top}^{(2)}(M̃) has the obvious meaning and properties.

3.5 Analytic L^{2}-Torsion

In this section we introduce the analytic version of L^{2}-torsion and compute it for universal coverings of closed hyperbolic manifolds.

3.5.1 Definition of Analytic L^{2}-Torsion

The next definition of analytic L^{2}-torsion is taken from Lott [316] and Mathai [358]. The notion of analytic determinant class is due to Burghelea-Friedlander-Kappeler-McDonald [84, Definition 4.1 on page 800]. Recall the definition of the Γ -function as a holomorphic function

Γ (s) = ∫_{0}^{∞} t^{s−1}e^{−t} dt  (3.127)

for ℜ(s) > 0, where ℜ(s) denotes the real part of the complex number s. It has a meromorphic extension to ℂ whose set of poles is {n|n ∈ ℤ, n ≤ 0} and which satisfies Γ (s + 1) = s · Γ (s). All poles have order one. We have Γ (z) ≠ 0 for all z ∈ ℂ − {0, −1, −2, …} and Γ (n + 1) = n! for n ∈ ℤ, n ≥ 0, where we use the standard convention 0! = 1.

Definition 3.128 (Analytic L^{2}-torsion). Let M be a cocompact free proper G-manifold without boundary and with G-invariant Riemannian metric. Define

3.5 Analytic L^{2}-Torsion 179

θ_{p}(M)(t) := ∫_{F} tr_{ℂ}(e^{−tΔ_{p}}(x, x)) dvol,  (3.129)

where e^{−tΔ_{p}}(x, y) denotes the heat kernel associated to the Laplacian acting on p-forms and F is a fundamental domain for the G-action. We call M of analytic determinant class if for any 0 ≤ p ≤ dim(M) and for some (and hence all) ϵ > 0

∫_{ϵ}^{∞} t^{−1} · (θ_{p}(M)(t) − b_{p}^{(2)}(M)) dt < ∞.

In this case we define the analytic L^{2}-torsion of M for any choice of ϵ > 0 by

ρ_{an}^{(2)}(M) := \frac{1}{2} · ∑_{p≥0}(−1)^{p} · p · (\frac{d}{ds} \frac{1}{Γ (s)} ∫_{0}^{ϵ} t^{s−1} · (θ_{p}(M)(t) − b_{p}^{(2)}(M)) dt|_{s=0}

+ ∫_{ϵ}^{∞} t^{−1} · (θ_{p}(M)(t) − b_{p}^{(2)}(M)) dt) .

Some comments are necessary in order to show that this definition makes sense and how it is motivated.

The expression \frac{d}{ds} \frac{1}{Γ (s)} ∫_{0}^{ϵ} t^{s−1} · (θ_{p}(M)(t) − b_{p}^{(2)}(M)) dt|_{s=0} is to be un-

derstood in the sense that \frac{1}{Γ (s)} ∫_{0}^{ϵ} t^{s−1} · (θ_{p}(M)(t) − b_{p}^{(2)}(M)) dt is holomorphic for ℜ(s) > dim(M)/2 and has a meromorphic extension to ℂ with no pole in 0 [316, Section 3]. This fact is shown by comparing the heat kernel on M with the heat kernel on G\M [316, Lemma 4] and using the corresponding statement for G\M.

Definition 3.128 is independent of the choice of ϵ by the following calculation. In the sequel we abbreviate

θ_{p}^{⊥}(t) := θ_{p}(M)(t) − b_{p}^{(2)}(M).  (3.130)

We compute for 0 < ϵ ≤ δ

\frac{d}{ds} \frac{1}{Γ (s)} ∫_{ϵ}^{δ} t^{s−1} · θ_{p}^{⊥}(t) dt|_{s=0}

= \frac{d}{ds} s · \frac{1}{Γ (s + 1)} · ∫_{ϵ}^{δ} t^{s−1} · θ_{p}^{⊥}(t) dt|_{s=0}

= \frac{d}{ds}s|_{s=0} · \frac{1}{Γ (s + 1)} · ∫_{ϵ}^{δ} t^{s−1} · θ_{p}^{⊥}(t) dt|_{s=0}

+0 · \frac{d}{ds} \frac{1}{Γ (s + 1)} · ∫_{ϵ}^{δ} t^{s−1} · θ_{p}^{⊥}(t) dt|_{s=0}

= ∫_{ϵ}^{δ} t^{−1} · θ_{p}^{⊥}(t) dt.  (3.131)

We first explain the relation of the definition above to the classical Ray-Singer torsion (3.9) for a closed manifold M. We have for s ∈ ℂ with ℜ(s) > 0 and real number λ > 0

\frac{1}{Γ (s)} · ∫_{0}^{∞} t^{s−1}e^{−λt} dt = \frac{1}{Γ (s)} · ∫_{0}^{∞}(λ^{−1}u)^{s−1}e^{−λ·(λ^{−1}u)}λ^{−1} du
= \frac{1}{Γ (s)} · λ^{−s} · ∫_{0}^{∞} u^{s−1}e^{−u} du
= λ^{−s}.  (3.132)

From (3.132) we get in the setting of Ray-Singer torsion the following equation, where λ runs over the eigenvalues of the Laplace operator in dimension p listed with multiplicity.

∑_{λ>0} λ^{−s} = ∑_{λ>0} \frac{1}{Γ (s)} · ∫_{0}^{∞} t^{s−1}e^{−tλ} dt

= \frac{1}{Γ (s)} · ∫_{0}^{∞} t^{s−1} · ∑_{λ>0} e^{−λt} dt

= \frac{1}{Γ (s)} · ∫_{0}^{∞} t^{s−1} · (tr_{ℂ} (e^{−tΔ_{p}}) − dim_{ℂ}(H_{p}(M; V))) dt. (3.133)

Hence we can rewrite the analytic Ray-Singer torsion (3.9), where θ_{p}^{⊥} is tr_{ℂ} (e^{−tΔ_{p}}) − dim_{ℂ}(H_{p}(M; V))

ρ_{an}(M; V) := \frac{1}{2} · ∑_{p≥0}(−1)^{p} · p · \frac{d}{ds} \frac{1}{Γ (s)} · ∫_{0}^{∞} t^{s−1} · θ_{p}^{⊥} dt |_{s=0} . (3.134)

Notice that in the setting of Ray-Singer torsion one has only to deal with convergence problems connected with the asymptotics of the large eigenvalues of the Laplacian, whereas in the L^{2}-setting there is an additional convergence problem connected with small eigenvalues and which causes us to require the condition to be of determinant class. Therefore one cannot define the L^{2}- torsion directly using expression (3.134) because there is no guarantee that θ_{p}^{⊥} decays fast enough in the L^{2}-setting. In the Ray-Singer situation or, more generally, under the assumption that the Laplacian Δ_{p} has a gap in the spectrum at zero, θ_{p}^{⊥} decays exponentially as we will see in Lemma 3.139 (5). If θ_{p}^{⊥} decays exponentially, then (3.131) is also true for a real number ϵ > 0 and δ = ∞. Hence we can rewrite in the setting of Ray-Singer torsion

ρ_{an}(M; V) := \frac{1}{2} · ∑_{p≥0}(−1)^{p} · p · (\frac{d}{ds} \frac{1}{Γ (s)} ∫_{0}^{ϵ} t^{s−1} · θ_{p}^{⊥}(t) dt |_{s=0}

+ ∫_{ϵ}^{∞} t^{−1} · θ_{p}(t)^{⊥} dt ) .  (3.135)

This expression (3.135) for the analytic Ray-Singer torsion is the one which can be extended to the L^{2}-setting.

Next we want to show

Theorem 3.136. Let M be a cocompact free proper G-manifold without boundary with invariant Riemannian metric. Then

(1) We have

b_{p}^{(2)}(M) = F (0) = lim_{t→∞} θ_{p}(M)(t);

(2) M is of analytic determinant class in the sense of Definition 3.128 if and only if it is of determinant class in the sense of Definition 3.120;

(3) The following statements are equivalent:

(a) The Laplace operator Δ_{p} has a gap in its spectrum at zero;

(b) α_{p}^{Δ}(M) = ∞^{+};

(c) There exists ϵ > 0 such that F_{p}^{Δ}(M)(λ) = F_{p}^{Δ}(M)(0) holds for 0 < λ < ϵ;

(d) There exists ϵ > 0 and a constant C(ϵ) such that θ_{p}(M)(t) ≤ C(ϵ) ·

e^{−ϵt} holds for t > 0;

(4) Suppose that α_{p}^{Δ}(M) ≠ ∞^{+}. Then

α_{p}^{Δ}(M) = lim inf_{t→∞} \frac{− ln(θ_{p}(M)(t) − b_{p}^{(2)}(M))}{ln(t)}.

We will prove Theorem 3.136 in Subsection 3.5.2 after we have dealt with the Laplace transform of a density function. Theorem 3.136 (2) has already been proved in [84, Proposition 5.6 on page 815].

3.5.2 The Laplace Transform of a Density Function

Let F : [0, ∞) → [0, ∞) be a density function (see Definition 2.7). Its Laplace transform is defined by

θ_{F} (t) = ∫_{0}^{∞} e^{−tλ} dF (λ).  (3.137)

This definition is motivated by

Lemma 3.138. Let M be a cocompact free proper G-manifold without boundary with invariant Riemannian metric. Then the Laplace transform in the sense of (3.137) of the spectral density function F_{p}^{Δ}(M) (see Definition 2.64) is θ_{p}(M) defined in (3.129).

Proof. We get from (1.65) and [9, Proposition 4.16 on page 63]

θ_{F_{p}^{Δ}}(t) = ∫_{0}^{∞} e^{−tλ} dF_{p}^{Δ}(λ)
= tr_{N (G)} (e^{−t(Δ_{p})_{min}})
= θ_{p}(M). ⊓ ⊔

The asymptotic behaviour of the Laplace transform θ_{F} (t) for t → ∞ can be read of from the asymptotic behaviour of F (λ) for λ → 0 and vice versa as explained in the next lemma (cf. [240, Appendix])

Lemma 3.139. Let F: [0, ∞) → [0, ∞) be a density function with Laplace transform θ_{F} (t). Then

(1) We have for λ > 0

F (λ) ≤ θ_{F} (t) · e^{tλ};

(2) Suppose that for all t > 0 there is a constant C(t) such that F (λ) ≤ C(t) · e^{tλ} holds for all λ ≥ 0. Then we get for t > 0

θ_{F} (t) = t · ∫_{0}^{∞} e^{−tλ} · F (λ) dλ;

(3) The Laplace transform θ_{F} (t) is finite for all t > 0 if and only if for all t > 0 there is a constant C(t) such that F (λ) ≤ C(t) · e^{tλ} holds for all λ ≥ 0;

(4) Suppose that θ_{F} (t) < ∞ for all t > 0. Then

F (0) = lim_{t→∞} θ_{F} (t).

(5) Suppose that θ_{F} (t) < ∞ for all t > 0. Let ϵ > 0 be a real number. Then F (λ) = F (0) for all λ < ϵ if and only if there is a constant C(ϵ) such that θ_{F} (t) − F (0) ≤ C(ϵ) · e^{−ϵt} holds for t > 0;

(6) Suppose that F (λ) > F (0) for all λ > 0 and that θ_{F} (t) < ∞ for all t > 0. Then

lim inf_{λ→0} \frac{F (λ) − F (0)}{ln(λ)} = lim inf_{t→∞} \frac{− ln(θ_{F} (t) − F (0))}{ln(t)};

(7) Suppose that F and G are dilatationally equivalent density functions. Then ∫_{0+}^{1} ln(λ) dF (λ) > −∞ if and only if ∫_{0+}^{1} ln(λ) dG(λ) > −∞;

(8) Suppose that θ_{F} (t) < ∞ for all t > 0. Then ∫_{0+}^{1} ln(λ) dF (λ) > −∞ if and only if ∫_{1}^{∞} t^{−1} · (θ_{F} (t) − F (0)) dt < ∞;

Proof. (1) Integration by part (see (3.16)) yields for 0 < ϵ < K < ∞

∫_{ϵ+}^{K} e^{−tλ} dF (λ) = e^{−tK} · F (K) − e^{−tϵ} · F (ϵ) + t · ∫_{ϵ}^{K} e^{−tλ} · F (λ) dλ.

Since F is right continuous, Levi’s Theorem of Monotone Convergence implies

∫_{0+}^{K} e^{−tλ} dF (λ) = e^{−tK} · F (K) − F (0) + t · ∫_{0}^{K} e^{−tλ} · F (λ) dλ, (3.140)

and hence

∫_{0}^{K} e^{−tλ} dF (λ) = e^{−tK} · F (K) + t · ∫_{0}^{K} e^{−tλ} · F (λ) dλ.  (3.141)

We conclude F (λ) ≤ θ_{F} (t) · e^{tλ} for all t, λ > 0 from (3.141).

(2) By assumption we have F (λ) ≤ C(t/2) · e^{−t/2·λ}. This implies for t > 0

lim_{λ→∞} e^{−tλ} · F (λ) = 0.  (3.142)

Now assertion (2) follows from (3.141), (3.142) and Levi’s Theorem of Monotone Convergence.

(3) Suppose that F (λ) ≤ C(t/2) · e^{−t/2·λ}. Then θ_{F} (t) < ∞ follows from the following estimate based on assertion (2).

θ_{F} (t) = t · ∫_{0}^{∞} e^{−tλ} · F (λ) dλ
≤ t · ∫_{0}^{∞} e^{−tλ} · C(t/2) · e^{t/2·λ} dλ
= 2 · C(t/2) · ∫_{0}^{∞} t/2 · e^{−t/2·λ} dλ
= 2 · C(t/2).

The other implication follows from assertion (1).

(4) This follows from Lebesgue’s Theorem of Majorized Convergence.

(5) Suppose that F (ϵ) = F (0). Consider λ < ϵ. We conclude from assertions (1) and (2) for t > 2

θ_{F} (t) − F (0) = t · ∫_{0}^{∞} e^{−tλ} · (F (λ) − F (0)) dλ
= t · ∫_{ϵ}^{∞} e^{−tλ} · (F (λ) − F (0)) dλ
≤ t · ∫_{ϵ}^{∞} e^{−tλ} · F (λ) dλ
≤ t · ∫_{ϵ}^{∞} e^{−tλ} · θ_{F} (1) · e^{λ} dλ
= θ_{F} (1) · t · ∫_{ϵ}^{∞} e^{(−t+1)λ} dλ
= θ_{F} (1) · \frac{t}{t − 1} · e^{(−t+1)ϵ}
≤ θ_{F} (1) · 2 · e^{ϵ} · e^{−tϵ}.

This implies the existence of a constant C(ϵ) > 0 such that θ_{F} (t) − F (0) ≤ C(ϵ) · e^{−tϵ} holds for all t > 0.

Now suppose that θ_{F} (t)−F (0) ≤ C(ϵ)·e^{−tϵ} holds for t > 0. Fix μ ∈ (0, 1). Then we get for t > 0

(F (μ · ϵ) − F (0)) · (e^{(1−μ)ϵt} − 1) · e^{−ϵt} = (F (μ · ϵ) − F (0)) · (e^{−μϵt} − e^{−ϵt})
= t · ∫_{μϵ}^{ϵ} (F (μ · ϵ) − F (0)) · e^{−λt} dλ
≤ t · ∫_{μϵ}^{ϵ}(F (λ) − F (0)) · e^{−λt} dλ
≤ t · ∫_{0}^{∞}(F (λ) − F (0)) · e^{−λt} dλ
= (t · ∫_{0}^{∞} F (λ) · e^{−λt} dλ) − F (0).

We conclude from assertion (2)

(F (μ · ϵ) − F (0)) · (e^{(1−μ)ϵt} − 1) · e^{−ϵt} ≤ θ_{F} (t) − F (0)
≤ C(ϵ) · e^{−tϵ}.

This implies for all t > 0

(F (μ · ϵ) − F (0)) · (e^{(1−μ)ϵt} − 1) ≤ C(ϵ).

Hence F (μ · ϵ) − F (0) = 0 for all μ ∈ (0, 1) and assertion (5) is proved.

(6) We can assume without loss of generality that F (0) = 0, otherwise consider the new density function F (λ) − F (0). We first show

lim inf_{λ→0} \frac{ln(F (λ))}{ln(λ)} ≤ lim inf_{t→∞} \frac{− ln(θ_{F} (t))}{ln(t)}.  (3.143)

Obviously it suffices to treat the case, where the left-hand side is different from zero. Consider 0 < α such that α < lim inf_{λ→0} \frac{ln(F (λ))}{ln(λ)}. Then there is ϵ > 0 such that F (λ) ≤ λ^{α} for all λ ∈ (0, ϵ). We get from assertions (1) and (2) and (3.132) for t > 1

θ_{F} (t) = t · ∫_{0}^{ϵ} e^{−λt} · F (λ) dλ + t · ∫_{ϵ}^{∞} e^{−λt} · F (λ) dλ
≤ t · ∫_{0}^{ϵ} e^{−λt} · λ^{α} dλ + t · ∫_{ϵ}^{∞} e^{−λt} · θ_{F} (1) · e^{λ} dλ
≤ t · ∫_{0}^{∞} e^{−λt} · λ^{α} dλ + θ_{F} (1) · t · ∫_{ϵ}^{∞} e^{λ·(−t+1)} dλ
= Γ (α + 1) · t^{−α} + θ_{F} (1) · \frac{t}{t − 1} · e^{ϵ·(−t+1)}.

We conclude α ≤ lim inf_{t→∞} \frac{− ln(θ_{F} (t))}{ln(t)}. This proves (3.143). It remains to show

lim inf_{λ→0} \frac{ln(F (λ))}{ln(λ)} ≥ lim inf_{t→∞} \frac{− ln(θ_{F} (t))}{ln(t)}.  (3.144)

Obviously it suffices to treat the case, where the right-hand side is greater than zero. Fix 0 < α satisfying α < lim inf_{t→∞} \frac{− ln(θ_{F} (t))}{ln(t)}. Then we can find K > 0 such that θ_{F} (t) ≤ t^{−α} for t > K holds. We conclude from assertion (1) for t > K

F (λ) ≤ e^{tλ} · θ_{F} (t) ≤ e^{tλ} · t^{−α}.

If we take t = λ^{−1}, we get F (λ) ≤ e · λ^{α}. This implies

α ≤ lim inf_{λ→0} \frac{ln(F (λ))}{ln(λ)}.

This finishes the proof of (3.144) and hence of assertion (6).

(7) We conclude from Lemma 3.15 (1) that ∫_{0+}^{1} ln(λ) dF (λ) > −∞ is equivalent to ∫_{0+}^{1} \frac{F (λ)−F (0)}{λ} dλ < ∞ and analogous for G.

(8) Obviously the claim is true for F − F (0) if and only if it is true for F. Hence we can assume without loss of generality that F (0) = 0 in the sequel. Assertion (2) implies

∫_{1}^{∞} t^{−1} · θ_{F} (t) dt = ∫_{1}^{∞} (∫_{0}^{∞} e^{−tλ} · F (λ) dλ) dt.  (3.145)

We get

∫_{1}^{∞} (∫_{1}^{∞} e^{−tλ} · F (λ) dλ) dt < ∞  (3.146)

from the inequality F (λ) ≤ C(1) · e^{λ} of assertion (1). Using the inequality F (λ) ≤ C(t) · e^{t·λ} of assertion (1) and the standard results about commuting differentiation and integration we conclude

\frac{d}{dt} ∫_{0}^{1} \frac{e^{−tλ}}{−λ} · F (λ) dλ = ∫_{0}^{1} e^{−tλ} · F (λ) dλ.

This implies using Lebesgue’s Theorem of Majorized Convergence and Levi’s Theorem of Monotone Convergence

∫_{1}^{∞} (∫_{0}^{1} e^{−tλ} · F (λ) dλ) dt
= lim_{K→∞} ∫_{1}^{K} (∫_{0}^{1} e^{−tλ} · F (λ) dλ) dt
= lim_{K→∞} ∫_{0}^{1} \frac{e^{−Kλ}}{−λ} · F (λ) dλ − ∫_{0}^{1} \frac{e^{−1λ}}{−λ} · F (λ) dλ
= ∫_{0}^{1} lim_{K→∞} \frac{e^{−Kλ}}{−λ} · F (λ) dλ − ∫_{0}^{1} \frac{e^{−λ}}{−λ} · F (λ) dλ
= ∫_{0}^{1} \frac{e^{−λ}}{λ} · F (λ) dλ.  (3.147)

We have

e^{−1} · ∫_{0}^{1} \frac{F (λ)}{λ} dλ ≤ ∫_{0}^{1} \frac{e^{−λ}}{λ} · F (λ) dλ ≤ ∫_{0}^{1} \frac{F (λ)}{λ} dλ. (3.148)

We conclude from Lemma 3.15 (1), (3.145), (3.146), (3.147) and (3.148) that ∫_{1}^{∞} t^{−1} · θ_{F} (t) dt < ∞ is equivalent to ∫_{0+}^{1} ln(λ) dF (λ) > −∞. This finishes the proof of Lemma 3.139. ⊓ ⊔

Now Theorem 3.136 follows from Theorem 2.68, Lemma 3.138 and Lemma 3.139.

3.5.3 Comparison of Topological and Analytic L^{2}-Torsion

Next we cite the deep result of Burghelea, Friedlander, Kappeler and McDonald [84] that the topological and analytic L^{2}-torsion ρ_{top}^{(2)}(M) (see Definition 3.120) and ρ_{an}^{(2)}(M) (see Definition 3.128) agree. The main idea is to perform the Witten deformation of the Laplacian with a suitable Morse function and investigate the splitting of the de Rham complex according to small and large eigenvalues. We do not give the long and complicated proof here. The main technical tools are asymptotic expansions and a Mayer-Vietoris type formula for determinants (see [79], [80]). A survey is given in [81]. See also [83].

Theorem 3.149 (Analytic and topological L^{2}-torsion). Let M be a cocompact free proper G-manifold without boundary and with G-invariant Riemannian metric. Suppose that M is of analytic determinant class in the sense of Definition 3.128, or equivalently, of determinant class in the sense of Definition 3.120 (see Theorem 3.136 (2)). Then

ρ_{top}^{(2)}(M) = ρ_{an}^{(2)}(M).

3.5.4 Analytic L^{2}-Torsion for Hyperbolic Manifolds

In general it is easier to work with topological L^{2}-torsion because it has nice properties as stated for instance in Theorem 3.96, Theorem 3.100 and Theorem 3.106 and it can be computed combinatorially without investigating the spectral density functions of certain operators (see Theorem 3.172). However, there are some special cases where it is easier to deal with analytic L^{2}-torsion because the Riemannian metrics have special properties and there is a proportionality principle (see Theorem 3.183). Examples are hyperbolic manifolds which are treated in the next theorem due to Hess and Schick [254], the special case of dimension 3 is proved in [316, Proposition 16] and [358, Corollary 6.7]. Recall that the L^{2}-torsion of the universal covering of a closed even-dimensional Riemannian manifold is always trivial so that only the odd-dimensional case is interesting.

Consider the polynomial with integer coefficients for j ∈ {0, 1, 2, …, n−1}

P_{j}^{n}(ν) := \frac{∏_{i=0}^{n}(ν^{2} + i^{2})}{ν^{2} + (n − j)^{2}} = ∑_{k≥0}^{2n} K_{k,j}^{n} · ν^{2k}.  (3.150)

Define

C_{d} := ∑_{j=0}^{n−1}(−1)^{n+j+1} \frac{n!}{(2n)! · π^{n}} · ( \begin{matrix}2n \\ j \end{matrix} )

· ∑_{k=0}^{n} K_{k,j}^{n} · \frac{(−1)^{k+1}}{2k + 1} · (n − j)^{2k+1}.  (3.151)

The first values of C_{d} are computed in [254, Theorem 2]

C_{3} = \frac{1}{6π} ≈ 0.05305;
C_{5} = \frac{31}{45π^{2}} ≈ 0.06980;
C_{7} = \frac{221}{70π^{3}} ≈ 0.10182;
C_{39}  ≈ 2.4026 · 10^{7},

and the constants C_{d} are positive, strictly increasing and grow very fast, namely they satisfy [254, Proposition 6]

C_{2n+1} ≥ \frac{n}{2π} · C_{2n−1};
C_{2n+1} ≥ \frac{2n!}{3(2π)^{n}} .

Theorem 3.152 (Analytic L^{2}-torsion of hyperbolic manifolds). Let M be a closed hyperbolic d-dimensional manifold for odd d = 2n + 1. Let C_{d} > 0 be the constant introduced in (3.151). It can be written as C_{d} = π^{−n} · r_{d} for a rational number r_{d} > 0. Then

ρ^{(2)}(M̃) = (−1)^{n} · C_{d} · vol(M).

Proof. In the sequel we abbreviate

D := \frac{(4π)^{−n−1/2}}{Γ (n + 1/2)} = \frac{n!}{2 · π^{n+1} · (2n)!}.

Notice that M̃ can be identified with the hyperbolic space ℍ^{d} since M is by assumption hyperbolic and that there are no harmonic L^{2}-integrable p-forms on ℍ^{d} for p ≥ 0 by Theorem 1.62 Let Δ_{j}|: d^{j−1}(Ω_{c}^{j−1}(ℍ^{d}))^{⊥} → d^{j−1}(Ω_{c}^{j−1}(ℍ^{d}))^{⊥} be the operator induced by the Laplacian Δ_{j}: L^{2}Ω^{j}(ℍ^{d}) → L^{2}Ω^{j}(ℍ^{d}) (see Theorem 1.57) and e^{tΔ_{j}|}(x, x) be the kernel of the operator e^{tΔ_{j}|}. Recall that ℍ^{d} is homogeneous, i.e. for two points x and y in ℍ^{d} there is an isometry ℍ^{d} → ℍ^{d} mapping x to y. Hence tr_{ℂ}(e^{tΔ_{j}|}(x, x)) is independent of x ∈ ℍ^{d}. Put

L_{j} = \frac{d}{ds} \frac{1}{Γ (s)} ∫_{0}^{1} t^{s−1} · tr_{ℂ}(e^{tΔ_{j}|}(x, x)) dt|_{s=0}
+ ∫_{1}^{∞} t^{−1} · tr_{ℂ}(e^{tΔ_{j}|}(x, x)) dt,

where the first summand is to be understood as before, namely it is a holomorphic function for large ℜ(s) and has a meromorphic extension to ℂ without pole in 0. Now a calculation analogous to the proof of Lemma 3.30 and using the Hodge-de Rham decomposition (see Theorem 1.57) and the fact that the Laplacian is compatible with the Hodge star-operator (1.48) shows

\frac{ρ_{an}^{(2)}(M)}{vol(M)} = ∑_{j=0}^{n}(−1)^{j+1}L_{j}.  (3.153)

The following equality is taken from [316, Proposition 15] where it is derived as a special case of [371]

tr_{ℂ}(e^{tΔ_{j}|}(x, x)) = D · ( \begin{matrix}2n \\ j \end{matrix} ) · ∫_{−∞}^{∞} e^{−t(ν^{2}+(n−j)^{2})} · P_{j}^{n}(ν) dν,

where the polynomial P_{j}^{n}(ν) and its coefficients K_{k,j}^{n} are defined in (3.150). We compute

tr_{ℂ}(e^{tΔ_{j}|}(x, x)) = D · ( \begin{matrix}2n \\ j \end{matrix} ) · ∑_{k=0}^{n} K_{k,j}^{n}e^{−t(n−j)^{2}} · t^{−k−1/2} · Γ (k + 1/2).

Hence we can write

L_{j} = D · ( \begin{matrix}2n \\ j \end{matrix} ) · ∑_{k=0}^{n} K_{k,j}^{n} · Γ (k + 1/2) ·

( \frac{d}{ds} \frac{1}{Γ (s)} ∫_{0}^{1} e^{−t(n−j)^{2}} · t^{s−k−3/2} dt|_{s=0}
+ ∫_{1}^{∞} e^{−t(n−j)^{2}} · t^{−k−3/2} dt) .  (3.154)

A computation analogous to the one in (3.131) shows for (n − j) > 0

\frac{d}{ds} \frac{1}{Γ (s)} ∫_{1}^{∞} e^{−t(n−j)^{2}} · t^{s−k−3/2} dt|_{s=0}
= ∫_{1}^{∞} e^{−t(n−j)^{2}} · t^{−k−3/2} dt.  (3.155)

One easily checks for j < n

\frac{d}{ds} \frac{1}{Γ (s)} ∫_{0}^{∞} e^{−t(n−j)^{2}} · t^{s−k−3/2} dt|_{s=0}
= \frac{d}{ds} \frac{s}{Γ (s + 1)} · (n − j)^{−2s+2k+1} · Γ (s − k − 1/2)|_{s=0}
= (n − j)^{2k+1} · Γ (−k − 1/2).  (3.156)

We conclude from (3.154), (3.155) and (3.156) for j ≠ n

L_{j} = D · ( \begin{matrix}2n \\ j \end{matrix} ) · ∑_{k=0}^{n} K_{k,j}^{n} · Γ (k + 1/2) · Γ (−k − 1/2) · (n − j)^{2k+1}

= D · ( \begin{matrix}2n \\ j \end{matrix} ) · ∑_{k=0}^{n} K_{k,j}^{n} · (−1)^{k+1} · \frac{2π}{2k + 1} · (n − j)^{2k+1}.  (3.157)

We get

L_{n} = 0  (3.158)

from the following calculation for α > 0

\frac{d}{ds} \frac{1}{Γ (s)} ∫_{0}^{1} t^{s−1−α} dt|_{s=0} + ∫_{1}^{∞} t^{−1−α} dt

= \frac{d}{ds} \frac{s}{Γ (s + 1)} · \frac{1}{s − α}|_{s=0} + \frac{1}{α}
= \frac{1}{−α} + \frac{1}{α}
= 0.

We get from (3.153), (3.157) and (3.158)

ρ_{an}^{(2)}(M) = (−1)^{n} · C_{d} · vol(M).  (3.159)

This finishes the proof of Theorem 3.152. ⊓ ⊔

Theorem 3.152 will be extended to closed Riemannian manifolds whose universal coverings are symmetric spaces of non-compact type in Theorem 5.12.

3.6 L^{2}-Torsion of Manifolds with Boundary

In this section we discuss what happens for a cocompact free proper Gmanifold M with G-invariant Riemannian metric whose boundary ∂M is the disjoint union of the (possibly empty) G-spaces ∂_{0}M and ∂_{1}M. Choose an equivariant smooth triangulation (K; K_{0}, K_{1}) → (M; ∂_{0}M, ∂_{1}M). We have introduced the L^{2}-Hodge-de Rham isomorphism A_{K}^{p}: H_{(2)}^{p}(M, ∂_{0}M) ⟶^{≅} H_{(2)}^{p}(K, K_{0}) in Theorem 1.89. We define analogously to the case where the boundary is empty the notions of determinant class and of topological L^{2}- torsion

ρ_{top}^{(2)}(M, ∂_{0}M) = ρ^{(2)}(K, K_{0}) − ∑_{p≥0}(−1)^{p} ·
ln (det (A_{K}^{p}: H_{(2)}^{p}(M, ∂_{0}M) ⟶^{≅} H_{(2)}^{p}(K, K_{0}))) . (3.160)

We define the notions of analytic determinant class and of analytic L^{2}-torsion ρ_{an}^{(2)}(M, ∂_{0}M) as in Definition 3.128 but now imposing Dirichlet boundary conditions on ∂_{0}M and Neumann boundary conditions on ∂_{1}M, i.e. we use the Laplacian Δ_{p}: L^{2}Ω^{p}(M) → L^{2}Ω^{p}(M) which is the closure of the operator Δ_{p}: Ω_{2}^{p}(M) → Ω_{c}^{p}(M) where Ω_{2}^{p}(M) ⊂ Ω_{c}^{p}(M) was introduced in 1.83. Analogously we obtain the notions of topological Reidemeister torsion ρ_{top}(M, ∂_{0}M; V) and of analytic Reidemeister torsion or Ray-Singer torsion ρ_{an}(M, ∂_{0}M; V). We say that the G-invariant Riemannian metric is a product near the boundary, if there are ϵ > 0 and a G-invariant neighborhood U of ∂M in M together with an isometric G-diffeomorphism ∂M × [0, ϵ) → U inducing the identity on ∂M, where ∂M is equipped with the G-invariant Riemannian metric induced from M, [0, ϵ) with the standard Riemannian metric and ∂M × [0, ϵ) with the product Riemannian metric.

Theorem 3.161 (L^{2}-torsion of manifolds with boundary). Let M be a cocompact free proper G-manifold M with G-invariant Riemannian metric whose boundary ∂M is the disjoint union of the (possibly empty) G-spaces ∂_{0}M and ∂_{1}M. Suppose that the G-invariant Riemannian metric is a product near the boundary. Suppose that (M, ∂_{0}M), (M, ∂M) and ∂M are of determinant class. Then

ρ_{an}^{(2)}(M, ∂_{0}M) = ρ_{top}^{(2)}(M, ∂_{0}M) + \frac{ln(2)}{4} · χ(G\∂M);
ρ_{an}(G\M, G\∂_{0}M; V) = ρ_{top}(G\M, G\∂_{0}M; V)

+\frac{ln(2)}{4} · χ(G\∂M) · dim_{ℝ}(V).

Proof. The first equation is proved by Burghelea, Friedlander and Kappeler [82, Theorem 4.1 on page 34] and the second independently by Lück [327, Theorem 4.5 on page 266] and Vishik [505]. ⊓ ⊔

3.6 L^{2}-Torsion of Manifolds with Boundary 191

In the case of the analytic Ray-Singer-torsion the general boundary correction term (without the condition that the metric is a product near the boundary) is given in [126]. One would expect to get the same correction term in the L^{2}-case.

Remark 3.162. We have seen in the context of L^{2}-Betti numbers (see Theorem 1.89) and of Novikov-Shubin invariants (see (2.84)) that the combinatorial and analytic versions also agree if M has a boundary. In the context of L^{2}-torsion there are two main differences, there is a correction term involving the Euler characteristic of the boundary and the assumption that the Riemannian metric is a product near the boundary. The additional Euler characteristic term is related to the observation that the boundary ∂M is a zero set in M and hence on the analytical side certain integration processes on the double M ∪_{∂M} M cannot feel the boundary ∂M. On the other hand equivariant cells sitting in the boundary ∂M do contribute to the cellular chain complex of M ∪_{∂M} M and affect therefore the topological side. The condition that the metric is a product near the boundary cannot be dropped for both Reidemeister torsion and L^{2}-torsion because otherwise examples of Lück and Schick [346, Appendix A] based on [60] show that the formulas become wrong. The condition that the metric is a product near the boundary ensures that the double M ∪_{∂M} M inherits a G-invariant Riemannian metric and that one can rediscover the information about (M, ∂M) by inspecting the manifold without boundary M ∪_{∂M} M but now with the obvious G×ℤ/2action induced by the flip map (see also Section 2.6).

The necessity of the condition that the Riemannian metric is a product near the boundary also shows that the next result is rather delicate since one has to chop the manifold into compact pieces with boundary and the relevant comparison formulas and glueing formulas are not a priori true any more since this chopping cannot be done such that metrics are product metrics near the boundaries.

The next result is proved in [346, Theorem 0.5].

Theorem 3.163. L^{2}-torsion of hyperbolic manifolds with boundary. Let \overline{M} be a compact connected manifold with boundary of odd dimension d = 2n + 1 such that the interior M comes with a complete hyperbolic metric of finite volume. Then ˜ \overline{M} is of determinant class and

ρ_{top}^{(2)}(˜\overline{M}) = ρ_{an}^{(2)}(M̃) = (−1)^{n} · C_{d} · vol(M),

where ρ_{an}^{(2)}(M̃) is defined as in the cocompact case (which makes sense since M̃ is homogeneous and has some cocompact free proper action of some discrete group by isometries and M has finite volume) and C_{d} > 0 is the dimension constant of 3.151.

Poincaré duality still holds for manifolds with boundary provided that M is orientable (cf. Theorem 3.93 (3)) i.e.

ρ_{an}^{(2)}(M, ∂_{0}M) = (−1)^{dim(M)+1} · ρ_{an}^{(2)}(M, ∂_{1}M);  (3.164)
ρ_{top}^{(2)}(M, ∂_{0}M) = (−1)^{dim(M)+1} · ρ_{top}^{(2)}(M, ∂_{1}M),  (3.165)

and analogously for Reidemeister torsion. We mention the following glueing formula. Define ρ^{(2)}(C^{∗}) of a finite Hilbert N (G)-cochain complex C^{∗} of determinant class by

ρ^{(2)}(C^{∗}) := −\frac{1}{2} · ∑_{p∈ℤ}(−1)^{p} · p · ln(det(Δ_{p})
= ∑_{p∈ℤ}(−1)^{p} · ln(det(c^{p})).  (3.166)

Theorem 3.167 (Glueing formula for L^{2}-torsion). Let M and N be cocompact free proper G-manifolds with G-invariant Riemannian metrics which are products near the boundary. Their boundaries come with decompositions ∂M = ∂_{0}M ∐ ∂_{1}M ∐ ∂_{2}M and ∂N = ∂_{0}N ∐ ∂_{1}N ∐ ∂_{2}N. Let f: ∂_{2}M → ∂_{2}N be an isometric G-diffeomorphism. Let M ∪_{f} N be the cocompact free proper G-manifold with G-invariant Riemannian metric obtained by glueing M and N together along f. Suppose that (M, ∂_{0}M), (M, ∂M), (N, ∂_{0}N), (N, ∂N), ∂M and ∂N are of determinant class. Then (M ∪_{f} N, ∂_{0}M ∐ ∂_{0}N) is of determinant class. We obtain a long weakly exact sequence of finitely generated Hilbert N (G)-modules

… → H_{(2)}^{p−1}(∂_{2}M) → H_{(2)}^{p}(M ∪_{f} N, ∂_{0}M ∐ ∂_{0}N)
→ H_{(2)}^{p}(M, ∂_{0}M) ⊕ H_{(2)}^{p}(N, ∂_{0}N) → H_{(2)}^{p}(∂_{2}M) → …

where we use the Hilbert structures coming from the Riemannian metrics and the maps are given by comparing the corresponding weakly exact L^{2}- cohomology sequence associated to equivariant smooth triangulations with the various L^{2}-Hodge-de Rham isomorphisms. We view it as a weakly acyclic Hilbert N (G)-cochain complex LHS^{∗} with H_{(2)}^{0}(M ∪_{f} N, ∂_{0}M ∐ ∂_{0}N) in dimension zero. Then LHS^{∗} is weakly det-L^{2}-acyclic and

ρ_{top}^{(2)}(M ∪_{f} N, ∂_{0}M ∐ ∂_{0}N) = ρ_{top}^{(2)}(M, ∂_{0}M) + ρ_{top}^{(2)}(N, ∂_{0}N) − ρ_{top}^{(2)}(∂_{2}M)

−ρ^{(2)}(LHS^{∗});

ρ_{an}^{(2)}(M ∪_{f} N, ∂_{0}M ∐ ∂_{0}N) = ρ_{an}^{(2)}(M, ∂_{0}M) + ρ_{an}^{(2)}(N, ∂_{0}N) − ρ_{an}^{(2)}(∂_{2}M)

−ρ^{(2)}(LHS^{∗}) − \frac{ln(2)}{2} · χ(G\∂_{2}M).

The analogous result is true for the topological Reidemeister torsion and analytic Reidemeister torsion.

3.7 Combinatorial Computations of L^{2}-Invariants 193

Proof. The claim about the analytic L^{2}-torsion is proved in [82, Theorem 4.4 on page 67]. The claim for the topological L^{2}-torsion can also be derived from Theorem 3.35 (1) applied to the exact sequence of Hilbert N (G)-chain complexes 0 → C_{∗}(∂_{2}M) → C_{∗}(M, ∂_{0}M) ⊕ C_{∗}(N, ∂_{0}N) → C_{∗}(M ∪_{f} N, ∂_{0}M ∐ ∂_{0}N) → 0 and the fact that the L^{2}-torsion of the weakly exact long homology sequence is the negative of the torsion of the weakly exact long cohomology sequence because ρ^{(2)}(C_{∗}) = ρ^{(2)}((C_{∗})^{∗}) holds for a finite Hilbert N (G)-chain complex C_{∗}. The claim for the analytic L^{2}-torsion is then a consequence of Theorem 3.161.

The claim for the topological and analytical Reidemeister torsion is proved in [327, Proposition 3.11 on page 290 and Proposition 5.9 on page 313]. ⊓ ⊔

3.7 Combinatorial Computations of L^{2}-Invariants

In this section we want to give a more combinatorial approach to the L^{2}- invariants such as L^{2}-Betti numbers, Novikov-Shubin invariants and L^{2}- torsion. The point is that it is in general very hard to compute the spectral density function of some morphism of finitely generated Hilbert N (G)- modules. However in the geometric situation these morphisms are induced by matrices over the integral group ring ℤG. We want to exploit this information to get an algorithm which produces a sequence of rational numbers which converges to the L^{2}-Betti number or the L^{2}-torsion and whose members are computable in an algorithmic way.

Let A ∈ M(n, m, ℂG) be a (n, m)-matrix over ℂG. It induces by right multiplication a ℂG-homomorphism of left ℂG-modules

R_{A}: ⨁_{i=1}^{n} ℂG → ⨁_{i=1}^{m} ℂG, x ↦ xA

and by completion a bounded G-equivariant operator

R_{A}^{(2)} : ⨁_{i=1}^{n} l^{2}(G) → ⨁_{i=1}^{m} l^{2}(G).

Notice for the sequel that R_{AB} = R_{B} ∘ R_{A} holds. We define an involution of rings on ℂG by

\overline{∑_{g∈G} λ_{g} · g} := ∑_{g∈G} \overline{λ_{g}} · g^{−1},  (3.168)

where \overline{λ_{g}} is the complex conjugate of λ_{g}. Denote by A^{∗} the (m, n)-matrix obtained from A by transposing and applying the involution above to each entry. As the notation suggests, the bounded G-equivariant operator R_{A^{∗}}^{(2)} is the adjoint (R_{A}^{(2)})^{∗} of the bounded G-equivariant operator R_{A}^{(2)}. Define the ℂG-trace of an element u = ∑_{g∈G} λ_{g} · g ∈ ℂG by

tr_{ℂG}(u) := λ_{e} ∈ ℂ  (3.169)

for e the unit element in G. This extends to a square (n, n)-matrix A over ℂG by

tr_{ℂG}(A) := ∑_{i=1}^{n} tr_{ℂG}(a_{i,i}).  (3.170)

We get directly from the definitions that the ℂG-trace tr_{ℂG}(A) agrees with the von Neumann trace tr_{N (G)}(R_{A}^{(2)}) introduced in Definition 1.2.

Let A ∈ M(n, m, ℂG) be a (n, m)-matrix over ℂG. In the sequel let K be any positive real number satisfying

K ≥ ||R_{A}^{(2)}||_{∞},

where ||R_{A}^{(2)}||_{∞} is the operator norm of the bounded G-equivariant operator R_{A}^{(2)}. For u = ∑_{g∈G} λ_{g} · g ∈ ℂG define ||u||_{1} by ∑_{g∈G} |λ_{g}|. Then a possible choice for K is given by:

K = \sqrt{(2n − 1)m} · max \{||a_{i,j}||_{1} | 1 ≤ i ≤ n, 1 ≤ j ≤ m\} .

The bounded G-equivariant operator 1 − K^{−2} · (R_{A}^{(2)})^{∗}R_{A}^{(2)} : ⨁_{i=1}^{n} l^{2}(G) → ⨁_{i=1}^{n} l^{2}(G) is positive. Let (1 − K^{−2} · AA^{∗})^{p} be the p-fold product of matri-

ces and let (1 − K^{−2} · (R_{A}^{(2)})^{∗}R_{A}^{(2)})^{p} be the p-fold composition of operators.

Definition 3.171. The characteristic sequence of a matrix A ∈ M(n, m, ℂG) and a non-negative real number K satisfying K ≥ ||R_{A}^{(2)}||_{∞} is the sequence of real numbers

c(A, K)_{p} := tr_{ℂG} ((1 − K^{−2} · AA^{∗})^{p}) = tr_{N (G)} ((1 − K^{−2} · (R_{A}^{(2)})^{∗}R_{A}^{(2)})^{p}) .

We have defined b^{(2)}(R_{A}^{(2)}) in Definition 2.1 and det(R_{A}^{(2)}) in Definition 3.11.

Theorem 3.172. (Combinatorial computation of L^{2}-invariants). Let A ∈ M(n, m, ℂG) be a (n, m)-matrix over ℂG. Denote by F the spectral density function of R_{A}^{(2)}. Let K be a positive real number satisfying K ≥ ||R_{A}^{(2)}||_{∞}. Then

(1) The characteristic sequence c(A, K)_{p} is a monotone decreasing sequence of non-negative real numbers; (2) We have

b^{(2)}(R_{A}^{(2)}) = lim_{p→∞} c(A, K)_{p};

(3) Define β(A) ∈ [0, ∞] by

β(A) := sup {β ∈ [0, ∞) | lim_{p→∞} p^{β} · (c(A, K)_{p} − b^{(2)}(R_{A}^{(2)})) = 0} .

If α(R_{A}^{(2)}) < ∞, then α(R_{A}^{(2)}) ≤ β(A) and if α(R_{A}^{(2)}) ∈ {∞, ∞^{+}}, then β(A) = ∞;

(4) The sum of positive real numbers

∑_{p=1}^{∞} \frac{1}{p} · (c(A, K)_{p} − b^{(2)}(R_{A}^{(2)}))

converges if and only if R_{A}^{(2)} is of determinant class and in this case

2·ln(det(R_{A}^{(2)})) = 2·(n−b^{(2)}(R_{A}^{(2)}))·ln(K)−∑_{p=1}^{∞} \frac{1}{p}·(c(A, K)_{p} − b^{(2)}(R_{A}^{(2)}));

(5) Suppose α(R_{A}^{(2)}) > 0. Then R_{A}^{(2)} is of determinant class. Given a real number α satisfying 0 < α < α(R_{A}^{(2)}), there is a real number C such that we have for all L ≥ 1

0 ≤ c(A, K)_{L} − b^{(2)}(R_{A}^{(2)}) ≤ \frac{C}{L^{α}}

and

0 ≤ −2 · ln(det(R_{A}^{(2)})) + 2 · (n − b^{(2)}(R_{A}^{(2)})) · ln(K)

− ∑_{p=1}^{L} \frac{1}{p} · (c(A, K)_{p} − b^{(2)}(R_{A}^{(2)})) ≤ \frac{C}{L^{α}} .

Remark 3.173. Before we give the proof of Theorem 3.172, we discuss its meaning. Let X be a finite free G-CW -complex. Describe the p-th differential c_{p}: C_{p}(X) → C_{p−1}(X) of its cellular ℤG-chain complex with respect to a cellular basis by the matrix A_{p} ∈ M(n_{p}, n_{p−1}, ℤG). Then R_{A_{p}}^{(2)} is just

the p-th differential c_{p}^{(2)} of the cellular Hilbert N (G)-chain complex C_{∗}^{(2)}(X) and for the p-th Laplace operator Δ_{p}: C_{p}^{(2)}(X) → C_{p}^{(2)}(X) of (1.17) we get R_{A_{p}A_{p}^{∗}+A_{p+1}^{∗}A_{p+1}}^{(2)}. This implies

b_{p}^{(2)}(X) = b^{(2)}(R_{A_{p}A_{p}^{∗}+A_{p+1}^{∗}A_{p+1}}^{(2)});  (3.174)
α_{p}(X) = α(R_{A_{p}}^{(2)});  (3.175)
α_{p}^{Δ}(X) = α(R_{A_{p}A_{p}^{∗}+A_{p+1}^{∗}A_{p+1}}^{(2)});  (3.176)
ρ^{(2)}(C_{∗}^{(2)}(X)) = − ∑_{p≥0}(−1)^{p} · det(R_{A_{p}})  (3.177)

= −\frac{1}{2} · ∑_{p≥0}(−1)^{p} · p · det(R_{A_{p}A_{p}^{∗}+A_{p+1}^{∗}A_{p+1}}^{(2)}), (3.178)

where ρ^{(2)}(C_{∗}^{(2)}(X)) is introduced in Definition 3.29 and agrees with ρ^{(2)}(X) if X is det-L^{2}-acyclic (see Definition 3.91) and we have used Lemma 1.18 and Lemma 3.30. Notice that in all cases the relevant L^{2}-invariant is given by the corresponding L^{2}-invariant of a morphism of the shape R_{B}^{(2)} for some matrix over ℤG. Hence Theorem 3.172 applies to the geometric situation.

Each term of the characteristic sequences c(A, K)_{p} can be computed by an algorithm as long as the word problem for G has a solution. Because of Theorem 3.172 (1) and (2) one can use the following strategy to show the vanishing of b_{p}^{(2)}(X) provided one knows that there is an integer such that n · b_{p}^{(2)}(X) ∈ ℤ (see the Strong Atiyah Conjecture 10.2). One computes c(A, K)_{p} for p = 1, 2, 3, … and hopes that for some p its value becomes smaller than 1/n. This would imply b_{p}^{(2)}(X) = 0.

Notice that the knowledge of the Novikov-Shubin invariants gives information about the speed of convergence of the relevant sequences or sums converging to the L^{2}-Betti number or L^{2}-torsion because of Theorem 3.172 (5).

We need the following lemma for the proof of Theorem 3.172.

Lemma 3.179. If F (λ) is the spectral density function of (R_{A}^{(2)})^{∗}R_{A}^{(2)} for

A ∈ M(n, m, ℂG) and K satisfies K ≥ ||R_{A}^{(2)}||_{∞} = \sqrt{||(R_{A}^{(2)})^{∗}R_{A}^{(2)}||_{∞}}, then we get for all λ ∈ [0, 1]

(1 − λ)^{p} · (F (K^{2} · λ) − F (0)) ≤ c(A, K)_{p} − b^{(2)}(R_{A}^{(2)})
≤ F (K^{2} · λ) − F (0) + (1 − λ)^{p} · (n − F (0)).

Proof. We have for μ ∈ [0, ||R_{A}^{(2)}||_{∞}^{2}]

(1 − λ)^{p} · χ_{[0,λ]}(K^{−2} · μ) ≤ (1 − K^{−2} · μ)^{p} ≤ χ_{[0,λ]}(K^{−2} · μ) + (1 − λ)^{p}.

Hence we get by integrating over μ

∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}}(1 − λ)^{p} · χ_{[0,λ]}(K^{−2} · μ) dF

≤ ∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}}(1 − K^{−2} · μ)^{p} dF

≤ ∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}} χ_{[0,λ]}(K^{−2} · μ) + (1 − λ)^{p} dF.

We have

∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}}(1 − λ)^{p} · χ_{[0,λ]}(K^{−2} · μ) dF = (1 − λ)^{p} · (F (K^{2} · λ) − F (0));

∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}}(1 − K^{−2} · μ)^{p} dF = tr_{N (G)} ((1 − K^{−2} · (R_{A}^{(2)})^{∗}R_{A}^{(2)})^{p})
− dim(ker((R_{A}^{(2)})^{∗}R_{A}^{(2)}));
∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}} χ_{[0,λ]}(K^{−2} · μ) + (1 − λ)^{p} dF = F (K^{2} · λ) − F (0)
+(1 − λ)^{p} · (F (||R_{A}^{(2)}||_{∞}^{2}) − F (0)).

This finishes the proof of Lemma 3.179. ⊓ ⊔ Now we can give the proof of Theorem 3.172.

Proof. (1) The bounded G-equivariant operator

1 − K^{−2} · (R_{A}^{(2)})^{∗}R_{A}^{(2)} : ⨁_{i=1}^{n} l^{2}(G) → ⨁_{i=1}^{n} l^{2}(G)

is positive and satisfies

0 ≤ 1 − K^{−2} · (R_{A}^{(2)})^{∗}R_{A}^{(2)} ≤ 1.

This implies for 0 ≤ p ≤ q

0 ≤ (1 − K^{−2} · (R_{A}^{(2)})^{∗}R_{A}^{(2)})^{q} ≤ (1 − K^{−2} · (R_{A}^{(2)})^{∗}R_{A}^{(2)})^{p} ≤ 1

and the first assertion follows as the trace is monotone.

(2) If we apply Lemma 3.179 to the value λ = 1 − \sqrt[p]{\frac{1}{p}} we obtain for all positive integers p

0 ≤ c(A, K)_{p} − b^{(2)}(R_{A}^{(2)}) ≤ F (K^{2} · (1 − \sqrt[p]{\frac{1}{p}})) − F (0) + \frac{n − F (0)}{p}.

We get lim_{x→0+} x · ln(x) = 0 from l’Hospital’s rule. This implies

lim_{p→∞} 1 − \sqrt[p]{\frac{1}{p}} = 0.

Since the spectral density function is right continuous, assertion (2) of Theorem 3.172 follows.

(3) Let β and α be any real numbers satisfying

0 < β < α < α(R_{A}^{(2)}).

Choose a real number γ satisfying

\frac{β}{α} < γ < 1.

We conclude from Lemma 3.179 for λ = p^{−γ}

0 ≤ c(A, K)_{p} − b^{(2)}(R_{A}^{(2)}) ≤ F (K^{2} · p^{−γ}) − F (0) + (1 − p^{−γ})^{p} · (n − F (0)).

By the definition of α(R_{A}^{(2)}) there is δ > 0 such that we have for 0 < λ < δ

F (λ) − F (0) ≤ λ^{α}.

The last two inequalities imply for p satisfying K^{2}p^{−γ} < δ

0 ≤ p^{β} · (c(A, K)_{p} − b^{(2)}(R_{A}^{(2)}))
≤ p^{β} · ((K^{2} · p^{−γ})^{α} + (1 − p^{−γ})^{p} · (n − F (0))) .  (3.180)

We get using l’Hospital’s rule

lim_{x→∞} x · ln(1 − x^{−γ}) = −∞;
lim_{x→∞} \frac{ln(x)}{x ln(1 − x^{−γ})} = 0;
lim_{x→∞} (\frac{β ln(x)}{x ln(1 − x^{−γ})} + 1) = 1;
lim_{x→∞} β ln(x) + x ln(1 − x^{−γ}) = −∞;
lim_{x→∞} x^{β}(1 − x^{−γ})^{x} = 0.

Since β − γα < 0 holds we have lim_{x→∞} x^{β}(K^{2} · x^{−γ})^{α} = 0. Hence we get

lim_{p→∞} p^{β} · ((K^{2} · p^{−γ})^{α} + (1 − p^{−γ})^{p} · (n − F (0))) = 0.

This implies using the inequality (3.180)

lim_{p→∞} p^{β} · (c(A, K)_{p} − b^{(2)}(R_{A}^{(2)})) = 0.

We have shown β ≤ β(A). Since β was an arbitrary number satisfying 0 < β < α(R_{A}^{(2)}), assertion (3) follows.

(4) We get the chain of equations where a sum or integral is put to be −∞ if it does not converge, and {E_{λ} | λ} is the spectral family of (R_{A}^{(2)})^{∗}R_{A}^{(2)}

2 · (n − b^{(2)}(R_{A}^{(2)})) · ln(K) − ∑_{p=1}^{∞} \frac{1}{p} · (c(A, K)_{p} − b^{(2)}(R_{A}^{(2)}))

= 2 · (n − b^{(2)}(R_{A}^{(2)})) · ln(K) −

∑_{p=1}^{∞} \frac{1}{p} · (tr_{N (G)} ((1 − K^{−2} · (R_{A}^{(2)})^{∗}R_{A}^{(2)})^{p}) − b^{(2)}(R_{A}^{(2)}))

= 2 · (n − b^{(2)}(R_{A}^{(2)})) · ln(K)

− ∑_{p=1}^{∞} \frac{1}{p} · tr_{N (G)} (∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}} (1 − K^{−2} · λ)^{p} dE_{λ}) .

The trace is linear, monotone and ultra-weakly continuous.

= 2 · (n − b^{(2)}(R_{A}^{(2)})) · ln(K) − ∑_{p=1}^{∞} \frac{1}{p} · ∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}} (1 − K^{−2} · λ)^{p} d tr_{N (G)}(E_{λ})

= 2 · (n − b^{(2)}(R_{A}^{(2)})) · ln(K) − ∑_{p=1}^{∞} \frac{1}{p} · ∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}} (1 − K^{−2} · λ)^{p} dF

We can put the sum under the integral sign because of Levi’s Theorem of Monotone Convergence since (1 − K^{−2} · λ)^{p} is non-negative for 0 < λ ≤ ||R_{A}^{(2)}||_{∞}^{2} ≤ K^{2}.

= 2 · (n − b^{(2)}(R_{A}^{(2)})) · ln(K) − ∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}} ∑_{p=1}^{∞} \frac{1}{p} · (1 − K^{−2} · λ)^{p} dF

The Taylor series − ∑_{p=1}^{∞} \frac{1}{p} ·(1 − μ)^{p} of ln(μ) about 1 converges for |1−μ| < 1.

= 2 · (n − b^{(2)}(R_{A}^{(2)})) · ln(K) + ∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}} ln(K^{−2} · λ) dF
= 2 · (n − b^{(2)}(R_{A}^{(2)})) · ln(K) + ∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}} ln(λ) dF − ∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}} ln(K^{2}) dF
= 2 · (n − b^{(2)}(R_{A}^{(2)})) · ln(K) + ∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}} ln(λ) dF − ln(K^{2}) · (n − b^{(2)}(R_{A}^{(2)}))
= ∫_{0+}^{||R_{A}^{(2)}||_{∞}^{2}} ln(λ) dF = ∫_{0+}^{∞} ln(λ) dF
= ln(det(R_{A}^{(2)})^{∗}R_{A}^{(2)}) = 2 · ln(det(R_{A}^{(2)})),

where the last equation follows from Lemma 3.15 (4). This finishes the proof of assertion (4).

(5) Let α be any number satisfying α < α(R_{A}^{(2)}). Then we conclude from assertion (3) lim_{p→∞} p^{α} (c(A, K)_{p} − b^{(2)}(R_{A}^{(2)})) = 0. Let C be any positive

number such that p^{α} (c(A, K)_{p} − b^{(2)}(R_{A}^{(2)})) ≤ C holds for all p. We conclude

0 ≤ c(A, K)_{p} − b^{(2)}(R_{A}^{(2)}) ≤ \frac{C}{p^{α}} .

Next we estimate using assertion (4)

0 ≤ −2 · ln(det(R_{A}^{(2)})) + 2(n − b^{(2)}(R_{A}^{(2)})) · ln(K)

− ∑_{p=1}^{L} \frac{1}{p} · (c(A, K)_{p} − b^{(2)}(R_{A}^{(2)}))

= ∑_{p=L+1}^{∞} \frac{1}{p} · (c(A, K)_{p} − b^{(2)}(R_{A}^{(2)}))

= ∑_{p=L+1}^{∞} p^{−1−α} · (p^{α} · (c(A, K)_{p} − b^{(2)}(R_{A}^{(2)})))

≤ C · ∑_{p=L+1}^{∞} p^{−1−α}

≤ C · ∫_{L}^{∞} x^{−1−α}dx
= \frac{C}{α} · L^{−α}.

This finishes the proof of Theorem 3.172. ⊓ ⊔

Remark 3.181. We conjecture that the inequality α(R_{A}^{(2)}) ≤ β(A) of Theorem 3.172.3 is an equality where we do not distinguish between ∞ and ∞^{+} as value for α(R_{A}^{(2)}). If the spectral density function F of R_{A}^{(2)} has the limit property (see Definition 2.41), then this is true.

We will give more explicit calculations for 3-manifolds later which depend only on a presentation of the fundamental group (see Theorem 4.9).

3.8 Miscellaneous

The following result is taken from [106, Proposition 6.4 on page 149].

Lemma 3.182. Let M be a simply connected Riemannian manifold and f: M → ℝ be a function which is invariant under the isometries of M. Then there is a constant C(f) with the property that for any cocompact free proper action of a discrete group G by isometries and any fundamental domain F

∫_{F} f dvol_{M} = C(f) · vol(G\M)

holds.

Lemma 3.182 is obvious if M is homogeneous since then G acts transitively on M and hence any function on M which is invariant under the isometries of M is constant. Lemma 3.182 implies the following proportionality principle.

Theorem 3.183 (Proportionality Principle for L^{2}-invariants). Let M be a simply connected Riemannian manifold. Then there are constants B_{p}^{(2)}(M) for p ≥ 0, A_{p}^{(2)}(M) for p ≥ 1 and T^{(2)}(M) depending only on the Riemannian manifold M with the following property: For any discrete group G with a cocomapct free proper action on M by isometries the following holds

b_{p}^{(2)}(M; N (G)) = B_{p}^{(2)}(M) · vol(G\M);
α_{p}^{(2)}(M; N (G)) = A_{p}^{(2)}(M);
ρ^{(2)}(M; N (G)) = T^{(2)}(M) · vol(G\M),

where for the third equality we assume that M with this G-action (and hence for all cocompact free proper actions) is of determinant class.

Proof. Obviously the following function

M → ℝ, x ↦ tr_{ℝ}(E_{λ}^{(δ^{p+1}d^{p})_{min}^{⊥}}(x, x))

is invariant under isometries since d^{p} and δ^{p} are compatible with isometries and hence the endomorphisms E_{λ}^{(δ^{p+1}d^{p})_{min}^{⊥}}(x, x) and E_{λ}^{(δ^{p+1}d^{p})_{min}^{⊥}}(ϕ(x), ϕ(x)) are conjugate for any isometry ϕ: M → M and x ∈ X. Let C_{p}(M)(λ) be the constants associated to this function for each λ ≥ 0 in Lemma 3.182. We conclude from Lemma 3.182 and Lemma 2.66

F_{p}(M; N (G))(λ) = C_{p}(M)(λ) · vol(G\M),

where F (M; N (G)) is the analytic spectral density function associated to the cocompact free proper G-manifold M with G-invariant Riemannian metric. Notice that C_{p}(M)(λ) is independent of G. Now the claim for the L^{2}- Betti numbers and the Novikov-Shubin invariants follow. Analogously we conclude that there is also a function Θ_{p}(M)(t) depending only on M such that

∫_{F} tr_{ℝ}(e^{−tΔ_{p}}(x, x)) dvol_{x} = Θ_{p}(M)(t) · vol(G\M)

holds. Now the claim for the L^{2}-torsion follows. ⊓ ⊔

Another proportionality principle for L^{2}-Betti numbers will be given in Theorem 7.34 and Corollary 7.37.

Remark 3.184. We call a finite free G-CW -complex X NS-L^{2}-acyclic if b_{p}^{(2)}(X) = 0 and α_{p}(X) > 0 for all p ≥ 0. Notice that NS-L^{2}-acyclic implies det-L^{2}-acyclic because of Theorem 3.14 (4). We conclude from Remark 3.90 that all the results of Section 3.4 remain true if one replaces det-L^{2}-acyclic by NS-L^{2}-acyclic. The point is that the property NS-L^{2}-acyclic is enherited like the property det-L^{2}-acyclic.

The next result is well-known.

Lemma 3.185. A closed hyperbolic manifold does not carry a non-trivial S^{1}-action.

Proof. Suppose that the closed hyperbolic manifold M carries a non-trivial S^{1}-action. Since the universal covering of M is the hyperbolic space and hence contractible, M̃ is det-L^{2}-acyclic and ρ^{(2)}(M̃) = 0 by Corollary 3.111. This contradicts Theorem 1.62 if dim(M) is even and Theorem 3.152 if n is odd. ⊓ ⊔

One may ask whether Theorem 3.111 extends from S^{1}-actions to S^{1}- foliations, namely

Question 3.186. (S^{1}-foliations and L^{2}-torsion for closed aspherical manifolds). Let M be an aspherical closed manifold which admits an S^{1}-foliation. Is then M̃ of determinant class and

b_{p}^{(2)}(M̃ = 0  for all p;
α_{p}(M̃) ≥ 1  for all p;
ρ^{(2)}(M̃) = 0 ?

If the answer is positive, one would get a negative answer to the (to the author’s knowledge) open problem whether there is a S^{1}-foliation on a closed hyperbolic manifold.

The notion of spectral density function makes also sense for proper G-CW -complexes of finite type because in this setting the cellular L^{2}-chain complex is still defined as a Hilbert N (G)-chain complex as explained in Section 1.6. Hence the notion of det-L^{2}-acyclic and of L^{2}-torsion ρ^{(2)}(X) for a proper finite G-CW -complex in the sense of Definition 3.91 and the notion of topological L^{2}-torsion ρ_{top}^{(2)}(M) for a cocompact proper G-manifold M with G-invariant Riemannian metric in the sense of Definition 3.120 are defined. Theorem 3.93 (1) remains true if one replaces Wh(G) by K_{1}(ℚG)/T (G), where T (G) is the subgroup generated by elements represented by automorphisms of the shape ±ℚ[ϕ]: ℚ[G/H] → ℚ[G/H] for finite subgroups H ⊂ G and G-maps ϕ: G/H → G/H, and the Whitehead torsion τ(f) by the corresponding element in K_{1}(ℚG)/T (G) given by the ℚG-chain homotopy equivalence ℚ⊗_{ℤ}C_{∗}(f): ℚ⊗_{ℤ}C_{∗}(X) → ℚ⊗_{ℤ}C_{∗}(Y). Notice that the map induced by the logarithm of the Fuglede-Kadison determinant Φ^{G}: K_{1}(ℚG)/T (G) → ℝ is definitely non-trivial. But it may still be true that Φ^{G}(τ(f)) is always trivial for a G-homotopy equivalence f and hence that ρ^{(2)}(X) is a G-homotopy invariant for det-L^{2}-acyclic finite proper G-CW -complexes. This can be proved, provided that G is of det ≥ 1-class or if G belongs to the class G (see Definition 13.9). Theorem 3.93 (2), (3), (4), (5) and (6) carry over word by word for a det-L^{2}-acyclic finite proper G-CW -complex. The proof of Poincaré duality stated in Theorem 3.93 (3) is non-trivial because the Poincaré ℚGchain homotopy equivalence ℚ ⊗_{ℤ} C^{n−∗}(M) → ℚ ⊗_{ℤ} C_{∗}(M) has in general non-trivial Whitehead torsion in K_{1}(ℚG)/T (G) [327, Definition 3.19 and Example 3.25], but one can show that its image under the homomorphism K_{1}(ℚG)/T (G) → ℝ given by the logarithm of the Fuglede-Kadison determinant is trivial. The notion of analytic L^{2}-torsion ρ_{an}^{(2)}(M) for a cocompact proper G-manifold with G-invariant Riemannian metric introduced in Definition 3.128 still makes sense. Without having checked the details we claim that Theorem 3.149 of Burghela, Friedlander, Kappeler and McDonald [84] is still true if one drops the assumption free.

One can try to get an improved L^{2}-torsion working in K-theory, instead of applying the Fuglede-Kadison determinant from the very beginning. Fix a set W of morphisms of Hilbert N (G)-modules f: l^{2}(G)^{n} → l^{2}(G)^{n} with the following properties: (i) Any isomorphism l^{2}(G)^{n} → l^{2}(G)^{n} belongs to W, (ii) any element in W is a weak isomorphism. (iii) If two of the morphisms f, g and g ∘ f belong to W, then also the third. (iv) If both f ∘ g and g ∘ f belong to W, then f and g belong to W. (v) If two of the morphisms

f_{1} : l^{2}(G)^{m} → l^{2}(G)^{m}, f_{2} : l^{2}(G)^{n} → l^{2}(G)^{n} and ( \begin{matrix}f_{1} & g \\ 0 & f_{2} \end{matrix} ) belong to W, then

all three. Define K_{1}^{W}(N (G)) to be the abelian group whose generators are classes [f] of endomorphisms f: l^{2}(G)^{n} → l^{2}(G)^{n} belonging to W such that the following relations are satisfied: (i) [id: l^{2}(G) → l^{2}(G)] = 0, (ii) [g ∘ f] =

[f] + [g] for f, g : l^{2}(G)^{n} → l^{2}(G)^{n} in W and (iii) [( \begin{matrix}f_{1} & g \\ 0 & f_{2} \end{matrix} )] = [f_{1}] + [f_{2}] for

f_{1}, f_{2} ∈ W. Let K̃_{1}^{W}(N (G)) be the quotient of K_{1}^{W}(N (G)) by the subgroup generated by [− id: l^{2}(G) → l^{2}(G)].

We call a finite Hilbert N (G)-chain complex C_{∗} W-acyclic if there is a weak chain contraction (γ_{∗}, u_{∗}) in the sense of Definition 3.38, where we now require that u_{p}: C_{p} → C_{p} belongs to W. Then we can associate to a finite W-acyclic Hilbert N (G)-chain complex C_{∗} analogously to the formulas appearing in Lemma 3.41 an element

ρ^{W}(C_{∗}) := [(uc + γ)_{odd}] − [u_{odd}] ∈ K̃_{1}^{W}(N (G)).  (3.187)

By the same arguments we used for the L^{2}-torsion one shows that ρ^{W}(C_{∗}) is well-defined, i.e. independent of the choice of (γ_{∗}, u_{∗}) and that a lot of the good properties remain true. For instance, if 0 → C_{∗} → D_{∗} → E_{∗} → 0 is an exact sequence of finite Hilbert N (G)-chain complexes and two of them are W-acyclic, then all three are W-acyclic and we get analogously to Lemma 3.68

ρ^{W}(C_{∗}) − ρ^{W}(D_{∗}) + ρ^{W}(E_{∗}) = ρ^{W}(C_{∗}, D_{∗}, E_{∗}).  (3.188)

The problem is to find out how much information K_{1}^{W}(N (G)) contains depending of the various choices of W.

If we take for instance W to be the class of isomorphisms, K̃_{1}^{W}(N (G)) is just the ordinary K̃_{1}(N (G)) of the ring N (G) which has been computed in [344] (see also Subsection 9.2.2). However, W-acyclic means in this situation that C_{∗} is contractible and this condition is extremely restrictive as we will see in Chapter 12 on the Zero-in-the-Spectrum-conjecture. If we take W to be the class of weak isomorphisms of determinant class, then the logarithm of the Fuglede-Kadison determinant gives a homomorphism det: K̃_{1}^{W}(N (G)) → ℝ which maps ρ^{W}(C_{∗}) to ρ^{(2)}(C_{∗}). This homomorphism is split surjective but we do not know whether it is bijective. Finally we discuss the case where W consists of all weak isomorphisms. Then K_{1}^{W}(N (G)) can be identified with K_{1}^{inj}(N (G)) (see Definition 9.16). If G contains ℤ^{n} as subgroup of finite index, K_{1}^{W}(N (G)) has been computed in [345] (see also Section 9.3) and there it is shown that ρ^{W}(C_{∗}) contains essentially the same information as the classical Alexander polynomial. If G is finitely generated and does not contain ℤ^{n} as subgroup of finite index, then K_{1}^{W}(N (G)) = 0 [344] (see also Section 9.3) and hence ρ^{W}(C_{∗}) carries no information. This shows why a condition such as of determinant class has to appear to get a meaningful invariant.

Lott [321] defines analytically delocalized L^{2}-torsion for the universal covering M̃ of a closed Riemannian manifold M, which gives a number for each conjugacy class of the fundamental group π_{1}(M). The value of the conjugacy class of the unit element is the L^{2}-torsion. This invariant is presently only defined under certain technical convergence assumptions. At least for universal coverings of closed hyperbolic manifolds of odd dimension the delocalized L^{2}-torsion is defined and the marked length spectrum can be recovered from it.

Formulas for the variation of the L^{2}-torsion under varying the Riemannian metric are given for instance in [316, page 480] and [346, section 7].

We will give further computations of L^{2}-torsion for universal coverings of compact 3-manifolds in Theorem 4.3, of knot complements in Section 4.3 and of closed locally symmetric spaces in Theorem 5.12 and in (5.13).

We will discuss the behaviour of L^{2}-Betti numbers, Novikov-Shubin invariants and L^{2}-torsion of groups under quasi-isometry and measure equivalence in Section 7.5.

In Chapter 11 we will deal with the Conjecture 11.3 which makes a prediction about the parity of the L^{2}-torsion of the universal covering of a closed manifold M, provided M is aspherical or carries a Riemannian metric with negative sectional curvature.

We will relate the L^{2}-torsion of the universal covering of an aspherical closed orientable manifold to its simplicial volume in Chapter 14.

Further references about L^{2}-torsion are [92], [93], [137], [138], [139], [360], [362], [363].

Exercises

3.1. Let C_{∗} be a finite free ℤ-chain complex. Choose a ℤ-basis for C_{p} and for H_{p}(C_{∗})/ tors(H_{p}(C_{∗})) for each p ∈ ℤ. They induce ℝ-bases and thus Hilbert space structures on C_{p} ⊗_{ℤ} ℝ and on H_{p}(C_{∗} ⊗_{ℤ} ℝ) ≅ H_{p}(C_{∗})⊗_{ℤ} ℝ for all p ∈ ℤ. Let ρ^{ℤ}(C_{∗}) be the real number ρ(C_{∗} ⊗_{ℤ} ℝ) whose definition is the obvious variation of (3.5). Prove

ρ^{ℤ}(C_{∗}) = ∑_{p∈ℤ}(−1)^{p} · ln (|tors(H_{p}(C_{∗}))|) .

3.2. Let X and Y be finite CW -complexes. Let j_{i}: X_{i} → X and k_{i}: Y_{i} → Y be inclusions of CW -subcomplexes for i = 0, 1, 2 such that X = X_{1} ∪ X_{2}, X_{0} = X_{1} ∩ X_{2}, Y = Y_{1} ∪ Y_{2} and Y_{0} = Y_{1} ∩ Y_{2} holds. Let f: X → Y be a map which induces homotopy equivalences f_{i}: X_{i} → Y_{i} for i = 0, 1, 2. Show that f is a homotopy equivalence and

τ(f̃) = k_{1∗}(τ(f̃_{1})) + k_{2∗}(τ(f̃_{2})) − k_{0∗}(τ(f̃_{0})).

3.3. Let f: X → Y and g: Y → Z be homotopy equivalences of finite CW -complexes. Prove

τ({ \atop g ∘ f}) = τ(g̃) + g_{∗}(τ(f̃)).

3.4. Let f: X′ → X and g: Y′ → Y be homotopy equivalences of connected finite CW -complexes. Denote by k_{X}: X → X × Y and k_{Y} : Y → X × Y the canonical inclusions for some choice of base points in X and Y. Prove

τ({ \atop f × g}) = χ(X) · k_{Y ∗}(τ(g̃)) + χ(Y) · k_{X∗}(τ(f̃)).

3.5. Show for S^{n} equipped with some Riemannian metric

ρ_{top}(S^{n}; ℝ) = \frac{(1 + (−1)^{n+1})}{2} · ln(vol(S^{n})).

3.6. Equip ℝ with the standard Riemannian metric and ℤ-operation. Denote by V the trivial orthogonal 1-dimensional ℤ-representation whose underlying vector space is ℝ with the standard Hilbert space structure. Let ζ_{R}(s) = ∑_{n≥1} n^{−s} be the Riemannian Zeta-function. Show for the Zetafunction ζ_{1}(ℝ; V)(s) defined in (3.8)

ζ_{1}(ℝ; V)(s) = 2 · (2π)^{−2s} · ζ_{R}(2s).

3.7. Define a function F: ℝ → [0, ∞) by F (λ) = \frac{1}{− ln(− ln(λ))·ln(λ)} for 0 < λ ≤ e^{−e}, F (λ) = e^{−1} for λ ≥ e^{−e} and F (λ) = 0 for λ ≤ 0. Show that F is a density function with

lim_{λ→0+} ln(λ) · F (λ) = 0;
∫_{0+}^{1} ln(λ) dF = −∞.

3.8. Let A be a (k, k)-matrix over ℂ[ℤ^{n}]. Denote its determinant over ℂ[ℤ^{n}] by det_{ℂ[ℤ^{n}]}(A) ∈ ℂ[ℤ^{n}]. Let R_{A}: l^{2}(ℤ^{n})^{k} → l^{2}(ℤ^{n})^{k} and R_{det_{ℂ[ℤ^{n}]}(A)} : l^{2}(ℤ^{n}) → l^{2}(ℤ^{n}) be the morphism given by right multiplication with A and det_{ℂ[ℤ]}(A). Show that the following statements are equivalent: i.) R_{A} is a weak isomorphism, ii.) R_{det_{ℂ[ℤ^{n}]}(A)} is a weak isomorphism and iii.) det_{ℂ[ℤ^{n}]}(A) ≠ 0. Show that in this case det(R_{A}) = det(R_{det_{ℂ[ℤ^{n}]}(A)}).

3.9. Let X be a finite free ℤ-CW -complex. Since ℂ[ℤ] is a principal ideal domain [15, Proposition V.5.8 on page 151 and Corollary V.8.7 on page 162], we can write

H_{p}(X; ℂ) = ℂ[ℤ]^{n_{p}} ⊕ (⨁_{i_{p}=1}^{s_{p}} ℂ[ℤ]/((z − a_{p,i_{p}})^{r_{p,ip}}))

for a_{p,i_{p}} ∈ ℂ and n_{p}, s_{p}, r_{p,i_{p}} ∈ ℤ with n_{p}, s_{p} ≥ 0 and r_{p,i_{p}} ≥ 1 where z ∈ ℤ is a fixed generator.

Prove for p ≥ 0

b_{p}^{(2)}(X) = n_{p};
α_{p+1}(X) = \{ \begin{matrix}min\{\frac{1}{r_{p,ip}} | i_{p} = 1, 2 …, s_{p}, |a_{p,i_{p}}| = 1\} \\ if s_{p} ≥ 1 and \{i_{p} = 1, 2 …, s_{p}, |a_{p,i_{p}}| = 1\} ≠ ∅ \\ ∞^{+} & otherwise \end{matrix} .

Show that X is det-L^{2}-acyclic if and only if n_{p} = 0 for all p ≥ 0.

Prove that ρ^{(2)}(X) cannot be read off from the ℂ[ℤ]-modules H_{p}(X; ℂ) for all p ∈ ℤ, but that ρ^{(2)}(X) is determined by the ℤ[ℤ]-modules H_{p}(X; ℂ) for p ∈ ℤ and can be written as ln(|a|) − ln(|b|) for algebraic integers a, b ∈ ℂ with |a|, |b| ≥ 1.

3.10. Give an example of finite Hilbert N (G)-chain complexes C_{∗} and D_{∗} which are both of determinant class and a weak homology equivalence f_{∗}: C_{∗} → D_{∗} such that f_{p} is of determinant class for all p ∈ ℤ, but neither f_{∗} is of determinant class nor H_{p}^{(2)}(f_{∗}) is of determinant class for all p ∈ ℤ. (cf. Theorem 3.35 (5) and Lemma 3.44).

3.11. Let (C_{∗,∗}, d_{∗,∗}) be a bicomplex of finitely generated Hilbert N (G)- chain modules such that C_{p,q} = 0 for |p|, |q| ≥ N holds for some number N. Suppose for p ∈ ℤ that the chain complex C_{p,∗} given by the p-th column and the chain complex C_{∗,q} given by the q-th row are det-L^{2}-acyclic. Let T_{∗} be the associated total chain complex with T_{n} = ⨁_{p+q=n} C_{p,q}. Show that T_{∗} is det-L^{2}-acyclic and

∑_{p∈ℤ}(−1)^{p} · ρ^{(2)}(C_{p,∗}) = ρ^{(2)}(T_{∗}) = ∑_{q∈ℤ}(−1)^{q} · ρ^{(2)}(C_{∗,q}).

3.12. Show that the following statements are equivalent:

(1) The map Φ^{G}: Wh(G) → ℝ of (3.92) is trivial for all groups G;

(2) For all finitely generated groups G and G-homotopy equivalences f: X → Y of det-L^{2}-acyclic free finite G-CW -complexes ρ^{(2)}(X) = ρ^{(2)}(Y) holds;

(3) For all finitely generated groups G and G-homotopy equivalences f: M → N of det-L^{2}-acyclic cocompact free proper G-manifolds with G-invariant Riemannian metric and without boundary ρ_{an}^{(2)}(M) = ρ_{an}^{(2)}(N) holds.

3.13. Show that the composition of the obvious map given by induction ⨁_{H⊂G,|H|<∞} Wh(H) → Wh(G) with ϕ^{G}: Wh(G) → ℝ is trivial.

3.14. Let G be a countable group. Show that the following sets are countable

{b_{p}^{(2)}(X; N (G)) | X connected free G-CW -complex of finite type, p ≥ 0};
{α_{p}^{(2)}(X; N (G)) | X connected free G-CW -complex of finite type, p ≥ 1};
{ρ^{(2)}(X; N (G)) | X det -L^{2}-acyclic connected finite free G-CW -complex}.

Show that the first set is closed under addition in ℝ^{≥0}, and that the third set is an additive subgroup of ℝ provided that the third set is non-empty.

3.15. Show that the following sets are countable

{b_{p}^{(2)}(X̃) | X connected CW -complex of finite type, p ≥ 0};
{α_{p}(X̃) | X connected CW -complex of finite type, p ≥ 1};
{ρ^{(2)}(X̃) | X det -L^{2}-acyclic connected finite CW -complex}.

3.16. Give an example of a fibration of connected finite CW -complexes which is not simple, i.e. θ(p) ≠ 0.

3.17. Let f: X → X be a selfmap of a connected finite CW -complex and let T_{f} be its mapping torus. Let L be the colimit of the system

… ⟶^{π_{1}(f)} π_{1}(X) ⟶^{π_{1}(f)} π_{1}(X) ⟶^{π_{1}(f)} …

There is a canonical epimorphism ι: π_{1}(X) → L and an automorphism μ : L → L satisfying μ ∘ ι = ι ∘ π_{1}(f). Show that π_{1}(T_{f}) is isomorphic to the semidirect product L⋊_{μ}ℤ. Prove that ι is bijective and π_{1}(T_{f}) ≅ π_{1}(X)⋊_{π_{1}(f)} ℤ, provided that π_{1}(f) is bijective.

3.18. Let p: E → S^{1} be a fibration with fiber S^{n} ∨ S^{n} for n ≥ 2. Suppose that π_{1}(S^{1}) = ℤ acts on H_{n}(S^{n} ∨ S^{n}; ℤ) by an automorphism H_{1}(f) with determinant 1. Let tr ∈ ℤ be the trace of H_{1}(f). Show that all L^{2}-Betti numbers of T̃_{f} vanish, α_{1}(T̃_{f}) = 1, α_{p}(T̃_{f}) = ∞^{+} for p ≠ 1, n + 1 and that precisely one of the following cases occurs

(1) H_{1}(f) is periodic. Then tr ∈ {−1, 0, +1} or H_{1}(f) = ± id and we have α_{n+1}(T̃_{f}) = 1 and ρ^{(2)}(T̃_{f}) = 0;

(2) H_{1}(f) is parabolic, i.e. H_{1}(f) is not periodic and all complex eigenvalues have norm 1. Then we have tr ∈ {−2, +2}, α_{n+1}(T̃_{f}) = 1/2 and ρ^{(2)}(T̃_{f}) = 0;

(3) H_{1}(f) is hyperbolic, i.e. there is one (and hence two) complex eigenvalue whose norm is not 1. Then | tr | > 2, α_{n+1}(T̃_{f}) = ∞^{+} and ρ^{(2)}(T̃_{f}) =

ln (\frac{tr}{2} + \sqrt{\frac{tr^{2}}{4} − 1}).

3.19. Give examples of two fibrations F → E_{0} → B and F → E_{1} → B of connected finite CW -complexes with the same fiber and base space such that the fiber is simply connected, Ẽ_{0} and Ẽ_{1} are of determinant class and have trivial L^{2}-Betti numbers, but Ẽ_{0} and Ẽ_{1} have both different Novikov-Shubin invariants and different L^{2}-torsion.

3.20. Let F: [0, ∞) → [0, ∞) be a density function with F (0) = 0 for which there is K > 0 with F (λ) = F (K). Let θ_{F} be its Laplace transform. Fix ϵ > 0. Show:

(1) We have

θ_{F} (t) = ∑_{n≥0} a_{n}t^{n}

for a power series ∑_{n≥0} a_{n}t^{n} which converges for all t ∈ ℝ and satisfies a_{0} = F (K);

(2) The function

\frac{1}{Γ (s)} · ∫_{0}^{ϵ} t^{s−1}θ_{F} (t) dt

is holomorphic for ℜ(s) > 0 and has a meromorphic extension to ℂ which has poles at s = −1, −2, −3, …, all of order 1;

(3) Suppose 0 < α(F). The function

\frac{1}{Γ (s)} · ∫_{ϵ}^{∞} t^{s−1} · θ_{F} (t) dt

is holomorphic for ℜ(s) < α(F). We have

\frac{d}{ds} \frac{1}{Γ (s)} · ∫_{ϵ}^{∞} t^{s−1}θ_{F} (t) dt|_{s=0} = ∫_{ϵ}^{∞} t^{−1} · θ_{F} (t) dt < ∞;

(4) Suppose 0 < α(F). Then the function ∫_{0}^{∞} λ^{−s} dF (λ) is holomorphic for ℜ(s) < α(F) and we get

\frac{d}{ds} ∫_{0}^{∞} λ^{−s} dF (λ)|_{s=0} = ∫_{0}^{∞} ln(λ) dF (λ) < ∞;

(5) We have for 0 < ℜ(s) < α(F)

\frac{1}{Γ (s)} · ∫_{0}^{∞} t^{s−1} · θ_{F} (t) dt = ∫_{0}^{∞} λ^{−s} dF (λ);

(6) If 0 < α(F), we get

\frac{d}{Dr} \frac{1}{Γ (s)} ∫_{0}^{ϵ} t^{s−1} · θ_{F} (t) du|_{s=0} + ∫_{ϵ}^{∞} t^{−1} · θ_{F} (t) dt

= \frac{d}{ds} \frac{1}{Γ (s)} ∫_{0}^{∞} t^{s−1} · θ_{F} (t) dt|_{s=0}

= ∫_{0}^{∞} ln(λ) dF (λ) < ∞.

3.21. Show directly using Definition 3.128 that ρ_{an}^{(2)}(S̃^{1}) = 0.

3.22. Compute ρ_{an}([0, 1]) = \frac{ln(2)}{2} and ρ_{top}([0, 1]) = 0 for [0, 1] equipped with the standard Riemannian metric directly from the definitions without using Theorem 3.161 but using the facts for the Riemannian Zeta-function ξ_{R}(s) that ξ_{R}(0) = −1/2 and \frac{d}{ds}ξ_{R}(0) = − ln(2)/2.

3.23. Let M be a cocompact proper free G-manifold of even dimension which is orientable. Suppose that M is det-L^{2}-acyclic. Show that ∂M is det-L^{2}- acyclic and

ρ^{(2)}(M) = \frac{ρ^{(2)}(∂M)}{2}.

3.24. Compute the characteristic sequence c(A, K)_{p} of the (1, 1)-matrix (z − λ) over ℂ[ℤ] for z ∈ ℤ a fixed generator and real numbers λ ≥ 0 and K = λ+1 and conclude

∑_{p=1}^{∞} \frac{1}{p} · (\frac{(2 · λ)^{p}}{(1 + λ)^{2p}} · ∑_{k=0}^{[p/2]} 4^{−k} · \frac{p!}{(p − 2k)! · k! · k!})

= \{ \begin{matrix}2 · ln(λ + 1) − 2 · ln(|λ|) & if λ ≥ 1 \\ 2 · ln(λ + 1) & if λ ≤ 1 \end{matrix} .

4. L^{2}-Invariants of 3-Manifolds

Introduction

In this section we compute all L^{2}-Betti numbers and the L^{2}-torsion and give some values and estimates for the Novikov-Shubin invariants for the universal covering of a compact connected orientable 3-manifold. This will use both the general properties of these L^{2}-invariants which we have developed in the preceding chapters and the geometry of 3-manifolds. In particular our computations will be based on Thurston’s Geometrization Conjecture. The necessary input of the theory of 3-manifolds will be given in Section 4.1 and the actual computations and sketches of their proofs in Section 4.2. In our opinion they combine analytic, geometric and topological methods in a beautiful way. Moreover, these computations will give evidence for various general conjectures about L^{2}-invariants such as Conjecture 2.82 about the positivity and rationality of Novikov-Shubin invariants, the Strong Atiyah Conjecture 10.2, the Singer Conjecture 11.1, Conjecture 11.3 about the parity of the L^{2}-torsion of the universal covering of an aspherical closed manifold and the zero-in-the-spectrum Conjecture 12.1.

4.1 Survey on 3-Manifolds

In this section we give a brief survey about connected compact orientable 3-manifolds. For more information we refer for instance to [252], [466], [491], [492].

In the sequel 3-manifold means connected compact orientable 3-manifold possibly with boundary. A 3-manifold M is prime if for any decomposition of M as a connected sum M_{1}#M_{2}, M_{1} or M_{2} is homeomorphic to S^{3}. It is irreducible if every embedded 2-sphere bounds an embedded 3-disk. Any prime 3-manifold is either irreducible or is homeomorphic to S^{1} × S^{2} [252, Lemma 3.13]. A 3-manifold M has a prime decomposition, i.e. one can write M as a connected sum

M = M_{1}#M_{2}# … #M_{r},

where each M_{j} is prime, and this prime decomposition is unique up to renumbering and orientation preserving homeomorphism [252, Theorems 3.15, 3.21].

By the Sphere Theorem [252, Theorem 4.3], an irreducible 3-manifold is aspherical if and only if it is a 3-disk or has infinite fundamental group.

Given a 3-manifold M, a compact connnected orientable surface F which is properly embedded in M, i.e. ∂M ∩ F = ∂F, or embedded in ∂M, is called incompressible if it is not a 2-sphere and the inclusion F → M induces an injection on the fundamental groups. One says that ∂M is incompressible in M if and only if ∂M is empty or any component C of ∂M is incompressible in the sense above. An irreducible 3-manifold is Haken if it contains an embedded orientable incompressible surface. If H_{1}(M) is infinite, which is implied if ∂M contains a surface other than S^{2}, and M is irreducible, then M is Haken [252, Lemma 6.6 and 6.7].

The fundamental group plays a dominant role in the theory of 3-manifolds, as explained by the next results. Let M be a 3-manifold with incompressible boundary whose fundamental group admits a splitting α: π_{1}(M) → Γ_{1} ∗ Γ_{2}. Kneser’s Conjecture, whose proof can be found in [252, chapter 7], says that there are manifolds M_{1} and M_{2} with Γ_{1} and Γ_{2} as fundamental groups and a homeomorphism M → M_{1}#M_{2} inducing α on the fundamental groups. Kneser’s conjecture fails even in the closed case in dimensions ≥ 5 by results of Cappell [87], [88] and remains true in dimension 4 stably but not unstably [298], [299].

Let (f, ∂f): (M, ∂M) → (N, ∂N) is a map of (compact connected orientable) Haken 3-manifolds such that π_{1}(f, x): π_{1}(M, x) → π_{1}(N, f(x)) and π_{1}(∂f, y): π_{1}(∂M, y) → π_{1}(∂N, f(y)) are isomorphisms for any choice of base points x ∈ M and y ∈ ∂M, then f is homotopic to a homeomorphism. This is a result of Waldhausen [252, Corollary 13.7 on page 148], [507]. One can read off from the fundamental group of a 3-manifold M whether M is the total space of a fiber bundle [252, Chapter 11]. One knows which finite groups or abelian groups occur as fundamental groups of 3-manifolds [252, Chapter 9]. Notice that for n ≥ 4 any finitely presented group is the fundamental group of a closed connected orientable n-dimensional manifold but not any finitely presented group occurs as the fundamental group of a compact connected 3-manifold. For instance the fundamental group of a 3-manifold whose prime factors are all non-exceptional is residually finite [253]. (The notion of exceptional 3-manifold will be introduced in Section 4.2.)

Recall that a manifold (possible with boundary) is called hyperbolic if its interior admits a complete Riemannian metric whose sectional curvature is constant −1. We use the definition of Seifert fibered 3-manifold or briefly Seifert manifold given in [466], which we recommend as a reference on Seifert manifolds. If a 3-manifold M has infinite fundamental group and empty or incompressible boundary, then it is Seifert if and only if it admits a finite covering \overline{M} which is the total space of a S^{1}-principal bundle over a compact orientable surface [466, page 436]. The work of Casson and Gabai shows that an irreducible 3-manifold with infinite fundamental group π is Seifert if and only if π contains a normal infinite cyclic subgroup [213, Corollary 2 on page 395].

A geometry on a 3-manifold M is a complete locally homogeneous Riemannian metric on its interior. The universal cover of the interior has a complete homogeneous Riemannian metric, meaning that the isometry group acts transitively [476]. Thurston has shown that there are precisely eight simply connected 3-dimensional geometries having compact quotients, namely S^{3}, ℝ^{3}, S^{2} × ℝ, H^{2} × ℝ, Nil, { \atop SL_{2}(ℝ)}, Sol and H^{3}. If a closed 3-manifold admits a geometric structure modelled on one of these eight geometries then the geometry involved is unique. In terms of the Euler class e of the Seifert bundle and the Euler characteristic χ of the base orbifold, the geometric structure of a closed Seifert manifold M is determined as follows [466, Theorem 5.3]

χ > 0 χ = 0 χ < 0
e = 0 S^{2} × ℝ ℝ^{3} H^{2} × ℝ
e ≠ 0 S^{3} Nil { \atop SL_{2}(ℝ)}

If M has a S^{3}-structure then π_{1}(M) is finite. In all other cases M is finitely covered by the total space \overline{M} of an S^{1}-principal bundle over an orientable closed surface F. Moreover, e(M) = 0 if and only if e(\overline{M}) = 0, and the Euler characteristic χ of the base orbifold of M is negative, zero or positive according to the same condition for χ(\overline{M}/S^{1}) [466, page 426, 427 and 436].

Next we summarize what is known about Thurston’s Geometrization Conjecture for irreducible 3-manifolds with infinite fundamental groups. (Again, our 3-manifolds are understood to be compact, connected and orientable.) Johannson [278] and Jaco and Shalen [276] have shown that given an irreducible 3-manifold M with incompressible boundary, there is a finite family of disjoint, pairwise-nonisotopic incompressible tori in M which are not isotopic to boundary components and which split M into pieces that are Seifert manifolds or are geometrically atoroidal, meaning that they admit no embedded incompressible torus (except possibly parallel to the boundary). A minimal family of such tori is unique up to isotopy, and we will say that it gives a toral splitting of M. We will say that the toral splitting is a geometric toral splitting if the geometrically atoroidal pieces which do not admit a Seifert structure are hyperbolic. Thurston’s Geometrization Conjecture for irreducible 3-manifolds with infinite fundamental groups states that such manifolds have geometric toral splittings.

For completeness we mention that Thurston’s Geometrization Conjecture says for a closed 3-manifold with finite fundamental group that its universal covering is homeomorphic to S^{3}, the fundamental group of M is a subgroup of SO(4) and the action of it on the universal covering is conjugated by a homeomorphism to the restriction of the obvious SO(4)-action on S^{3}. This implies, in particular, the Poincaré Conjecture that any homotopy 3-sphere is homeomorphic to S^{3}.

Suppose that M is Haken. The pieces in its toral splitting are certainly Haken. Let N be a geometrically atoroidal piece. The Torus Theorem says that N is a special Seifert manifold or is homotopically atoroidal, i.e. any subgroup of π_{1}(N) which is isomorphic to ℤ × ℤ is conjugate to the fundamental group of a boundary component. McMullen following Thurston has shown that a homotopically atoroidal Haken manifold is a twisted I-bundle over the Klein bottle (which is Seifert), or is hyperbolic [368]. Thus the only case in which Thurston’s Geometrization Conjecture for an irreducible 3-manifold M with infinite fundamental group is still open is when M is a closed non-Haken irreducible 3-manifold with infinite fundamental group which is not Seifert. The conjecture states that such a manifold is hyperbolic.

4.2 L^{2}-Invariants of 3-Manifolds

In this section we state the values of the various L^{2}-invariants for universal coverings of compact connected orientable 3-manifolds. Notice that the assumption orientable is not a serious restriction, since any non-orientable 3-manifold has a connected two-sheeted covering which is orientable and we know how the L^{2}-invariants behave under finite coverings (see Theorem 1.35 (9), Theorem 2.55 (6) and Theorem 3.96 (5)). Recall that we have already computed the L^{2}-invariants for the universal covering of a compact connected orientable surface F (see Example 1.36, Example 2.70 and Theorem 3.105). Notice in the context of L^{2}-torsion that the universal covering of a compact orientable surface F is L^{2}-acyclic if and only if F is T^{2} or S^{1} × D^{1}.

Let us say that a prime 3-manifold is exceptional if it is closed and no finite covering of it is homotopy equivalent to a Haken, Seifert or hyperbolic 3-manifold. No exceptional prime 3-manifolds are known, and Thurston’s Geometrization Conjecture and Waldhausen’s Conjecture that any 3-manifold is finitely covered by a Haken manifold imply that there are none. Notice that any exceptional manifold has infinite fundamental group.

Theorem 4.1 (L^{2}-Betti numbers of 3-manifolds). Let M be the connected sum M_{1}# … #M_{r} of (compact connected orientable) prime 3-manifolds M_{j} which are non-exceptional. Assume that π_{1}(M) is infinite. Then the L^{2}- Betti numbers of the universal covering M̃ are given by

b_{0}^{(2)}(M̃) = 0;
b_{1}^{(2)}(M̃) = (r − 1) − ∑_{j=1}^{r} \frac{1}{| π_{1}(M_{j}) |} + |\{C ∈ π_{0}(∂M) | C ≅ S^{2}\}| − χ(M);

b_{2}^{(2)}(M̃) = (r − 1) − ∑_{j=1}^{r} \frac{1}{| π_{1}(M_{j}) |} + |\{C ∈ π_{0}(∂M) | C ≅ S^{2}\}|;
b_{3}^{(2)}(M̃) = 0.

In particular, M̃ has trivial L^{2}-cohomology if and only if M is homotopy equivalent to RP^{3}#RP^{3} or a prime 3-manifold with infinite fundamental group whose boundary is empty or a union of tori.

Proof. We give a sketch of the strategy of proof. Details can be found in [322, Sections 5 and 6]. Since the fundamental group is infinite, we get b_{0}^{(2)}(M̃) = 0 from Theorem 1.35 (8). If M is closed, we get b_{3}^{(2)}(M̃) = 0 because of Poincaré duality (see Theorem 1.35 (3)). If M has boundary, it is homotopy equivalent to a 2-dimensional CW -complex and hence b_{3}^{(2)}(M̃) = 0. It remains to compute the second L^{2}-Betti number, because the first one is then determined by the Euler-Poincaré formula of Theorem 1.35 (2).

Using the formula for connected sums of Theorem 1.35 (6) we reduce the claim to prime 3-manifolds. Since a prime 3-manifold is either irreducible or S^{1} ×S^{2}, it remains to treat the irreducible case. If the boundary is compressible, we use the Loop Theorem [252, Theorem 4.2 on page 39] to reduce the claim to the incompressible case. By doubling M we can reduce the claim further to the case of an irreducible 3-manifold with infinite fundamental group and incompressible torus boundary. Because of the toral splitting and the assumptions about Thurston’s Geometrization Conjecture it suffices to show that the L^{2}-Betti numbers vanish if M is Seifert with infinite fundamental group or is hyperbolic with incompressible torus boundary. All these steps use the weakly exact Mayer-Vietoris sequence for L^{2}-(co)homology (see Theorem 1.21). In the Seifert case we can assume by the multiplicative property (see Theorem 1.35 (9)) that M is a S^{1}-principal bundle over a 2-dimensional manifold. Then we can apply Theorem 1.40.

The hyperbolic case follows directly from Theorem 1.62 provided that the manifold has no boundary. One of the hard parts in the proof is to reduce the case of a hyperbolic 3-manifold with incompressible torus boundary to the closed case by a careful analysis of the manifold near its boundary using explicit models and the fact that the volume is finite. ⊓ ⊔

Let χ_{virt}(π_{1}(M)) be the ℚ-valued virtual group Euler characteristic of the group π_{1}(M) in the sense of [69, IX.7], [509]. (This will be the same as the L^{2}-Euler characteristic of π_{1}(M) as explained in Remark 6.81). Then the conclusion in Theorem 4.1 is equivalent to

b_{1}^{(2)}(M̃) = −χ_{virt}(π_{1}(M));
b_{2}^{(2)}(M̃) = χ(M) − χ_{virt}(π_{1}(M)).

This is proved in [322, page 53 - 54].

Next we state what is known about the values of the Novikov-Shubin invariants of 3-manifolds (see [322, Theorem 0.1]).

Theorem 4.2 (Novikov-Shubin invariants of 3-manifolds). Let M be the connected sum M_{1}# … #M_{r} of (compact connected orientable) prime 3-manifolds M_{j} which are non-exceptional. Assume that π_{1}(M) is infinite. Then (1) We have α_{p}(M̃) > 0 for p ≥ 1;

(2) Let the Poincaré associate P (M) be the connected sum of the M_{j}’s which

are not 3-disks or homotopy 3-spheres. Then α_{p}({ \atop P (M)}) = α_{p}(M̃) for p ≤ 2. We have α_{1}(M̃) = ∞^{+} except for the following cases:

(a) α_{1}(M̃) = 1 if P (M) is S^{1} × D^{2}, S^{1} × S^{2} or homotopy equivalent to RP^{3}#RP^{3};

(b) α_{1}(M̃) = 2 if P (M) is T^{2} × I or a twisted I-bundle over the Klein bottle K;

(c) α_{1}(M̃) = 3 if P (M) is a closed ℝ^{3}-manifold;

(d) α_{1}(M̃) = 4 if P (M) is a closed Nil-manifold;

(e) α_{1}(M̃) = ∞ if P (M) is a closed Sol-manifold;

(3) If M is a closed hyperbolic 3-manifold then α_{2}(M) = 1. If M is a closed Seifert 3-manifold then α_{2}(M) is given in terms of the Euler class e of the bundle and the Euler characteristic χ of the base orbifold by

χ > 0 χ = 0 χ < 0
e = 0 ∞^{+} 3  1
e ≠ 0 ∞^{+} 2  1

If M is a Seifert 3-manifold with boundary then α_{2}(M) = ∞^{+} if M = S^{1} × D^{2}, α_{2}(M) = 2 if M is T^{2} × I or a twisted I-bundle over K, and α_{2}(M) = 1 otherwise. If M is a closed Sol-manifold then α_{2}(M) ≥ 1.

(4) If ∂M contains an incompressible torus then α_{2}(M̃) ≤ 2. If one of the M_{j}’s is closed with infinite fundamental group and does not admit an ℝ^{3}, S^{2} × ℝ or Sol-structure, then α_{2}(M̃) ≤ 2.

(5) If M is closed then α_{3}(M̃) = α_{1}(M̃). If M is not closed then α_{3}(M̃) = ∞^{+}.

Proof. The strategy of the proof is similar to the one of Theorem 4.1 using now Theorem 2.20, Theorem 2.55, Theorem 2.61, Theorem 2.68 and Theorem 3.183 together with explicit computations of heat kernels on the various spaces occuring in Thurston’s list of eight geometries with compact quotients in dimension 3. Details can be found in [322, Sections 5 and 6]. ⊓ ⊔

Finally we state the values for the L^{2}-torsion (see [346, Theorem 0.6]).

Theorem 4.3 (L^{2}-torsion of 3-manifolds). Let M be a compact connected orientable prime 3-manifold with infinite fundamental group such that the boundary of M is empty or a disjoint union of incompressible tori. Suppose that M satisfies Thurston’s Geometrization Conjecture, i.e. there is a geometric toral splitting along disjoint incompressible 2-sided tori in M whose pieces are Seifert manifolds or hyperbolic manifolds. Let M_{1}, M_{2}, …, M_{r} be the hyperbolic pieces. They all have finite volume [385, Theorem B on page 52]. Then M̃ is det-L^{2}-acyclic and

4.3 L^{2}-Invariants of Knot Complements 217

ρ^{(2)}(M̃) = − \frac{1}{6π} · ∑_{i=1}^{r} vol(M_{i}).

In particular, ρ^{(2)}(M̃) is 0 if and and only if there are no hyperbolic pieces.

Proof. The strategy of the proof is similar to the one of Theorem 4.1 using now Theorem 3.96, Theorem 3.105, Theorem 3.161 and Theorem 3.163. ⊓ ⊔

4.3 L^{2}-Invariants of Knot Complements

Let K ⊂ S^{3} be a knot, i.e. a smooth embedding of S^{1} into S^{3}. Let N(K) be a closed tubular neighboorhood. Notice that N(K) is diffeomorphic to S^{1}×D^{2}. Define the knot complement M(K) to be the 3-manifold S^{3} −int(N(K)). The complement of the trivial knot is S^{1} × D^{2}.

Lemma 4.4. The knot complement M(K) of a non-trivial knot is an irreducible compact connected oriented 3-manifold whose boundary is an incompressible torus T^{2}.

Proof. Everything is obvious except for the fact that the boundary is incompressible and M(K) is irreducible. Incompressibility is for instance proved in [75, Proposition 3.17 on page 39]. Next we show irreducibility. Let S^{2} ⊂ M(K) be an embedded 2-sphere. In particular we can think of S^{2} as embedded in S^{3}. By the Alexander-Schönflies Theorem [75, Theorem 1.8 on page 5], [381] there are embbeded balls D_{1}^{3} and D_{2}^{3} in S^{3} such that S^{3} = D_{1}^{3} ∪ D_{2}^{3} and ∂D_{1}^{3} = ∂D_{2}^{3} = D_{1}^{3} ∩ D_{1}^{3} = S^{2}. Since the knot is connected and does not meet the embedded S^{2}, it is contained in one of the balls, let us say D_{2}^{3}. Then D_{1}^{3} is a ball embedded in M(K) whose boundary is the given S^{2}. ⊓ ⊔

In particular M(K) is Haken and Theorem 4.3 applies to M(K) for a nontrivial knot. If we choose a different tubular neighborhood, then the corresponding knot complements are diffeomorphic. This implies that ρ^{(2)}({ \atop M(K)}) is defined for all knots and depends only on the ambient isotopy class of the knot.

Definition 4.5 (L^{2}-torsion of a knot). Define the L^{2}-torsion of a knot K ⊂ S^{3} to be the real number

ρ^{(2)}(K) := ρ^{(2)}({ \atop M(K)}).

Notice that ρ^{(2)}(M̃) = 0 for the trivial knot by Theorem 3.96 (5). We get from Theorem 4.3.

Theorem 4.6. Let K be a non-trivial knot. Then the boundary of M(K) is incompressible and there is a geometric toral splitting of M(K) along disjoint incompressible 2-sided tori in M whose pieces are Seifert manifolds or hyperbolic manifolds. Let M_{1}, M_{2}, …, M_{r} be the hyperbolic pieces. They all have finite volume. We have

ρ^{(2)}(K) = − \frac{1}{6π} · ∑_{i=1}^{r} vol(M_{i});

The next result follows from [225, Corollary 4.2 on page 696], [395, Lemma 5.5 and Lemma 5.6 on page 102]. For the notions of connected sum (sometimes also called product) of knots and cabling of knots were refer for instance to [75, 2.7 on page 19 and 2.9 on page 20]. The Alexander polynomial Δ(K) of a knot is explained for instance in [75, Definition 8.10 on page 109], [374], [376], [497].

Theorem 4.7. (1) Let K be a knot. Then ρ^{(2)}(K) = 0 if and only if K is obtained from the trivial knot by applying a finite number of times the operation “connected sum” and “cabling”;

(2) A knot is trivial if and only if both its L^{2}-torsion ρ^{(2)}(M̃) and its Alexander polynomial Δ(K) are trivial.

This shows that invariants of Reidemeister torsion type, namely the L^{2}- torsion and the Alexander polynomial, detect whether a knot is trivial.

There is the following conjecture due to Kashaev [286] and H. and J. Murakami [395, Conjecture 5.1 on page 102].

Conjecture 4.8 (Volume Conjecture). Let K be a knot and denote by J_{N}(K) the normalized colored Jones polynomial at the primitive N-th root of unity as defined in (see [395]). Then

ρ^{(2)}(K) = \frac{−1}{3} · lim_{N→∞} \frac{ln(|J_{N}(K)|)}{N}.

Kashaev states his conjecture in terms of the sum of the volumes of the hyperbolic pieces in the geometric toral splitting and H. and J. Murakami in terms of the simplicial volume. These lead to equivalent conjectures by Theorem 4.6 and Theorem 14.18 (3). Maybe there is a link between Conjecture 4.8 above and Question 13.73.

The Volume Conjecture 4.8 implies by Theorem 4.7 (2) the version of Vassiliev’s conjecture that a knot K is trivial if and only if every Vassiliev finite type invariant of K agrees with the one of the trivial knot. The point is that the colored Jones polynomials and the Alexander polynomial are determined by the Vassiliev finite type invariants.

4.4 Miscellaneous

The combinatorial computations of Theorem 3.172 and Remark 3.173 enables us to compute the L^{2}-torsion of the universal covering of a 3-manifold (and hence in view of Theorem 4.3 the sum of the volumes of its hyperbolic pieces in its geometric toral splitting) directly from a presentation of its fundamental group. Namely, we have (see [330, Theorem 2.4 on page 84])

Theorem 4.9. Let M be a compact connected orientable irreducible 3-manifold with infinite fundamental group G. Let

G = ⟨s_{1}, s_{2}, … s_{g} | R_{1}, R_{2}, … R_{r}⟩

be a presentation of G. Let the (r, g)-matrix

F = ( \begin{matrix}\frac{∂R_{1}}{∂s_{1}} & … & \frac{∂R_{1}}{∂s_{g}} \\ ⋮ & ... & ⋮ \\ \frac{∂R_{r}}{∂s_{1}} & … & \frac{∂R_{r}}{∂s_{g}} \end{matrix} )

be the Fox matrix of the presentation (see [75, 9B on page 123], [200], [330, page 84]). Now there are two cases:

(1) Suppose ∂M is non-empty. We make the assumption that ∂M is a union of incompressible tori and that g = r + 1. Then M is det-L^{2}-acyclic. Define A to be the (g −1, g −1)-matrix with entries in ℤG obtained from the Fox matrix F by deleting one of the columns. Let α be any real number satisfying 0 < α < \frac{2·α_{2}(M̃)}{α_{2}(M̃)+2};

(2) Suppose ∂M is empty. We make the assumption that a finite covering of M is homotopy equivalent to a hyperbolic, Seifert or Haken 3-manifold and that the given presentation comes from a Heegaard decomposition. Then M is det-L^{2}-acyclic and g = r. Define A to be the (g − 1, g − 1)- matrix with entries in ℤG obtained from the Fox matrix F by deleting one of the columns and one of the rows. Let α be any real number satisfying 0 < α < \frac{2·α_{2}(M̃)}{α_{2}(M̃)+1}.

Let K be any positive real number satisfying K ≥ ∥R_{A}^{(2)}∥. A possible choice for K is the product of (g − 1)^{2} and the maximum over the word length of those relations R_{i} whose Fox derivatives appear in A.

Then the sum of non-negative rational numbers

∑_{p=1}^{L} \frac{1}{p} · tr_{ℤG} ((1 − K^{−2} · AA^{∗})^{p})

converges for L → ∞ to the real number 2 · ρ^{(2)}(M̃) + 2(g − 1) · ln(K). More precisely, there is a constant C > 0 such that we get for all L ≥ 1

0 ≤ 2 · ρ^{(2)}(M̃) + 2(g − 1) · ln(K) − ∑_{p=1}^{L} \frac{1}{p} · tr_{ℤG} ((1 − K^{−2} · AA^{∗})^{p}) ≤ \frac{C}{L^{α}} .⊓ ⊔

The case of an automorphism of a compact connected orientable surface, which yields a 3-manifold by taking its mapping torus is discussed in Subsection 7.4.2

Exercises

4.1. Let M be a compact connected orientable 3-manifold. Show that it is aspherical if and only its prime decompositions has precisely one factor which is not a homotopy sphere and this factor is either D^{3} or an irreducible 3-manifold with infinite fundamental group.

4.2. Show that two connected closed 3-manifolds possessing the same geometry have the same L^{2}-Betti numbers and Novikov-Shubin invariants, provided their fundamental groups are infinite. Which of the eight geometries can be distinguished from one another by the knowledge of all L^{2}-Betti numbers and Novikov-Shubin invariants of the universal coverings?

4.3. Let M be a closed connected orientable 3-manifold which possesses a geometry. Show that the relevant geometry can be read off from the fundamental group π as follows:

(1) ℍ^{3}: π is not virtually cyclic and contains no subgroup isomorphic to ℤ ⊕ ℤ;

(2) S^{3}: π is finite;

(3) S^{2} × ℝ: π is virtually cyclic and infinite;

(4) ℝ^{3}: π is contains ℤ^{3} as subgroup of finite index;

(5) Nil: π contains a subgroup of finite index G which can be written as an extension 1 → ℤ → G → ℤ^{2} → 1 but π does not contain ℤ^{3} as subgroup of finite index;

(6) ℍ^{2}×ℝ: π contains a subgroup of finite index which is isomorphic to ℤ×G for some group G and π is not solvable;

(7) { \atop Sl_{2}(ℝ)}: π is not solvable, contains ℤ ⊕ ℤ as subgroup and contains no subgroup of finite index which is isomorphic to ℤ × G for some group G;

(8) Sol: π is not virtually abelian and contains a subgroup G of finite index which is an extension 0 → ℤ^{2} → G → ℤ → 0.

4.4. Compute the L^{2}-Betti numbers and the Novikov-Shubin invariants of the universal covering of a compact connected 3-manifold whose fundamental group is finite.

4.5. Let M be the connected sum M_{1}# … #M_{r} of compact connected orientable 3-manifolds M_{j} which are non-exceptional and prime. Show that then for some p ≥ 0 the Laplace operator (Δ_{p})_{min} of Definition 2.64 acting on smooth p-forms on the universal covering M̃ has zero in its spectrum.

4.6. Give a proof of the conclusion which is stated in the last sentence of Theorem 4.1.

4.7. Let f: F → F be a selfhomeomorphism of a closed orientable surface of genus ≤ 1. Show that T̃_{f} is det-L^{2}-acyclic and ρ^{(2)}(T̃_{f}) = 0.

5. L^{2}-Invariants of Symmetric Spaces

Introduction

In this chapter we state the values of the L^{2}-Betti numbers, the Novikov-Shubin invariants and the L^{2}-torsion for universal coverings of closed locally symmetric spaces. We give a brief survey about locally symmetric and symmetric spaces in Section 5.1 and state the values in Section 5.2 and 5.3. These computations will give evidence for various general conjectures about L^{2}-invariants such as Conjecture 2.82 about the positivity and rationality of Novikov-Shubin invariants, the Strong Atiyah Conjecture 10.2, the Singer Conjecture 11.1, Conjecture 11.3 about the parity of the L^{2}-torsion of the universal covering of an aspherical closed manifold and the zero-in-the-spectrum Conjecture 12.1.

5.1 Survey on Symmetric Spaces

In this section we collect some basic facts about symmetric spaces so that the reader will be able to understand the results on the computations of L^{2}- invariants of Section 5.2 and 5.3. A reader who is familiar with symmetric spaces should pass directly to Section 5.2.

Let M be a complete Riemannian manifold and let x ∈ M be a point. A normal neighborhood of M at x is an open neighborhood V of x in M such that there is an open neighborhood U of 0 in the tangent space T_{x}M with the properties that for any u ∈ U and t ∈ [−1, 1] also tu belongs to U and the exponential map exp_{x}: T_{x}M → M induces a diffeomorphism U → V. The geodesic symmetry of a normal neighborhood V at x is the diffeomorphism s_{p}: V → V which sends exp_{x}(u) to exp_{x}(−u) for u ∈ U.

Definition 5.1 (Locally symmetric space). A complete Riemannian manifold M is called a locally symmetry space if for any x ∈ M there exists a normal neighborhood V of x such that the geodesic symmetry is an isometry.

A complete Riemannian manifold is a locally symmetric space if and only if the sectional curvature is invariant under parallel transports with respect to the Levi-Civita connection [251, Theorem 1.3 in IV.1 on page 201].

Definition 5.2 (Symmetric space). A complete Riemannian manifold is called (globally) symmetric space if for each x ∈ M there is an isometric diffeomorphism t_{x}: M → M which is an involution, i.e. t_{x} ∘ t_{x} = id, and has x as isolated fixed point, i.e. t_{x}(x) = x and there is a neighborhood W of x in M such that y ∈ W, t_{x}(y) = y implies x = y.

Examples of symmetric spaces are S^{n}, ℝℙ^{n}, ℂℙ^{n}, ℝ^{n}, ℍ^{n}.

A symmetric space is always locally symmetric [251, Lemma 3.1 in IV.3 on page 205]. On the other hand any simply connected locally symmetric space is a symmetric space [251, Theorem 5.6. in IV.5 on page 222]. In particular the universal covering of a locally symmetric space is a symmetric space.

Let M be a symmetric space. Denote by Isom(M) the group of isometries M → M. This group inherits the structure of a topological group by the compact-open topology coming from the topology of M. Denote for a Lie group L its identity component by L^{0}. We get from [251, Lemma 3.2 in IV.3 on page 205, Theorem 3.3 in IV.3 on page 208]

Theorem 5.3. The group Isom(M) has the unique structure of an (analytic) Lie group. Given a point x ∈ M, let Isom(M)_{x}^{0} be the stabilizer of Isom(M)^{0} at x, i.e. the subgroup of elements f ∈ Isom(M)^{0} with f(x) = x. Then Isom(M)^{0} acts transitively on M, Isom(M)_{x}^{0} is compact and we get a (analytic) diffeomorphism

\overline{ev}_{x}: Isom(M)^{0}/ Isom(M)_{x}^{0} ⟶^{≅} M, f · Isom(M)_{x}^{0} ↦ f(x).

The Killing form of a Lie algebra 𝔤 is defined by

B: 𝔤 × 𝔤 → ℝ, (a, b) ↦ tr_{ℝ}(ad(a) ad(b)),

where ad(x): 𝔤 → 𝔤 denotes the adjoint representation sending z to [x, z] and tr_{ℝ} is the trace of an endomorphism of a finite dimensional real vector space. The Lie algebra 𝔤 is semisimple if the Killing form is non-degenerate. This is equivalent to the condition that 𝔤 contains no non-trivial solvable ideals, where solvable means that the commutators series ends at {0} [293, page 668]. A Lie algebra 𝔤 is simple if it is non-abelian and contains no proper ideals.

The Killing form on a semisimple Lie algebra 𝔤 is strictly negative-definite if and only if there is a connected compact Lie group G whose Lie algebra is 𝔤 [251, Corollary 6.7 in II.6 on page 133].

An involution of Lie algebras θ: 𝔤 → 𝔤 is a Cartan involution if for the associated decomposition 𝔤 = 𝔨⊕𝔭 with 𝔨 = ker(θ−id) and 𝔭 = ker(θ+id) the Killing form B is strictly negative-definite on 𝔨 and strictly positive-definite on 𝔭. In this context 𝔤 = 𝔨⊕𝔭 is called Cartan decomposition. Any semisimple Lie algebra 𝔤 has a Cartan decomposition and two Cartan decompositions are conjugate [251, III.7]. Notice that 𝔨 is a subalgebra and 𝔭 a vector subspace.

Example 5.4. A connected linear reductive Lie group G is a closed connected subgroup of GL(n, ℝ) or GL(n, ℂ), which is stable under taking transpose conjugate matrices. Examples are SL(n, ℂ) ⊂ GL(n, ℂ) and SO(n) ⊂ GL(n, ℝ). Such a group G is a Lie group. It is semisimple if and only if its center is finite. Taking inverse conjugate transpose matrices induces an involution of Lie groups Θ: G → G. The group K = {g ∈ G | Θ(g) = g} turns out to be a maximal compact subgroup. Denote by 𝔤 the Lie algebra of G and by θ: 𝔤 → 𝔤 the differential of Θ at the identity matrix. Notice that 𝔤𝔩_{n}(ℝ) is M_{n}(ℝ) and 𝔤𝔩_{n}(ℂ) is M_{n}(ℂ) and that 𝔤 is a subalgebra. The involution of Lie algebras θ sends a matrix to its negative conjugate transpose. The Lie bracket of 𝔤 is given by taking commutators. The involution θ is a Cartan involution. In the Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔭 the subalgebra 𝔨 is the Lie algebra of K ⊂ G.

Let M be a symmetric space. In the sequel we abbreviate G = Isom(M)^{0} and K = Isom(M)_{x}^{0}. Let σ: G → G be given by conjugation with the geodesic symmetry s_{x} ∈ K. Let 𝔤 be the Lie algebra of G. Put

𝔨 = {a ∈ 𝔤 | T_{1}σ(a) = a};
𝔭 = {a ∈ 𝔤 | T_{1}σ(a) = −a}.

We get a decomposition of Lie algebras 𝔤 = 𝔨 ⊕ 𝔭. Notice that 𝔨 is the kernel of the differential at 1 ∈ G of the evaluation map ev_{x}: G → M which sends f to f(x).

Definition 5.5 (Type of a symmetric space). The symmetric space M = G/K is of compact type if 𝔤 is semisimple and has strictly negative-definite Killing form. It is of non-compact type if 𝔤 is semisimple and σ is a Cartan involution. It is called of Euclidean type if 𝔭 is an abelian ideal in 𝔤.

A symmetric space M is of compact type if and only Isom(M) is compact and semisimple. A symmetric space M of compact type is a compact manifold.

If M = G/K for G = Isom(M)^{0} and K = Isom(M)_{x}^{0} is a symmetric space of non-compact type, then K is connected and is a maximal compact subgroup in G and M is diffeomorphic to ℝ^{n} [251, Theorem 1. in Chaper VI on page 252]. A symmetric space M is of non-compact type if and only if the Lie algebra of Isom(M) is semisimple and has no compact ideal [157, Proposition 2.1.1 on page 69], [251, page 250]. On the other hand, given a connected semisimple Lie group G with finite center such that its Lie algebra has no compact ideal, the homogeneous space G/K for a maximal compact subgroup K ⊂ G equipped with a G-invariant Riemannian metric is a symmetric space of non-compact type with G = Isom(M)^{0} and K = Isom(M)_{x}^{0} [157, Section 2.2 on page 70].

A θ-stable Cartan subalgebra 𝔥 ⊂ 𝔤 of a semisimple Lie algebra 𝔤 is a maximal abelian θ-stable abelian subalgebra. All θ-stable Cartan subalgebras have the same dimension. Hence we can define the complex rank rk_{ℂ}(𝔤) by the dimension of a θ-stable Cartan subalgebra of 𝔤. [293, page 128f]. (This should not be confused with the real rank which is the dimension of the term 𝔞 in the Iwasawa decomposition 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫.) Recall that a Lie group G is semisimple if its Lie algebra is semisimple. The complex rank of a semisimple Lie group rk_{ℂ}(G) is defined to be the complex rank of its Lie algebra 𝔤. The complex rank of a connected compact Lie group is the same as the dimension of a maximal torus.

Definition 5.6. Let M = G/K be a symmetric space such that G = Isom(M)^{0} is semisimple. Define its fundamental rank

f-rk(M) := rk_{ℂ}(G) − rk_{ℂ}(K).

This is the notion of rank which will be relevant for our considerations. It should not be confused with the following different notion.

A Riemannian submanifold N ⊂ M of a complete Riemannian manifold M is totally geodesic if for any geodesic γ: ℝ → M, for which γ(0) ∈ N and γ′(0) lies in the tangent space T_{γ(0)}N, the image of γ lies in N. Recall that a Riemannian manifold M is flat if the sectional curvature is identically zero.

Definition 5.7. Let M be a symmetric space. Its rank is the maximal dimension of a flat totally geodesic complete submanifold of M.

The rank of M in the sense of Definition 5.7 is always greater or equal to the fundamental rank. This follows from [56, Formula (3) in III.4 on page 99].

The next two results are taken from [251, Proposition 4.2 in V.4 on page 244, Theorem 3.1 in V.3 on page 241].

Theorem 5.8. Let M be a simply connected symmetric space. Then it can be written as a product

M = M_{cp} × M_{Eucl} × M_{ncp},

where M_{cp} is of compact type, M_{Eucl} of Euclidean type and M_{ncp} of noncompact type.

Theorem 5.9. Let M be a symmetric space. Then

(1) If M is of compact type, then the sectional curvature of M is nonnegative: sec(M) ≥ 0;

(2) If M is of Euclidean type, then M is flat: sec(M) = 0;

(3) If M is of non-compact type, then the sectional curvature of M is nonpositive: sec(M) ≤ 0;

5.2 L^{2}-Invariants of Symmetric Spaces of Non-Compact Type 227

Lemma 5.10. A simply connected symmetric space M is contractible if and only if in the decomposition M = M_{cp} × M_{Eucl} × M_{ncp} of Theorem 5.8 the factor M_{cp} of compact type is trivial. In this case M is diffeomorphic to ℝ^{n}.

Proof. The factor M_{cp} is a closed manifold and hence contractible if and only if M_{cp} is a point. Since M_{Eucl} and M_{ncp} carry Riemannian metrics of nonpositive sectional curvature, they are diffeomorphic to ℝ^{n} for appropriate n by Hadamard’s Theorem. ⊓ ⊔

There is an important duality between symmetric spaces of non-compact type and symmetric spaces of compact type [251, V.2]. Let M = G/K be a symmetric space of non-compact type. Let 𝔤 and 𝔨 be the Lie algebras of G and K and let 𝔤 = 𝔨 ⊕ 𝔭 be the Cartan decomposition. The complexification G_{ℂ} of G is the simply connected Lie group with the complexification ℂ⊗_{ℝ}𝔤 of 𝔤 as Lie algebra. Obviously 𝔤^{d} = 𝔨⊕i·𝔭 is a real subalgebra of ℂ⊗_{ℝ} 𝔤. Let G^{d} be the corresponding analytic subgroup of the complexification G_{ℂ} of G. Then G^{d} is a compact group. Let K′ ⊂ G^{d} be the subgroup corresponding to 𝔨 ⊂ 𝔤^{d}. The dual symmetric space is defined to be M^{d} = G^{d}/K′ with respect to the G^{d}-invariant Riemannian metric for which multiplication with i induces an isometry T_{1K}G/K → T_{1K}G^{d}/K′. M. Olbricht pointed out to us that one can assume without loss of generality that G is linear, i.e. G ⊂ GL(n, ℝ). Put G^{d} to be the analytic subgroup in GL(n, ℂ) corresponding to 𝔤^{d}. Then K is also a subgroup of G^{d} and M^{d} agrees with G^{d}/K. The symmetric space M^{d} is of compact type. Analogously one can associate to a symmetric space of compact type M a symmetric space of non-compact type M^{d}. In both cases (M^{d})^{d} = M. The following example is taken from [251, Example 1 in V.2 on page 238].

Example 5.11. Denote by by SO(p, q) the group of real (p + q)-(p + q)- matrices of determinant 1 which leave the quadratic form −x_{1}^{2} − … − x_{p}^{2} + x_{p+1}^{2} + … x_{p+q}^{2} invariant. Denote by SO(p, q)^{0} the identity component of SO(p, q). Let SO(n) be the Lie group {A ∈ GL(n, ℝ) | AA^{t} = I, det(A) = 1}. This agrees with SO(0, n). There are obvious embeddings of SO(p) × SO(q) into both SO(p, q)^{0} and SO(p + q). Equip SO(p, q)^{0}/SO(p) × SO(q) and SO(p + q)/SO(p) × SO(q) with a SO(p, q)^{0}-invariant and SO(p + q)- invariant Riemannian metric. These are uniquely determined up to scaling with a constant. Then SO(p, q)^{0}/SO(p)×SO(q) is a symmetric space of noncompact type and SO(p+q)/SO(p)×SO(q) is a symmetric space of compact type. They are dual to one another possibly after scaling the Riemannian metric with a constant.

5.2 L^{2}-Invariants of Symmetric Spaces of Non-Compact Type

In this section we state the values of the L^{2}-Betti numbers, the Novikov-Shubin invariants and the L^{2}-torsion of the universal covering M̃ of a closed Riemannian manifold M provided that M̃ is a symmetric space of noncompact type. Notice that a symmetric space comes with a preferred Riemannian metric so that its L^{2}-torsion is defined without the assumption that all L^{2}-Betti numbers of M̃ vanish. Recall that it does not matter whether we work with the topological version (see Definition 3.120) or with the analytic version (see Definition 3.128) of L^{2}-torsion because of Theorem 3.149.

Theorem 5.12 (L^{2}-invariants of symmetric spaces). Let M be a closed Riemannian manifold whose universal covering M̃ is a symmetric space of non-compact type. Let f-rk(M̃) := rk_{ℂ}(G) − rk_{ℂ}(K) be the fundamental rank of the universal covering M̃ for G = Isom(M̃)^{0} and K ⊂ G a maximal compact subgroup. Then

(1) We have b_{p}^{(2)}(M̃) ≠ 0 if and only if f-rk(M̃) = 0 and 2p = dim(M). Let M̃^{d} be the to M̃ dual symmetric space. If f-rk(M̃) = 0, then dim(M) is even and for 2p = dim(M) we get

0 < b_{p}^{(2)}(M̃) = (−1)^{p} · χ(M) = \frac{vol(M)}{vol(M̃^{d})} · χ(M̃^{d});

(2) We have α_{p}(M̃) ≠ ∞^{+} if and only if f-rk(M̃) > 0 and p belongs to

[\frac{dim(M)−f-rk(M̃)}{2} + 1, \frac{dim(M)+f-rk(M̃)}{2}]. If α_{p}(M̃) ≠ ∞^{+}, then α_{p}(M̃) = f-rk(M̃). The number dim(M) − f-rk(M̃) is even and positive if M is not the onepoint-space {∗};

(3) We have ρ^{(2)}(M̃) ≠ 0 if and only if f-rk(M̃) = 1;

(4) Suppose that f-rk(M̃) = 1. Then M̃ = X_{0} × X_{1}, where X_{0} is a symmetric space of non-compact type with f-rk(X_{0}) = 0 and X_{1} = X_{p,q} := SO(p, q)^{0}/SO(p)×SO(q) for p, q odd or X_{1} = SL(3, ℝ)/SO(3). We have

ρ^{(2)}(M̃) = vol(M) · T^{(2)}(M̃),

where the number T^{(2)}(M̃) is given below.

(a) If X_{0}^{d} is the symmetric space dual to X_{0}, then dim(X_{0}) is even, χ(X_{0}^{d}) > 0 and we have

T^{(2)}(M̃) := (−1)^{dim(X_{0})/2} · \frac{χ(X_{0}^{d})}{vol(X_{0}^{d})} · T^{(2)} (X_{1}) ;

(b) Let C_{n} be the positive constant introduced in (3.151). Denote by (ℍ^{n})^{d} the symmetric space dual to the hyperbolic space ℍ^{n}. (Then (ℍ^{n})^{d} is the sphere S^{n} with the Riemannian metric whose sectional curvature is constant 1). We have

T^{(2)} (X_{p,q}) = (−1)^{\frac{pq−1}{2}} ·χ ((X_{p−1,q−1})^{d})· \begin{matrix}vol & ((ℍ^{p+q−1})^{d}) \\ vol & ((X_{p,q})^{d}) \end{matrix} ·C_{p+q−1}

and

χ ((X_{p−1,q−1})^{d}) = \{ \begin{matrix}2 · ( \frac{p+q−2}{2} ; \frac{p−1}{2} ) & p, q > 1 \\ 1 & p = 1, q > 1 \end{matrix} ;

(c) We have

T^{(2)}(SL(3, ℝ)/SO(3)) = \frac{2π}{3 vol ((SL(3, ℝ)/SO(3))^{d})}.

Here are some explanations. Of course M̃ inherits its Riemannian metric from the one on M. The symmetric spaces X_{p,q} = SO(p, q)^{0}/SO(p) × SO(q) and SL(3, ℝ)/SO(3) are equipped with the Riemannian metrics coming from M̃ = X_{0} × X_{1}. These Riemannian metrics are SO(p, q)^{0}-invariant and SL(3, ℝ)-invariant. Two such invariant Riemannian metrics on X_{p,q} = SO(p, q)^{0}/SO(p) × SO(q) and SL(3, ℝ)/SO(3) differ only by scaling with a constant. Notice that X_{1,q} is isometric to ℍ^{q} after possibly scaling the metric with a constant. If we scale the Riemannian metric on M by a constant C, then vol(M) is scaled by C^{dim(M)} and T^{(2)}(M̃) by C^{− dim(M)}. Hence ρ^{(2)}(M̃) is unchanged.

The result for L^{2}-Betti numbers has been proved in [54]. The computations for the Novikov-Shubin invariants have been carried out in [314, Section 11], [404, Theorem 1.1], partial results have already been obtained in [316, VII.B]. (The reader should be aware of the fact that range of the finiteness of α_{p}(M̃) is misprinted in [314, Section 11] and correct in [316, VII.B] and [404, Theorem 1.1].) Notice that there is a shift by one in the range where α_{p}(M̃) ≠ ∞^{+} in Theorem 5.12 (2) in comparison with [404, Theorem 1.1], since the definition of α_{p}(M̃) here and in [404, Theorem 1.1] differ by 1 concerning the index p. The result about L^{2}-torsion is proved in [404, Theorem 1.1, Proposition 1.3 and Proposition 1.4]. The basic input is the Harish-Chandra Plancherel Theorem (see [248], [293, Theorem 3.11 in Chapter XIII on page 511]) and (𝔤, K)-cohomology.

5.3 L^{2}-Invariants of Symmetric Spaces

Let M be a simply connected symmetric space which is not necessarily of noncompact type. We conclude from Theorem 3.183 that there are constants B_{p}^{(2)}(M) for p ≥ 0, A_{p}^{(2)}(M) for p ≥ 1 and T^{(2)}(M) depending only on the Riemannian structure on M such that for any cocompact free proper action of a group G by isometries

b_{p}^{(2)}(M; N (G)) = B_{p}^{(2)}(M) · vol(G\M);
α_{p}^{(2)}(M; N (G)) = A_{p}^{(2)}(M);
ρ^{(2)}(M; N (G)) = T^{(2)}(M) · vol(G\M).

We want to compute these numbers explicitly. Recall that M splits as a product M_{cp} × M_{Eucl} × M_{ncp}, where M_{cp}, M_{Eucl} and M_{ncp} respectively are symmetric spaces of compact type, Euclidean type and non-compact type respectively (see Theorem 5.8). Notice that these numbers have already been computed for M_{ncp} in Theorem 5.12 and that M_{Eucl} is isometric to ℝ^{n} with the standard Euclidean Riemannian metric [218, Theorem 3.82 on page 131]. We want to extend these computations to M. Suppose that G_{0} and G_{1} respectively are groups with a cocompact free proper action by isometries on M_{Eucl} and M_{ncp} respectively. We conclude from the product formula for L^{2}-torsion (see Theorem 3.93 (4) and (3.126)) that ρ^{(2)}(M_{cp} × M_{Eucl} × M_{ncp}; N ({1} × G_{0} × G_{1})) is given by

χ(M_{cp}) · ρ^{(2)}(M_{ncp}; N (G_{1})) + χ(G_{1}\M_{ncp}) · ρ^{(2)}(M_{cp}; N ({1})),

if M_{Eucl} = ∗ and by 0 if M_{Eucl} ≠ ∗. Since by Hirzebruch’s proportionality principle [262]. \frac{χ(G_{1}\M_{ncp})}{vol(G_{1}\M_{ncp})} = \frac{χ(M_{ncp}^{d})}{vol(M_{ncp}^{d})}, we conclude

T^{(2)}(M) = \{ \begin{matrix}\frac{χ(M_{ncp}^{d})}{vol(M_{ncp}^{d})} · T^{(2)}(M_{cp}) & if M_{Eucl} = \{∗\}, f-rk(M_{ncp}) = 0; \\ \frac{χ(M_{cp})}{vol(M_{cp})} · T^{(2)}(M_{ncp}) & if M_{Eucl} = \{∗\}, f-rk(M_{ncp}) = 1; \\ 0 & otherwise. \end{matrix}(5.13)

Notice that we have already given the value of T^{(2)}(M_{ncp}) in Theorem 5.12 (4). Similar one gets from the product formula for the Novikov-Shubin invariants (see Theorem 2.55 (3)), where the necessary assumption about the limit property follows from the computations in [404]

A_{p}(M) = \{ \begin{matrix}a & if b_{p−i}(M_{cp}) ≠ 0 for some integer i satisfying \\ & \frac{dim(M_{ncp})−f-rk(M_{ncp})}{2} + 1 ≤ i and \\ & i ≤ \frac{dim(M_{ncp})+f-rk(M_{ncp})}{2} + dim(M_{Eucl}); \\ ∞^{+} & otherwise. \end{matrix} (5.14)

where a = f-rk(M_{ncp}) + dim(M_{Eucl}). We get from the product formula for L^{2}-Betti numbers (see Theorem 1.35 (4))

B_{p}^{(2)}(M) = \{ \begin{matrix}b & if f-rk(M_{ncp}) = 0 and M_{Eucl} = \{∗\} \\ 0 & otherwise \end{matrix} ;  (5.15)

where b = \frac{b_{(p−dim(Mncp)/2)}(M_{cp})}{vol(M_{cp})} · (−1)^{dim(M_{ncp})/2} · \frac{χ(M_{ncp}^{d})}{vol(M_{ncp}^{d})}.

Suppose that the closed locally symmetric space M has negative sectional curvature. Then its universal covering M̃ must be of non-compact type with fundamental rank f-rk(M̃) ≤ 1 since its rank in the sense of Definition 5.7 is always greater or equal to the fundamental rank [56, Formula (3) in III.4 on page 99] and is equal to 1 for a Riemannian metric with negative sectional curvature. We conclude from Lemma 5.10, Theorem 5.12 and equations (5.13), (5.14) and (5.15)

Corollary 5.16. Let M be an aspherical closed Riemannian manifold whose universal covering M̃ is a symmetric space. Then

(1) We have b_{p}^{(2)}(M̃) = 0 if 2p ≠ dim(M). If M̃ has negative sectional curvature and has even dimension, then b_{dim(M)/2}^{(2)}(M̃) > 0, f-rk(M̃) = 0;

(2) One of the L^{2}-Betti numbers b_{p}^{(2)}(M̃) is different from zero or one of the Novikov Shubin invariants α_{p}(M̃) is different from ∞^{+};

(3) If ρ^{(2)}(M̃) ≠ 0, then M̃ is of non-compact type, dim(M) is odd and we

have

(−1)^{\frac{dim(M)−1}{2}} · ρ^{(2)}(M) > 0.

If M carries a metric of negative sectional curvature and has odd dimension, then

(−1)^{\frac{dim(M)−1}{2}} · ρ^{(2)}(M) > 0.

If M carries a metric of negative sectional curvature and has even dimension, then

α_{p}(M̃) = ∞^{+} for p ≥ 1.

Corollary 5.16 implies that various conjectures for aspherical closed manifolds (see Section 11.1.3 and Subsection 12.2.2) turn out to be true in the case that the universal covering is a symmetric space.

5.4 Miscellaneous

Consider a closed Riemannian manifold M with non-positive sectional curvature such that M̃ is det-L^{2}-acyclic. Suppose that N is a closed manifold which is homotopy equivalent to M. The Whitehead group of the fundamental group of a closed Riemannian manifold with non-positive sectional curvature is known to be zero [192, page 61]. (Actually Farrell and Jones prove the stronger statement that for any closed aspherical topological manifold N with dim(N) ≠ 3, 4 and π_{1}(M) ≅ π_{1}(N) any homotopy equivalence M → N is homotopic to a homeomorphism [195, Theorem 0.1].) Hence also Ñ is det-L^{2}-acyclic and ρ^{(2)}(M̃) = ρ^{(2)}(Ñ) by Theorem 3.96 (1).

Notice that this applies by Lemma 5.9, Lemma 5.10 and Theorem 5.12 (2) to the case, where M is an aspherical closed manifold which admits the structure of a locally symmetric space and satisfies b_{p}^{(2)}(M̃) = 0 for p ≥ 0. In particular we conclude from Lemma 5.10, from Theorem 5.12 (1) and (3) and from (5.13) that for two aspherical closed locally symmetric Riemannian manifolds M and N with isomorphic fundamental groups ρ^{(2)}(M̃) = ρ^{(2)}(Ñ) holds.

Let M be a closed topological manifold with dim(M) ≠ 3, 4. Then M carries the structure of a locally symmetric Riemannian manifold whose universal covering is a symmetric space of non-compact type if and only if M is aspherical and π_{1}(M) is isomorphic to a cocompact discrete subgroup of a linear semisimple Lie group with finitely many path components [195, Theorem 0.2].

The famous rigidity result of Mostow [388] says that two closed locally symmetric spaces M and N with non-positive sectional curvature are isometrically diffeomorphic if and only if π_{1}(M) ≅ π_{1}(N) and vol(M) = vol(N) hold, provided that Ñ is irreducible, i.e. not a product of two Riemannian manifolds of positive dimension, and dim(N) ≥ 3. The following rigidity result is proved in [20, Theorem 1 on page i]. Let M be a closed locally symmetric space such that its rank (in the sense of Definition 5.7) is greater or equal to 2 and its universal covering is irreducible. Let N be a closed Riemannian manifold with non-positive sectional curvature. Suppose that π_{1}(M) ≅ π_{1}(N) and vol(M) = vol(N). Then M and N are isometrically diffeomorphic.

A classification of symmetric spaces is given in [251, Chapter X]. More information about Lie groups, Lie algebras and symmetric spaces can be found for instance in [17], [19], [20], [157], [158], [251], [268] and [293].

Exercises

5.1. Let M be a symmetric space and let x ∈ M. Let t_{x}: M → M be an isometric involution which has x as isolated fixed point. Show that there is a normal neighborhood U of x such that t induces the geodesic symmetry on U.

5.2. Show that the following groups are connected linear reductive Lie groups:

GL(n, ℂ) = {A ∈ M_{n}(ℂ) | A invertible};
SL(n, ℂ) = {A ∈ GL(n, ℂ) | det(A) = 1};
U(n) = \{A ∈ GL(n, ℂ) | A\overline{A}^{t} = 1\};
SU(n) = \{A ∈ GL(n, ℂ) | A\overline{A}^{t} = 1, det(A) = 1\};
SO(n, ℂ) = {A ∈ GL(n, ℂ) | AA^{t} = 1, det(A) = 1};
SO(n) = {A ∈ GL(n, ℝ) | AA^{t} = 1}.

Show that their Lie algebras are given by

𝔤𝔩(n, ℂ) = M_{n}(ℂ);
𝔰𝔩(n, ℂ) = {A ∈ M_{n}(ℂ) | tr(A) = 0};
𝔲(n) = \{A ∈ M_{n}(ℂ) | A + \overline{A}^{t} = 0\};
𝔰𝔲(n) = \{A ∈ M_{n}(ℂ) | A + \overline{A}^{t} = 0 and tr(A) = 0\};
𝔰𝔬(n, ℂ) = {A ∈ M_{n}(ℂ) | A + A^{t} = 0};
𝔰𝔬(n) = {A ∈ M_{n}(ℝ) | A + A^{t} = 0}.

Show that GL(n, ℂ) for n ≥ 1 and SO(2) are not semisimple. Show that SL(n, ℂ), SO(n, ℂ) and SU(n) for n ≥ 2, and SO(n) for n ≥ 3 are semisimple. Prove that U(n), SU(n) and SO(n) is the group K = {g ∈ G | Θ(g) = g} for G = GL(n, ℂ), SL(n, ℂ) and SO(n, ℂ) and is in particular a maximal compact subgroup.

5.3. Let G be a connected linear reductive Lie group. Show that s: 𝔤×𝔤 → ℝ sending (A, B) to the real part ℜ(tr(AB)) of tr(AB) is an inner product on the real vector space 𝔤. Show for any a ∈ 𝔤 that the adjoint ad(A)^{∗} of ad(A): 𝔤 → 𝔤 with respect to this inner product is − ad(A). Conclude for the Killing form B that −B(A, θ(A)) = tr(ad(A) ad(A)^{∗}) holds and that hence the symmetric bilinear form B(A, θ(A)) on 𝔤 is strictly negative-definite, where θ is defined in Example 5.4. Conclude that θ is a Cartan involution.

5.4. Show that a connected linear reductive Lie group is semisimple if and only if its center is finite.

5.5. Let G and H be connected linear reductive Lie groups. Show that G × H is again a connected linear reductive Lie group. Prove that G × H is semisimple if and only if both G and H are semisimple.

5.6. Show that S^{n}, ℝℙ^{n} and ℂℙ^{n} are symmetric spaces of compact type, ℝ^{n} is a symmetric space of Euclidean type and ℍ^{n} is a symmetric space of non-compact type.

5.7. Let G_{p}(ℝ^{p+q}) be the topological space of oriented p-dimensional linear subspaces of ℝ^{p+q}. Show that it carries the structure of a symmetric space of compact type. Determine its dual symmetric space.

5.8. Let M and N be closed Riemannian manifolds whose universal coverings M̃ and Ñ are symmetric spaces of non-compact type. Suppose that π_{1}(M) ≅ π_{1}(N). Show that then f-rk(M̃) = f-rk(Ñ).

5.9. Let M be a closed locally symmetric Riemannian manifold. Show that π_{1}(M) is amenable if and only if M̃_{ncp} = {∗}.

5.10. Construct two closed Riemannian manifolds M and N whose universal coverings are symmetric spaces of non-compact type such that π_{1}(M) and π_{1}(N) are not isomorphic but all L^{2}-Betti numbers, all Novikov-Shubin invariants and the L^{2}-torsion of M̃ and Ñ with respect to the π_{1}(M)- and π_{1}(N)-action agree.

5.11. Let M be an aspherical closed locally symmetric Riemannian manifold of even dimension dim(M) = 2p. Show that (−1)^{p} · χ(M) ≥ 0. Prove that (−1)^{p} · χ(M) > 0 if and only if M̃_{Eucl} = {∗} and f-rk(M̃_{ncp}) = 0.

5.12. Let M be an aspherical closed locally symmetric Riemannian manifold. Show that scaling the Riemannian metric by a constant does not change ρ^{(2)}(M̃) if and only if ρ^{(2)}(M̃) = ρ^{(2)}(Ñ) holds for any closed Riemannian manifold N which is homotopy equivalent to M.

6. L^{2}-Invariants for General Spaces with Group Action

Introduction

In this chapter we will extend the definition of L^{2}-Betti numbers for free G-CW -complexes of finite type to arbitrary G-spaces. Of course then the value may be infinite, but we will see that in surprisingly many interesting situations the value will be finite or even zero. This will be applied to problems in geometry, topology, group theory and K-theory.

The first elementary observation is that the ℂ-category of finitely generated Hilbert N (G)-modules is isomorphic to the ℂ-category of finitely generated projective N (G)-modules. Notice that the second category does not involve any functional analytic structure of N (G), only the ring structure comes in. The second observation is that the ring N (G) is semihereditary, i.e. any finitely generated N (G)-submodule of a projective module is projective again. This implies that the ℂ-category of finitely presented N (G)- modules is abelian. Thus a finitely generated Hilbert N (G)-chain complex C_{∗} defines a finitely generated projective N (G)-chain complex denoted by ν^{−1}(C_{∗}) and the homology of ν^{−1}(C_{∗}) consists of finitely presented N (G)- modules. Any finitely presented N (G)-module M splits as TM ⊕PM, where PM is finitely generated projective. Then the L^{2}-homology of C_{∗} corresponds to PH_{∗}(ν^{−1}(C_{∗})) = ν^{−1}(H^{(2)}(C_{∗})). These facts will be explained and proved in Section 6.2 and we will compare them with the approach of Farber [182] in Section 6.8.

Now the main technical result of Section 6.1 is that the von Neumann dimension, which is a priori defined for finitely generated projective N (G)- modules, has an extension to arbitrary N (G)-modules which takes values in [0, ∞] and is uniquely determined by three important and desired properties, namely, Additivity, Cofinality and Continuity. This will allow us to define in Section 6.5 L^{2}-Betti numbers for an arbitrary G-space X by the extended von Neumann dimension of the N (G)-module H_{p}^{G}(X; N (G)), which is the homology of the N (G)-chain complex N (G) ⊗_{ℤG} C_{∗}^{sing}(X). This is the L^{2}- Betti number defined in Section 1.2 if X happens to be a free G-CW -complex of finite type. Thus we can define the L^{2}-Betti number of an arbitrary group G by applying this construction to the classifying space EG. Notice that after we have established Assumption 6.2 for the von Neumann algebra and the von Neumann dimension, no more functional analysis will appear, the rest is pure homological algebra and ring theory. We mention the slogan that the von Neumann algebra N (G) is very similar to the ring ℤ of integers with two exceptions, namely, N (G) is not Noetherian, unless G is finite, and N (G) has non-trivial zero-divisors, unless G is finite. We recommend the reader to test statements and results about modules and their dimensions over N (G) by the corresponding ones for modules over ℤ.

This algebraic approach is very convenient because it is very flexible and constructions like taking kernels, cokernels, quotients and homology are of course available. One may also try to extend the notion of L^{2}-Betti numbers within the category of Hilbert N (G)-modules, the von Neumann dimension is defined for any Hilbert N (G)-module. The problem is that one would have to take the Hilbert completion of the cellular chain complex. To get Hilbert N (G)-chain modules one would have to restrict to proper G-CW - complexes. More serious problems occur with the differentials. Only under very restrictive conditions the differentials become bounded operators in the case of a proper G-CW -complex (not necessarily of finite type), which for instance are not satisfied for the bar-model of EG. If the differentials are not bounded one could take their minimal closures, but then it becomes very difficult or impossible to do certain constructions and establish certain proofs. The same problems arise with the chain maps induced by G-maps of G-spaces.

We show in Section 6.3 for an injective group homomorphism i: H → G that induction with the induced ring homomorphism i: N (H) → N (G) is faithfully flat and compatible with von Neumann dimension. This will be used all over the place when one wants to pass from the universal covering of a space X to a regular covering associated to an injective group homomorphism π_{1}(X) → G. We will also prove that N (G) ⊗_{ℂG} ℂ is non-trivial if and only if G is amenable. This may be viewed as an extension of the result of Brooks [68] that the Laplacian on functions on the universal covering of a closed Riemannian manifold M has zero in its spectrum if and only if π_{1}(M) is amenable.

In Section 6.4 we investigate the von Neumann dimension for amenable G. A survey of amenable groups is presented in Subsection 6.4.1. The von Neumann algebra N (G) is known to be flat over ℂG only for virtually cyclic groups (and conjecturally these are the only ones), but from a dimension point of view it looks like a flat module over ℂG, more precisely, the von Neumann dimension of Tor_{p}^{ℂG}(N (G), M) vanishes for all ℂG-modules and p ≥ 1, provided that G is amenable. This implies that the p-th L^{2}-Betti number of a G-space X for amenable G can be read off from the ℂG-module H_{p}(X; ℂ), namely, it is the von Neumann dimension of N (G) ⊗_{ℂG} H_{p}(X; ℂ). Since H_{p}(EG; ℂ) vanishes for p ≥ 1, one obtains the result of Cheeger and Gromov [107] that all the L^{2}-Betti numbers of an amenable group G vanish. The dimension-flatness of the von Neumann algebra N (G) over ℂG for amenable G will play also an important role in applications to G-theory in Subsection 9.5.3.

In Section 6.5 we will prove the basic properties of the L^{2}-Betti numbers for arbitrary G-spaces such as homotopy or more generally homology invariance, Künneth formula and behaviour under induction and restriction. We will investigate its behaviour under fibrations and S^{1}-actions. In Section 6.6 we will introduce the L^{2}-Euler characteristic and prove its basic properties like Euler-Poincaré formula, homology invariance, Künneth formula, sum formula and behaviour under induction and restriction. We will compare the various L^{2}-Euler characteristics of the fixed point sets of a finite proper G-CW -complex with the equivariant Euler characteristic which takes value in the so called Burnside group. This will be applied in particular to the classifying space E(G, FIN). The definition of the Burnside group A(G) of a group G, which should not be confused with the Burnside group B(m, n) appearing in group theory, is analogous to the definition of the Burnside ring of a finite group, but the Burnside group A(G) for infinite G does not inherit an internal multiplication.

To get a quick overview one should read through Theorem 6.7, Theorem 6.24, Theorem 6.37 and then start immediately with Section 6.5. To understand the basics of this Chapter 6 only some knowledge about Section 1.1 from the preceding chapters is necessary.

We briefly will mention in Section 6.8. that a similar approach can be used to extend the notion of Novikov-Shubin invariants to arbitray G-spaces.

This chapter is based on the papers [333] and [334] which have been motivated by the paper of Cheeger and Gromov [107].

6.1 Dimension Theory for Arbitrary Modules

In this section we show that the von Neumann dimension for finitely generated projective N (G)-modules has a unique extension to all N (G)-modules which has nice properties like Additivity, Cofinality and Continuity. Ring will always mean associative ring with unit and R-module will mean left R-module unless explicitly stated differently.

Recall that the dual M^{∗} of a left or right respectively R-module M is the right or left respectively R-module hom_{R}(M, R) where the R-multiplication is given by (fr)(x) = f(x)r or (rf)(x) = rf(x) respectively for f ∈ M^{∗}, x ∈ M and r ∈ R.

Definition 6.1 (Closure of a submodule). Let M be an R-submodule of N. Define the closure of M in N to be the R-submodule of N

\overline{M} = \{x ∈ N | f(x) = 0 for all f ∈ N^{∗} with M ⊂ ker(f)\}.

For an R-module M define the R-submodule TM and the quotient R-module PM by

TM := {x ∈ M | f(x) = 0 for all f ∈ M^{∗}};
PM := M/TM.

We call a sequence of R-modules L ⟶^{i} M ⟶^{q} N weakly exact if \overline{im(i)} = ker(q).

Notice that TM is the closure of the trivial submodule in M. It can also be described as the kernel of the canonical map i(M): M → (M^{∗})^{∗} which sends x ∈ M to the map M^{∗} → R, f ↦ f(x). Notice that TPM = 0, PPM = PM, M^{∗} = (PM)^{∗} and that PM = 0 is equivalent to M^{∗} = 0.

Assumption 6.2. We assume that there is a dimension function dim which assigns to any finitely generated projective R-module P a non-negative real number

dim(P) ∈ [0, ∞)

with the following properties:

(1) If P and Q are finitely generated projective R-modules, then

P ≅_{R} Q ⇒ dim(P) = dim(Q);
dim(P ⊕ Q) = dim(P) + dim(Q);

(2) Let K ⊂ Q be a submodule of the finitely generated projective R-module Q. Then its closure \overline{K} (see Definition 6.1) is a direct summand in Q and

dim(\overline{K}) = sup\{dim(P) | P ⊂ K finitely generated projective submodule\}.

Let N (G) be the group von Neumann algebra of the discrete group G (see Definition 1.1). Let P be any finitely generated projective N (G)-module. (Here we view N (G) just as a ring, there is no Hilbert structure involved in P). Choose any (n, n)-matrix A ∈ M_{n}(N (G)) such that A^{2} = A and the image of the N (G)-linear map induced by right multiplication with A

r_{A}: N (G)^{n} → N (G)^{n}, x ↦ xA

is isomorphic as an N (G)-module to P. (It is not necessary but possible to require A = A^{∗}.) Define the von Neumann dimension

dim_{N (G)}(P) := tr_{N (G)}(A),  (6.3)

where tr_{N (G)} : M_{n}(N (G)) → ℂ sends A to the sum of the von Neumann traces (see Definition 1.2) of its diagonal entries. This is independent of the choice of A by the following standard argument.

Suppose B ∈ M_{p}(N (G)) is a second square matrix with B^{2} = B and im(r_{B}) ≅ P. By possibly taking the direct sum with a zero square-matrix we can achieve without changing tr_{N (G)}(A) and tr_{N (G)}(B) and the isomorphism class of im(r_{A}) and im(r_{B}) that n = p and that im(r_{1−A}) and im(r_{1−B}) are isomorphic. Let C ∈ M_{n}(N (G)) be an invertible matrix such that r_{C} maps im(r_{A}) to im(r_{B}) and im(r_{1−A}) to im(r_{1−B}). Then r_{B} ∘ r_{C} = r_{B} ∘ r_{C} ∘ r_{A} = r_{C} ∘ r_{A} and hence CBC^{−1} = A. This implies

tr_{N (G)}(B) = tr_{N (G)}(C^{−1}CB) = tr_{N (G)}(CBC^{−1}) = tr_{N (G)}(A). (6.4)

We will explain in Theorem 6.24 that this notion is essentially the same as the notion of the von Neumann dimension of a finitely generated Hilbert N (G)-module of Definition 1.10.

The proof of the following Theorem 6.5 will be given in Section 6.2. Notice that in the extension of the von Neumann dimension from finitely generated projective N (G)-modules to arbitrary N (G)-modules the functional analytic aspects only enter in the proof of Theorem 6.5 the rest is purely algebraic ring and module theory.

Theorem 6.5. The pair (N (G), dim_{N (G)}) satisfies Assumption 6.2.

Definition 6.6 (Extended dimension). If (R, dim) satisfies Assumption 6.2, we define for an R-module M its extended dimension

dim′(M) := sup{dim(P) | P ⊂ M finitely generated projective submodule}
∈ [0, ∞].

We will later drop the prime in dim′ (see Notation 6.11).

The next result is one of the basic results of this chapter.

Theorem 6.7. (Dimension function for arbitrary N (G)-modules). Suppose that (R, dim) satisfies Assumption 6.2. Then

(1) R is semihereditary, i.e. any finitely generated submodule of a projective module is projective;

(2) If K ⊂ M is a submodule of the finitely generated R-module M, then M/\overline{K} is finitely generated projective and \overline{K} is a direct summand in M;

(3) If M is a finitely generated R-module, then PM is finitely generated

projective and

M ≅ PM ⊕ TM;

(4) The dimension dim′ has the following properties:

(a) Extension Property If M is a finitely generated projective R-module, then

dim′(M) = dim(M);

(b) Additivity If 0 → M_{0} ⟶^{i} M_{1} ⟶^{p} M_{2} → 0 is an exact sequence of R-modules, then

dim′(M_{1}) = dim′(M_{0}) + dim′(M_{2}),

where for r, s ∈ [0, ∞] we define r +s by the ordinary sum of two real numbers if both r and s are not ∞, and by ∞ otherwise; (c) Cofinality

Let {M_{i} | i ∈ I} be a cofinal system of submodules of M, i.e. M = ⋃_{i∈I} M_{i} and for two indices i and j there is an index k in I satisfying M_{i}, M_{j} ⊂ M_{k}. Then

dim′(M) = sup{dim′(M_{i}) | i ∈ I};

(d) Continuity If K ⊂ M is a submodule of the finitely generated R-module M, then

dim′(K) = dim′(\overline{K});

(e) If M is a finitely generated R-module, then

dim′(M) = dim(PM);
dim′(TM) = 0;

(f) The dimension dim′ is uniquely determined by the Extension Property, Additivity, Cofinality and Continuity.

Proof. (1) Let M ⊂ P be a finitely generated R-submodule of the projective R-module P. Choose a homomorphism q: R^{n} → P with im(q) = M. Then ker(q) = \overline{ker(q)}. Hence ker(q) is by Assumption 6.2 a direct summand. This shows that M is projective.

(2) Choose an epimorphism q: R^{n} → M. One easily checks that q^{−1}(\overline{K}) = \overline{q^{−1}(K)} and that R^{n}/q^{−1}(\overline{K}) and M/\overline{K} are R-isomorphic. By Assumption 6.2 R^{n}/\overline{q^{−1}(K)} and hence M/\overline{K} are finitely generated projective.

(3) This follows from (2) by taking K = 0. (4a) If P ⊂ M is a finitely generated projective R-submodule of the finitely generated projective R-module M, we conclude

dim(P) ≤ dim(M)  (6.8)

from the following calculation based on Assumption 6.2

dim(P) ≤ dim(\overline{P}) = dim(M) − dim(M/\overline{P}) ≤ dim(M).

This implies dim(M) = dim′(M). (4b) Let P ⊂ M_{2} be a finitely generated projective submodule. We obtain an exact sequence 0 → M_{0} → p^{−1}(P) → P → 0. Since p^{−1}(P) ≅ M_{0} ⊕ P, we conclude

dim′(M_{0}) + dim(P) ≤ dim′(p^{−1}(P)) ≤ dim′(M_{1}).

Since this holds for all finitely generated projective submodules P ⊂ M_{2}, we get

dim′(M_{0}) + dim′(M_{2}) ≤ dim′(M_{1}).  (6.9)

Let Q ⊂ M_{1} be finitely generated projective. Let \overline{i(M_{0}) ∩ Q} be the closure of i(M_{0}) ∩ Q in Q. We obtain exact sequences

\begin{aligned}0 & → & i(M_{0}) ∩ Q & → & Q & → & p(Q) & → & 0; \\ 0 & → & \overline{i(M_{0}) ∩ Q} & → & Q & → & Q/\overline{i(M_{0}) ∩ Q} & → & 0. \end{aligned}

By Assumption 6.2 \overline{i(M_{0}) ∩ Q} is a direct summand in Q. We conclude

dim(Q) = dim(\overline{i(M_{0}) ∩ Q}) + dim(Q/\overline{i(M_{0}) ∩ Q}).

From Assumption 6.2 and the fact that there is an epimorphism from p(Q) onto the finitely generated projective R-module Q/\overline{i(M_{0}) ∩ Q}, we conclude

dim(\overline{i(M_{0}) ∩ Q}) = dim′(i(M_{0}) ∩ Q);
dim(Q/\overline{i(M_{0}) ∩ Q}) ≤ dim′(p(Q)).

Since obviously dim′(M) ≤ dim′(N) holds for R-modules M and N with M ⊂ N, we get

dim(Q) = dim(\overline{i(M_{0}) ∩ Q}) + dim(Q/\overline{i(M_{0}) ∩ Q})
≤ dim′(i(M_{0}) ∩ Q) + dim′(p(Q))
≤ dim′(M_{0}) + dim′(M_{2}).

Since this holds for all finitely generated projective submodules Q ⊂ M_{1}, we get

dim′(M_{1}) ≤ dim′(M_{0}) + dim′(M_{2}).  (6.10)

Now assertion (4b) follows from (6.9) and (6.10). (4c) If P ⊂ M is a finitely generated projective submodule, then there is an index i ∈ I with P ⊂ M_{i} by cofinality. (4d) Choose an epimorphism q: R^{n} → M. Since q^{−1}(\overline{K}) = \overline{q^{−1}(K)}, we get from Assumption 6.2 and assertion (4a)

dim′(q^{−1}(K)) = dim′(\overline{q^{−1}(K)}) = dim′(q^{−1}(\overline{K})).

If L is the kernel of q, we conclude from assertions (4a) and (4b)

dim′(q^{−1}(\overline{K})) = dim′(L) + dim′(\overline{K});
dim′(q^{−1}(K)) = dim′(L) + dim′(K);
dim′(q^{−1}(K)) ≤ dim(R^{n}) < ∞.

This proves assertion (4d). (4e) This follows from (4d) applied to the special case K = 0 and assertions (3), (4a) and (4b).

(4f) Let dim′′ be another function satisfying Extension Property, Additivity, Cofinality and Continuity. We want to show for an R-module M

dim′′(M) = dim′(M).

Since assertion (4e) is a consequence of assertions (3), (4a), (4b) and (4d), it holds also for dim′′. Hence we get dim′(M) = dim′′(M) for any finitely generated R-module. Since the system of finitely generated submodules of a module is cofinal, the claim follows from Cofinality. This finishes the proof of Theorem 6.7. ⊓ ⊔

Notation 6.11. In view of Theorem 6.7 we will not distinguish between dim′ and dim in the sequel. ⊓ ⊔

Example 6.12. Let R be a principal ideal domain. Then any finitely generated projective R-module P is isomorphic to R^{n} for a unique n ≥ 0 and we consider the dimension dim(P) := n. Notice that for a submodule M ⊂ R^{n} we have

\overline{M} = \{x ∈ R^{n} | r · x ∈ M for appropriate r ∈ R, r ≠ 0\}.

One easily checks that Assumption 6.2 is satisfied. Let F be the quotient field of R. Then we get for any R-module M for the dimension defined in Definition 6.6

dim(M) = dim_{F} (F ⊗_{R} M),

where dim_{F} (F ⊗_{R} M) is the dimension of the F -vector space F ⊗_{R} M.

Notice that dim(P) is finite for a projective R-module P if and only if P is finitely generated. This is the crucial difference to the case, where we consider the von Neumann algebra N (G) and the extension dim_{N (G)} of Definition 6.6 of the von Neumann dimension dim_{N (G)} of (6.3). If for instance H_{1}, H_{2}, … is a sequence of finite subgroups of G, then ⨁_{i=1}^{∞} N (G)⊗_{ℂH_{i}} ℂ is a projective not finitely generated N (G)-module and

dim_{N (G)} (⨁_{i=1}^{∞} N (G) ⊗_{ℂH_{i}} ℂ) = ∑_{i=1}^{∞} \frac{1}{|H_{i}|},

and this infinite sum may converge to a finite real number.

Recall that a directed set I is a set with a partial ordering ≤ such that for two elements i_{0} and i_{1} there exists an element i with i_{0} ≤ i and i_{1} ≤ i. We can consider I as a category with I as set of objects, where the set of morphisms from i_{0} to i_{1} consists of precisely one element, if i_{0} ≤ i_{1}, and is empty otherwise. A directed system or an inverse system respectively {M_{i} | i ∈ I} of R-modules indexed by I is a functor from I into the category of R-modules which is covariant or contravariant respectively. We denote by colim_{i∈I} M_{i} the colimit of the directed and by lim_{i∈I} M_{i} the limit of the inverse system {M_{i} | i ∈ I} which is again an R-module. We mention that colimit is sometimes also called inductive limit or direct limit and that limit is sometimes also called inverse limit or projective limit in the literature. The colimit is characterized by the following property. For any i ∈ I there is an Rhomomorphism ψ_{i}: M_{i} → colim_{i∈I} M_{i} such that for any two elements i_{0} and i_{1} in I with i_{0} ≤ i_{1} we have ψ_{i_{1}} ∘ϕ_{i_{0},i_{1}} = ψ_{i_{0}}, where ϕ_{i_{0},i_{1}} : M_{i_{0}} → M_{i_{1}} comes from functoriality. For any R-module N together with R-homomorphisms f_{i}: M_{i} → N such that for any two elements i_{0} and i_{1} in I with i_{0} ≤ i_{1} we have f_{i_{1}} ∘ϕ_{i_{0},i_{1}} = f_{i_{0}}, there is precisely one R-homomorphism f: colim_{i∈I} M_{i} → N satisfying f ∘ ψ_{i} = f_{i} for all i ∈ I. The limit of an inverse system has an analogous characterization, one has to reverse all arrows.

Next we investigate the behaviour of dimension under colimits indexed by a directed set.

Theorem 6.13 (Dimension and colimits). Let {M_{i} | i ∈ I} be a directed system of R-modules over the directed set I. For i ≤ j let ϕ_{i,j}: M_{i} → M_{j} be the associated morphism of R-modules. For i ∈ I let ψ_{i}: M_{i} → colim_{i∈I} M_{i} be the canonical morphism of R-modules. Then

(1) We get for the dimension of the R-module given by the colimit colim_{i∈I} M_{i}

dim (colim_{i∈I} M_{i}) = sup {dim(im(ψ_{i})) | i ∈ I};

(2) Suppose for each i ∈ I that there is i_{0} ∈ I with i ≤ i_{0} such that dim(im(ϕ_{i,i_{0}})) < ∞ holds. Then

dim (colim_{i∈I} M_{i})
= sup {inf {dim(im(ϕ_{i,j}: M_{i} → M_{j})) | j ∈ I, i ≤ j} | i ∈ I} .

Proof. (1) Recall that the colimit colim_{i∈I} M_{i} is ∐_{i∈I} M_{i}/ ∼ for the equivalence relation for which M_{i} ∋ x ∼ y ∈ M_{j} holds precisely if there is k ∈ I with i ≤ k and j ≤ k with the property ϕ_{i,k}(x) = ϕ_{j,k}(y). With this description one easily checks

colim_{i∈I} M_{i} = ⋃_{i∈I} im(ψ_{i}: M_{i} → colim_{j∈I} M_{j}).

Now apply Cofinality (see Theorem 6.7 (4c)).

(2) It remains to show for i ∈ I

dim(im(ψ_{i})) = inf {dim(im(ϕ_{i,j}: M_{i} → M_{j})) | j ∈ I, i ≤ j} . (6.14)

By assumption there is i_{0} ∈ I with i ≤ i_{0} such that dim(im(ϕ_{i,i_{0}})) is finite. Let K_{i_{0},j} be the kernel of the map im(ϕ_{i,i_{0}}) → im(ϕ_{i,j}) induced by ϕ_{i_{0},j} for i_{0} ≤ j and K_{i_{0}} be the kernel of the map im(ϕ_{i,i_{0}}) → im(ψ_{i}) induced by ψ_{i_{0}}. Then K_{i_{0}} = ⋃_{j∈I,i_{0}≤j} K_{i_{0},j} and hence by Cofinality (see Theorem 6.7 (4c))

dim(K_{i_{0}}) = sup{dim(K_{i_{0},j}) | j ∈ I, i_{0} ≤ j}.

Since dim(im(ϕ_{i,i_{0}})) is finite, we get from Additivity (see Theorem 6.7 (4b))

dim(im(ψ_{i}))

= dim (im (ψ_{i_{0}}|_{im(ϕ_{i,i_{0}} )} : im(ϕ_{i,i_{0}}) → colim_{i∈I} M_{i}))
= dim(im(ϕ_{i,i_{0}})) − dim(K_{i_{0}})
= dim(im(ϕ_{i,i_{0}})) − sup{dim(K_{i_{0},j}) | j ∈ I, i_{0} ≤ j}
= inf {dim(im(ϕ_{i,i_{0}})) − dim(K_{i_{0},j}) | j ∈ I, i_{0} ≤ j}
= inf {dim (im(ϕ_{i_{0},j}|_{im(ϕ_{i,i_{0}} )} : im(ϕ_{i,i_{0}}) → im(ϕ_{i,j}))) | j ∈ I, i_{0} ≤ j}
= inf {dim (im(ϕ_{i,j})) | j ∈ I, i_{0} ≤ j} .  (6.15)

Given j_{0} ∈ J with i ≤ j_{0}, there is j ∈ I with i_{0} ≤ j and j_{0} ≤ j. We conclude dim(im(ϕ_{i,j_{0}})) ≥ dim(im(ϕ_{i,j})) from Additivity (see Theorem 6.7 (4b)). This implies

inf{dim(im(ϕ_{i,j})) | j ∈ J, i ≤ j}
= inf{dim(im(ϕ_{i,j})) | j ∈ J, i_{0} ≤ j}. (6.16)

Now (6.14) follows from (6.15) and (6.16). This finishes the proof of Theorem 6.13. ⊓ ⊔

Example 6.17. The condition in Theorem 6.13 (2) above that for each i ∈ I there is i_{0} ∈ I with i ≤ i_{0} and dim(im(ϕ_{i,i_{0}})) < ∞ is necessary as the following example shows. Take I = ℕ. Define M_{j} = ⨁_{n=j}^{∞} R and ϕ_{j,k}: ⨁_{m=j}^{∞} R → ⨁_{m=k}^{∞} R to be the obvious projection for j ≤ k. Then dim(im(ϕ_{j,k})) = ∞ for all j ≤ k, but colim_{i∈I} M_{i} is trivial and hence has dimension zero.

Next we state the version of Theorem 6.13 about dimension and colimits for limits over inverse systems. Since we do not need it elsewhere, we do not give its proof which is, however, much harder than the one of Theorem 6.13.

Theorem 6.18 (Dimension and limits). Let {M_{i} | i ∈ I} be an inverse system of N (G)-modules over the directed set I. For i ≤ j let ϕ_{i,j}: M_{j} → M_{i} be the associated morphism of N (G)-modules. For i ∈ I let ψ_{i}: lim_{i∈I} M_{i} → M_{i} be the canonical map. Suppose that there is a countable sequence i_{1} ≤ i_{2} ≤ … such that for each j ∈ I there is n ≥ 0 with j ≤ i_{n}. Let dim_{N (G)} be the extension of Definition 6.6 of the von Neumann dimension of (6.3). Then

(1)

dim_{N (G)} (lim_{i∈I} M_{i}) = sup {dim_{N (G)}(im(ψ_{i}: lim_{i∈I} M_{i} → M_{i}))| i ∈ I};

(2) Suppose that for each index i ∈ I there is an index i_{0} ∈ I with i ≤ i_{0} and dim_{N (G)}(im(ϕ_{i,i_{0}})) < ∞. Then

dim_{N (G)} (lim_{i∈I} M_{i}) = sup {inf{dim_{N (G)}(im(ϕ_{i,j}))| j ∈ I, i ≤ j} | i ∈ I} .

Example 6.19. In this example we want to show that the condition that there is a countable sequence i_{1} ≤ i_{2} ≤ … such that for each j ∈ I there is n ≥ 0 with j ≤ i_{n} appearing in Theorem 6.18 is necessary. (Compare this with the claim in [107, page 210] that equation (A1) in [107, page 210] holds for arbitrary (not necessarily countable) intersections of Γ -weakly closed submodules.) Let G = ℤ. Then N (ℤ) = L^{∞}(S^{1}) by Example 1.4. For u ∈ S^{1} let (z −u) be the N (ℤ)-ideal which is generated by the function S^{1} → ℂ sending z ∈ S^{1} ⊂ ℂ to z − u. Next we prove

⋂_{u∈S^{1}}(z − u) = 0.

Consider f ∈ ∩_{u∈S^{1}}(z − u). Define s ∈ [0, 1] to be the supremum over all r ∈ [0, 1] such that |f(exp(2πit))| ≤ 1 holds for almost all t ∈ [0, r]. For almost all means for all elements with the exception of the elements of a subset of Lebesgue measure zero. Notice that the definition of s makes sense, its value does not change if we add to f a measurable function which vanishes outside a set of measure zero. We want to show s = 1 by contradiction. Suppose s < 1. Put u = exp(2πis). Since f belongs to (z − u) there is g ∈ L^{∞}(S^{1}) with f(z) = (z − u) · g(z). Choose ϵ > 0 with ||g||_{∞} ≤ ϵ^{−1} and s + ϵ ≤ 1. Then we have |f(z)| ≤ 1 for almost all elements z in {exp(2πit) | s − ϵ ≤ t ≤ s + ϵ}. This implies that |f(z)| ≤ 1 holds for almost all elements z in {exp(2πit) | 0 ≤ t ≤ s + ϵ}. We conclude s ≥ s + ϵ, a contradiction. We get s = 1. Hence we have ||f||_{∞} ≤ 1 for all f ∈ ⋂_{u∈S^{1}}(z − u). This implies that ⋂_{u∈S^{1}}(z − u) is zero.

This shows that the obvious N (ℤ)-map L^{∞}(S^{1}) → ∏_{u∈S^{1}} L^{∞}(S^{1})/(z−u) is injective. From Additivity (see Theorem 6.7 (4b)) we conclude

dim_{N (ℤ)} ( ∏_{u∈S^{1}} L^{∞}(S^{1})/(z − u)) ≥ 1.

Notice that ∏_{u∈S^{1}} L^{∞}(S^{1})/(z−u) is the limit over the obvious inverse system {∏_{u∈J} L^{∞}(S^{1})/(z − u)) | J ∈ I}, where I is the set of finite subsets of S^{1} ordered by inclusion, and that dim_{N (ℤ)} (∏_{u∈J} L^{∞}(S^{1})/(z − u)) = 0 holds for all J ∈ I.

Also the condition that for each index i ∈ I there is an index j ∈ I with dim_{N (ℤ)}(im(ϕ_{i,j})) < ∞ appearing in Theorem 6.18 (2) is necessary as the following example shows. Take I = ℕ. Define M_{j} = ∏_{n=j}^{∞} N (G) and ϕ_{j,k}: ∏_{m=k}^{∞} N (G) → ∏_{m=j}^{∞} N (G) to be the canonical inclusion for j ≤ k.

Then for all j ≤ k we get dim_{N (ℤ)}(im(ϕ_{j,k})) = ∞, but lim_{i∈I} M_{i} is trivial and hence has dimension zero.

Theorem 6.18 (2) has only been stated for R = N (G) and the extended von Neumann dimension dim_{N (G)}, in contrast to Theorem 6.13 about dimension and colimits. Indeed Theorem 6.18 fails for R = ℤ with the ordinary rank of abelian groups as dimension. For instance the limit of the inverse system

ℤ ⊃ 2 · ℤ ⊃ 2^{2} · ℤ ⊃ 2^{3} · ℤ ⊃ …

is zero and hence has dimension 0, but the image of any structure map 2^{j}·ℤ → 2^{i} · ℤ has dimension 1.

Definition 6.20 (Extended von Neumann dimension). Let M be an N (G)-module. Define its extended von Neumann dimension

dim_{N (G)}(M) ∈ [0, ∞]

by the extension of Definition 6.6 of the von Neumann dimension of (6.3).

We will extend in Theorem 8.29 the dimension function for N (G)-modules to U(G)-modules, where U(G) is the algebra of affiliated operators, or, equivalently, the Ore localization of N (G) with respect to all non-zero-divisors in N (G).

6.2 Comparison of Modules and Hilbert Modules

In this section we want to show that the category of finitely generated projective N (G)-modules (with inner product) and the category of finitely generated Hilbert N (G)-modules are equivalent as ℂ-categories (with involution) and that this equivalence preserves weakly exact and exact sequences and dimension (see Theorem 6.24). This will be the key step to come from the operator theoretic approach to a purely algebraic approach to L^{2}-Betti numbers. Recall that we use the convention that groups and rings act from the left unless stated explicitly differently.

We need some notations to formulate the main result of this section. A ℂ-category C is a category such that for any two objects the set of morphisms between them carries the structure of a complex vector space for which composition of morphisms is bilinear and C has a (strict) sum which is compatible with the complex vector space structures above. A (strict) involution on a ℂ-category C is an assignment which associates to each morphism f: x → y a morphism f^{∗}: y → x and has the following properties

(f^{∗})^{∗} = f;
(λ · f + μ · g)^{∗} = \overline{λ} · f^{∗} + \overline{μ} · g^{∗};
(f ∘ g)^{∗} = g^{∗} ∘ f^{∗};
(f ⊕ g)^{∗} = f^{∗} ⊕ g^{∗},

where f, g are morphisms, λ and μ complex numbers. There is a canonical structure of a ℂ-category with involution on the category of finitely generated Hilbert N (G)-modules {fin. gen. Hilb. N (G)-mod.} (see Definition 1.5), where the involution is given by taking adjoint operators. We call an endomorphism f in C selfadjoint if f = f^{∗}. We call an isomorphism f in C unitary if f^{∗} = f^{−1}. A functor of ℂ-categories (with involution) is a functor compatible with the complex vector space structures on the set of morphisms between two objects and the sums (and the involutions). A natural equivalence T of functors of ℂ-categories with involution is called unitary if the evaluation of T at each object is a unitary isomorphism. An equivalence of ℂ-categories (with involution) is a functor of such categories such that there is a functor of such categories in the other direction with the property that both compositions are (unitarily) naturally equivalent to the identity. If a functor F of ℂ-categories (with involution) induces a bijection on the sets of (unitary) isomorphism classes of objects and for any two objects x and y it induces a bijection between the set of morphisms from x to y to the set of morphisms from F (x) to F (y), then it is an equivalence of ℂ-categories (with involution) and vice versa. (cf [351, Theorem 1 in IV.9 on page 91]).

Given a finitely generated projective (left) N (G)-module P, an inner product on P is a map μ: P × P → N (G) satisfying (cf. [514, Definition 15.1.1 on page 232])

(1) μ is N (G)-linear in the first variable;

(2) μ is symmetric in the sense μ(x, y) = μ(y, x)^{∗};

(3) μ is positive-definite in the sense that μ(p, p) is a positive element in N (G), i.e. of the form a^{∗}a for some a ∈ N (G), and μ(p, p) = 0 ⇔ p = 0;

(4) The induced map \overline{μ}: P → P^{∗} = hom_{N (G)}(P, N (G)), defined by \overline{μ}(y)(x) = μ(x, y), is bijective.

Notice that we have already introduced an N (G)-right module structure on P^{∗} given by (fr)(x) = f(x)r for r ∈ N (G). Using the involution on N (G), we can define also a left N (G)-module structure by rf(x) = f(x)r^{∗}. Then \overline{μ} is an isomorphism of left N (G)-modules. Moreover, \overline{μ} agrees with

the composition P ⟶^{i(P)} (P^{∗})^{∗} ⟶^{\overline{μ}^{∗}} P^{∗}, where i(P): P → (P^{∗})^{∗} is the bijection which sends x ∈ P to the map P^{∗} → N (G), f ↦ f(x)^{∗}. Let {fin. gen. proj. N (G)-mod. with ⟨ ⟩} be the ℂ-category with involution, whose objects are finitely generated projective N (G)-modules with inner product (P, μ) and whose morphisms are N (G)-linear maps. We get an involution on it if we specify f^{∗}: (P_{1}, μ_{1}) → (P_{0}, μ_{0}) for f: (P_{0}, μ_{0}) → (P_{1}, μ_{1}) by requiring μ_{1}(f(x), y) = μ_{0}(x, f^{∗}(y)) for all x ∈ P_{0} and y ∈ P_{1}. In other words, we define f^{∗} := \overline{μ_{0}}^{−1} ∘f^{∗} ∘\overline{μ_{1}} where the second f^{∗} refers to the N (G)- map f^{∗} = hom_{N (G)}(f, id): P_{1}^{∗} → P_{0}^{∗}. In the sequel we will use the symbol f^{∗} for both f^{∗}: P_{1} → P_{0} and f^{∗}: P_{1}^{∗} → P_{0}^{∗}.

Let {N (G)^{n}} ⊂ {fin. gen. proj. N (G)-mod. with ⟨ ⟩} be the full subcategory whose objects are (N (G)^{n}, μ_{st}), where the standard inner product is given by

μ_{st} : N (G)^{n} × N (G)^{n} → N (G), (x, y) ↦ ∑_{i=1}^{n} x_{i}y_{i}^{∗}.  (6.21)

Denote by {l^{2}(G)^{n}} ⊂ {fin. gen. Hilb. N (G)-mod.} the full subcategory whose objects are l^{2}(G)^{n}. We define an isomorphism of ℂ-categories with involution

ν: {N (G)^{n}} → {l^{2}(G)^{n}}  (6.22)

as follows. It sends an object N (G)^{n} to the object l^{2}(G)^{n}. Let f: N (G)^{m} → N (G)^{n} be an N (G)-homomorphism of (left) N (G)-modules. Choose an (m, n)-matrix A ∈ M(m, n, N (G)) such that f sends x to xA. Define

ν(f): l^{2}(G)^{m} → l^{2}(G)^{n}, y ↦ (A^{∗}y^{t})^{t}

where y^{t} is obtained from y by transposing and applying elementwise the involution l^{2}(G) → l^{2}(G) which sends ∑_{g∈G} λ_{g} · g to ∑_{g∈G} \overline{λ_{g}} · g, the matrix A^{∗} is obtained from A by transposing and applying the involution ∗: N (G) → N (G) to each entry, and Ay^{t} is understood in the sense of matrices and plugging y_{j} into an element a: l^{2}(G) → l^{2}(G) in N (G). The involutions appearing in the definition above come from the fact that we have decided to consider always left group actions and left module structures and have defined N (G) to be B(l^{2}(G))^{G}. We will extend ν to finitely generated N (G)-modules with inner product by a completion process below.

Lemma 6.23. For any finitely generated projective N (G)-module P there is an N (G)-map p: N (G)^{n} → N (G)^{n} with p∘p = p and p^{∗} = p such that im(p) is N (G)-isomorphic to P. Any finitely generated projective N (G)-module P has an inner product. Two finitely generated projective N (G)-modules with inner product are unitarily N (G)-isomorphic if and only if the underlying N (G)-modules are N (G)-isomorphic.

Proof. If P is finitely generated projective, we can find an N (G)-map q: N (G)^{n} → N (G)^{n} with q∘q = q and an N (G)-isomorphism f: P ⟶^{≅} im(q). Let p: N (G)^{n} → N (G)^{n} be the N (G)-map for which ν(p) is the orthogonal projection onto the image of ν(q). Then p satisfies p ∘ p = p, p^{∗} = p and im(p) = im(q). The standard inner product μ_{st} on N (G)^{n} defined in (6.21) restricts to an inner product on im(p), also denoted by μ_{st}, and this restriction can be pulled back to an inner product on P by the isomorphism f: P → im(p). It remains to show for an N (G)-map p: N (G)^{n} → N (G)^{n} with p ∘ p = p and p^{∗} = p that for any inner product μ on im(p) there is a unitary isomorphism g: (im(p), μ) → (im(p), μ_{st}).

Let f: P → P be the N (G)-isomorphism (\overline{μ_{st}})^{−1} ∘\overline{μ}. It satisfies μ(x, y) = μ_{st}(x, f(y)) for all x, y ∈ P. Since μ_{st}(x, f(x)) = μ(x, x) ≥ 0 holds, f is positive with respect to μ_{st}. Consider the composition i ∘ f ∘ p: N (G)^{n} → N (G)^{n} where i: P → N (G)^{n} is the inclusion, which is the adjoint of p: N (G)^{n} → P with respect to μ_{st}. This composition is positive with respect to μ_{st} on N (G)^{n}. Recall that we can consider ℂG as a dense subset of l^{2}(G) in the obvious way and as a subset of N (G) if we identify g ∈ G with the element R_{g^{−1}} : l^{2}(G) → l^{2}(G) in N (G) which is given by right multiplication with g^{−1}. Under this identification one easily checks for u, v ∈ ℂG^{n}

tr_{N (G)}(μ_{st}(u, i ∘ f ∘ p(v))) = ⟨u, ν(i ∘ f ∘ p)(v)⟩_{l^{2}(G)^{n}}.

Hence ⟨u, ν(i ∘ f ∘ p)(u)⟩_{l^{2}(G)^{n}} ≥ 0 for all u ∈ ℂG^{n} and therefore for all u ∈ l^{2}(G)^{n}. So ν(i ∘ f ∘ p) is positive. Let g′: N (G)^{n} → N (G)^{n} be defined by the property that ν(g′): l^{2}(G)^{n} → l^{2}(G)^{n} is positive and ν(g′) ∘ ν(g′) = ν(i ∘ f ∘ p). Define g: P → P by p ∘ g′ ∘ i. Then g is invertible, is selfadjoint with respect to μ_{st} and g^{2} = f. Now the claim follows from

μ_{st}(g(x), g(y)) = μ_{st}(x, g^{2}(y)) = μ_{st}(x, f(y)) = μ(x, y).⊓ ⊔

Given a finitely generated projective N (G)-module (P, μ) with inner product μ, we obtain a pre-Hilbert structure on P by tr_{N (G)} ∘μ: P × P → ℂ for tr_{N (G)} the standard trace (see Definition 1.2). Let ν(P, μ) be the associated Hilbert space. The group G acts from the left by unitary operators on P and hence on ν(P, μ) by putting g · x := R_{g^{−1}} · x for x ∈ P and R_{g^{−1}} : l^{2}(G) → l^{2}(G) ∈ N (G) given by right multiplication with g^{−1}. This is a finitely generated Hilbert N (G)-module since one can find another finitely generated projective N (G)-module with inner product (P_{0}, μ_{0}) and a unitary isomorphism (P, μ)⊕(P_{0}, μ_{0}) → (N (G)^{n}, μ_{st}) (see Lemma 6.23) which induces an isometric G-isomorphism ν(P, μ)⊕ν(P_{0}, μ_{0}) → ν(N (G)^{n}, μ_{st}) = l^{2}(G)^{n}. Let (P, μ) and (P′, μ′) be finitely generated projective N (G)-modules with inner product and let f : P → P′ be an N (G)-homomorphism. Then f extends to a morphism of finitely generated Hilbert N (G)-modules ν(f) : ν(P, μ) → ν(P′, μ′).

Theorem 6.24 (Modules over N (G) and Hilbert N (G)-modules).

(1) We obtain an equivalence of ℂ-categories with involution

ν: {fin. gen. proj. N (G)-mod. with ⟨ ⟩} → {fin. gen. Hilb. N (G)-mod.};

(2) The forgetful functor

F: {fin. gen. proj. N (G)-mod. with ⟨ ⟩} → {fin. gen. proj. N (G)-mod.}

is an equivalence of ℂ-categories. We obtain functors of ℂ-categories

ν ∘ F^{−1} : {fin. gen. proj. N (G)-mod.} → {fin. gen. Hilb. N (G)-mod.}
F ∘ ν^{−1} : {fin. gen. Hilb. N (G)-mod.} → {fin. gen. proj. N (G)-mod.}

which are unique up to natural equivalence of functors of ℂ-categories and inverse to one another up to natural equivalence. We will denote ν ∘ F^{−1} and F ∘ ν^{−1} by ν and ν^{−1} again; (3) The functors ν and ν^{−1} preserve exact sequences and weakly exact se-

quences;

(4) If P is a finitely generated projective N (G)-module, then

dim_{N (G)}(P) = dim_{N (G)}(ν(P))

for the two notions of dim_{N (G)} defined in Definition 1.10 and in (6.3).

Proof. (1) The idempotent completion Idem(C) of a ℂ-category C with involution has as objects (V, p) selfadjoint idempotents p: V → V. A morphism from (V_{0}, p_{0}) to (V_{1}, p_{1}) is a morphism f: V_{0} → V_{1} satisfying p_{1} ∘ f ∘ p_{0} = f. The identity on (V, p) is given by p: (V, p) → (V, p). The idempotent completion Idem(C) inherits from C the structure of a ℂ-category with involution in the obvious way. There are unitary equivalences of ℂ-categories with involutions

IM: Idem({N (G)^{n}}) → {fin. gen. proj. N (G)-mod. with ⟨ ⟩};
IM: Idem({l^{2}(G)^{n}}) → {fin. gen. Hilb. N (G)-mod.},

which sends (N (G)^{n}, p) or (l^{2}(G)^{n}, p) to the image of p where the inner product μ_{st} on im(p) is given by restricting the standard inner product μ_{st} on N (G)^{n}. The functor ν defined in (6.22) induces an isomorphism

Idem(ν): Idem({N (G)^{n}}) → Idem({l^{2}(G)^{n}}).

The following diagram commutes

Idem({N (G)^{n}})  ⟶^{Idem(ν)}  Idem({l^{2}(G)^{n}}
IM↓  ^{IM}↓
{fin. gen. proj. N (G)-mod. with ⟨ ⟩} ⟶^{ν} {fin. gen. Hilb. N (G)-mod.}

up to unitary natural equivalence which is induced from the inclusion N (G)^{n} → l^{2}(G)^{n} sending (a_{1}, …, a_{n}) to (\overline{a_{1}^{∗}(e)}, …, \overline{a_{n}^{∗}(e)}), where for u = ∑_{g∈G} λ_{g} · g ∈ l^{2}(G) we put \overline{u} := ∑_{g∈G} \overline{λ_{g}} · g. Now (1) follows.

(2) follows from Lemma 6.23 and assertion (1).

(3) A sequence U ⟶^{f} V ⟶^{g} W of finitely generated Hilbert N (G)-modules is weakly exact at V if and only if the following holds: g ∘ f = 0 and for any finitely generated Hilbert N (G)-modules P and Q and morphisms u: V → P and v: Q → V with u ∘ f = 0 and g ∘ v = 0 we get u ∘ v = 0. It is exact at V if and only if the following holds: g ∘ f = 0 and for any finitely generated Hilbert N (G)-module P and morphism v: P → V with g ∘ v = 0 there is a morphism u: P → U satisfying f ∘ u = v. The analogous statements are true if one considers finitely generated projective N (G)-modules instead of finitely generated Hilbert N (G)-modules. Now ν and ν^{−1} obviously preserve these criterions for weak exactness and exactness and the claim follows.

(4) This follows from the definitions. ⊓ ⊔ Next we can give the proof of Theorem 6.5.

Proof. Part (1) of Assumption 6.2 is obvious, it remains to prove Part (2). In the sequel we use Theorem 6.24 and the functors ν and ν^{−1} appearing there.

First we show that any finitely generated submodule M ⊂ P of a finitely generated projective N (G)-module is finitely generated projective. Namely, choose an N (G)-map f: N (G)^{n} → P with im(f) = M. Let p: N (G)^{n} → N (G)^{n} be the N (G)-map for which ν(p): l^{2}(G)^{n} → l^{2}(G)^{n} is the orthogonal projection onto the kernel of ν(f): ν(N (G)^{n}) = l^{2}(G)^{n} → ν(P). Then p is an idempotent with im(p) = ker(f) and hence M is projective.

Let P = {P_{i} | i ∈ I} be the directed system of finitely generated projective N (G)-submodules of K. Notice that P is indeed directed by inclusion since the submodule of P generated by two finitely generated projective submodules is again finitely generated and hence finitely generated projective. Let j_{i}: P_{i} → Q be the inclusion. Equip Q and each P_{i} with a fixed inner product (Lemma 6.23). Let pr_{i}: ν(Q) → ν(Q) be the orthogonal projection satisfying im(pr_{i}) = \overline{im(ν(j_{i}))} and pr: ν(Q) → ν(Q) be the orthogonal projection satisfying im(pr) = \overline{⋃_{i∈I} im(pr_{i})}. Next we show

im(ν^{−1}(pr)) = \overline{K}.  (6.25)

Let f: Q → N (G) be an N (G)-map with K ⊂ ker(f). Then f ∘ j_{i} = 0 and therefore ν(f) ∘ ν(j_{i}) = 0 for all i ∈ I. We get im(pr_{i}) ⊂ ker(ν(f)) for all i ∈ I. Since the kernel of ν(f) is closed we conclude im(pr) ⊂ ker(ν(f)). This shows im(ν^{−1}(pr)) ⊂ ker(f) and hence im(ν^{−1}(pr)) ⊂ \overline{K}. As K ⊂ ker(id −ν^{−1}(pr)) = im(ν^{−1}(pr)), we conclude \overline{K} ⊂ im(ν^{−1}(pr)). This finishes the proof of (6.25). In particular \overline{K} is a direct summand in Q.

Next we prove

dim′_{N (G)}(K) = dim_{N (G)}(\overline{K}).  (6.26)

The inclusion j_{i} induces a weak isomorphism ν(P_{i}) → im(pr_{i}) of finitely generated Hilbert N (G)-modules. If we apply the Polar Decomposition Theorem to it we obtain a unitary N (G)-isomorphism from ν(P_{i}) to im(pr_{i}). This implies dim_{N (G)}(P_{i}) = tr_{N (G)}(pr_{i}). Therefore it remains to prove

tr_{N (G)}(pr) = sup{tr_{N (G)}(pr_{i}) | i ∈ I}.  (6.27)

Let ϵ > 0 and x ∈ ν(Q) be given. Choose i(ϵ) ∈ I and x_{i(ϵ)} ∈ im(pr_{i(ϵ)}) with || pr(x) − x_{i(ϵ)}|| ≤ ϵ/2. Since im(pr_{i(ϵ)}) ⊂ im(pr_{i}) ⊂ im(pr) we get for all i ≥ i(ϵ)

|| pr(x) − pr_{i}(x)|| ≤ || pr(x) − pr_{i(ϵ)}(x)||
≤ || pr(x) − pr_{i(ϵ)}(x_{i(ϵ)})|| + || pr_{i(ϵ)}(x_{i(ϵ)}) − pr_{i(ϵ)}(x)||
= || pr(x) − x_{i(ϵ)}|| + || pr_{i(ϵ)}(x_{i(ϵ)} − pr(x))||
≤ || pr(x) − x_{i(ϵ)}|| + || pr_{i(ϵ)} || · ||x_{i(ϵ)} − pr(x)||
≤ 2 · || pr(x) − x_{i(ϵ)}||
≤ ϵ.

Now (6.27) and hence (6.26) follow from Theorem 1.9 (2). This finishes the proof of Theorem 6.5. ⊓ ⊔

We will sometimes use

Lemma 6.28. (1) Let f: P → Q be an N (G)-map of finitely generated projective N (G)-modules. If f is a weak isomorphism, then P and Q are N (G)-isomorphic;

(2) Let f: P → Q be an N (G)-map of finitely generated projective N (G)- modules with dim_{N (G)}(P) = dim_{N (G)}(Q). Then the following assertions are equivalent:

(a) f is injective;

(b) f has dense image;

(c) f is a weak isomorphism;

(3) A projective N (G)-module P is trivial if and only if dim_{N (G)}(P) = 0;

(4) Let M be a finitely presented N (G)-module. Then dim_{N (G)}(M) = 0 if

and only if there is an exact sequence 0 → N (G)^{n} ⟶^{f} N (G)^{n} → M → 0 for some N (G)-map f which is positive, i.e. f = h^{∗}h for some N (G)-map h: N (G)^{n} → N (G)^{n}.

Proof. (1) and (2) follow from Lemma 1.13 and Theorem 6.24.

(3) Since P is the colimit of its finitely generated submodules, it suffices to prove the claim for a finitely generated projective N (G)-module by Theorem 6.7 (1) and (4c). Now apply Theorem 1.12 (1) and Theorem 6.24.

(4) Suppose that dim_{N (G)}(M) = 0. Since N (G) is semihereditary (see Theorem 6.7 (1)) and M is finitely presented, there is an exact sequence 0 → P ⟶^{g} N (G)^{n} ⟶^{q} M → 0 of N (G)-modules such that P is finitely generated projective. We get from Additivity (see Theorem 6.7 (4b)) that dim_{N (G)}(P) = dim_{N (G)}(N (G)^{n}). Since by assertion (2) g is a weak isomorphism, ν(g) is a weak isomorphism by Theorem 6.24 (3). Let f: N (G)^{n} → N (G)^{n} be the N (G)-homomorphism for which ν(f) is the positive part in the polar decomposition of ν(g). Now 0 → N (G)^{n} ⟶^{f} N (G)^{n} → M → 0 is exact by Theorem 6.24 (3). The other implication follows from Additivity (see Theorem 6.7 (4b)). ⊓ ⊔

6.3 Induction and the Extended von Neumann Dimension

In this section we show that induction for the ring homomorphism N (H) → N (G) induced by an inclusion of groups i: H → G is faithfully flat and compatible with the extended von Neumann dimension. This will be important when we will compare L^{2}-invariants of a regular covering with the ones of the universal covering of a given space.

Recall from Theorem 6.5 and Theorem 6.7 that dim_{N (G)} introduced in Definition 6.20 satisfies Cofinality, Additivity and Continuity and that for a finitely generated N (G)-module M we get a finitely generated projective N (G)-module PM such that M = PM ⊕ TM and dim_{N (G)}(M) = dim_{N (G)}(PM).

We have associated to an injective group homomorphism i: H → G a ring homomorphism i: N (H) → N (G) (see Definition 1.23). Given an N (H)- module M, the induction with i is the N (G)-module i_{∗}M = N (G)⊗_{N (H)} M. Obviously i_{∗} is a covariant functor from the category of N (H)-modules to the category of N (G)-modules, preserves direct sums and the properties “finitely generated” and “projective” and sends N (H) to N (G).

Theorem 6.29. Let i: H → G be an injective group homomorphism. Then

(1) Induction with i is a faithfully flat functor from the category of N (H)- modules to the category of N (G)-modules, i.e. a sequence of N (H)- modules M_{0} → M_{1} → M_{2} is exact at M_{1} if and only if the induced sequence of N (G)-modules i_{∗}M_{0} → i_{∗}M_{1} → i_{∗}M_{2} is exact at i_{∗}M_{1};

(2) For any N (H)-module M we have:

dim_{N (H)}(M) = dim_{N (G)}(i_{∗}M).

Proof. It is enough to show for any N (H)-module M

dim_{N (H)}(M) = dim_{N (G)}(i_{∗}M);  (6.30)
Tor_{p}^{N (H)}(N (G), M) = 0  for p ≥ 1;  (6.31)
i_{∗}M = 0 ⇒ M = 0.  (6.32)

We begin with the case where M is finitely generated projective. Let A ∈ M_{n}(N (H)) be a matrix such that A = A^{∗}, A^{2} = A and the image of the N (H)-linear map R_{A}: N (H)^{n} → N (H)^{n} induced by right multiplication with A is N (H)-isomorphic to M. Let i_{∗}A be the matrix in M_{n}(N (G)) obtained from A by applying i to each entry. Then i_{∗}A = (i_{∗}A)^{∗}, (i_{∗}A)^{2} = i_{∗}A and the image of the N (G)-linear map R_{i_{∗}A}: N (G)^{n} → N (G)^{n} induced by right multiplication with i_{∗}A is N (G)-isomorphic to i_{∗}M. Hence we get from the definition (6.3)

dim_{N (H)}(M) = tr_{N (H)}(A);
dim_{N (G)}(i_{∗}M) = tr_{N (G)}(i_{∗}A).

Since tr_{N (G)}(i(a)) = tr_{N (H)}(a) holds for a ∈ N (H) (see Lemma 1.24 (1)), we get (6.30). Since for a finitely generated projective N (H)-module we have M = 0 ⇔ dim_{N (H)}(M) = 0 and i_{∗}M is a finitely generated projective N (G)- module, (6.32) follows from (6.30). If M is projective, (6.31) is obviously true.

Next we treat the case where M is finitely presented. Then M splits as M = TM ⊕ PM where PM is finitely generated projective and TM is finitely presented (see Theorem 6.7 (3)). By Lemma 6.28 (4) we obtain an exact sequence 0 → N (H)^{n} ⟶^{f} N (H)^{n} ⟶^{q} TM → 0 with f = f^{∗}. If we apply the right exact functor given by induction with i to it, we get an exact sequence N (G)^{n} ⟶^{i_{∗}f} N (G)^{n} → i_{∗}TM → 0 with (i_{∗}f)^{∗} = i_{∗}f. Since we know (6.30) and (6.31) already for finitely generated projective N (H)- modules, we conclude from Theorem 6.7 (4b) and (4e) and the definition of Tor that (6.30) and (6.31) hold for the finitely presented N (H)-module M provided that we can show that i_{∗}f is injective.

We have i_{∗}(ν(f)) = ν(i_{∗}f), where i_{∗}(ν(f)) was introduced in Definition 1.23. Because ν respects weak exactness (see Theorem 6.24 (3)), ν(f) has dense image since N (H)^{n} ⟶^{f} N (H)^{n} → 0 is weakly exact. Then one easily checks that ν(i_{∗}f) = i_{∗}(ν(f)) has dense image since ℂG ⊗_{ℂH} l^{2}(H) is a dense subspace of l^{2}(G). Since the kernel of a bounded operator of Hilbert spaces is the orthogonal complement of the image of its adjoint and ν(i_{∗}f) is selfadjoint, ν(i_{∗}f) is injective. Since ν^{−1} respects exactness (see Theorem 6.24 (3)) i_{∗}f is injective.

Next we show (6.32) for finitely presented M. It suffices to show for an N (H)-map N (H)^{m} → N (H)^{n} that g is surjective if i_{∗}g is surjective. Since the functors ν^{−1} and ν of Theorem 6.24 are exact we have to show for an H-equivariant bounded operator h: l^{2}(H)^{m} → l^{2}(H)^{n} that h is surjective if i_{∗}h: l^{2}(G)^{m} → l^{2}(G)^{n} is surjective. Let {E_{λ} | λ ≥ 0} be the spectral family of the positive operator h ∘ h^{∗}. Then {i_{∗}E_{λ} | λ ≥ 0} is the spectral family of the positive operator i_{∗}h ∘ (i_{∗}h)^{∗}. Notice that h or i_{∗}h respectively is surjective if and only if E_{λ} = 0 or i_{∗}E_{λ} = 0 respectively for some λ > 0. Because E_{λ} = 0 or i_{∗}E_{λ} = 0 respectively is equivalent to tr_{N (H)}(E_{λ}) = 0 or tr_{N (G)}(i_{∗}E_{λ}) = 0 respectively and tr_{N (H)}(E_{λ}) = tr_{N (G)}(i_{∗}E_{λ}) holds by Lemma 1.24 (1), the claim follows. This finishes the proof of (6.30), (6.31) and (6.32) in the case where M is finitely presented.

Next we explain how we can derive the case of a finitely generated N (H)- module M from the case of a finitely presented N (H)-module. Notice that from now on the argument is purely algebraic, no more functional analysis will enter. If f: N (H)^{n} → M is an epimorphism with kernel K, then M is the colimit over the directed system of N (H)-modules N (H)^{n}/K_{j} indexed by the set {K_{j} | j ∈ J} of finitely generated submodules of K. Now (6.30) follows from Additivity and Cofinality (see Theorem 4 (4b) and (4c)) or directly from Theorem 6.13 about dimension and colimits since i_{∗} commutes with colimits. The functor Tor commutes in both variables with colimits over directed systems [94, Proposition VI.1.3. on page 107] and hence (6.31) follows.

Next we prove (6.32) for finitely generated M. As above we write M = colim_{j∈J} N (H)^{n}/K_{j}. Since each structure map N (H)^{n}/K_{j_{0}} → N (H)^{n}/K_{j_{1}} is an epimorphism of finitely generated N (H)-modules, we have M = 0 if and only if for each j_{0} ∈ J there is j_{1} ∈ J such that N (H)^{n}/K_{j_{1}} vanishes. Recall that i_{∗} commutes with colimits and is right exact. Hence i_{∗}M = colim_{j∈J} i_{∗}(N (H)^{n}/K_{j}) and i_{∗}M = 0 if and only if for each j_{0} ∈ J there is j_{1} ∈ J such that i_{∗}(N (H)^{n}/K_{j_{1}}) vanishes. Since N (H)^{n}/K_{j_{1}} is finitely presented, we know already that N (H)^{n}/K_{j_{1}} = 0 if and only if i_{∗}(N (H)^{n}/K_{j_{1}}) = 0. Hence i_{∗}M = 0 if and only if M = 0.

The argument that (6.30), (6.31) and (6.32) are true for all N (H)-modules if they are true for all finitely generated N (H)-modules is analogous to the proof above that they are true for all finitely generated N (H)-modules if they are true for all finitely presented N (H)-modules. Namely, repeat the argument for the directed system of finitely generated submodules of a given N (H)-module. This finishes the proof of Theorem 6.29. ⊓ ⊔

The proof of Theorem 6.29 would be obvious if we knew that N (G) viewed as an N (H)-module were projective. Notice that this is a stronger statement than the one proved in Theorem 6.29. One would have to show that the higher Ext-groups instead of the Tor-groups vanish to get this stronger statement. However, the proof for the Tor-groups does not go through directly since the Ext-groups are not compatible with colimits.

The rather easy proof of the next lemma can be found in [333, Lemma 3.4 on page 149].

Lemma 6.33. Let H ⊂ G be a subgroup. Then

(1) dim_{N (G)} (N (G) ⊗_{ℂG} ℂ[G/H]) = |H|^{−1}, where |H|^{−1} is defined to be zero if H is infinite;

(2) If G is infinite and V is a ℂG-module which is finite dimensional over

ℂ, then

dim_{N (G)}(N (G) ⊗_{ℂG} V) = 0.

6.4 The Extended Dimension Function and Amenable Groups

In this section we deal with the special case of an amenable group G. The main result (Theorem 6.37) will be that N (G) is flat over ℂG from the point of view of the extended dimension function, although N (G) is flat over ℂG in the strict sense only for very few groups, conjecturally exactly for virtually cyclic groups (see Conjecture 6.49).

6.4.1 Survey on Amenable Groups

In this subsection we give a brief survey about amenable and elementary amenable groups. Let l^{∞}(G, ℝ) be the space of bounded functions from G to ℝ with the supremum norm. Denote by 1 the constant function with value 1.

Definition 6.34 (Amenable group). A group G is called amenable if there is a (left) G-invariant linear operator μ: l^{∞}(G, ℝ) → ℝ with μ(1) = 1 which satisfies for all f ∈ l^{∞}(G, ℝ)

inf{f(g) | g ∈ G} ≤ μ(f) ≤ sup{f(g) | g ∈ G}.

The latter condition is equivalent to the condition that μ is bounded and μ(f) ≥ 0 if f(g) ≥ 0 for all g ∈ G. Let AM be the class of amenable groups.

The class EAM of elementary amenable groups is defined as the smallest class of groups which has the following properties:

(1) It contains all finite and all abelian groups;

(2) It is closed under taking subgroups;

(3) It is closed under taking quotient groups;

(4) It is closed under extensions, i.e. if 1 → H → G → K → 1 is an exact sequence of groups and H and K belong to EAM, then also G ∈ EAM;

(5) It is closed under directed unions, i.e. if {G_{i} | i ∈ I} is a directed system of subgroups such that G = ⋃_{i∈I} G_{i} and each G_{i} belongs to EAM, then G ∈ EAM. (Directed means that for two indices i and j there is a third index k with G_{i}, G_{j} ⊂ G_{k}.)

We give an overview of some basic properties of these notions. A group G is amenable if and only if each finitely generated subgroup is amenable [419, Proposition 0.16 on page 14]. The class of amenable groups satisfies the conditions appearing in the Definition 6.34 of elementary amenable groups, namely, it contains all finite and all abelian groups, and is closed under taking subgroups, forming factor groups, group extensions, and directed unions [419, Proposition 0.15 and 0.16 on page 14]. Hence any elementary amenable group is amenable. Grigorchuk [228] has constructed a finitely presented group which is amenable but not elementary amenable. A group which contains the free group on two letters ℤ ∗ ℤ as a subgroup is not amenable [419, Example 0.6 on page 6]. There exist examples of finitely presented non-amenable groups which do not contain ℤ ∗ ℤ as a subgroup [407]. Any countable amenable group is a-T-menable, i.e. it admits an affine, isometric action on a real inner product space which is metrically proper in the sense that lim_{g→∞} ||gv|| = ∞ holds for every v ∈ V [33]. For a-T-menable groups the Baum-Connes-Conjecture has been proved by Higson and Kasparov [257]. An infinite a-T-menable group and in particular an infinite amenable group does not have Kazhdan’s property T.

A useful geometric characterization of amenable groups is given by the Følner condition [35, Theorem F.6.8 on page 308].

Lemma 6.35. A group G is amenable if and only if it satisfies the Følner condition, i.e. for any finite set S ⊂ G with s ∈ S ⇒ s^{−1} ∈ S and ϵ > 0 there exists a finite non-empty subset A ⊂ G such that for its S-boundary ∂_{S}A = {a ∈ A | there is s ∈ S with as ∉ A} we have

|∂_{S}A| ≤ ϵ · |A|.

Another version of the Følner criterion is that a finitely presented group G is amenable if and only if for any positive integer n, any connected closed n-dimensional Riemannian manifold M with fundamental group π_{1}(M) = G and any ϵ > 0 there is a domain Ω ⊂ M̃ with (n − 1)-measurable boundary such that the (n − 1)-measure of ∂Ω does not exceed ϵ times the measure of Ω [35, Theorem F.6.8 on page 308]. Such a domain can be constructed by an appropriate finite union of translations of a fundamental domain if G is amenable.

The fundamental group of a closed connected manifold is not amenable if M admits a Riemannian metric of non-positive curvature which is not constant zero [16]. Any finitely generated group which is not amenable has exponential growth [35, Proposition F.6.24 on page 318]. A group G is called good in the sense of Freedmann if the so called π_{1}-null disk lemma holds for G which implies the topological s-Cobordism Theorem for 4-manifolds with G as fundamental group. Any group with subexponential growth is good [204, Theorem 0.1 on page 511] and amenable. No amenable group is known which is not good and it may be true that the classes of good groups and amenable groups coincide. The following conditions on a group G are equivalent:

(1) G is amenable;

(2) The canonical map from the full C^{∗}-algebra of G to the reduced C^{∗}- algebra of G is an isomorphism [420, Theorem 7.3.9 on page 243];

(3) The reduced C^{∗}-algebra of G is nuclear [303], provided that G is countable;

(4) G is finite or for any connected free G-CW -complex X the first Novikov-Shubin invariant α_{1}(X) is not ∞^{+} (see Theorem 2.55 (5b));

(5) For a closed connected Riemannian manifold M with G = π_{1}(M) the Laplacian Δ_{0} : L^{2}Ω^{0}(M̃) → L^{2}Ω^{0}(M̃) acting on functions on the universal covering has zero in its spectrum, provided that G is finitely presented (see [68] or Section 2.7);

(6) H^{1}(G, l^{2}(G)) is not Hausdorff [242, Corollary III.2.4 on page 188];

(7) N (G) ⊗_{ℂG} ℂ is not trivial (see Lemma 6.36).

Conjecture 6.48 implies another characterization of amenability in terms of G_{0}(ℂG), namely [ℂG] = 0 would hold in G_{0}(ℂG) if and only if G is not amenable (see Conjecture 9.67).

It is sometimes easier to deal with elementary amenable groups than with amenable groups since elementary amenable groups have a useful description in terms of transfinite induction (see Lemma 10.40).

For more information about amenable groups we refer to [419].

6.4.2 Amenability and the Coinvariants of the Group von Neumann Algebra

Lemma 6.36. N (G)⊗_{ℂG}ℂ[G/H] is trivial if and only if H is non-amenable

Proof. Since N (G) ⊗_{ℂG} ℂ[G/H] and N (G) ⊗_{N (H)} N (H) ⊗_{ℂH} ℂ are N (G)- isomorphic, we conclude from Theorem 6.29 (1) that its suffices to prove the claim in the special case G = H.

Let S be a set of generators of G. Then ⨁_{s∈S} ℂG \underrightarrow{⨁_{s∈S} r_{s−1}} ℂG ⟶^{ϵ} ℂ → 0 is exact where ϵ(∑_{g∈G} λ_{g} · g) = ∑_{g∈G} λ_{g} and r_{u} denotes right multipli-

cation with u ∈ ℂG. We obtain an exact sequence ⨁_{s∈S} N (G) \underrightarrow{⨁_{s∈S} r_{s−1}} N (G) ⟶^{ϵ} N (G) ⊗_{ℂG} ℂ → 0 since the tensor product is right exact. Hence

N (G) ⊗_{ℂG} ℂ is trivial if and only if ⨁_{s∈S} N (G) \underrightarrow{⨁_{s∈S} r_{s−1}} N (G) is surjective. This is equivalent to the existence of a finite subset T ⊂ S such that

⨁_{s∈T} N (G) \underrightarrow{⨁_{s∈T} r_{s−1}} N (G) is surjective. Let G_{0} ⊂ G be the subgroup generated by T. Then the map above is induction with the inclusion of G_{0} ⊂ G

applied to ⨁_{t∈T} N (G_{0}) \underrightarrow{⨁_{t∈T} r_{t−1}} N (G_{0}). Hence we conclude from Theorem 6.29 (1) that N (G)⊗_{ℂG} ℂ is trivial if and only if N (G_{0})⊗_{ℂG_{0}} ℂ is trivial for some finitely generated subgroup G_{0} ⊂ G. The group G is amenable if and only if each of its finitely generated subgroups is amenable [419, Proposition 0.16 on page 14]. Hence we can assume without loss of generality that G is finitely generated and S is finite. Moreover, we can also assume that S is symmetric, i.e. s ∈ S implies s^{−1} ∈ S.

Because the functor ν of Theorem 6.24 (3) is exact, N (G)⊗_{ℂG} ℂ is trivial

if and only if the operator f: ⨁_{s∈S} l^{2}(G) \underrightarrow{⨁_{s∈S} r_{s−1}} l^{2}(G) is surjective. This is equivalent to the bijectivity of the operator

\frac{1}{2 · |S|}f ∘ f^{∗}: l^{2}(G) \underrightarrow{id − ∑_{s∈S} \frac{1}{|S|} ·r_{s}} l^{2}(G).

It is bijective if and only if the spectral radius of the operator l^{2}(G) \underrightarrow{∑_{s∈S} \frac{1}{|S|} ·r_{s}} l^{2}(G) is different from 1. Since this operator is convolution with a probability distribution whose support contains S, namely

P: G → [0, 1], g ↦ \{ \begin{matrix}|S|^{−1}, & g ∈ S \\ 0, & g ∉ S \end{matrix}

the spectral radius is 1 precisely if G is amenable, by a result of Kesten [290].

⊓ ⊔

6.4.3 Amenability and Flatness Properties of the Group von Neumann Algebra over the Group Ring

In this subsection we investigate the flatness properties of the group von Neumann algebra N (G) over the complex group ring ℂG. The next statement says roughly that for an amenable group the group von Neumann algebra is flat over the group ring from the point of view of dimension theory.

Theorem 6.37. (Dimension-flatness of N (G) over ℂG for amenable G). Let G be amenable and M be a ℂG-module. Then

dim_{N (G)} (Tor_{p}^{ℂG}(N (G), M)) = 0  for p ≥ 1,

where we consider N (G) as an N (G)-ℂG-bimodule.

Proof. In the first step we show for a finitely presented N (G)-module M

dim_{N (G)} (Tor_{1}^{ℂG}(N (G), M)) = 0.  (6.38)

Choose a finite presentation ℂG^{m} ⟶^{f} ℂG^{n} ⟶^{p} M → 0. For an element u = ∑_{g} λ_{g} ·g in l^{2}(G) define its support by supp(u) := {g ∈ G | λ_{g} ≠ 0} ⊂ G. Let B = (b_{i,j}) ∈ M(m, n, ℂG) be the matrix describing f, i.e. f is given by right multiplication with B. Define the finite subset S of G by

S := {g ∈ G | g or g^{−1} ∈ ⋃_{i,j} supp(b_{i,j})}.

Let f^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n} be the bounded G-equivariant operator induced by f. Denote by K the G-invariant linear subspace of l^{2}(G)^{m} which is the image of the kernel of f under the canonical inclusion k: ℂG^{m} → l^{2}(G)^{m}. Next we show for the closure \overline{K} of K

\overline{K} = ker(f^{(2)}).  (6.39)

Obviously \overline{K} ⊂ ker(f^{(2)}). Let pr: l^{2}(G)^{m} → l^{2}(G)^{m} be the orthogonal projection onto the closed G-invariant subspace \overline{K}^{⊥} ∩ ker(f^{(2)}). The von Neumann dimension of im(pr) is zero if and only if pr itself is zero (see Theorem 1.9 (3)). Hence (6.39) will follow if we can prove

tr_{N (G)}(pr) = 0.  (6.40)

Let ϵ > 0 be given. We conclude from Lemma 6.35 that there is a finite non-empty subset A ⊂ G satisfying

\frac{m · (|S| + 1) · |∂_{S}A|}{|A|} ≤ ϵ,  (6.41)

where ∂_{S}A is defined by {a ∈ A | there is s ∈ S with as ∉ A}. Define

Δ := {g ∈ G | g ∈ ∂_{S}A or gt ∈ ∂_{S}A for some t ∈ S}
= ∂_{S}A ⋃ (⋃_{t∈S} ∂_{S}A · t) .

Let pr_{A}: l^{2}(G) → l^{2}(G) be the projection which sends ∑_{g∈G} λ_{g} · g to ∑_{g∈A} λ_{g} · g. Define pr_{Δ}: l^{2}(G) → l^{2}(G) analogously. Next we show for s ∈ S and u ∈ l^{2}(G)

pr_{A} ∘r_{s}(u) = r_{s} ∘ pr_{A}(u)  if pr_{Δ}(u) = 0,  (6.42)

where r_{s}: l^{2}(G) → l^{2}(G) is right multiplication with s. Since t ∈ S implies t^{−1} ∈ S, we get the following equality of subsets of G

{g ∈ G | gs ∈ A, g ∉ Δ} = {g ∈ A | g ∉ Δ}.

Now (6.42) follows from the following calculation for u = ∑_{g∈G,g∉Δ} λ_{g} · g ∈ l^{2}(G)

pr_{A} ∘r_{s}(u) = pr_{A} ( ∑_{g∈G,g∉Δ} λ_{g} · gs) = ∑_{g∈G,gs∈A,g∉Δ} λ_{g} · gs

= ∑_{g∈A,g∉Δ} λ_{g} · gs = ( ∑_{g∈A,g∉Δ} λ_{g} · g) · s = r_{s} ∘ pr_{A}(u).

We have defined S such that each entry in the matrix B describing f is a linear combination of elements in S. Hence (6.42) implies

(⨁_{j=1}^{n} pr_{A}) ∘ f^{(2)}(u) = f^{(2)} ∘ (⨁_{i=1}^{m} pr_{A}) (u)

if pr_{Δ}(u_{i}) = 0 for i = 1, 2 …, m.

Notice that the image of ⨁_{i=1}^{m} pr_{A} lies in ℂG^{m}. We conclude

⨁_{i=1}^{m} pr_{A}(u) ∈ K  if u ∈ ker(f^{(2)}), pr_{Δ}(u_{i}) = 0 for i = 1, 2 …, m.

This shows

(pr ∘ ⨁_{i=1}^{m} pr_{A}) (ker(f^{(2)}) ∩ ⨁_{i=1}^{m} ker(pr_{Δ})) = 0.

Since ker(pr_{Δ}) has complex codimension |Δ| in l^{2}(G) and |Δ| ≤ (|S| + 1) · |∂_{S}A|, we conclude for the complex dimension dim_{ℂ} of complex vector spaces

dim_{ℂ} ((pr ∘ ⨁_{i=1}^{m} pr_{A}) (ker(f^{(2)}))) ≤ m · (|S| + 1) · |∂_{S}A|. (6.43)

Since pr ∘ pr_{A} is an endomorphism of Hilbert spaces with finite dimensional image, it is trace-class and its trace tr_{ℂ}(pr ∘ pr_{A}) is defined. We get

tr_{N (G)}(pr) = \frac{tr_{ℂ} (pr ∘ ⨁_{i=1}^{m} pr_{A})}{|A|}  (6.44)

from the following computation for e ∈ G ⊂ l^{2}(G) the unit element

tr_{N (G)}(pr) = ∑_{i=1}^{m}⟨pr_{i,i}(e), e⟩

= \frac{1}{|A|} ∑_{i=1}^{m} |A| · ⟨pr_{i,i}(e), e⟩

= \frac{1}{|A|} ∑_{i=1}^{m} ∑_{g∈A}⟨pr_{i,i}(g), g⟩

= \frac{1}{|A|} ∑_{i=1}^{m} ∑_{g∈A}⟨pr_{i,i} ∘ pr_{A}(g), g⟩

= \frac{1}{|A|} ∑_{i=1}^{m} ∑_{g∈G}⟨pr_{i,i} ∘ pr_{A}(g), g⟩

= \frac{1}{|A|} ∑_{i=1}^{m} tr_{ℂ}(pr_{i,i} ∘ pr_{A})

= \frac{1}{|A|} tr_{ℂ} (pr ∘(⨁_{i=1}^{m} pr_{A})) .

If H is a Hilbert space and f: H → H is a bounded operator with finite dimensional image, then tr_{ℂ}(f) ≤ ||f|| · dim_{ℂ} (f(im(f))). Since the image of pr is contained in ker(f^{(2)}) and pr and pr_{A} have operator norm 1, we conclude

tr_{ℂ} (pr ∘ ⨁_{i=1}^{m} pr_{A}) ≤ dim_{ℂ} ((pr ∘ ⨁_{i=1}^{m} pr_{A}) (ker(f^{(2)}))) . (6.45)

Equations (6.41), (6.43), (6.44) and (6.45) imply

tr_{N (G)}(pr) ≤ ϵ.

Since this holds for all ϵ > 0, we get (6.40) and hence (6.39) is true.

Let pr\overline{K}: l^{2}(G)^{m} → l^{2}(G)^{m} be the projection onto \overline{K}. Let i: ker(f) → ℂG^{m} be the inclusion. It induces a map

id_{N (G)} ⊗_{ℂG}i: N (G) ⊗_{ℂG} ker(f) → N (G)^{m},

where here and in the sequel we will identify N (G) ⊗_{ℂG} ℂG^{m} = N (G)^{m} by the obvious isomorphism. Next we want to show

im(ν^{−1}(pr\overline{K})) = \overline{im(id_{N (G)} ⊗_{ℂG}i)}.  (6.46)

Let x ∈ ker(f). Then

l ∘ (id −ν^{−1}(pr\overline{K})) ∘ (id_{N (G)} ⊗_{ℂG}i)(1 ⊗ x) = (id − pr\overline{K}) ∘ k ∘ i(x),(6.47)

where k: ℂG^{m} → l^{2}(G)^{m} and l: N (G)^{m} → l^{2}(G)^{m} are the obvious inclusions. Since (id − pr\overline{K}) is trivial on K we get (id − pr\overline{K}) ∘ k ∘ i = 0. Now we conclude from (6.47) that im(id_{N (G)} ⊗_{ℂG}i) ⊂ ker(id −ν^{−1}(pr\overline{K})) and hence im(id_{N (G)} ⊗_{ℂG}i) ⊂ im(ν^{−1}(pr\overline{K})) holds. This shows \overline{im(id_{N (G)} ⊗_{ℂG}i)} ⊂ im(ν^{−1}(pr\overline{K})). It remains to prove for any N (G)-map g: N (G)^{m} → N (G) with im(id_{N (G)} ⊗_{ℂG}i) ⊂ ker(g) that g ∘ ν^{−1}(pr\overline{K}) is trivial. Obviously K ⊂ ker(ν(g)). Since ker(ν(g)) is a closed subspace, we get \overline{K} ⊂ ker(ν(g)). We conclude ν(g) ∘ pr\overline{K} = 0 and hence g ∘ ν^{−1}(pr\overline{K}) = 0. This finishes the proof of (6.46).

Since ν^{−1} preserves exactness by Theorem 6.24 (3) and id_{N (G)} ⊗_{ℂG}f = ν^{−1}(f^{(2)}), we conclude from (6.39) and (6.46) that the sequence

N (G) ⊗_{ℂG} ker(f) ⟶^{id_{N (G)} ⊗_{ℂG}i} N (G)^{m} ⟶^{id_{N (G)} ⊗_{ℂG}f} N (G)^{m}

is weakly exact. Continuity of the dimension function (see Theorem 6.7 (4d)) implies

dim_{N (G)} (ker(id_{N (G)} ⊗_{ℂG}f)/ im(id_{N (G)} ⊗_{ℂG}i)) = 0.

Since Tor_{1}^{ℂG}(N (G), M) = ker(id_{N (G)} ⊗_{ℂG}f)/ im(id_{N (G)} ⊗_{ℂG}i) holds, we have proved (6.38) provided that M is finitely presented.

Next we prove (6.38) for arbitrary ℂG-modules M. Obviously M is the union of its finitely generated submodules. Any finitely generated module N is a colimit over a directed system of finitely presented modules, namely, choose an epimorphism from a finitely generated free module F to N with kernel K. Since K is the union of its finitely generated submodules, N is the colimit of the directed system F/L where L runs over the finitely generated submodules of K. The functor Tor commutes in both variables with colimits over directed systems [94, Proposition VI.1.3. on page 107]. Now (6.38) follows from Additivity and Cofinality (see Theorem 6.7 (4c) and 4b)) or directly from Theorem 6.13 about dimension and colimits.

Next we show for any ℂG-module, that dim_{N (G)} (Tor_{p}^{ℂG}(N (G), M)) = 0 holds for all p ≥ 1 by induction over p. The induction step p = 1 has been proved above. Choose an exact sequence 0 → N → F → M → 0 of N (G)- modules such that F is free. Then we obtain form the associated long exact sequence of Tor-groups an isomorphism Tor_{p}^{ℂG}(N (G), M) ≅ Tor_{p−1}^{ℂG}(N (G), N) and the induction step follows. This finishes the proof of Theorem 6.37. ⊓ ⊔

Conjecture 6.48. (Amenability and dimension-flatness of N (G) over ℂG). A group G is amenable if and only if for each ℂG-module M

dim_{N (G)} (Tor_{p}^{ℂG}(N (G), M)) = 0  for p ≥ 1

holds.

6.5 L^{2}-Betti Numbers for General Spaces with Group Actions 263

Theorem 6.37 gives the “only if”-statement. Evidence for the “if”-state-ment comes from the following observation. Suppose that G contains a free group ℤ ∗ ℤ of rank 2 as subgroup. Notice that S^{1} ∨ S^{1} is a model for B(ℤ ∗ ℤ). Its cellular ℂ[ℤ ∗ ℤ]-chain complex yields an exact sequence 0 → ℂ[ℤ ∗ ℤ]^{2} → ℂ[ℤ ∗ ℤ] → ℂ → 0, where ℂ is equipped with the trivial ℤ ∗ ℤ-action. We conclude from Additivity (see Theorem 6.7 (4b)), Theorem 6.29 (1) and Lemma 6.36

dim_{N (ℤ∗ℤ)} (Tor_{1}^{ℂ[ℤ∗ℤ]}(N (ℤ ∗ ℤ), ℂ)) = 1;
N (G) ⊗_{N (ℤ∗ℤ)} Tor_{1}^{ℂ[ℤ∗ℤ]}(N (ℤ ∗ ℤ), ℂ)) = Tor_{1}^{ℂG}(N (G), ℂG ⊗_{ℂ[ℤ∗ℤ]} ℂ).

We conclude from Theorem 6.29 (2)

dim_{N (G)} (Tor_{1}^{ℂG}(N (G), ℂG ⊗_{ℂ[ℤ∗ℤ]} ℂ)) ≠ 0.

One may ask for which groups the von Neumann algebra N (G) is flat as a ℂG-module. This is true if G is virtually cyclic. There is some evidence for the conjecture

Conjecture 6.49 (Flatness of N (G) over ℂG). The group von Neumann algebra N (G) is flat over ℂG if and only if G is virtually cyclic, i.e. G is finite or contains ℤ as normal subgroup of finite index.

6.5 L^{2}-Betti Numbers for General Spaces with Group Actions

In this section we extend the notion of L^{2}-Betti numbers for free G-CW - complexes of finite type introduced in Definition 1.30 to arbitrary G-spaces, where we will use the extended dimension function dim_{N (G)} of Definition 6.20. We prove the main properties of these notions.

Definition 6.50 (L^{2}-Betti numbers). Let X be a (left) G-space. Equip N (G) with the obvious N (G)-ℤG-bimodule structure. The singular homology H_{p}^{G}(X; N (G)) of X with coefficients in N (G) is the homology of the N (G)- chain complex N (G) ⊗_{ℤG} C_{∗}^{sing}(X), where C_{∗}^{sing}(X) is the singular chain complex of X with the induced ℤG-structure. Define the p-th L^{2}-Betti number of X by

b_{p}^{(2)}(X; N (G)) := dim_{N (G)}(H_{p}^{G}(X; N (G))) ∈ [0, ∞],

where dim_{N (G)} is the extended dimension function of Definition 6.20.

If G and its action on X are clear from the context, we often omit N (G) in the notation above.

Define for any (discrete) group G its p-th L^{2}-Betti number by

b_{p}^{(2)}(G) := b_{p}^{(2)}(EG, N (G)).

We first show that this new definition and the old one agree when both apply. For that purpose we will need the following two lemmas. The proof of the first one can be found for instance in [333, Lemma 4.2 on page 152].

Lemma 6.51. Let X be a G-CW -complex and C_{∗}(X) be its cellular ℤGchain complex. Then there exists a ℤG-chain homotopy equivalence

f_{∗}(X): C_{∗}(X) → C_{∗}^{sing}(X),

which is, up to ℤG-homotopy, uniquely defined and natural in X. In particular we get a natural isomorphism of N (G)-modules

H_{p}(N (G) ⊗_{ℤG} C_{∗}(X)) ⟶^{≅} H_{p}^{G}(X; N (G)) := H_{p}(N (G) ⊗_{ℤG} C_{∗}^{sing}(X)).

Lemma 6.52. Let C_{∗} be a finitely generated Hilbert N (G)-chain complex. Then there is an isomorphism, natural in C_{∗},

h_{p}(C_{∗}): ν^{−1}(H_{p}^{(2)}(C_{∗})) ⟶^{≅} PH_{p}(ν^{−1}(C_{∗})),

where ν^{−1} is the functor appearing in Theorem 6.24.

Proof. We define h_{p}(C_{∗}) by the following commutative diagram whose columns are exact and whose middle and lower vertical arrows are isomorphisms by Theorem 6.7 (2) and Theorem 6.24 (3)

0  0
↑  ↑
ν^{−1}(H_{p}^{(2)}(C_{∗})) ⟶^{h_{p}(C_{∗})} PH_{p}(ν^{−1}(C_{∗}))
ν^{−1}(q)↑  ^{r}↑
ν^{−1}(ker(c_{p})) ⟶^{ν^{−1}(i)} ker(ν^{−1}(c_{p}))
ν^{−1}(j)↑  ^{l}↑
ν^{−1}(\overline{im(c_{p+1})}) ⟶^{ν^{−1}(k)} \overline{im(ν^{−1}(c_{p+1}))}
↑  ↑
0  0

where i: ker(c_{p}) → C_{p}, j: \overline{im(c_{p+1})} → ker(c_{p}), k: \overline{im(c_{p+1})} → C_{p} and l are the obvious inclusions and q and r the obvious projections. Then h_{p}(C_{∗}) is an isomorphism by the five-lemma. ⊓ ⊔

Lemma 6.53. Let X be a G-CW -complex of finite type. Then Definition 1.30 and Definition 6.50 of L^{2}-Betti numbers b_{p}^{(2)}(X; N (G)) agree.

Proof. We can identify N (G) ⊗_{ℤG} C_{∗}(X) with ν^{−1}(C_{∗}^{(2)}(X)). Now apply Theorem 6.7 (4e), Theorem 6.24 (4), Lemma 6.51 and Lemma 6.52. ⊓ ⊔

Next we collect some basic properties of L^{2}-Betti numbers.

Theorem 6.54. L^{2}-Betti numbers for arbitrary spaces).

(1) Homology invariance We have for a G-map f: X → Y

(a) Suppose for n ≥ 1 that for each subgroup H ⊂ G the induced map f^{H}: X^{H} ⟶ Y^{H} is ℂ-homologically n-connected, i.e. the map H_{p}^{sing}(f^{H}; ℂ): H_{p}^{sing}(X^{H}; ℂ) → H_{p}^{sing}(Y^{H}; ℂ) induced by f^{H} on singular homology with complex coefficients is bijective for p < n and surjective for p = n. Then the induced map f_{∗}: H_{p}^{G}(X; N (G)) ⟶ H_{p}^{G}(Y; N (G)) is bijective for p < n and surjective for p = n and we get

b_{p}^{(2)}(X) = b_{p}^{(2)}(Y)  for p < n;
b_{p}^{(2)}(X) ≥ b_{p}^{(2)}(Y)  for p = n;

(b) Suppose that for each subgroup H ⊂ G the induced map f^{H}: X^{H} → Y^{H} is a ℂ-homology equivalence, i.e. H_{p}^{sing}(f^{H}; ℂ) is bijective for p ≥ 0. Then for all p ≥ 0 the map f_{∗}: H_{p}^{G}(X; N (G)) → H_{p}^{G}(Y; N (G)) induced by f is bijective and we get

b_{p}^{(2)}(X) = b_{p}^{(2)}(Y) for p ≥ 0;

(2) Comparison with the Borel construction Let X be a G-CW -complex. Suppose that for all x ∈ X the isotropy group G_{x} is finite or satisfies b_{p}^{(2)}(G_{x}) = 0 for all p ≥ 0. Then

b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(EG × X; N (G))  for p ≥ 0;

(3) Invariance under non-equivariant ℂ-homology equivalences Suppose that f: X → Y is a G-equivariant map of G-CW -complexes such that the induced map H_{p}^{sing}(f; ℂ) on singular homology with complex coefficients is bijective. Suppose that for all x ∈ X the isotropy group G_{x} is finite or satisfies b_{p}^{(2)}(G_{x}) = 0 for all p ≥ 0, and analogously for all y ∈ Y. Then we have for all p ≥ 0

b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(Y; N (G));

(4) Independence of equivariant cells with infinite isotropy Let X be a G-CW -complex. Let X[∞] be the G-CW -subcomplex consisting of those points whose isotropy subgroups are infinite. Then we get for all p ≥ 0

b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(X, X[∞]; N (G));

(5) Künneth formula Let X be a G-space and Y be an H-space. Then X × Y is a G × H-space and we get for all n ≥ 0

b_{n}^{(2)}(X × Y) = ∑_{p+q=n} b_{p}^{(2)}(X) · b_{q}^{(2)}(Y),

where we use the convention that 0 · ∞ = 0, r · ∞ = ∞ for r ∈ (0, ∞] and r + ∞ = ∞ for r ∈ [0, ∞];

(6) Restriction Let H ⊂ G be a subgroup of finite index [G : H]. Then

(a) Let M be an N (G)-module and res_{G}^{H} M be the N (H)-module obtained from M by restriction. Then

dim_{N (H)}(res_{G}^{H} M) = [G : H] · dim_{N (G)}(M),

where [G : H] · ∞ is understood to be ∞;

(b) Let X be a G-space and let res_{G}^{H} X be the H-space obtained from X by restriction. Then

b_{p}^{(2)}(res_{G}^{H} X; N (H)) = [G : H] · b_{p}^{(2)}(X; N (G));

(7) Induction Let i: H → G be an inclusion of groups and let X be an H-space. Let i: N (H) → N (G) be the induced ring homomorphism (see Definition 1.23). Then

H_{p}^{G}(G ×_{H} X; N (G)) = i_{∗}H_{p}^{H}(X; N (H));
b_{p}^{(2)}(G ×_{H} X; N (G)) = b_{p}^{(2)}(X; N (H));

(8) Zero-th homology and L^{2}-Betti number Let X be a path-connected G-space. Then

(a) There is an N (G)-isomorphism H_{0}^{G}(X; N (G)) ⟶^{≅} N (G) ⊗_{ℂG} ℂ;

(b) b_{0}^{(2)}(X; N (G)) = |G|^{−1}, where |G|^{−1} is defined to be zero if the order |G| of G is infinite;

(c) H_{0}^{G}(X; N (G)) is trivial if and only if G is non-amenable;

Proof. (1) The proof of assertion (1) which generalizes Theorem 1.35 (1) can be found in [333, Lemma 4.8 on page 153].

(2) Because of Additivity (see Theorem 6.7 (4b)) it suffices to prove that the dimension of the kernel and the cokernel of the map induced by the projection

pr_{∗}: H_{p}^{G}(EG × X; N (G)) → H_{p}^{G}(X; N (G))

are trivial. Notice that X is the directed colimit of its finite G-CW -sub-complexes. Since H_{p}^{G}(−, N (G)) is compatible with directed colimits and directed colimits preserve exact sequences, we can assume by Theorem 6.13 about dimension and colimits that X itself is finite. By induction over the number of equivariant cells, the long exact homology sequence and Additivity (see Theorem 6.7 (4b)) the claim reduces to the case where X is of the shape G/H. Because of assertion (7) it suffices to prove for the map pr_{∗}: H_{p}^{H}(EH; N (H)) → H_{p}^{H}({∗}; N (H)) that its kernel and cokernel have trivial dimension, provided that H is finite or b_{p}^{(2)}(H; N (H)) = 0 for p ≥ 0. This follows from assertion (8a).

(3) Since EG × X and EG × Y are free G-CW -complexes and id ×f: EG × X → EG × Y induces an isomorphism on homology with ℂ-coefficients, the claim follows from assertions (1b) and (2).

(4) We get an exact sequence of cellular ℤG-chain complexes 0 → C_{∗}(X[∞]) → C_{∗}(X) → C_{∗}(X, X[∞]) → 0 which is a split exact sequence of ℤG-modules in each dimension. Hence it stays exact after applying N (G) ⊗_{ℤG} −, and we get a long exact sequence of N (G)-homology modules

… → H_{p}^{G}(X[∞]; N (G)) → H_{p}^{G}(X; N (G)) → H_{p}^{G}(X, X[∞]; N (G))
→ H_{p−1}^{G}(X[∞]; N (G)) → ….

Because of Additivity (see Theorem 6.7 (4b)) it suffices to prove that dim_{N (G)}(N (G) ⊗ C_{p}(X[∞])) = 0 for all p ≥ 0. This follows from Additivity (see Theorem 6.7 (4b)), Cofinality (see Theorem 6.7 (4c)) and Lemma 6.33 (1), since C_{p}(X[∞]) is a direct sum of ℤG-modules of the shape ℤ[G/H] for infinite subgroups H ⊂ G.

(5) This assertion is a generalization of Theorem 1.35 (4).

For any G-space Z there is a G-CW -complex Z′ together with a G-map f: Z → Z′ such that f^{H} is a weak homotopy equivalence and hence induces an isomorphism on singular homology with complex coefficients for all H ⊂ G [326, Proposition 2.3 iii) on page 35]. Hence we can assume without loss of generality that X is a G-CW -complex and Y is a H-CW -complex by assertion (1b).

The G-CW -complex X is the colimit of the directed system {X_{i} | i ∈ I} of its finite G-CW -subcomplexes. Analogously Y is the colimit of the directed system {Y_{j} | j ∈ J} of its finite H-CW -subcomplexes. Since we are working in the category of compactly generated spaces, the G×H-CW -complex X×Y is the colimit of the directed system {X_{i} ×Y_{j} | (i, j) ∈ I ×J}. Since homology and tensor products are compatible with colimits over directed systems, we conclude from Theorem 6.13 (2) about dimension and colimits b_{p}^{(2)}(X; N (G))

= sup {inf {dim_{N (G)} (im(ϕ_{i_{0},i_{1}} : H_{p}^{G}(X_{i_{0}}; N (G)) → H_{p}^{G}(X_{i_{1}}; N (G))))
| i_{1} ∈ I, i_{0} ≤ i_{1}} | i_{0} ∈ I};  (6.55)

b_{p}^{(2)}(Y; N (G))

= sup {inf {dim_{N (H)} (im(ψ_{j_{0},j_{1}} : H_{p}^{H}(Y_{j_{0}}; N (H)) → H_{p}^{H}(Y_{j_{1}}; N (H))))
| j_{1} ∈ J, j_{0} ≤ j_{1}} | j_{0} ∈ J};  (6.56)

b_{p}^{(2)}(X × Y; N (G × H))

= sup {inf {dim_{N (G×H)} (im(μ_{(i_{0},j_{0}),(i_{1},j_{1})} :

H_{p}^{G×H}(X_{i_{0}} × Y_{j_{0}}; N (G × H)) → H_{p}^{G×H}(X_{i_{1}} × Y_{j_{1}}; N (G × H))))

| (i_{1}, j_{1}) ∈ I × J, i_{0} ≤ i_{1}, j_{0} ≤ j_{1}} | (i_{0}, j_{0}) ∈ I × J} , (6.57)

where here and in the sequel ϕ_{i_{0},i_{1}}, ψ_{j_{0},j_{1}} and μ_{(i_{0},j_{0}),(i_{1},j_{1})} are the maps induced by the obvious inclusions.

Next we show for n ≥ 0 and i_{0}, i_{1} ∈ I with i_{0} ≤ i_{1} and j_{0}, j_{1} ∈ J with j_{0} ≤ j_{1}

dim_{N (G×H)} (im (μ_{(i_{0},j_{0}),(i_{1},j_{1})} : H_{n}^{G×H}(X_{i_{0}} × Y_{j_{0}}; N (G × H))
→ H_{n}(X_{i_{1}} × X_{j_{1}}; N (G × H))))
= ∑_{p+q=n} dim_{N (G)} (im(ϕ_{i_{0},i_{1}} : H_{p}^{G}(X_{i_{0}}; N (G)) → H_{p}^{G}(X_{i_{1}}; N (G))))
· dim_{N (H)} (im(ψ_{j_{0},j_{1}} : H_{q}^{H}(Y_{j_{0}}; N (G)) → H_{q}^{H}(Y_{j_{1}}; N (G)))) . (6.58)

For k = 0, 1 let C_{∗}[k] be the cellular ℤG-chain complex of the G-CW -pair (X_{i_{k}}, X_{i_{k}}[∞]), where X_{i_{k}}[∞] has been introduced in Theorem 6.54 (4). For k = 0, 1 let D_{∗}[k] be the cellular ℤH-chain complex of the H-CW -pair (Y_{j_{k}}, Y_{j_{k}}[∞]). Then C_{∗}[k] ⊗_{ℤ} D_{∗}[k] is the cellular ℤ[G × H]-chain complex of the G × H-CW -pair (X × Y, (X × Y)[∞]). Notice that C_{p}[k] is a direct sum of ℤG-modules of the shape ℤ[G/H] for finite subgroups H ⊂ G. Hence the G-CW -complex structure on (X_{i_{k}}, X_{i_{k}}[∞]) induces the structure of a Hilbert N (G)-chain complex on C_{∗}^{(2)}[k] := l^{2}(G)⊗_{ℤG}C_{∗}[k]. The analogous statements are true for D_{∗}^{(2)}[k] := l^{2}(H)⊗_{ℤG}D_{∗}[k] and l^{2}(G×H)⊗_{ℤ[G×H]}(C_{∗}[k]⊗_{ℤ}D_{∗}[k]). Notice that the last Hilbert N (G × H)-chain complex is isomorphic to the Hilbert tensor product C_{∗}^{(2)}[k] ⊗ D_{∗}^{(2)}[k]. We conclude from Theorem 6.7 (2) and (4d), Theorem 6.24 and Theorem 6.54 (4)

dim_{N (G)} (im(ϕ_{i_{0},i_{1}} : H_{p}^{G}(X_{i_{0}}; N (G)) → H_{p}^{G}(X_{i_{1}}; N (G)))
= dim_{N (G)} (clos(im(ϕ_{i_{0},i_{1}} : H_{p}^{(2)}(C_{∗}^{(2)}[0]) → H_{p}^{(2)}(C_{∗}^{(2)}[1])))); (6.59)
dim_{N (H)} (im(ψ_{j_{0},j_{1}} : H_{p}^{H}(Y_{j_{0}}; N (H)) → H_{p}^{H}(Y_{j_{1}}; N (H))))
= dim_{N (H)} (clos(im(ψ_{j_{0},j_{1}} : H_{p}^{(2)}(D_{∗}^{(2)}[0]) → H_{p}^{(2)}(D_{∗}^{(2)}[1])))); (6.60)
dim_{N (G×H)} (im(μ_{(i_{0},j_{0}),(i_{1},j_{1})} : H_{p}^{G×H}(X_{i_{0}} × Y_{j_{0}}; N (G × H))
→ H_{p}^{G×H}(X_{i_{1}} × Y_{j_{1}}; N (G × H))))
= dim_{N (G×H)} (clos(im(μ_{(i_{0},j_{0}),(i_{1},j_{1})} : H_{p}^{(2)}(C_{∗}^{(2)}[0] ⊗ D_{∗}^{(2)}[0])
→ H_{p}^{(2)}(C_{∗}^{(2)}[1] ⊗ D_{∗}^{(2)}[1])))) .  (6.61)

There is a commutative diagram of Hilbert N (G × H)-modules

⨁_{p+q=n} H_{p}^{(2)}(C_{∗}^{(2)}[0]) ⊗ H_{q}((D_{∗}^{(2)}[0]) ⟶^{≅} H_{n}^{(2)}(C_{∗}^{(2)}[0] ⊗ D_{∗}^{(2)}[0])
⨁_{p+q=n} ϕ_{i_{0},i_{1}} ⊗ψ_{j_{0},j_{1}} ↓  ^{μ_{(i_{0},j_{0}),(i_{1},j_{1})}}↓
⨁_{p+q=n} H_{p}^{(2)}(C_{∗}^{(2)}[1]) ⊗ H_{q}^{(2)}((D_{∗}^{(2)}[1]) ⟶^{≅} H_{n}^{(2)}(C_{∗}^{(2)}[1] ⊗ D_{∗}^{(2)}[1])

where the horizontal isomorphisms are the ones appearing in Lemma 1.22. Thus we obtain an isomorphism of closure of the images of the two vertical maps. The closure of the image of a Hilbert tensor product of two morphisms is the Hilbert tensor product of the closures of the individual images. Now (6.58) follows from Theorem 1.12 (5), (6.59), (6.60) and (6.61). Finally assertion (5) follows from (6.55), (6.56), (6.57) and (6.58).

(6) generalizes Theorem 1.35 (9).

We begin with the proof of assertion (6a) Notice for the sequel that res_{G}^{H} N (G) is N (H)-isomorphic to N (H)^{[G:H]}. Since M is the colimit of the directed system of its finitely generated N (G)-submodules, it suffices to consider the case where M is finitely generated because of Cofinality (see Theorem 6.7 (4c)). Since for a finitely generated N (G)-module we have T(res_{G}^{H} M) = res_{G}^{H}(TM) and hence P(res_{G}^{H} M) = res_{G}^{H}(PM), we conclude from Theorem 6.7 (3) and (4e) that it suffices to treat the case of a finitely generated projective N (G)-module M. The functor ν appearing in Theorem 6.24 is compatible with restriction for N (G)-modules and restriction from G to H for Hilbert modules. Using Theorem 6.24 (4) we reduce the claim to the assertion that for any finitely generated Hilbert N (G)-module V we have

dim_{N (H)}(res_{G}^{H} V) = [G : H] · dim_{N (G)}(V).

This has already been proved in Theorem 1.12 (6).

Next we prove assertion (6b). We obtain an isomorphism of N (H)-ℂG-bimodules N (H) ⊗_{ℂH} ℂG ⟶^{≅} N (G) by sending u ⊗ g to i_{∗}(u) ∘ R_{g^{−1}}. It induces an N (H)-isomorphism

N (H) ⊗_{ℂH} C_{∗}(res_{G}^{H} X) ⟶^{≅} res_{G}^{H}(N (G) ⊗_{ℂG} C_{∗}(X)).

Now assertion (6b) follows from assertion (6a).

(7) is a generalization of Theorem 1.35 (10) and follows from Theorem 6.29.

(8) This follows from Lemma 6.33 and Lemma 6.36 and generalizes Theorem 1.35 (8). This finishes the proof of Theorem 6.54. ⊓ ⊔

We will investigate the class B_{∞} of groups G, for which b_{p}^{(2)}(G) = 0 holds for all p ≥ 0, in Section 7.1.

Remark 6.62. Let M be a closed Riemannian manifold. We have already mentioned in Section 2.7 that the analytic Laplace operator Δ_{p}: L^{2}Ω^{p}(M̃) → L^{2}Ω^{p}(M̃) on the universal covering does not have zero in its spectrum if and only if we get for its analytic p-th L^{2}-Betti number b_{p}^{(2)}(M̃) = 0 and for its analytic p-th Novikov-Shubin invariant α_{p}^{Δ}(M̃) = ∞^{+}. In the case p = 0 this is equivalent to the condition that H_{0}^{G}(X; N (G)) = 0 by Theorem 1.59, Lemma 2.11 (8) Theorem 2.68 and Theorem 6.24 (3). The latter condition is by Theorem 6.54 (8c) equivalent to the condition that π_{1}(M) is amenable. Thus we rediscover (and in some sense generalize to arbitrary G-spaces) the result of Brooks [68] that for a closed Riemannian manifold M the Laplacian Δ_{0} : L^{2}Ω^{0}(M̃) → L^{2}Ω^{0}(M̃) acting on functions on the universal covering has zero in its spectrum if and only if π_{1}(M) is amenable. Notice that both Brook’s and our proof are based on Kesten [290].

The next result is a generalization of Theorem 1.39 and was conjectured for a closed aspherical manifold fibering over the circle in [237, page 229].

Theorem 6.63 (Vanishing of L^{2}-Betti numbers of mapping tori). Let f: X → X be a cellular selfmap of a connected CW -complex X and let π_{1}(T_{f}) ⟶^{ϕ} G ⟶^{ψ} ℤ be a factorization of the canonical epimorphism into epimorphisms ϕ and ψ. Suppose for given p ≥ 0 that b_{p}^{(2)}(G ×_{ϕ∘i} X̃; N (G)) < ∞ and b_{p−1}^{(2)}(G ×_{ϕ∘i} X̃; N (G)) < ∞ holds, where i: π_{1}(X) → π_{1}(T_{f}) is the map induced by the obvious inclusion of X into T_{f}. Let \overline{T_{f}} be the covering of T_{f} associated to ϕ, which is a free G-CW -complex. Then we get

b_{p}^{(2)}(\overline{T_{f}}; N (G)) = 0.

Proof. The proof is analogous to that of Theorem 1.39 except the following changes. Instead of Theorem 1.35 (1) and Theorem 1.35 (9) one has to refer to Theorem 6.54 (1b) and Theorem 6.54 (6b) and one has to give a different argument for the existence of a constant C which is independent of n and satisfies

b_{p}^{(2)}(\overline{T_{f^{n}}}; N (G_{n})) ≤ C.  (6.64)

In the proof of Theorem 1.39 we have taken C = β_{p} + β_{p−1}, where β_{p} is the number of p-cells in X. Next we explain why we can take for (6.64) the constant

C = b_{p}^{(2)}(G ×_{ϕ∘i} X̃; N (G)) + b_{p−1}^{(2)}(G ×_{ϕ∘i} X̃; N (G)).

First one checks that the image of ϕ ∘ i: π_{1}(X) → G is contained in ψ^{−1}(0) and hence in G_{n}. One constructs an exact sequence of ℤG_{n}-chain complexes

0 → C_{∗}(G_{n} ×_{ϕ∘i} X̃) → C_{∗}(\overline{T_{f^{n}}}) → ΣC_{∗}(G_{n} ×_{ϕ∘i} X̃) → 0.

Tensoring with N (G_{n}) and taking the long homology sequence yields the exact sequence of N (G_{n})-modules

H_{p}(N (G_{n}) ⊗_{ℤG_{n}} C_{∗}(G_{n} ×_{ϕ∘i} X̃)) → H_{p}(N (G_{n}) ⊗_{ℤG_{n}} C_{∗}(\overline{T_{f^{n}}}))
→ H_{p−1}(N (G_{n}) ⊗_{ℤG_{n}} C_{∗}(G_{n} ×_{ϕ∘i} X̃)).

Additivity (see Theorem 6.7 (4b)) and Theorem 6.54 (7) imply

b_{p}^{(2)}(\overline{T_{f^{n}}}; N (G_{n})) ≤ b_{p}^{(2)}(G_{n} ×_{ϕ∘i} X̃; N (G_{n}))

+b_{p−1}^{(2)}(G_{n} ×_{ϕ∘i} X̃; N (G_{n}));

b_{p}^{(2)}(G_{n} ×_{ϕ∘i} X̃; N (G_{n})) = b_{p}^{(2)}(G ×_{ϕ∘i} X̃; N (G));
b_{p−1}^{(2)}(G_{n} ×_{ϕ∘i} X̃; N (G_{n})) = b_{p−1}^{(2)}(G ×_{ϕ∘i} X̃; N (G)).

This finishes the proof of (6.64) and thus of Theorem 6.63. ⊓ ⊔

The next result is a generalization of Theorem 1.40. Its proof is analogous to that one of Theorem 1.40.

Theorem 6.65. (L^{2}-Betti numbers and S^{1}-actions). Let X be a connected S^{1}-CW -complex. Suppose that for one orbit S^{1}/H (and hence for all orbits) the inclusion into X induces a map on π_{1} with infinite image. (In particular the S^{1}-action has no fixed points.) Let X̃ be the universal covering of X with the canonical π_{1}(X)-action. Then we get for all p ≥ 0

b_{p}^{(2)}(X̃) = 0.

The next Lemma 6.66 is a generalization of Lemma 1.41. Although it is more general — we have dropped the finiteness conditions on the CW - complexes F, E and B — its proof is now simpler because our algebraic approach to dimension theory and L^{2}-Betti numbers is so flexible that we can use standard methods from algebra and topology such as the Serre spectral sequence for fibrations and local coefficients.

Lemma 6.66. Let F ⟶^{i} E → B be a fibration of connected CW -complexes. Let ϕ: π_{1}(E) → G be a group homomorphisms and let i_{∗}: π_{1}(F) → π_{1}(E) be the homomorphism induced by the inclusion i. Suppose that for a given integer d ≥ 0 the L^{2}-Betti number b_{p}^{(2)}(G ×_{ϕ∘i_{∗}} F̃; N (G)) vanishes for all p ≤ d. Then the L^{2}-Betti number b_{p}^{(2)}(G ×_{ϕ} Ẽ; N (G)) vanishes for all p ≤ d.

Proof. There is a spectral homology sequence of N (G)-modules which converges to H_{p+q}^{G}(G ×_{ϕ} Ẽ; N (G)) and whose E^{1}-term is

E_{p,q}^{1} = H_{q}^{G}(G ×_{ϕ∘i_{∗}} F̃; N (G)) ⊗_{ℤπ_{1}(B)} C_{p}(B̃),

where the right π_{1}(B)-action on H_{q}^{G}(G×_{ϕ∘i_{∗}} F̃; N (G)) is induced by the fiber transport. Since

b_{q}^{(2)}(G ×_{ϕ∘i_{∗}} F̃; N (G)) = dim_{N (G)} (H_{q}^{G}(G ×_{ϕ∘i_{∗}} F̃; N (G))) = 0

holds for q ≤ d by assumption and C_{p}(B̃) is a free ℤπ_{1}(B)-module, we conclude from Additivity (see Theorem 6.7 (4b)) and Cofinality (see Theorem 6.7 (4c)) that dim_{N (G)}(E_{p,q}^{1}) = 0 holds for all p ≥ 0 and q ≤ d. Hence b_{p}^{(2)}(G ×_{ϕ} Ẽ; N (G)) = 0 holds for all p ≤ d. ⊓ ⊔

Theorem 6.67 (L^{2}-Betti numbers and fibrations). Let F ⟶^{i} E ⟶^{p} B be a fibration of connected CW -complexes. Let p_{∗}: π_{1}(E) ⟶^{ϕ} G ⟶^{ψ} π_{1}(B) be a factorization of the map induced by p into epimorphisms ϕ and ψ. Let i_{∗}: π_{1}(F) → π_{1}(E) be the homomorphism induced by the inclusion i. Suppose for a given integer d ≥ 1 that b_{p}^{(2)}(G ×_{ϕ∘i_{∗}} F̃; N (G)) = 0 for p ≤ d − 1 and b_{d}^{(2)}(G ×_{ϕ∘i_{∗}} F̃; N (G)) < ∞ holds. Suppose that π_{1}(B) contains an element of infinite order or finite subgroups of arbitrarily large order. Then b_{p}^{(2)}(G ×_{ϕ} Ẽ; N (G)) = 0 for p ≤ d.

Proof. Since b_{q}^{(2)}(G×_{ϕ∘i_{∗}} F̃; N (G)) = 0 holds for q ≤ d−1 by assumption and C_{p}(B̃) is a free ℤπ_{1}(B)-module, we conclude by the same spectral sequence argument as in the proof of Lemma 6.66

b_{p}^{(2)}(G ×_{ϕ} Ẽ; N (G)) = 0  0 ≤ p ≤ d − 1

and

b_{d}^{(2)}(G ×_{ϕ} Ẽ; N (G))
= dim_{N (G)} (E_{0,d}^{2})
= dim_{N (G)} (H_{0}^{π_{1}(B)}(B̃; H_{d}^{G}(G ×_{ϕ∘i_{∗}} F̃; N (G)))
= dim_{N (G)} (H_{d}^{G}(G ×_{ϕ∘i_{∗}} F̃; N (G)) ⊗_{ℤπ_{1}(B)} ℤ) ,  (6.68)

where the right π_{1}(B)-action on H_{d}^{G}(G×_{ϕ∘i_{∗}} F̃; N (G)) is induced by the fiber transport. It remains to prove

b_{d}^{(2)}(G ×_{ϕ} Ẽ; N (G)) = 0.  (6.69)

Let Δ ⊂ π_{1}(B) be a subgroup which is either ℤ or finite. Let X be S^{1} if Δ = ℤ and a connected finite 2-dimensional CW -complex X with π_{1}(X) = Δ if Δ is finite. In any case we can choose a map f: X → B which induces on fundamental groups the inclusion of Δ into π_{1}(B). Let F ⟶^{i_{0}} E_{0} ⟶^{p_{0}} X be the fibration obtained from F ⟶^{i} E ⟶^{p} B by applying the pullback construction with respect to f. Let \overline{f}: E_{0} → E be the canonical map. Since there is an obvious N (G)-epimorphism from H_{d}^{G}(G ×_{ϕ∘i_{∗}} F̃; N (G)) ⊗_{ℤΔ} ℤ to H_{d}^{G}(G ×_{ϕ∘i_{∗}} F̃; N (G)) ⊗_{ℤπ_{1}(B)} ℤ, we get from Additivity (see Theorem 6.7 (4b))

dim_{N (G)} (H_{d}^{G}(G ×_{ϕ∘i_{∗}} F̃; N (G)) ⊗_{ℤΔ} ℤ)
≥ dim_{N (G)} (H_{d}^{G}(G ×_{ϕ∘i_{∗}} F̃; N (G)) ⊗_{ℤπ_{1}(B)} ℤ) . (6.70)

Define G_{0} = ψ^{−1}(Δ). Then ϕ and ψ induce epimorphisms ϕ_{0} and ψ_{0} which make the following diagram commute

π_{1}(E_{0}) ⟶^{ϕ_{0}} G_{0} ⟶^{ψ_{0}} π_{1}(X)
\overline{f}_{∗}↓  ↓  ^{f_{∗}}↓
π_{1}(E) ⟶^{ϕ} G ⟶^{ψ} π_{1}(B)

Notice that the fiber transport is compatible with pullbacks. From Theorem 6.29, from (6.68) applied to p and ϕ and applied to p_{0} and ϕ_{0} and from (6.70) we conclude

b_{d}^{(2)}(G ×_{ϕ} Ẽ; N (G)) ≤ b_{d}^{(2)}(G_{0} ×_{ϕ_{0}} Ẽ_{0}; N (G_{0})).  (6.71)

We firstly consider the case where Δ = ℤ and X = S^{1}. Then E_{0} is up to homotopy the mapping torus of the selfhomotopy equivalence F → F given by the fiber transport of p_{0} along a generator of π_{1}(S^{1}) and hence b_{d}^{(2)}(G_{0} ×_{ϕ} Ẽ_{0}; N (G_{0})) = 0 by Theorem 6.63. Now (6.69) follows from (6.71). Next we consider the case where Δ is finite. Let F ⟶^{i_{1}} E_{1} ⟶^{p_{1}} X̃ be the fibration obtained from p_{0} : E_{0} → X by the pullback construction with respect to the universal covering X̃ → X. Put G_{1} = ψ^{−1}(1). This is a subgroup of G_{0} of index |Δ|. Let ϕ_{1} : π_{1}(E_{1}) → G_{1} be the epimorphism induced by ϕ_{0}. Notice that the canonical map E_{1} → E_{0} is a |Δ|-sheeted covering. We conclude from Theorem 6.54 (7) and (6b) and from applying (6.68) to \overline{p_{1}} and ϕ_{1} : π_{1}(E_{1}) → G_{1}

|Δ| · b_{d}^{(2)}(G_{0} ×_{ϕ} Ẽ_{0}; N (G_{0})) = b_{d}^{(2)}(res_{G_{0}}^{G_{1}}(G_{0} ×_{ϕ_{0}} Ẽ_{0}); N (G_{1}))
= b_{d}^{(2)}(G_{1} ×_{ϕ_{1}} Ẽ_{1}; N (G_{1}))
= dim_{N (G_{1})} (H_{d}^{G_{1}}(G_{1} ×_{ϕ_{1}∘(i_{1})∗} F̃; N (G_{1})))
= b_{d}^{(2)}(G_{1} ×_{ϕ_{1}∘(i_{1})∗} F̃; N (G));
= b_{d}^{(2)}(G ×_{ϕ∘i_{∗}} F̃; N (G)).

This implies together with (6.71)

b_{d}^{(2)}(G ×_{ϕ} Ẽ; N (G)) ≤ \frac{b_{d}^{(2)}(G ×_{ϕ∘i_{∗}} F̃; N (G))}{|Δ|}.

Since by assumption π_{1}(B) either contains ℤ as a subgroup or finite subgroups Δ of arbitrarily large order, (6.69) follows. This finishes the proof of Theorem 6.67. ⊓ ⊔

As far as the assumption about π_{1}(B) in Theorem 6.67 is concerned, we mention that there is for any prime number p > 10^{75} an infinite finitely generated group all of whose proper subgroups are finite of order p [405]. To the author’s knowledge it is not known whether there is an infinite finitely presented group with finite exponent. For more information about this so called Burnside problem we refer to [469]. Using Theorem 6.29 one obtains the following special case of Theorem 6.67.

Corollary 6.72. Let F ⟶^{i} E ⟶^{p} B be a fibration of connected CW -complexes such that i_{∗}: π_{1}(F) → π_{1}(E) is injective, π_{1}(F) is infinite and b_{1}^{(2)}(F̃) < ∞. Suppose that π_{1}(B) contains an element of infinite order or finite subgroups of arbitrarily large order. Then

b_{0}(Ẽ) = b_{1}(Ẽ) = 0.

Corollary 6.72 has been conjectured in [237, page 235] for fibrations F → E → B of connected aspherical CW -complexes of finite type with non-trivial fundamental groups. It will play a role in applications of L^{2}-Betti numbers to deficiencies of finitely presented groups in Section 7.3.

The next Theorem shows that for amenable G the p-th L^{2}-Betti number of a G-space X depends only on the ℂG-modules given by the singular homology of X with complex coefficients.The key ingredient will be Theorem 6.37 which shows that the von Neumann algebra of an amenable group is not flat over the complex group ring in the strict sense but is “dimension-flat”.

Theorem 6.73. Let G be an amenable group and X be a G-space. Then

b_{p}^{(2)}(X; N (G)) = dim_{N (G)} (N (G) ⊗_{ℂG} H_{p}^{sing}(X; ℂ)) ,

where H_{p}^{sing}(X; ℂ) is the ℂG-module given by the singular homology of X with complex coefficients.

Proof. We will show for an arbitrary ℂG-chain complex C_{∗} with C_{p} = 0 for p < 0

dim_{N (G)} (H_{p}(N (G) ⊗_{ℂG} C_{∗})) = dim_{N (G)} (N (G) ⊗_{ℂG} H_{p}(C_{∗})) . (6.74)

We begin with the case where C_{∗} is projective. Then there is a universal coefficient spectral sequence converging to H_{p+q}(N (G) ⊗_{ℂG} C_{∗}) [518, Theorem 5.6.4 on page 143] whose E^{2}-term is E_{p,q}^{2} = Tor_{p}^{ℂG}(N (G), H_{q}(C_{∗})). Now Additivity (see Theorem 6.7 (4b)) together with Theorem 6.37 imply (6.74) if C_{∗} is projective.

Next we prove (6.74) in the case where C_{∗} is acyclic. If C_{∗} is 2-dimensional, we conclude dim_{N (G)} (H_{p}(N (G) ⊗_{ℂG} C_{∗})) = 0 for p ≥ 0 using the long exact Tor-sequences of the exact sequence 0 → C_{2} → C_{1} → C_{0} → 0, Additivity (see Theorem 6.7 (4b)) and Theorem 6.37. Now the claim for any acyclic ℂG-chain complexes C_{∗} with C_{p} = 0 for p < 0 follows from by inspecting the various short exact sequences 0 → im(c_{p+1}) → C_{p} ⟶^{c_{p}} im(c_{p}) → 0.

In the general case one chooses a projective ℂG-chain complex P_{∗} together with a ℂG-chain map f_{∗}: P_{∗} → C_{∗} which induces an isomorphism on homology. Notice that the mapping cylinder cyl(f_{∗}) is ℂG-chain homotopy equivalent to C_{∗} and the mapping cone cone(f_{∗}) of f_{∗} is acyclic. Hence (6.74) is true for P_{∗} and cone(f_{∗}). Thus we get (6.74) for C_{∗} from Additivity (see Theorem 6.7 (4b)) and the long exact homology sequences of the short exact sequence of N (G)-chain complexes which we obtain by applying N (G) ⊗_{ℂG} − to the short exact (in each dimension split exact) ℂG-sequence 0 → P_{∗} → cyl(f_{∗}) → cone(f_{∗}) → 0. This finishes the proof of Theorem 6.73.

⊓ ⊔

We conclude from Theorem 6.54 (8b) and Theorem 6.73

Corollary 6.75. If G is an infinite amenable group, then b_{p}^{(2)}(G) = 0 for all p ≥ 0.

We will investigate the class B_{∞} of groups G, for which b_{p}^{(2)}(G) = 0 holds for all p ≥ 0, in Section 7.1.

Remark 6.76. Next we compare our approach with the one of Cheeger and Gromov [107, section 2], where Corollary 6.75 has already been proved in [107, Theorem 0.2 on page 191]. We begin with the case of a countable simplicial complex X with free simplicial G-action. Then for any exhaustion X_{0} ⊂ X_{1} ⊂ X_{2} ⊂ … ⊂ X by G-equivariant simplicial subcomplexes for which G\X is compact, the p-th L^{2}-Betti number in the sense and notation of [107, 2.8 on page 198] is given by

b_{p}^{(2)}(X : G) = lim_{j→∞} lim_{k→∞} dim_{N (G)} (im (\overline{H}_{(2)}^{p}(X_{k} : G) ⟶^{i_{j,k}^{∗}} \overline{H}_{(2)}^{p}(X_{j} : G))) ,

where i_{j,k}: X_{j} → X_{k} is the inclusion for j ≤ k. We get an identification \overline{H}_{(2)}^{p}(X_{j} : G) = H_{(2)}^{p}(X_{j}; N (G)) from Lemma 1.76. Notice that for a G-map f: Y → Z of G-CW -complexes of finite type H_{p}^{(2)}(Y; N (G)) can be identified with H_{(2)}^{p}(Y; N (G)) and analogously for Z and that under these identifications H_{(2)}^{p}(f) = (H_{(2)}^{p}(f))^{∗} (see Lemma 1.18). Moreover, we conclude

from Additivity (see Theorem 6.7 (4b)) that dim_{N (G)} (im (H_{(2)}^{p}(f))) =

dim_{N (G)} (im (H_{p}^{(2)}(f))). Now we get from Lemma 6.52, Lemma 6.53 and Theorem 6.7

dim_{N (G)} (im (\overline{H}_{(2)}^{p}(X_{k} : G) ⟶^{i_{j,k}^{∗}} \overline{H}_{(2)}^{p}(X_{j} : G)))
= dim_{N (G)} (im (H_{p}^{G}(X_{j}; N (G)) ⟶^{(i_{j,k})_{∗}} H_{p}^{G}(X_{k}; N (G)))) .

Hence we conclude from Theorem 6.13 about dimension and colimits that the definitions in [107, 2.8 on page 198] and in Definition 6.50 for a countable free simplicial complex X with free simplicial G-action agree

b_{p}^{(2)}(X : G) = b_{p}^{(2)}(X; N (G)).  (6.77)

If G is countable and X is a countable simplicial complex with simplicial (not necessarily free) G-action, then by [107, Proposition 2.2 on page 198] and by (6.77)

b_{p}^{(2)}(X : G) = b_{p}^{(2)}(EG × X : G) = b_{p}^{(2)}(EG × X; N (G)). (6.78)

Cheeger and Gromov [107, Section 2] define L^{2}-cohomology and L^{2}-Betti numbers of a G-space X by considering the category whose objects are Gmaps f: Y ⟶ X for a simplicial complex Y with cocompact free simplicial G-action and then using inverse limits to extend the classical notions for finite free G-CW -complexes such as Y to X. Their approach is technically more complicated for instance because they work with cohomology instead of homology and therefore have to deal with inverse limits instead of directed limits (see Theorem 6.13 and Theorem 6.18). Our approach is closer to standard notions, the only non-standard part is the verification of the properties of the extended dimension function (Theorem 6.7). Moreover, the notion of Cheeger and Gromov [107, Section 2] does only give b_{p}^{(2)}(EG × X; N (G)), where we can and do also consider b_{p}^{(2)}(X; N (G)).

6.6 L^{2}-Euler Characteristic

In this section we introduce the notion of L^{2}-Euler characteristic and investigate its relation to the equivariant Euler characteristic of a cocompact proper G-CW -complex and the Burnside group.

6.6 L^{2}-Euler Characteristic 277

6.6.1 Definition and Basic Properties of L^{2}-Euler Characteristic

If X is a G-CW -complex, denote by I(X) the set of its equivariant cells. For a cell c ∈ I(X) let (G_{c}) be the conjugacy class of subgroups of G given by its orbit type and let dim(c) be its dimension. Denote by |G_{c}|^{−1} the inverse of the order of any representative of (G_{c}), where |G_{c}|^{−1} is to be understood to be zero if the order is infinite.

Definition 6.79 (L^{2}-Euler characteristic). Let G be a group and let X be a G-space. Define

h^{(2)}(X; N (G)) := ∑_{p≥0} b_{p}^{(2)}(X; N (G)) ∈ [0, ∞];
χ^{(2)}(X; N (G)) := ∑_{p≥0}(−1)^{p} · b_{p}^{(2)}(X; N (G)) ∈ ℝ if h^{(2)}(X; N (G)) < ∞;
m(X; G) := ∑_{c∈I(X)} |G_{c}|^{−1} ∈ [0, ∞]  if X is a G-CW -complex;
h^{(2)}(G) := h^{(2)}(EG; N (G)) ∈ [0, ∞];
χ^{(2)}(G) := χ^{(2)}(EG; N (G)) ∈ ℝ  if h^{(2)}(G) < ∞.

We call χ^{(2)}(X; N (G)) and χ^{(2)}(G) the L^{2}-Euler characteristic of X and G.

The condition h^{(2)}(X; N (G)) < ∞ ensures that the sum which appears in the definition of χ^{(2)}(X; N (G)) converges absolutely and that the following results are true. The reader should compare the next theorem with [107, Theorem 0.3 on page 191]. We will investigate the class B_{∞} of groups whose L^{2}-Betti number all vanish in Section 7.1.

Theorem 6.80 (L^{2}-Euler characteristic).

(1) Generalized Euler-Poincaré formula Let X be a G-CW -complex with m(X; G) < ∞. Then

h^{(2)}(X; N (G)) < ∞;
∑_{c∈I(X)}(−1)^{dim(c)} · |G_{c}|^{−1} = χ^{(2)}(X; N (G));

(2) Sum formula Consider the following G-pushout

X_{0} ⟶^{i_{1}} X_{1}
i_{2}↓  ^{j_{1}}↓
X_{2} ⟶^{j_{2}} X

such that i_{1} is a G-cofibration. Suppose that h^{(2)}(X_{i}; N (G)) < ∞ for i = 0, 1, 2. Then

h^{(2)}(X; N (G)) < ∞;
χ^{(2)}(X; N (G)) = χ^{(2)}(X_{1}; N (G)) + χ^{(2)}(X_{2}; N (G)) − χ^{(2)}(X_{0}; N (G));

(3) Amenable groups Suppose that G is amenable and that X is a G-CW -complex with m(X; G) < ∞. Then

∑_{c∈I(X)}(−1)^{dim(c)} · |G_{c}|^{−1} = ∑_{p≥0}(−1)^{p} · dim_{N (G)} (N (G) ⊗_{ℂG} H_{p}(X; ℂ)) ,

where H_{p}(X; ℂ) is the ℂG-module given by the cellular or the singular homology of X with complex coefficients. In particular ∑_{c∈I(X)}(−1)^{dim(c)} · |G_{c}|^{−1} depends only the isomorphism class of the ℂG-modules H_{n}(X; ℂ) for all n ≥ 0;

(4) Comparison with the Borel construction Let X be a G-CW -complex. If for all c ∈ I(X) the group G_{c} is finite or b_{p}^{(2)}(G_{c}) = 0 for all p ≥ 0, then

b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(EG × X; N (G))  for p ≥ 0;
h^{(2)}(X; N (G)) = h^{(2)}(EG × X; N (G));
χ^{(2)}(X; N (G)) = χ^{(2)}(EG × X; N (G)), if h^{(2)}(X; N (G)) < ∞;
∑_{c∈I(X)}(−1)^{dim(c)} · |G_{c}|^{−1} = χ^{(2)}(EG × X; N (G)), if m(X; G) < ∞;

(5) Invariance under non-equivariant ℂ-homology equivalences Suppose that f: X → Y is a G-equivariant map of G-CW -complexes with m(X; G) < ∞ and m(Y; G) < ∞, such that the induced map H_{p}(f; ℂ) on homology with complex coefficients is bijective. Suppose that for all c ∈ I(X) the group G_{c} is finite or b_{p}^{(2)}(G_{c}) = 0 for all p ≥ 0, and analogously for all d ∈ I(Y). Then

χ^{(2)}(X; N (G)) = ∑_{c∈I(X)}(−1)^{dim(c)} · |G_{c}|^{−1}
= ∑_{d∈I(Y)}(−1)^{dim(d)} · |G_{d}|^{−1}
= χ^{(2)}(Y; N (G));

(6) Künneth formula Let X be a G-CW -complex and Y be an H-CW -complex. Then we get for the G × H-CW -complex X × Y

m(X × Y; N (G × H)) = m(X; G) · m(Y; H);
h^{(2)}(X × Y; N (G × H)) = h^{(2)}(X; N (G)) · h^{(2)}(Y; N (H));
χ^{(2)}(X × Y; N (G × H)) = χ^{(2)}(X; N (G)) · χ^{(2)}(Y; N (H)),

if h^{(2)}(X; N (G)), h^{(2)}(Y; N (H)) < ∞,

where we use the convention that 0 · ∞ = 0 and r · ∞ = ∞ for r ∈ (0, ∞];

(7) Restriction Let H ⊂ G be a subgroup of finite index [G : H]. Let X be a G-space and let res_{G}^{H} X be the H-space obtained from X by restriction. Then

m(res_{G}^{H} X; H) = [G : H] · m(X; G);
h^{(2)}(res_{G}^{H} X; N (H)) = [G : H] · h^{(2)}(X; N (G));
χ^{(2)}(res_{G}^{H} X; N (H)) = [G : H] · χ^{(2)}(X; N (G)), if h^{(2)}(X; N (G)) < ∞,

where [G : H] · ∞ is understood to be ∞;

(8) Induction Let H ⊂ G be a subgroup and let X be an H-space. Then

m(G ×_{H} X; N (G)) = m(X; H);
h^{(2)}(G ×_{H} X; N (G)) = h^{(2)}(X; N (H));
χ^{(2)}(G ×_{H} X; N (G)) = χ^{(2)}(X; N (H)),  if h^{(2)}(X; N (H)) < ∞.

Proof. (1) Additivity (see Theorem 6.7 (4b)) and Lemma 6.33 (1) imply

b_{p}^{(2)}(X; N (G)) = dim_{N (G)} (H_{p} (N (G) ⊗_{ℤG} C_{p}(X)))
≤ dim_{N (G)} (N (G) ⊗_{ℤG} C_{p}(X))
= ∑_{c∈I(X),dim(c)=p} |G_{c}|^{−1}.

This shows h^{(2)}(X; N (G)) ≤ m(X; G). Additivity (see Theorem 6.7 (4b)) implies

χ^{(2)}(X; N (G)) = ∑_{p=0}^{∞}(−1)^{p} · dim_{N (G)} (N (G) ⊗_{ℤG} C_{p}(X; N (G))) ,

and thus assertion (1) follows.

(2) Next we prove that the given square induces a long exact Mayer-Vietoris sequence of N (G)-modules

… → H_{p+1}^{G}(X; N (G)) → H_{p}^{G}(X_{0}; N (G)) → H_{p}^{G}(X_{1}; N (G))⊕H_{p}^{G}(X_{2}; N (G))

→ H_{p}^{G}(X; N (G)) → ….

This is clear if the given pushout consists of G-CW -complexes with i_{1} an inclusion of G-CW -complexes, i_{2} a cellular G-map and X equipped with the G-CW -complex structure induced by the ones on X_{0}, X_{1} and X_{2}, and we use the cellular ℤG-chain complexes. Namely, in this situation we obtain a short exact sequence of ℤG-chain complexes 0 → C_{∗}(X_{0}) → C_{∗}(X_{1}) ⊕ C_{∗}(X_{2}) → C_{∗}(X) → 0 which stays exact after applying N (G) ⊗_{ℤG} − and we can take the associated long homology sequence. The general case follows from Lemma 6.51, Theorem 6.54 (1b) and the fact that one can construct such a G-pushout of G-CW -complexes together with G-maps from the cellular G-pushout to the given G-pushout such that the maps induce on each H-fixed point set a weak homotopy equivalence (see [326, Proposition 2.3 on page 35 and Lemma 2.13 on page 38]).

Notice that the alternating sums defining the L^{2}-Euler characteristics converge absolutely so that one can reorder the summands without changing the limit. Now assertion (2) follows from Additivity (see Theorem 6.7 (4b)).

(3) This follows from assertion (1) and Theorem 6.73.

(4) This follows from assertion (1) and Theorem 6.54 (2).

(5) This follows from assertion (1) and Theorem 6.54 (3).

(6) This follows from Theorem 6.54 (5).

(7) This follows from Theorem 6.54 (6).

(8) This follows from Theorem 6.54 (7). This finishes the proof of Theorem 6.80. ⊓ ⊔

Remark 6.81. Let X be a CW -complex which is virtually homotopy finite, i.e. there is a d-sheeted covering p: \overline{X} → X for some positive integer d such that \overline{X} is homotopy equivalent to a finite CW -complex. Define the virtual Euler characteristic following Wall [509]

χ_{virt}(X) := \frac{χ(\overline{X})}{d}.

One easily checks that this is independent of the choice of p: \overline{X} → X since the ordinary Euler characteristic is multiplicative under finite coverings. Moreover, we conclude from Theorem 6.80 (1) and (7) that for virtually homotopy finite X

m(X̃; N (π_{1}(X))) < ∞;
χ^{(2)}(X̃; N (π_{1}(X))) = χ_{virt}(X).

6.6.2 L^{2}-Euler Characteristic, Equivariant Euler Characteristic and the Burnside Group

In this subsection we introduce the Burnside group and the equivariant Euler characteristic and relate these notions to the L^{2}-Euler characteristic. The elementary proof of the following lemma is left to the reader.

Lemma 6.82. Let H and K be subgroups of G. Let NK be the normalizer of K in G and WK be its Weyl group NK/K. Then

(1) G/H^{K} = {gH | g^{−1}Kg ⊂ H};

(2) The map

ϕ: G/H^{K} → consub(H), gH ↦ g^{−1}Kg

induces an injection

WK\(G/H^{K}) → consub(H),

where consub(H) is the set of conjugacy classes in H of subgroups of H;

(3) The WK-isotropy group of gH ∈ G/H^{K} is (gHg^{−1}∩NK)/K ⊂ NK/K = WK;

(4) If H is finite, then G/H^{K} is a finite union of WK-orbits of the shape WK/L for finite subgroups L ⊂ WK. ⊓ ⊔

Definition 6.83. Define the Burnside group A(G) of a group G to be the Grothendieck group which is associated to the abelian monoid under disjoint union of G-isomorphism classes of proper cocompact G-sets S, i.e. G-sets S for which the isotropy group of each element in S and the quotient G\S are finite. ⊓ ⊔

Notice that A(G) is the free abelian group generated by G-isomorphism classes of orbits G/H for finite subgroups H ⊂ G and that G/H and G/K are G-isomorphic if and only if H and K are conjugate in G. If G is a finite group, A(G) is the classical Burnside ring [494, section 5], [495, chapter IV]. If G is infinite, then the cartesian product of two proper cocompact G-sets with the diagonal action is not necessarily cocompact anymore so that the cartesian product does not induce a ring structure on A(G). At least there is a bilinear map induced by the cartesian product A(G_{1}) ⊗ A(G_{2}) → A(G_{1} × G_{2}).

Definition 6.84. Let X be a finite proper G-CW -complex. Define its equivariant Euler characteristic

χ^{G}(X) := ∑_{c∈I(X)}(−1)^{dim(c)} · [G/G_{c}] ∈ A(G).

An additive invariant (A, a) for finite proper G-CW -complexes consists of an abelian group A and a function a which assigns to any finite proper G-CW -complex X an element a(X) ∈ A such that the following three conditions hold, i.) if X and Y are G-homotopy equivalent, then a(X) = a(Y), ii.) if X_{0}, X_{1} and X_{2} are G-CW -subcomplexes of X with X = X_{1} ∪ X_{2} and X_{0} = X_{1} ∩ X_{2}, then a(X) = a(X_{1}) + a(X_{2}) − a(X_{0}), and iii.) a(∅) = 0. We call an additive invariant (U, u) universal, if for any additive invariant (A, a) there is precisely one homomorphism ψ: U → A such that ψ(u(X)) = a(X) holds for all finite proper G-CW -complexes. One easily checks using induction over the number of equivariant cells Lemma 6.85. (A(G), χ^{G}) is the universal additive invariant for finite proper G-CW -complexes and we get for a finite proper G-CW -complex X

χ^{G}(X) = ∑_{(H),|H|<∞} χ(WH\(X^{H}, X^{>H})) · [G/H],

where χ(WH\(X^{H}, X^{>H})) is the ordinary Euler characteristic of the pair of finite CW -complexes WH\(X^{H}, X^{>H}).

Definition 6.86. Define for a finite subgroup K ⊂ G the L^{2}-character map

ch_{K}^{G} : A(G) → ℚ, [S] ↦ ∑_{i=1}^{r} |L_{i}|^{−1}

if WK/L_{1}, WK/L_{2},…, WK/L_{r} are the WK-orbits of S^{K}. Define the global L^{2}-character map by

ch^{G} := ∏_{(K)} ch_{K}^{G}: A(G) → ∏_{(K)} ℚ

where (K) runs over the conjugacy classes of finite subgroups of G. ⊓ ⊔

Lemma 6.87. Let X be a finite proper G-CW -complex and K ⊂ G be a finite subgroup. Then X^{K} is a finite proper WK-CW -complex and

χ^{(2)}(X^{K}; N (WK)) = ch_{K}^{G}(χ^{G}(X)).

Proof. The WK-space X^{K} is a finite proper WK-CW -complex because for finite H ⊂ G the WK-set G/H^{K} is proper and cocompact by Lemma 6.82 (4). Since the assignment which associates to a finite proper G-CW -complex X the element χ^{(2)}(X^{K}; N (WK)) in ℚ is an additive invariant by Theorem 6.80 (2) and (5), it suffices by Lemma 6.85 to check the claim for X = G/H for finite H ⊂ G. This follows from the conclusion of Lemma 6.33 (1) that χ^{(2)}(WK/L; N (WK)) = |L|^{−1} holds for finite L ⊂ WK. ⊓ ⊔

Notice that one gets from Lemma 6.82 the following explicit formula for the value of ch_{K}^{G}(G/H). Namely, define

L_{K}(H) := {(L) ∈ consub(H) | L conjugate to K in G}.

For (L) ∈ L_{K}(H) choose L ∈ (L) and g ∈ G with g^{−1}Kg = L. Then

g(H ∩ NL)g^{−1} = gHg^{−1} ∩ NK;
|(gHg^{−1} ∩ NK)/K|^{−1} = \frac{|K|}{|H ∩ NL|}.

This implies

ch_{K}^{G}(G/H) = ∑_{(L)∈L_{K}(H)} \frac{|K|}{|H ∩ NL|}.  (6.88)

Lemma 6.89. The global L^{2}-character map of Definition 6.86 induces a map denoted by

ch^{G} ⊗_{ℤ}ℚ : A(G) ⊗_{ℤ} ℚ → ∏_{(K)} ℚ.

It is injective. If G has only finitely many conjugacy classes of finite subgroups, then it is bijective.

Proof. Consider an element ∑_{i=1}^{n} r_{i} · [G/H_{i}] in the kernel of ch^{G} ⊗_{ℤ}ℚ. We show by induction on n that the element must be trivial. The induction beginning n = 0 is trivial, the induction step is done as follows. We can choose the numeration such that H_{i} subconjugated to H_{j} implies i ≥ j. We get from (6.88)

ch_{K}^{G}(G/H) = 1  if H = K;
ch_{K}^{G}(G/H) = 0  if K is not subconjugated to H in G.

This implies

ch_{H_{1}}^{G} (∑_{i=1}^{n} r_{i} · [G/H_{i}]) = r_{1}

and hence r_{1} = 0. Hence the global L^{2}-character map is injective. If G has only finitely many conjugacy classes of finite subgroups, then the source and target of ch^{G} ⊗_{ℤ}ℚ are rational vector spaces of the same finite dimension and hence ch^{G} ⊗_{ℤ}ℚ must be bijective. ⊓ ⊔

Remark 6.90. Suppose that there are only finitely many conjugacy classes (H_{1}), (H_{2}), …, (H_{r}) of finite subgroups in G. Without loss of generality we can assume that H_{i} subconjugated to H_{j} implies i ≥ j. With respect to the obvious ordered basis for the source and target, the map ch^{G} ⊗_{ℤ}ℚ is described by an upper triangular matrix A with ones on the diagonal. One can get an explicit inverse A^{−1} which again has ones on the diagonal. This leads to a characterization of the image of A(G) under the global L^{2}-character map χ^{G}. Namely, an element in η ∈ ∏_{i=1}^{r} ℚ lies in ch^{G}(A(G)) if and only if the following Burnside integrality conditions are satisfied

A^{−1}η ∈ ∏_{i=1}^{r} ℤ.  (6.91)

Now suppose that G is finite. Then the global L^{2}-character map is related to the classical character map by the factor |WK|^{−1}, i.e. we have for each subgroup K of G and any finite G-set S

ch_{K}^{G}(S) = |WK|^{−1} · |S^{K}|.  (6.92)

One easily checks that under the identification (6.92) the integrality conditions (6.91) correspond to the classical Burnside ring congruences for finite groups [494, section 5.8], [495, section IV.5]. ⊓ ⊔

Let E(G, FIN) be the classifying G-space for the family FIN of finite subgroups (see Definition 1.28).

Lemma 6.93. Suppose that there is a model for E(G, FIN) which is a finite G-CW -complex. Then there are only finitely many conjugacy classes of finite subgroups and for a finite subgroup K ⊂ G

ch_{K}^{G}(χ^{G}(E(G, FIN))) = χ^{(2)}(WK).

If G is amenable, then we get for a finite subgroup K ⊂ G

ch_{K}^{G}(χ^{G}(E(G, FIN))) = |WK|^{−1},

where |WK|^{−1} is to be understood as 0 for infinite WK.

Proof. We get from Theorem 6.80 (4) and Lemma 6.87 since E(G, FIN)^{K} is a model for E(WK, FIN) and E(WK) × E(WK, FIN) is a model for E(WK)

ch_{K}^{G}(χ^{G}(E(G, FIN))) = χ^{(2)}(E(WK, FIN); N (WK)) = χ^{(2)}(WK).

In the case where G is amenable apply Theorem 6.54 (8b) and Corollary 6.75. ⊓ ⊔

Example 6.94. Let 1 → ℤ^{n} → G → ℤ/p → 1 be an extension of groups for n ≥ 1 and a prime number p. The conjugation action of G on the normal subgroup ℤ^{n} factorizes through the projection G → ℤ/p to an operation ρ of ℤ/p onto ℤ^{n}. There exists a finite G-CW -complex model for E(G, FIN) by the following argument. If G contains a finite subgroup, then G is a semidirect product of ℤ^{n} and ℤ/p and one can construct a finite G-CW -complex as model for E(G, FIN) with ℝ^{n} as underlying space. Suppose that G contains no finite subgroup. Then H_{1}(G; ℚ) ≅_{ℚ} (ℤ^{n})^{ℤ/p} ⊗_{ℤ} ℚ ≠ {0} and hence G admits an epimorphism onto ℤ. Now one can show inductively over n that there is a finite model for BG.

We want to compute χ^{G}(E(G, FIN)). If this operation has a non-trivial fixed point, then WH is infinite for any finite subgroup H of G and we conclude from Lemma 6.89 and Theorem 6.93 that

χ^{G}(E(G, FIN)) = 0.

Now suppose that this operation ρ has no non-trivial fixed points. Let H_{0} be the trivial subgroup and H_{1}, H_{2}, …, H_{r} be a complete set of representatives of the conjugacy classes of finite subgroups. Each H_{i} is isomorphic to ℤ/p. One easily checks that there is a bijection

H^{1}(ℤ/p; ℤ_{ρ}^{n}) → {(H) | H ⊂ G, 1 < |H| < ∞}

and in particular r ≥ 1, where ℤ_{ρ}^{n} denotes the ℤ[ℤ/p]-module given by ℤ^{n} and ρ. We compute using (6.88)

ch_{H_{0}}^{G}(G/H_{0}) = 1;
ch_{H_{0}}^{G}(G/H_{j}) = \frac{1}{p} j = 1, 2, …, r;
ch_{H_{i}}^{G}(G/H_{j}) = 1 i = j, i, j = 1, 2, …, r;
ch_{H_{i}}^{G}(G/H_{j}) = 0 i ≠ j, i, j = 1, 2, …, r.

We conclude

χ^{G}(E(G, FIN)) = −\frac{r}{p} · [G/H_{0}] + ∑_{i=1}^{r} [G/H_{i}].

The integrality conditions of (6.91) become in this case

η_{0} − \frac{1}{p} · ∑_{i=1}^{r} η_{i} ∈ ℤ;
η_{i} ∈ ℤ  i = 1, 2 ,…, r.

6.7 Finitely Presented Torsion Modules and Novikov-Shubin Invariants

In this section we explain how the Novikov-Shubin invariants can be read off from H_{p}(X; N (G)) for a free G-CW -complex X of finite type.

Let M be a finitely presented N (G)-module. Then we can choose an exact sequence 0 → N (G)^{n} ⟶^{f} N (G)^{n} → TM → 0 for some positive morphism f by Lemma 6.28. Now define the Novikov-Shubin invariant

α(M) := α(ν(f)) ∈ [0, ∞] ∐{∞^{+}}  (6.95)

by the Novikov-Shubin invariant of the morphism ν(f): l^{2}(G)^{n} → l^{2}(G)^{n} (see Definition 2.8), where ν has been introduced in (6.22). This is independent of the choice of f because of homotopy invariance (see Theorem 2.19). Moreover, for any finitely generated projective N (G)-resolution C_{∗} of M we have α_{1}(ν(C_{∗})) = α(ν(c_{1})) = α(M). Obviously α(M) = α(TM). Recall that the Novikov-Shubin invariant of a morphism f: U → V of finitely generated Hilbert N (G)-modules introduced in Definition 2.8 measures the deviation of the image of f to be closed. Analogously, the Novikov-Shubin invariant of the cokernel of an N (G)-map g: P → Q of finitely generated projective N (G)-modules measures the difference between im(g) and \overline{im(g)}.

Let 0 → M_{0} → M_{1} → M_{2} → 0 be an exact sequence of N (G)-modules such that two of them are finitely presented and have von Neumann dimension zero. Then M_{i} is finitely presented and TM_{i} = M_{i} for i = 0, 1, 2 by Theorem 6.5 and Theorem 6.7 and we conclude from Theorem 2.20

\frac{1}{α(M_{1})} ≤ \frac{1}{α(M_{0})} + \frac{1}{α(M_{2})}.  (6.96)

In particular we get \frac{1}{α(M)} ≤ \frac{1}{α(N)}, if M and N are finitely presented N (G)-modules, dim_{N (G)}(N) = 0 and M ⊂ N. Moreover, Lemma (2.11)

(8) and Theorem 6.24 (3) imply that α(M) = ∞^{+} ⇔ M = 0 holds for a finitely presented N (G)-module M with dim_{N (G)}(M) = 0. Hence we may say that \frac{1}{α(M)} measures the size of finitely presented N (G)-modules M with dim_{N (G)}(M) = 0.

Lemma 6.97. Let X be a free G-CW -complex of finite type. Then its (p − 1)-th L^{2}-Betti number b_{p−1}^{(2)}(X; N (G)) of Definition 1.30 and its p-th Novikov-Shubin invariant α_{p}(X; N (G)) of Definition 2.54 can be read off from H_{p−1}^{G}(X; N (G)) by

b_{p−1}^{(2)}(X; N (G)) = dim_{N (G)}(H_{p−1}^{G}(X; N (G)));
α_{p}(X; N (G)) = α(H_{p−1}^{G}(X; N (G))).

Proof. The statement about the L^{2}-Betti numbers has already been proved in Lemma 6.53. We conclude from Lemma 2.11 (9), Theorem 6.7 (3) and Theorem 6.24

α_{p}(X; N (G)) = α (c_{p}^{(2)} : C_{p}^{(2)}(X) → C_{p−1}^{(2)}(X))
= α (c_{p}^{(2)}|_{ker(c_{p}^{(2)})^{⊥}}: ker(c_{p}^{(2)})^{⊥} → clos(im(c_{p}^{(2)})))
= α (coker (ν^{−1} (c_{p}^{(2)}|_{ker(c_{p}^{(2)})^{⊥}}: ker(c_{p}^{(2)})^{⊥} → clos(im(c_{p}^{(2)})))))
= α (\overline{im(ν^{−1}(c_{p}^{(2)}))}/ im(ν^{−1}(c_{p}^{(2)})))
= α (\overline{im(N (G) ⊗_{ℤG} c_{p})}/ im(N (G) ⊗_{ℤG} c_{p}))
= α(TH_{p−1}^{G}(X; N (G)))
= α(H_{p−1}^{G}(X; N (G))),

where coker denotes the cokernel. ⊓ ⊔ Notice that Lemma 6.97 gives a good explanation for Theorem 2.55 (1). The next lemma will be interesting in connection with the zero-in-the-

spectrum Conjecture 12.1 (see Lemma 12.3).

Lemma 6.98. Let X be a free G-CW -complex of finite type and let p ≥ 0 be an integer. Then H_{p}^{G}(X; N (G)) = 0 if and only if b_{p}^{(2)}(X; N (G)) = 0 and α_{p+1}(X; N (G)) = ∞^{+}.

Proof. Lemma 6.97 implies that b_{p}^{(2)}(X; N (G)) = 0 and α_{p+1}(X; N (G)) = ∞^{+} if and only if dim_{N (G)}(H_{p}(X; N (G))) = 0 and α(H_{p}(X; N (G))) = ∞^{+}. Hence H_{p}(X; N (G)) = 0 implies b_{p}^{(2)}(X; N (G)) = 0 and α_{p+1}(X; N (G)) = ∞^{+}. Now suppose that b_{p}^{(2)}(X; N (G)) = 0 and α_{p+1}(X; N (G)) = ∞^{+}. Because of Theorem 6.7 (3) and (4e) we conclude dim_{N (G)}(PH_{p}(X; N (G))) = 0 and α(TH_{p}(X; N (G))) = ∞^{+} and therefore PH_{p}(X; N (G)) = 0 and TH_{p}(X; N (G)) = 0. This shows H_{p}(X; N (G)) = 0. ⊓ ⊔

6.8 Miscellaneous

We will deal with the Atiyah Conjecture for a group G in Chapter 10. It says for a finite free G-CW -complex that b_{p}^{(2)}(X; N (G)) ∈ ℚ. If furthermore there is an integer d such that the order of any finite subgroup H ⊂ G divides d, then the strong Atiyah Conjecture 10.2 predicts d · b_{p}^{(2)}(X; N (G)) ∈ ℤ for a finite free G-CW -complex X. The following result is proved in [334, Theorem 5.2 on page 233]

Theorem 6.99. (1) Suppose that there is no bound on the order of finite subgroups of G. Then for any sequence β_{3}, β_{4}, … of elements in [0, ∞], there is a free simply connected G-CW -complex X satisfying

b_{p}^{(2)}(X; N (G)) = β_{p} for p ≥ 3.

If G is countable, one can arrange that X has countably many Gequivariant cells;

(2) Suppose that there is an integer d such that the order of any finite subgroup of G divides d and that the strong Atiyah Conjecture 10.2 holds for G. Then we get for any G-space X and p ≥ 0

d · b_{p}^{(2)}(X; N (G)) ∈ ℤ ∪ {∞}.

In [343] the notion of Novikov-Shubin invariants for free G-CW -complexes of finite type is extended to arbitrary G-spaces. In particular one can talk of the Novikov-Shubin invariants α_{p}(G) := α_{p}(EG; N (G)) of a group G. Recall that a group is locally finite, if any finitely generated subgroup is finite. In [343, Theorem 3.9 on page 174] it is proved that α_{p}(G) ≥ 1 for p ≥ 1 if G contains ℤ^{n} as normal subgroup for some n ≥ 1, and that α_{p}(G) ≥ 1 for p = 1, 2 if G contains a normal subgroup, which is infinite elementaryamenable and which does not contain an infinite locally finite subgroup. This implies in particular that α_{p}(X; N (G)) ≥ 1 for p = 1, 2 holds for a G-CW - complex X of finite type if G contains a normal subgroup, which is infinite elementary-amenable and which does not contain an infinite locally finite subgroup.

Farber [182] constructs a category E(N (G)) which contains the category {fin. gen. Hilb. N (G)-mod.} as a subcategory. The point is that E(N (G)) is an abelian category, it is an abelian extension of {fin. gen. Hilb. N (G)-mod.} in the sense of [205]. An object in E(N (G)) is a map of finitely generated Hilbert N (G)-modules (α: A′ → A). A morphism in E(N (G)) from (α: A′ → A) to (β: B′ → B) is an equivalence class of maps f: A → B such that there exists a map g: A′ → B′ with f ∘ α = β ∘ g. Here f and f′ are called equivalent if and only if f − f′ = β ∘ h for some morphism h: A → B′. Let {fin. pres. N (G)-mod.} be the category of finitely presented N (G)-modules. The two approaches are unified by the fact that there is an equivalence of abelian categories

\overline{ν^{−1}} : E(N (G)) → \{fin. pres. N (G)-mod.\}

which induces the equivalence appearing in Theorem 6.24

ν^{−1} : {fin. gen. Hilb. N (G)-mod.} → {fin. gen. proj. N (G)-mod.}.

The equivalence \overline{ν^{−1}} sends an object (α: A′ → A) to the cokernel of ν^{−1}(α).

Any von Neumann algebra is semihereditary (cf. Theorem 6.7 (1)). This follows from the facts that any von Neumann algebra is a Baer ∗-ring and hence in particular a Rickart C^{∗}-algebra [36, Definition 1, Definition 2 and Proposition 9 in Chapter 1.4] and that a C^{∗}-algebra is semihereditary if and only if it is Rickart [8, Corollary 3.7 on page 270].

The material of Subsection 6.6.2 can be extended from finite proper G-CW -complexes to G-endomorphisms f: X → X of finite proper G-CW - complexes by replacing Euler characteristics by Lefschetz invariants. Define the equivariant Lefschetz invariant

L^{G}(f) := ∑_{(H),|H|<∞} ∑_{p≥0}(−1)^{p} · tr_{ℤWH} (C_{p}(f^{H}, f^{>H}): C_{p}(X^{H}, X^{>H})
→ C_{p}(X^{H}, X^{>H})) · [G/H] ∈ A(G),  (6.100)

where tr_{ℤWH} if the trace defined for endomorphisms of finitely generated projective ℤWH-modules which is essentially determined by tr_{ℤWK}(R_{u}: ℤWH → ℤWH) = λ_{e} for u = ∑_{w∈WH} λ_{w} · w ∈ ℤWH. One can also define an L^{2}- Lefschetz invariant

L^{(2)}(f^{K}; N (WK))
:= ∑_{p≥0}(−1)^{p} · tr_{N (WK)} (id ⊗_{ℤWK}C_{p}(f^{K}): N (WK) ⊗_{ℤWK} C_{p}(X^{K})
→ N (WK) ⊗_{ℤWK} C_{p}(X^{K})) ∈ ℂ.  (6.101)

Notice that L^{G}(id: X → X) = χ^{G}(X) and L^{(2)}(id: X^{K} → X^{K}; N (WK)) = χ^{(2)}(X^{K}; N (WK)). We can extend Theorem 6.80 (1) and Lemma 6.87 to

ch_{K}^{G}(L^{G}(f)) = L^{(2)}(f^{K}; N (WK))  (6.102)
= ∑_{p≥0}(−1)^{p} · tr_{N (WK)} (PH_{p}(f^{K}; N (WK)):
PH_{p}^{WK}(X^{K}; N (WK)) → PH_{p}^{WK}(X^{K}; N (WK))) . (6.103)

For more sophisticated Lefschetz type invariants see for instance [301] and

Dimension functions and the role of the torsion part TM for an N (G)- module will be further explained in Chapter 8 when we extend the group von Neumann algebra N (G) to the algebra U(G) of affiliated operators. This corresponds for a principal ideal domain R to the passage to its quotient field.

The dimension-flatness of the von Neumann algebra N (G) over ℂG for amenable G in the sense of Theorem 6.37 will play an important role in computations about G_{0}(ℂG) in Subsection 9.5.3.

We state without proof the next result which yields another proof of the Künneth formula (see Theorem 6.54 (5)).

Theorem 6.104. Let G and H be two groups.

(1) Let 0 → M_{0} → M_{1} → M_{2} → 0 be an exact sequence of N (G)-modules and let N be an N (H)-module. Then

dim_{N (G×H)} (ker (N (G × H) ⊗_{N (G)⊗_{ℂ}N (H)} M_{0} ⊗_{ℂ} N →

N (G × H) ⊗_{N (G)⊗_{ℂ}N (H)} M_{1} ⊗_{ℂ} N)) = 0;

(2) Let M be an N (G) and N be an N (H)-module. Then

dim_{N (G×H)} (N (G × H) ⊗_{N (G)⊗_{ℂ}N (H)} M ⊗_{ℂ} N)
= dim_{N (G)}(M) · dim_{N (H)}(N)

with the convention 0 · ∞ = ∞ · 0 = 0.

Elek [173] defines a dimension for finitely generated KG-modules for a finitely generated amenable group G and a field K which satisfies additivity and sends KG to 1.

Exercises

6.1. Let M be a submodule of a projective N (G)-module. Prove that dim_{N (G)}(M) = 0 holds if and only if M = 0.

6.2. Let M be a finitely generated N (G)-module and ϵ > 0. Then there is a finitely generated projective N (G)-module Q together with an epimorphism p: Q → M such that dim_{N (G)}(ker(p)) ≤ ϵ.

6.3. Let M be a submodule of the finitely generated projective N (G)-module P. Given ϵ > 0, there is a submodule P′ ⊂ M which is a direct summand in P and satisfies dim_{N (G)}(M) ≤ dim_{N (G)}(P′) + ϵ.

6.4. Let M be a countably generated N (G)-module and M^{∗} be its dual N (G)-module hom_{N (G)}(M, N (G)). Prove dim_{N (G)}(M) = dim_{N (G)}(M^{∗}).

6.5. Let I be the set of finite subsets of S^{1} directed by inclusion. Define the directed system {N_{J} | J ∈ I} by N_{J} = L^{∞}(S^{1}) with the associated maps ϕ_{J,K}: L^{∞}(S^{1}) → L^{∞}(S^{1}) for J ⊂ K which are given by multiplication with ∏_{u∈K−J} \overline{z − u} for J ⊂ K, where \overline{z − u} ∈ N (ℤ) is given by the function S^{1} → ℂ sending z ∈ S^{1} to the complex conjugate of z −u. Define the N (ℤ)-module M = colim_{j∈I} N_{j}. Let M^{∗} be the dual N (ℤ)-module hom_{N (ℤ)}(M, N (ℤ)). Show

dim_{N (ℤ)}(M) = 1;
M^{∗} = 0;
dim_{N (ℤ)}(M^{∗}) = 0.

6.6. Show that N (ℤ) is not Noetherian. Deduce that N (G) is not Noetherian if G contains an element of infinite order.

6.7. Let G be a group and let A be a ring with ℤ ⊂ A ⊂ ℂ. Let P be a projective AG-module such that for some finitely generated AG-submodule M ⊂ P we have dim_{N (G)}(N (G) ⊗_{AG} P/M) = 0. Show that P is finitely generated.

6.8. Let M be an N (G)-module. Define T_{dim}M to be the union of all N (G)- submodules N ⊂ M with dim_{N (G)}(N) = 0. Show that T_{dim}M is the largest N (G)-submodule of M with vanishing von Neumann dimension and that this definition coincides with the Definition 6.1 of TM provided that M is finitely generated, but not for arbitrary N (G)-modules M.

6.9. Let G be a group with the property that for any ℂG-module M dim_{N (G)} (Tor_{p}^{ℂG}(N (G), M)) = 0 holds for p ≥ 1. Show that then any sub-

group H ⊂ G inherits this property, i.e. dim_{N (H)} (Tor_{p}^{ℂH}(N (H), N)) = 0 holds for any ℂH-module N and any p ≥ 1.

6.10. Show for a virtually cyclic group G that N (G) is flat over ℂG.

6.11. Let G be a group for which N (G) is flat over ℂG. Prove

(1) For any subgroup H ⊂ G the von Neumann algebra N (H) is flat over ℂH;

(2) If the group K contains G as a subgroup of finite index, then N (K) is flat over ℂK;

(3) H_{p}^{G}(EG; N (G)) = 0 for p ≥ 1;

(4) b_{p}^{(2)}(G) = 0 for p ≥ 1;

(5) If there is a CW -model for BG of finite type, then α_{p}(EG; N (G)) = ∞^{+} for p ≥ 1;

(6) G does not contain a subgroup which is ℤ^{n} or ∗_{i=1}^{n}ℤ for some n ≥ 2;

(7) G does not contain the fundamental group of an aspherical closed manifold whose universal covering is a symmetric space;

(8) G does not contain the fundamental group of a connected sum M_{1}# … #M_{r} of (compact connected orientable) non-exceptional prime 3-manifolds M_{j}.

6.12. Let G be a group such that there is no bound on the order of its finite subgroups. Construct for β ∈ [0, ∞] a countably generated projective ℤG-module P satisfying dim_{N (G)}(N (G) ⊗_{ℤG} P) = β. Moreover, construct for a sequence β_{3}, β_{4}, β_{5}, … of elements in [0, ∞] a free simply connected G-CW -complex X with b_{p}^{(2)}(X; N (G)) = β_{p} for p ≥ 3. If G happens to be countable, find X with only countably many equivariant cells.

6.13. Given a sequence β_{1}, β_{2}, β_{3}, … of elements in [0, ∞], construct a group G with b_{p}^{(2)}(G) = β_{p} for p ≥ 1.

6.14. Let X be a G-CW -complex with m(X; G) < ∞. Show that then X has at most countably many equivariant cells with finite isotropy groups.

6.15. Let F → E → B be a fibration of connected CW -complexes and ϕ: π_{1}(E) → G be a group homomorphism. Let \overline{F} → F and \overline{E} → E be the coverings with G as deck transformation group associated to the homomorphisms π_{1}(F) ⟶^{i_{∗}} π_{1}(E) ⟶^{ϕ} G and ϕ: π_{1}(E) → G. Suppose that B is a finite CW -complex and h^{(2)}(\overline{F}; N (G)) < ∞. Show

h^{(2)}(\overline{E}, N (G)) < ∞;
χ^{(2)}(\overline{E}, N (G)) = χ^{(2)}(\overline{F} , N (G)) · χ(B),

where χ(B) is the ordinary Euler characteristic of the finite CW -complex B.

6.16. Show by constructing a counterexample that the condition b_{p}^{(2)}(G ×_{ϕ∘i} X̃; N (G)) < ∞ and b_{p−1}^{(2)}(G ×_{ϕ∘i} X̃; N (G)) < ∞ in Theorem 6.63 and the condition b_{1}^{(2)}(F̃) < ∞ in Corollary 6.72 are necessary.

6.17. Let M be the connected sum M_{1}# … #M_{r} of (compact connected orientable) non-exceptional prime 3-manifolds M_{j}. Assume that π_{1}(M) is infinite. Show

b_{1}^{(2)}(M̃) = −χ^{(2)}(π_{1}(M));
b_{2}^{(2)}(M̃) = χ(M) − χ^{(2)}(π_{1}(M));
b_{p}^{(2)}(M̃) = 0  for p ≠ 1, 2.

6.18. Let G be a group for which there is a d-dimensional G-CW -model for E(G, FIN). Show b_{p}^{(2)}(G) = 0 for p > d.

6.19. Let G be a group for which there is a finite G-CW -model for E(G, FIN). Show that then the ℂG-module ℂ, which is given by ℂ with the trivial G-operation, has a finite projective ℂG-resolution P_{∗}. The class ∑_{n≥0}(−1)^{p} · [P_{n}] ∈ K_{0}(ℂG) is independent of the choice of P_{∗} and denoted by [ℂ]. Prove that it is the image of χ^{G}(E(G; FIN)) under the homomorphism A(G) → K_{0}(ℂG), which sends a finite proper G-set S to the class of the finitely generated projective ℂG-module ℂ[S] given by the complex vector space with S as basis.

6.20. Let X be a G-CW -complex which is contractible (after forgetting the group action). Suppose that each isotropy group G_{x} is finite or satisfies b_{p}^{(2)}(G_{x}) = 0 for p ≥ 0. Prove b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(G) for p ≥ 0.

6.21. Equip ℤ[ℤ/5]/(N) ≅ ℤ[exp(2πi/5)] with the obvious ℤ/5-action, where N = ∑_{i=0}^{4} t^{i} for t ∈ ℤ/5 a fixed generator and (N) is the ℤ[ℤ/5]-ideal generated by N. Let G be the associated semidirect product. Compute A(G) and χ^{G}(E(G, FIN)) explicitly.

6.22. Let F_{g} be the free group of rank g ≥ 1. Using the fact that ℂ[F_{g}] is a so called fir (= free ideal ring), i.e. any submodule of a free module is free again (see [117, Corollary 3 on page 68]), show that for a F_{g}-CW -complex X of a finite type each ℂ[F_{g}]-module H_{p}^{sing}(X; ℂ) has a 1-dimensional finite free resolution. Then prove that b_{p}^{(2)}(X; N (F_{g})) and α_{p+1}(X; N (F_{g})) depend only on the ℂ[F_{g}]-isomorphism class of H_{p}^{sing}(X; ℂ).

6.23. Show for the group G = ∏_{i=1}^{∞} ℤ∗ℤ that H_{p}^{G}(EG; N (G)) = 0 for p ≥ 0.

6.24. Give a counterexample to the following statement by inspecting the special case G = H = ℤ and X = Y = S̃^{1}: If X is a G-CW -complex of finite type and Y an H-CW -complex of finite type, then there is an isomorphism

H_{n}^{G×H}(X × Y; N (G × H))

≅ ⨁_{p+q=n} N (G × H) ⊗_{N (G)⊗_{ℂ}N (H)} (H_{p}^{G}(X; N (G)) ⊗_{ℂ} H_{q}^{H}(Y; N (H))) .

6.25. Prove Theorem 6.104.

6.26. Show that Theorem 6.54 (5) follows from Theorem 6.104.

7. Applications to Groups

Introduction

In this chapter we apply the results of Chapter 6 to questions about group theory, mainly about deficiency and Euler characteristic.

In Section 7.1 we investigate the class B_{1} and B_{∞} respectively of groups for which b_{p}^{(2)}(G) vanishes for p = 0, 1 and for p ≥ 0 respectively. The classes B_{1} and B_{∞} turn out to be surprisingly large. For instance we prove that a group containing a normal infinite amenable subgroup belongs to B_{∞}. We also show that a group G belongs to B_{1} if it is an extension 1 → H → G → K → 1 of an infinite finitely generated group H and a group K which is infinite elementary amenable or which contains an element of infinite order or which contains finite subgroups of arbitrary large order.

The motivation to investigate the class B_{1} is that a finitely presented group which belongs to B_{1} has the following two properties. Its deficieny satisfies def(G) ≤ 1, and for any closed oriented smooth 4-manifold M with π_{1}(M) = G we have | sign(M)| ≤ χ(M). This will be explained in Section 7.3. A survey on deficiency is presented in Subsection 7.3.1.

In Section 7.2 we discuss the L^{2}-Euler characteristic of a group. It vanishes if G belongs to B_{∞} and satisfies all the properties which are known for the classical Euler characteristic χ(BG) of a group for which BG has a finite CW - model. The classical Euler characteristic and the virtual Euler characteristic due to Wall are special cases of the L^{2}-Euler characteristic.

In Section 7.4 we define for a group automorphism f: G → G of a group with finite model for BG a real number ρ^{(2)}(f) using L^{2}-torsion. It behaves like the Euler characteristic of BG. Applied to π_{1}(f) for a pseudo-Anosov self-homeomorphism f: S → S of a closed hyperbolic oriented surface S it detects the volume of the hyperbolic 3-manifold given by the mapping torus of f.

7.1 Groups with Vanishing L^{2}-Betti Numbers

In this section we investigate the following classes of groups. Recall that we have introduced the L^{2}-Betti numbers of a group G in Definition 6.50.

Definition 7.1. Let d be a non-negative integer or d = ∞. Define the class of groups

B_{d} := {G | b_{p}^{(2)}(G) = 0 for 0 ≤ p ≤ d}.

Notice that B_{0} is the class of infinite groups by Theorem 6.54 (8b).

7.1.1 General Criterions for the Vanishing of the L^{2}-Betti Numbers of a Group

Theorem 7.2. Let d be a non-negative integer or d = ∞. Then

(1) The class B_{∞} contains all infinite amenable groups;

(2) If G contains a normal subgroup H with H ∈ B_{d}, then G ∈ B_{d};

(3) If G is the union of a directed system of subgroups {G_{i} | i ∈ I} such that each G_{i} belongs to B_{d}, then G ∈ B_{d};

(4) Suppose that there are groups G_{1} and G_{2} and group homomorphisms ϕ_{i}: G_{0} → G_{i} for i = 1, 2 such that ϕ_{1} and ϕ_{2} are injective, G_{0} belongs to B_{d−1}, G_{1} and G_{2} belong to B_{d} and G is the amalgamated product G_{1} ∗_{G_{0}} G_{2} with respect to ϕ_{1} and ϕ_{2}. Then G belongs to B_{d};

(5) Let 1 → H ⟶^{i} G ⟶^{p} K → 1 be an exact sequence of groups such that b_{p}^{(2)}(H) is finite for all p ≤ d. Suppose that K is infinite amenable or suppose that BK has finite d-skeleton and there is an injective endomorphism j: K → K whose image has finite index, but is not equal to K. Then G ∈ B_{d};

(6) Let 1 → H ⟶^{i} G ⟶^{p} K → 1 be an exact sequence of groups such that H ∈ B_{d−1}, b_{d}^{(2)}(H) < ∞ and K contains an element of infinite order or finite subgroups of arbitrary large order. Then G ∈ B_{d};

(7) Let 1 → H ⟶^{i} G ⟶^{p} K → 1 be an exact sequence of infinite countable groups such that b_{1}^{(2)}(H) < ∞. Then G ∈ B_{1}.

Proof. (1) This has already been proved in Corollary 6.75.

(2) This follows from Theorem 6.54 (7) and Lemma 6.66 applied to the fibration BH → BG → B(G/H) and the obvious isomorphism ϕ: π_{1}(BG) → G.

(3) Inspecting for instance the bar-resolution or the infinite join model for EG, one sees that EG is the colimit of a directed system of G-CW - subcomplexes of the form G ×_{G_{i}} EG_{i} directed by I. Hence

H_{p}^{G}(EG; N (G)) = colim_{i∈I} H_{p}^{G}(G ×_{G_{i}} EG_{i}; N (G)).

Now the claim follows from Theorem 6.13 about dimension and colimits and Theorem 6.54 (7) about compatibility of L^{2}-Betti numbers with induction.

(4) Using the Seifert-van Kampen Theorem one easily checks that there is a G-pushout

7.1 Groups with Vanishing L^{2}-Betti Numbers 295

G ×_{G_{0}} EG_{0} ⟶ G ×_{G_{1}} EG_{1}
↓  ↓
G ×_{G_{2}} EG_{2} ⟶  EG

Now apply Theorem 6.54 (7) and Additivity (see Theorem 6.7 (4b)) to the associated long exact homology sequence for H_{∗}^{G}(−, N (G)).

(5) This will be proved in Theorem 7.4 (5) and (7).

(6) This follows from Theorem 6.54 (7) and Theorem 6.67 applied to the fibration BH → BG → BK and the obvious isomorphism ϕ: π_{1}(BG) → G.

(7) This is proved by Gaboriau [214, Theorem 6.8]. ⊓ ⊔ Next we prove a fibered version of Theorem 7.2.

Definition 7.3. Let d be a non-negative integer or d = ∞. Define BQ_{d} to be the class of groups G such that for any extension of groups 1 → H → K → G → 1 with b_{p}^{(2)}(H) < ∞ for p ≤ d the group K belongs to B_{d}. Define BF_{d} to be the class of groups G with the property that for any fibration F → E → BG for which F is path-connected and b_{p}^{(2)}(F̃) < ∞ holds for p ≤ d, we have b_{p}^{(2)}(Ẽ) = 0 for p ≤ d.

Theorem 7.4. Let d be a non-negative integer or d = ∞. Then

(1) BF_{d} ⊂ BQ_{d} ⊂ B_{d};

(2) If G contains a normal subgroup H which belongs to BQ_{d} or BF_{d} respectively , then G belongs to BQ_{d} or BF_{d} respectively;

(3) If G is the union of a directed system of subgroups {G_{i} | i ∈ I} such that each G_{i} belongs to BQ_{d} or BF_{d} respectively, then G belongs BQ_{d} or BF_{d} respectively;

(4) Suppose that there are groups G_{1} and G_{2} and group homomorphisms ϕ_{i}: G_{0} → G_{i} for i = 1, 2 such that ϕ_{1} and ϕ_{2} are injective, G_{0} belongs to BQ_{d−1} or BF_{d−1} respectively, G_{1} and G_{2} belong to BQ_{d} or BF_{d} respectively and G is the amalgamated product G_{1} ∗_{G_{0}} G_{2} with respect to ϕ_{1} and ϕ_{2}. Then G belongs to BQ_{d} or BF_{d} respectively;

(5) Suppose that BG has finite d-skeleton and that there is an injective endomorphism j: G → G whose image has finite index, but is not equal to G. Then G belongs to BF_{d};

(6) The class BF_{d} contains all infinite elementary amenable groups.

(7) The class BQ_{d} contains all infinite amenable groups.

Proof. (1) This is obvious.

(2) This follows for BQ_{d} from Theorem 7.2 (2) applied to p^{−1}(H) ⊂ L for a given extension 1 → K → L ⟶^{p} G → 1 with b_{p}^{(2)}(K) < ∞ for p ≤ d. For BF_{d}, the pullback construction for BH → BG yields a fibration F → E_{0} → BH and by hypothesis b_{p}^{(2)}(Ẽ_{0}) = 0 for p ≤ d. Since F is path-connected, both E_{0} and E are path-connected. We obtain an exact sequence 1 → π_{1}(E_{0}) ⟶^{π_{1}(i)} π_{1}(E) → G/H → 1 for i: E_{0} → E the inclusion. The π_{1}(E_{0})-space Ẽ_{0} is the restriction of the π_{1}(E) space Ẽ with π_{1}(i) : π_{1}(E_{0}) → π_{1}(E). Now we get b_{p}^{(2)}(Ẽ) = 0 for p ≤ d from Theorem 6.54 (6b).

(3) This follows for BQ_{d} from Theorem 7.2 (3). For BF_{d}, we can arrange that BG is the directed union of CW -subcomplexes BG_{i}. Let F → E_{i} → BG_{i} be the restriction of a given fibration F → E → BG with b_{p}^{(2)}(F̃) < ∞ for p ≤ d. By inspecting the maps between the long homotopy sequences associated to these fibrations one shows that π_{1}(E_{i}) → π_{1}(E) is injective for all i ∈ I. Obviously E is the union of the E_{i}. Since any compact subset of BG is contained in a finite CW -subcomplex and hence in one of the BG_{i}, any compact subset of E is contained in one of the E_{i}. This implies

H_{p}^{π_{1}(E)}(Ẽ; N (π_{1}(E)))
= colim_{i∈I} H_{p}^{π_{1}(E)}(π_{1}(E) ×_{π_{1}(E_{i})} Ẽ_{i}; N (π_{1}(E))).  (7.5)

Because of Theorem 6.13 (1) about dimension and colimits and Theorem 6.54 (7) it suffices to show b_{p}^{(2)}(Ẽ_{i}) = 0 for p ≤ d and i ∈ I. But this follows from the assumption G_{i} ∈ BF_{d}.

(4) This follows for BQ_{d} from Theorem 7.2 (4) since for an epimorphism p: K → G we can write K as the amalgamated product p^{−1}(G_{1}) ∗_{p^{−1}(G_{0})} p^{−1}(G_{2}). The proof for BF_{d} is analogous (using [326, Lemma 1.26 on page 19]).

(5) Fix an integer n ≥ 1. Put G_{n} = im(j^{n}). If k is the index of im(j) in G, then k^{n} is the index of G_{n} in G. Let F → E → BG be a fibration with b_{p}^{(2)}(F̃) < ∞ for p ≤ d. We get a k^{n}-sheeted covering BG_{n} → BG. The pullback construction yields a fibration F → E_{n} → BG_{n} together with a k^{n}-sheeted covering E_{n} → E. We conclude from Theorem 6.54 (6b)

b_{p}^{(2)}(Ẽ) = \frac{b_{p}^{(2)}(Ẽ_{n})}{k^{n}}.  (7.6)

Let i_{p} be the number of p-cells in BG. Since G_{n} is isomorphic to G, we get from the Leray-Serre spectral sequence applied to E_{n} → BG_{n} and Additivity (see Theorem 6.7 (4b))

b_{p}^{(2)}(Ẽ_{n}) ≤ ∑_{q=0}^{p} b_{q}^{(2)}(F̃) · i_{p−q}.  (7.7)

Equations (7.6) and (7.7) imply

b_{p}^{(2)}(Ẽ) = \frac{∑_{q=0}^{p} b_{q}^{(2)}(F̃) · i_{p−q}}{k^{n}} .  (7.8)

Since k > 1 and (7.8) holds for all n ≥ 1 and ∑_{q=0}^{p} b_{q}^{(2)}(F̃) · i_{p−q} is finite for p ≤ d by assumption, assertion (5) follows.

(6) We first show G ∈ BF_{∞} provided that G is infinite and locally finite. Let F → E → BG be a fibration with b_{p}^{(2)}(F̃) < ∞ for p ≤ d. We still have (7.5), if we take {G_{i} | i ∈ I} as the system of finite subgroups of G. From Theorem 6.54 (7) and (6b) we conclude

b_{p}^{(2)}(π_{1}(E) ×_{π_{1}(E_{i})} Ẽ_{i}; N (π_{1}(E)) = b_{p}^{(2)}(Ẽ_{i}) = \frac{b_{p}^{(2)}(F̃)}{|G_{i}|}.

Since G is infinite and locally finite, |G_{i}| becomes arbitrary large for appropriate i ∈ I. Additivity (see Theorem 6.7 (4b)) and Theorem 6.13 (2) about dimension and colimits implies together with (7.5) that b_{p}^{(2)}(Ẽ) = 0 for p ≤ d.

We want to show that the class of elementary amenable groups is contained in FIN ∪ BF_{d}, where FIN is the class of finite groups. By Lemma 10.40 it suffices to show the following two claims. i.) If all finitely generated subgroups of G belong to FIN ∪ BF_{d}, then G ∈ FIN ∪ BF_{d}, and ii.) for any extension 1 → H → G → K → 1, for which H ∈ FIN ∪ BF_{d} and K contains ℤ^{n} as normal subgroup of finite index for some n ≥ 0, we have G ∈ FIN ∪ BF_{d}. We begin with i.). We have already shown for locally finite G that G belongs to FIN ∪ BF_{d}. It remains to treat the case where G is not locally finite. Then G can be written as the union of the directed system of its infinite finitely generated subgroups G_{i}. By induction hypothesis each G_{i} belongs to BF_{d}. Then G ∈ BF_{d} by assertion (3). Finally we prove ii.) If H is finite, G contains ℤ^{n} for some n ≥ 0 as normal subgroup of finite index and hence belongs to FIN ∪ BF_{d} by assertions (2) and (5). It remains to treat the case, where H is infinite and hence by induction hypothesis belongs to BF_{d}. If F_{0} is the fiber of the composition E → BG → BK, then we obtain a fibration F → F_{0} → BH. Hence b_{p}^{(2)}(F̃_{0}) = 0 for p ≤ d. From Lemma 6.66 we conclude b_{p}^{(2)}(Ẽ) = 0 for p ≤ d.

(7) Because of assertion (3) it suffices to treat the case of a finitely generated (and hence countable) amenable group. This case follows from [214, Theorem 6.6]. This finishes the proof of Theorem 7.4. ⊓ ⊔

The next question is related to Question 1.95 and arises from Theorem 7.4.

Question 7.9. (Vanishing of L^{2}-Betti numbers of groups and epimorphism of groups). Is B_{d} = BQ_{d} = BF_{d}?

7.1.2 The Vanishing of the L^{2}-Betti Numbers of Thompson’s Group

Finally we explain the following observation about Thompson’s group F. It is the group of orientation preserving dyadic PL-automorphisms of [0, 1] where dyadic means that all slopes are integral powers of 2 and the break points are contained in ℤ[1/2]. It has the presentation

F = ⟨x_{0}, x_{1}, x_{2}, … | x_{i}^{−1}x_{n}x_{i} = x_{n+1} for i < n⟩.

This group has some very interesting properties. Its classifying space BF is of finite type [70] but is not homotopy equivalent to a finite dimensional CW - complex since F contains ℤ^{n} as subgroup for all n ≥ 0 [70, Proposition 1.8]. It is not elementary amenable and does not contain a subgroup which is free on two generators [66], [85]. Hence it is a very interesting question whether F is amenable or not. We conclude from Theorem 7.2 (1) that a necessary condition for F to be amenable is that b_{p}^{(2)}(F) vanishes for all p ≥ 0. This motivates the following result.

Theorem 7.10. (L^{2}-Betti numbers of Thompson’s group). All L^{2}-Betti numbers b_{p}^{(2)}(F) of Thompson’s group F vanish. ⊓ ⊔

Proof. There is a subgroup F_{1} ⊂ F together with a monomorphism Φ: F_{1} → F_{1} such that F_{1} is isomorphic to F and F is the HNN-extension of F_{1} with respect to Φ with one stable letter [70, Proposition 1.7 on page 370]. From the topological description of HNN-extensions [350, page 180] we conclude that F is the fundamental group of the mapping torus T_{BΦ} of the map BΦ: BF_{1} → BF_{1} induced by Φ. The inclusion BF_{1} → T_{BΦ} induces on the fundamental groups the inclusion of F_{1} in F. One easily checks that the cellular ℤF -chain complex of the universal covering T̃_{BΦ} of T_{BΦ} is the mapping cone of a certain ℤF -chain map from ℤF ⊗_{ℤF_{1}} C_{∗}(EF_{1}) to itself. Since ℤF is free over ℤF_{1}, we conclude for p ≥ 1

H_{p}(ℤF ⊗_{ℤF_{1}} C_{∗}(EF_{1})) = ℤF ⊗_{ℤF_{1}} H_{p}(C_{∗}(EF_{1})) = 0.

This implies H_{p}(T̃_{BΦ}; ℤ) = 0 for p ≥ 2. Hence T_{BΦ} is a model for BF. Now the claim follows from Theorem 1.39. ⊓ ⊔

7.2 Euler Characteristics of Groups

We have introduced the L^{2}-Euler characteristic χ^{(2)}(G) of a group already in Definition 6.79. It encompasses the rational valued virtual Euler characteristic of Wall (see Remark 6.81). We have related it to the equivariant Euler characteristic of E(G, FIN) provided that there is a finite G-CW -model for E(G, FIN). Namely, Lemma 6.93 implies

χ^{(2)}(G) = ch_{{1}}^{G}(χ^{G}(E(G, FIN))),  (7.11)

where ch_{{1}}^{G}: A(G) → ℚ sends [G/H] to |H|^{−1}.

Lemma 7.12. If G belongs to B_{∞}, then χ^{(2)}(G) = 0.

Recall that we have given criterions for G ∈ B_{∞} in Theorem 7.2. Now it becomes clear why it is worth while to extend the classical notion of the Euler characteristic χ(G) := χ(BG) for groups G with finite BG to arbitrary groups. For instance it may very well happen for a group G with finite BG that G contains a normal group H which is not even finitely generated and has in particular no finite model for BH and which belongs to B_{∞} (for instance, H is amenable). Then the classical Euler characteristic is not defined any more for H, but we can still conclude that the classical Euler characteristic of G vanishes.

The standard product and amalgamation formulas for the classical Euler characteristic carry over to the L^{2}-Euler characteristic. Namely, let G_{0}, G_{1} and G_{2} be groups with h^{(2)}(G_{i}) < ∞ for i = 0, 1, 2 and ϕ_{i}: G_{0} → G_{1} be injective group homomorphisms for i = 1, 2. Then the direct product G_{1} ×G_{2} and the amalgamated product G_{1} ∗_{G_{0}} G_{2} with respect to the homomorphisms ϕ_{1} and ϕ_{2} satisfy

h^{(2)}(G_{1} ∗_{G_{0}} G_{2}) < ∞;
χ^{(2)}(G_{1} ∗_{G_{0}} G_{2}) = χ^{(2)}(G_{1}) + χ^{(2)}(G_{2}) − χ^{(2)}(G_{0});  (7.13)
h^{(2)}(G_{1} × G_{2}) < ∞;
χ^{(2)}(G_{1} × G_{2}) = χ^{(2)}(G_{1}) · χ^{(2)}(G_{1}).  (7.14)

This follows from Theorem 6.80 (2) and (6). More information about the classical Euler characteristic and the virtual Euler characteristic of a group can be found in [69, Chapter IX].

7.3 Deficiency of Groups

7.3.1 Survey on Deficiency of Groups

Definition 7.15 (Deficiency). Let G be a finitely presented group. Define its deficiency def(G) to be the maximum g(P) − r(P), where P runs over all presentations P of G and g(P) is the number of generators and r(P) is the number of relations of a presentation P.

Next we reprove the well-known fact that the maximum appearing in Definition 7.15 does exist.

Lemma 7.16. Let G be a group with finite presentation

P = ⟨s_{1}, s_{2}, …, s_{g} | R_{1}, R_{2}, …, R_{r}⟩

Let ϕ: G → K be any group homomorphism. Then g(P) − r(P) ≤ 1 − b_{0}^{(2)}(K ×_{ϕ} EG; N (K)) + b_{1}^{(2)}(K ×_{ϕ} EG; N (K))

− b_{2}^{(2)}(K ×_{ϕ} EG; N (K)).

Proof. Given a presentation P with g generators and r relations, let X be the associated finite 2-dimensional CW -complex. It has one 0-cell, g 1-cells, one for each generator, and r 2-cells, one for each relation. The attaching map of the 2-cell associated to a relation is a map from S^{1} to the 1-skeleton, which is a wedge of g 1-dimensional spheres, and given by the word defining the relation. There is an obvious isomorphism from π_{1}(X) to G so that we can choose a map f: X → BG which induces an isomorphism on the fundamental groups. It induces a 2-connected K-equivariant map \overline{f}: K ×_{ϕ} X̃ → K ×_{ϕ} ẼG. Theorem 6.54 (1a) implies

b_{p}^{(2)}(K ×_{ϕ} X̃; N (K)) = b_{p}^{(2)}(K ×_{ϕ} EG; N (K)) for p = 0, 1; (7.17)
b_{2}^{(2)}(K ×_{ϕ} X̃; N (K)) ≥ b_{2}^{(2)}(K ×_{ϕ} EG; N (K)).  (7.18)

We conclude from the L^{2}-Euler-Poincaré formula (see Theorem 6.80 (1)) and from (7.17) and (7.18)

g − r = 1 − χ^{(2)}(K ×_{ϕ} X̃; N (K))
= 1 − b_{0}^{(2)}(K ×_{ϕ} X̃; N (K)) + b_{1}^{(2)}(K ×_{ϕ} X̃; N (K))
−b_{2}^{(2)}(K ×_{ϕ} X̃; N (K))
≤ 1 − b_{0}^{(2)}(K ×_{ϕ} EG; N (K)) + b_{1}^{(2)}(K ×_{ϕ} EG; N (K))
−b_{2}^{(2)}(K ×_{ϕ} EG; N (K)).⊓ ⊔

Example 7.19. We give some examples of groups, where the deficiency is realized by the “obvious” presentation.

The free group F_{g} of rank g has the obvious presentation ⟨s_{1}, s_{2}, …, s_{g} | ∅⟩ and its deficiency is realized by this presentation, namely def(F_{g}) = g. This follows from Lemma 7.16 because of b_{0}^{(2)}(F_{g}) = 0 and b_{1}^{(2)}(F_{g}) = g − 1. One also can apply the analog of Lemma 7.16 for the classical Betti numbers instead of the L^{2}-Betti numbers since b_{0}(F_{g}) = 1 and b_{1}(F_{g}) = g.

If G is a finite group, def(G) ≤ 0 by Lemma 7.16 because we get for the classical Betti numbers b_{0}(G) = 1 and b_{1}(G) = 0 or because we get for the L^{2}-Betti numbers b_{0}^{(2)}(G) = |G|^{−1} and b_{1}^{(2)}(G) = 0. The deficiency of a cyclic group ℤ/n is 0, the obvious presentation ⟨s | s^{n}⟩ realizes the deficiency. It is not hard to check using homology with coefficients in the finite field 𝔽_{p} of prime order p that the deficiency of ℤ/n×ℤ/n is −1. The obvious presentation ⟨s, t | s^{n}, t^{n}, [s, t]⟩ realizes the deficiency, where [s, t] denotes the commutator sts^{−1}t^{−1}.

The inequality in Lemma 7.16 is actually an equality and in particular def(G) = 1 − χ(BG) if BG is a finite 2-dimensional CW -complex. If G is a torsion-free one-relator group, the 2-dimensional CW -complex associated with any presentation with one relation and g generators is aspherical and hence BG is a finite 2-dimensional CW -complex [350, chapter III §§9 -11] and G has deficiency g − 1.

We conjecture that for a torsion-free group having a presentation with g ≥ 2 generators and one non-trivial relation

b_{2}^{(2)}(G) = 0;
b_{1}^{(2)}(G) = def(G) − 1 = g − 2

holds (compare [237, page 235]). This would follow from the strong Atiyah Conjecture 10.2, which says that the L^{2}-Betti numbers of the universal covering of a finite CW -complex with torsion-free fundamental group are integers, by the following argument. Namely, the kernel of the second differential of the L^{2}-chain complex of BG is a proper Hilbert N (G)-submodule of l^{2}(G) so that its dimension b_{2}^{(2)}(G) is less than one and hence by the Atiyah Conjecture zero. Since G must be infinite and hence b_{0}^{(2)}(G) = 0 (see Theorem 6.54 (8b), the Euler-Poincaré formula (see Theorem 6.80 (1)) implies b_{1}^{(2)}(G) = g − 2.

The following result is a direct consequence of [177, Theorem 2.5]. It is proved using homology with coefficients in ℤ/2.

Theorem 7.20. Let M be a connected compact orientable 3-manifold with fundamental group π and prime decomposition

M = M_{1}#M_{2}# … #M_{r}.

Let s(M) be the number of prime factors M_{i} with non-empty boundary and t(M) be the number of prime factors which are S^{2}-bundles over S^{1}. Denote by χ(M) the Euler characteristic. Then

def(π_{1}(M)) = dim_{ℤ/2}(H_{1}(π; ℤ/2)) − dim_{ℤ/2}(H_{2}(π; ℤ/2))
= s(M) + t(M) − χ(M).⊓ ⊔

Example 7.21. One may expect that the deficiency is additive under free products. This is not true as the following example, which is taken from [264, Theorem 3 on page 162], shows. It plays an important role in the construction of a counterexample up to homotopy of the Kneser Conjecture in dimension four [298]. There, a closed connected orientable smooth 4-manifold M is constructed whose fundamental group is the free product of two nontrivial groups such that M is not homotopy equivalent to M_{0}#M_{1} unless M_{0} or M_{1} is homeomorphic to S^{4}. We mention that a stable version of the Kneser Conjecture remains true in dimension four [299], where stable means that one has to allow connected sums with copies of S^{2} × S^{2}.

Suppose that m_{i}, r_{i}, n_{i} and q_{i} for i = 0, 1 are integers satisfying

r_{i} > 1, r_{i}^{m_{i}} − 1 = n_{i}q_{i}, r_{i} ≡ 1 mod n_{i}, (m_{i}, n_{i}) ≠ 1, (q_{0}, q_{1}) = 1.

Then the group

G = (ℤ/m_{0} × ℤ/n_{0}) ∗ (ℤ/m_{1} × ℤ/n_{1})

has the “obvious” presentation

G = ⟨a_{0}, b_{0}, a_{1}, b_{1} | a_{0}^{m_{0}}, b_{0}^{n_{0}}, [a_{0}, b_{0}], a_{1}^{m_{1}}, b_{1}^{n_{1}}, [a_{1}, b_{1}]⟩.

But its deficiency is not realized by this presentation. Namely its deficiency is −1 and is realized by the following presentation

G = ⟨a_{0}, b_{0}, a_{1}, b_{1} |

a_{0}^{m_{0}} = 1, [a_{0}, b_{0}] = b_{0}^{r_{0}−1}, a_{1}^{m_{1}} = 1, [a_{1}, b_{1}] = b_{1}^{r_{1}−1}, b_{0}^{n_{0}} = b_{1}^{n_{1}}⟩.

To show that this is indeed a presentation of G, it suffices to show that the relation b_{0}^{n_{0}} = 1 follows from the other relations. We start by proving

inductively for k = 1, 2, … the relation a_{i}^{k}b_{i}a_{i}^{−k} = b_{i}^{−r_{1}^{k}} for i = 0, 1. The induction step follows from the calculation

a_{i}^{k+1}b_{i}a_{i}^{−(k+1)} = a_{i}a_{i}^{k}b_{i}a_{i}^{−k}a_{i}^{−1} = a_{i}b_{i}^{r_{1}^{k}}a^{−1} = (a_{i}b_{i}a_{i}^{−1})^{r_{i}^{k}} = (b_{i}^{r_{i}})^{r_{i}^{k}} = b_{i}^{r_{i}^{k+1}}.

This implies for k = m_{i} and i = 0, 1

(b_{i}^{n_{i}})^{q_{i}} = b_{i}^{r_{i}^{m_{i}} −1} = 1.

Since b_{0}^{n_{0}} = b_{1}^{n_{1}} holds, we conclude

(b_{0}^{n_{0}})^{q_{0}} = (b_{0}^{n_{0}})^{q_{1}} = 1.

Since q_{0} and q_{1} are prime, we get b_{0}^{n_{0}} = 1.

Notice that groups appearing in the Example 7.21 above contain torsion. It may still be true that the deficiency is additive under free products of torsionfree groups.

Finally we mention the following result [30, Theorem 2] which is in a certain sense complementary to our results (see Lemma 7.22 and Theorem 7.25). If G is a finitely presented group with def(G) ≥ 2, then G can be written as an amalgamated product G = A ∗_{C} B, where A, B and C are finitely generated, C is proper subgroup of both A and B and has index greater than two in A or B. In particular G contains a free subgroup of rank 2 and is not amenable. This implies that an amenable finitely presented group has deficiency less or equal to one (see also [28], [160, Corollary 2.5’]). This also follows from Theorem 7.25. We mention that not every group of this particular shape A ∗_{C} B has deficiency ≥ 2, take for example ℤ ∗_{3·ℤ} ℤ = ⟨s, t | t^{3} = s^{3}⟩ which has deficiency 1. This follows from Lemma 7.22 (1) and Theorem 7.2 (4).

Another test for bounds on deficiencies is given in [349] using Fox ideals.

7.3.2 Applications of L^{2}-Betti Numbers to Deficiency and to Signatures of 4-Manifolds

Lemma 7.22. Let G be a finitely presented group and ϕ: G → K a homomorphism such that b_{1}^{(2)}(K ×_{ϕ} EG; N (G)) = 0. Then

(1) def(G) ≤ 1;

(2) Let M be a closed oriented 4-manifold with G as fundamental group.

Then

| sign(M)| ≤ χ(M);

Proof. (1) Follows directly from Lemma 7.16.

(2) By the L^{2}-Signature Theorem (see [9]) applied to the regular covering \overline{M} → M associated to ϕ, the signature σ(M) is the difference of the von Neumann dimensions of two complementary subspaces of the space of L^{2}- integrable harmonic smooth 2-forms H_{(2)}^{p}(\overline{M}) and hence

| sign(M)| ≤ dim_{N (K)}(H_{(2)}^{p}(\overline{M})).

We conclude from the L^{2}-Hodge-de Rham Theorem 1.59 and Lemma 6.53

| sign(M)| ≤ b_{2}^{(2)}(\overline{M}, N (K)).  (7.23)

We get from the assumption and Poincaré duality (see Theorem 1.35 (3) together with Lemma 6.53) that b_{p}^{(2)}(K ×_{ϕ} EG; N (G)) = 0 for p = 1, 3. The Euler-Poincaré formula (see Theorem 1.35 (2)) implies

χ(M) = ∑_{j=0}^{2} b_{2j}^{(2)}(\overline{M}, N (K)).  (7.24)

Now assertion (2) follows from (7.23) and (7.24). ⊓ ⊔

Theorem 7.25. Let 1 → H ⟶^{i} G ⟶^{q} K → 1 be an exact sequence of infinite groups. Suppose that G is finitely presented and one of the following conditions is satisfied.

(1) b_{1}^{(2)}(H) < ∞;

(2) The ordinary first Betti number of H satisfies b_{1}(H) < ∞ and K belongs to B_{1};

Then

(i) def(G) ≤ 1; (ii) Let M be a closed oriented 4-manifold with G as fundamental group.

Then

| sign(M)| ≤ χ(M).

Proof. If condition (1) is satisfied, then b_{p}^{(2)}(G) = 0 for p = 0, 1 by Theorem 7.2 (7), and the claim follows from Lemma 7.22.

Suppose that condition (2) is satisfied. There is a spectral sequence converging to H_{p+q}^{K}(K ×_{q} EG; N (K)) with E^{2}-term

E_{p,q}^{2} = Tor_{p}^{ℂK}(H_{q}(BH; ℂ), N (K))

[518, Theorem 5.6.4 on page 143]. Since H_{q}(BH; ℂ) is ℂ with the trivial Kaction for q = 0 and finite dimensional as complex vector space by assumption for q = 1, we conclude dim_{N (K)}(E_{p,q}^{2}) = 0 for p + q = 1 from the assumption b_{1}^{(2)}(K) = 0 and Lemma 6.33 (2). This implies b_{1}^{(2)}(K ×_{q} EG; N (K)) = 0 and the claim follows from Lemma 7.22. ⊓ ⊔

Theorem 7.25 generalizes results in [162], [279], where also some other information is given. See also [250], [297]. We mention the result of Hitchin [263] that a connected closed oriented smooth 4-manifold which admits an Einstein metric satisfies the stronger inequality | sign(M)| ≤ \frac{2}{3} · χ(M).

7.4 Group Automorphisms and L^{2}-Torsion

In this section we explain that for a group automorphism f: G → G the L^{2}- torsion applied to the (G ⋊_{f} ℤ)-CW -complex E(G ⋊_{f} ℤ) gives an interesting new invariant, provided that G is of det ≥ 1-class and satisfies certain finiteness assumptions, for instance, that there is a finite G-CW -model for EG or more generally for E(G, FIN). We will investigate the basic properties of this invariant.

7.4.1 Automorphisms of Groups G with Finite Models for BG

Let G be a group. We assume that there is a finite CW -model for its classifying space BG and that G is of det ≥ 1-class (see Definition 3.112). By Theorem 13.3 (2) G is of det ≥ 1-class if G belongs to the class G which will be dealt with in Subsection 13.1.3. It contains all residually amenable groups and in particular all residually finite groups.

Suppose that f: G → G is an automorphism. Let G_{f} := G ⋊_{f} ℤ be the semidirect product of G and ℤ with respect to the automorphism f. By assumption BG has a finite CW -model. We pick one. Let Bf: BG → BG be the map induced by f which is up to homotopy uniquely determined by the property that π_{1}(f) is conjugate to f under the identification G ≅ π_{1}(BG). The mapping torus T_{Bf} is a finite CW -model for B(G ⋊_{f} ℤ), since there is a fibration BG → T_{Bf} → S^{1}. We conclude from Theorem 1.39 that

7.4 Group Automorphisms and L^{2}-Torsion 305

b_{p}^{(2)}({ \atop B(G ⋊_{f} ℤ)}) = 0 for p ≥ 0. Since G is of det ≥ 1-class by assumption, G ⋊_{f} ℤ is of det ≥ 1-class by Theorem 13.3 (3). Notice that the construction of B(G ⋊_{f} ℤ) is unique up to homotopy. We conclude from Lemma 13.6 that { \atop B(G ⋊_{f} ℤ)} is a det-L^{2}-acyclic finite free (G ⋊_{f} ℤ)-CW -complex

and the L^{2}-torsion ρ^{(2)}({ \atop B(G ⋊_{f} ℤ)}) is well-defined and depends only on the automorphism f: G → G.

Definition 7.26 (L^{2}-torsion of group automorphisms). Let f: G → G be a group automorphism. Suppose that there is a finite CW -model for BG and G is of det ≥ 1-class. Define the L^{2}-torsion of f by

ρ^{(2)}(f: G → G) := ρ^{(2)}({ \atop B(G ⋊_{f} ℤ)}) ∈ ℝ.

Next we present the basic properties of this invariant. Notice that its behaviour is similar to the Euler characteristic χ(G) := χ(BG).

Theorem 7.27. Suppose that all groups appearing below have finite CW - models for their classifying spaces and are of det ≥ 1-class.

(1) Suppose that G is the amalgamated product G_{1} ∗_{G_{0}} G_{2} for subgroups G_{i} ⊂ G and the automorphism f: G → G is the amalgamated product f_{1} ∗_{f_{0}} f_{2} for automorphisms f_{i}: G_{i} → G_{i}. Then

ρ^{(2)}(f) = ρ^{(2)}(f_{1}) + ρ^{(2)}(f_{2}) − ρ^{(2)}(f_{0});

(2) Let f: G → H and g: H → G be isomorphisms of groups. Then

ρ^{(2)}(f ∘ g) = ρ^{(2)}(g ∘ f).

In particular ρ^{(2)}(f) is invariant under conjugation with automorphisms;

(3) Suppose that the following diagram of groups

1 ⟶ G_{1} ⟶ G_{2} ⟶ G_{3} ⟶ 1
f_{1}↓  ^{f_{2}}↓  ^{id}↓
1 ⟶ G_{1} ⟶ G_{2} ⟶ G_{3} ⟶ 1

commutes, has exact rows and its vertical arrows are automorphisms. Then

ρ^{(2)}(f_{2}) = χ(BG_{3}) · ρ^{(2)}(f_{1});

(4) Let f: G → G be an automorphism of a group. Then for all integers

n ≥ 1

ρ^{(2)}(f^{n}) = n · ρ^{(2)}(f);

(5) Suppose that G contains a subgroup G_{0} of finite index [G : G_{0}]. Let f: G → G be an automorphism with f(G_{0}) = G_{0}. Then

ρ^{(2)}(f) = \frac{1}{[G : G_{0}]} · ρ^{(2)}(f|_{G_{0}});

(6) Let f: G → G be an automorphism of a group G. Then ρ^{(2)}(f) depends only on the map H_{p}^{(2)}(B̃f): H_{p}^{(2)}(B̃G) → H_{p}^{(2)}(B̃G) induced by f on the L^{2}-homology of the universal covering of BG. More precisely,

ρ^{(2)}(f) = ∑_{p≥0}(−1)^{p} · ln (det_{G⋊_{f} ℤ} (\overline{H_{p}^{(2)}(B̃f)})) ,

where j_{∗}H_{p}^{(2)}(B̃G) is the finitely generated Hilbert N (G ⋊_{f} ℤ)-Hilbert module, which is obtained from the N (G)-Hilbert module H_{p}^{(2)}(B̃G) by induction with the canonical inclusion j: G → G ⋊_{f} ℤ, and the mor-

phism \overline{H_{p}^{(2)}(B̃f)}: j_{∗}H_{p}^{(2)}(B̃G) → j_{∗}H_{p}^{(2)}(B̃G) is the completion of the map ℂ[G ⋊_{f} ℤ] ⊗_{ℂG} H_{p}^{(2)}(B̃G) → ℂ[G ⋊_{f} ℤ] ⊗_{ℂG} H_{p}^{(2)}(B̃G) which sends γ⊗u to γ⊗u−γt⊗H_{p}^{(2)}(B̃f)(u) for t ∈ ℤ a fixed generator, γ ∈ ℂ[G⋊_{f} ℤ] and u ∈ H_{p}^{(2)}(B̃G);

(7) We have ρ^{(2)}(f) = 0 if G satisfies one of the following conditions: (a) All L^{2}-Betti numbers of the universal covering of BG vanish;

(b) G contains an amenable infinite normal subgroup.

Proof. (1) One constructs finite CW -models BG_{i} for i = 0, 1, 2 and BG such that BG_{i} ⊂ BG is a CW -subcomplex for i = 0, 1, 2 and BG = BG_{1} ∪ BG_{2} and BG_{0} = BG_{1} ∩ BG_{2} and the inclusion BG_{i} → BG induces on the fundamental groups the inclusions G_{i} → G for i = 0, 1, 2. Then one constructs self-homotopy equivalences Bf_{i}: BG_{i} → BG_{i} for i = 0, 1, 2 and Bf: BG → BG such that Bf restricts to Bf_{i} on BG_{i} and Bf and Bf_{i} induce on the fundamental groups f and f_{i} for i = 0, 1, 2. Then the mapping torus T_{Bf} contains T_{Bf_{i}} for i = 0, 1, 2 as subcomplex, T_{Bf} = T_{Bf_{1}} ∪T_{Bf_{2}} and T_{Bf_{0}} = T_{Bf_{1}} ∩ T_{Bf_{2}} and the inclusion T_{Bf_{i}} → T_{Bf} induces an injection on the fundamental groups for i = 0, 1, 2. The sum formula (see Theorem 3.96)

(2) implies

ρ^{(2)}(T̃_{Bf}) = ρ^{(2)}({ \atop T_{Bf_{1}}}) + ρ^{(2)}({ \atop T_{Bf_{2}}}) − ρ^{(2)}({ \atop T_{Bf_{0}}}).

(2) This follows from the fact that for maps u: X → Y and v: Y → X of CW -complexes the mapping tori T_{u∘v} and T_{v∘u} are homotopy equivalent [326, (7.31) on page 129].

(3) There is an induced fibration B(G_{1} ⋊_{f_{1}} ℤ) → B(G_{2} ⋊_{f_{2}} ℤ) → BG_{3} such that the inclusion of the fiber into the total space induces the obvious injection on the fundamental groups. Now apply the fibration formula (see Theorem 3.100 and Remark 3.102).

(4) Since there is a n-sheeted covering T_{(Bf)^{n}} → T_{Bf}, the claim follows from multiplicativity of L^{2}-torsion under finite coverings (see Theorem 3.96 (5)).

(5) There is a finite covering with [G : G_{0}]-sheets T_{Bf_{0}} → T_{Bf} since there is a bijection from G/G_{0} to (G ⋊_{f} ℤ)/(G_{0} ⋊_{f|_{G_{0}}} ℤ). Now the assertion follows from multiplicativity of L^{2}-torsion under finite coverings (see Theorem 3.96 (5)).

(6) This follows from Theorem 3.106.

(7) This follows from assertion (6) and Theorem 7.2 (1) and (2). This finishes the proof of Theorem 7.27. ⊓ ⊔

7.4.2 Automorphisms of Surfaces

Let S be a compact connected orientable 2-dimensional manifold, possibly with boundary. Let f: S → S be an orientation preserving homeomorphism. The mapping torus T_{f} is a compact connected orientable 3-manifold whose boundary is empty or a disjoint union of 2-dimensional tori. Then T_{f} is an irreducible Haken manifold with infinite fundamental group and incompressible boundary if S is different from S^{2} and D^{2}. In this case T_{f} satisfies Thurston’s Geometrization Conjecture, i.e. there is a maximal family of embedded incompressible tori, which are pairwise not isotopic and not boundary parallel, such that it decomposes T_{f} into pieces, which are Seifert or hyperbolic. Let M_{1}, M_{2}, …, M_{r} be the hyperbolic pieces. They all have finite volume vol(M_{i}). (We have explained the notions and facts above in Section 4.1). Then Theorem 4.3 shows

Theorem 7.28. If S is S^{2}, D^{2}, or T^{2}, then ρ^{(2)}(f) = 0. Otherwise we get

ρ^{(2)}(π_{1}(f): π_{1}(S) → π_{1}(S)) = \frac{−1}{6π} · ∑_{i=1}^{r} vol(M_{i}).

Let S be a closed orientable hyperbolic surface and f: S → S be a selfhomeomorphism. It is called irreducible if f is not homotopic to an automorphism which leaves some essential closed 1-dimensional submanifold invariant. Essential means that none of the components is nullhomotopic in S. It is called periodic if it is homotopic (or, equivalently, isotopic) to a map g: S → S for which there is a positive integer n with g^{n} = id. The notion pseudo-Anosov in terms of transverse singular foliations can be found for instance in [96, page 95]. It is important to know that the following statements for an irreducible selfhomeomorphism f: S → S are equivalent: i.) f is pseudo-Anosov, ii.) f is not periodic and iii.) the mapping torus T_{f} is hyperbolic [96, Theorem 6.3], [369, Theorem 3.6 on page 47, Theorem 3.9 on page 50]. We know from Theorem 4.3 that ρ^{(2)}(π_{1}(f)) is −\frac{1}{6π} · vol(T_{f}) and hence different from zero, provided that T_{f} is hyperbolic. If T_{f} is not hyperbolic, f must be, up to homotopy, periodic and hence ρ^{(2)}(T̃_{f}) = 0 by Theorem 7.27 (4). Hence f is pseudo-Anosov if and only if ρ^{(2)}(f) < 0, and f is periodic if and only if ρ^{(2)}(f) = 0.

Recall that we can read off ρ^{(2)}(f) := ρ^{(2)}(T̃_{f}) from H_{1}^{(2)}(\overline{f}) by Theorem 3.106 and thus, because of Theorem 4.9, from the map induced by f on the fundamental group π_{1}(F). So we have in principle a procedure to decide whether f is pseudo-Anosov by inspecting the map induced on the fundamental group.

7.4.3 A Combinatorial Approach for the L^{2}-Torsion of an Automorphism of a Finitely Generated Free Group

In this Subsection we discuss how the combinatorial approach to L^{2}-torsion of Subsection 3.7 specializes in the case of an automorphism f: F → F of the free group F on r letters s_{1}, s_{2}, …, s_{r}. Write G = F ⋊_{f} ℤ for the semidirect product associated to f. Let t ∈ ℤ be a generator and denote the corresponding element in G also by t. Define a (r, r)-matrix A over ℤF by

A = (\frac{∂}{∂s_{j}} f(s_{i}))_{1≤i,j≤r} ,

where \frac{∂}{∂s_{j}} denotes the Fox derivative. Choose a real number K > 0 which is greater than or equal to the operator norm of the morphism

r_{I−tA}: ⨁_{i=1}^{r} l^{2}(G) → ⨁_{i=1}^{r} l^{2}(G)

given by right multiplication with (I − tA). A possible choice for K is

K = \sqrt{(2r − 1)r} · max\{||1 − tA_{i,j}||_{1} | 1 ≤ i, j ≤ r\},

where || ∑_{g∈F ⋊_{f} ℤ} λ_{g} · g||_{1} is defined by ∑_{g∈G} |λ_{g}|. Denote by A^{∗} the matrix obtained from A by transposing and applying the standard involution ℤF → ℤF, which sends ∑_{u∈F} λ_{u} · u to ∑_{u∈F} λ_{u} · u^{−1}, to each entry. Denote by

tr_{ℤG}: ℤG → ℤ, ∑_{g∈G} λ_{g} · g ↦ λ_{e}

the standard trace on ℤG, where λ_{e} is the coefficient of the unit element e in G. It extends to square matrices over ℤG by taking the sum of the traces of the diagonal entries. Define

c(A, K)_{p} = tr_{ℤG} ((1 − K^{−2} · (1 − tA)(1 − A^{∗}t^{−1}))^{p}) .

Theorem 7.29. In the setting above the sequence c(A, K)_{p} is a monotone decreasing sequence of non-negative real numbers, and the L^{2}-torsion of f satisfies

ρ^{(2)}(f) = − r · ln(K) + \frac{1}{2} · ∑_{p=1}^{∞} \frac{1}{p} · c(A, K)_{p} ≤ 0.

Proof. Choose ⋁_{i=1}^{r} S^{1} with the obvious base point x as model for BF. Realize f as an endomorphism Bf: BF → BF respecting the base point. Fix a base point x̃ ∈ B̃F which is sent under the projection B̃F → BF to x. Choose a lift B̃f: B̃F → B̃F uniquely determined by the property that it sends x̃ to itself. In the sequel we identify F, π_{1}(BF) and the group of deck transformations of B̃F with respect to the given base points x and x̃. Then B̃f is (f: F → F)-equivariant. The cellular ℤF -chain complex of B̃F looks like

⨁_{i=1}^{r} ℤF \underrightarrow{∑_{i=1}^{r} r_{s_{i}−1}} ℤF

where r_{s_{i}−1} : ℤF → ℤF sends u to u(s_{1} − 1). The matrix A defines a (ℤf: ℤF → ℤF)-equivariant homomorphism

r̂_{A}: ⨁_{i=1}^{r} ℤF → ⨁_{i=1}^{r} ℤF,

(∑_{u∈F} λ_{u,j} · u)_{1≤j≤r} ↦ (∑_{u∈F} ∑_{j} λ_{u,j} · f(u)A_{j,k})_{1≤k≤r}

and analogously for the unit (r, r)-matrix I. The (ℤf: ℤF → ℤF)-equivariant chain map C_{∗}(B̃f): C_{∗}(B̃F) → C_{∗}(B̃F) looks like

⨁_{i=1}^{r} ℤF \underrightarrow{∑_{i=1}^{r} r_{s_{i}−1}} ℤF
r̂_{A}↓  r̂_{I} ↓
⨁_{i=1}^{r} ℤF \underrightarrow{∑_{i=1}^{r} r_{s_{i}−1}} ℤF

Then the cellular ℤG-chain complex of T̃_{Bf} = B̃G is the mapping cone of the following chain endomorphism of ℤG ⊗_{ℤF} C_{∗}(B̃F)

⨁_{i=1}^{r} ℤG \underrightarrow{∑_{i=1}^{r} r_{s_{i}−1}} ℤG
r_{I−tA}↓  ^{r_{1−t}}↓
⨁_{i=1}^{r} ℤG \underrightarrow{∑_{i=1}^{r} r_{s_{i}−1}} ℤG

where r_{I−tA} and r_{1−t} are given by right multiplication with the square matrices I − tA and (1 − t) over ℤG. It is det-L^{2}-acyclic after tensoring with l^{2}(G) since G belongs to the class G (see Definition 13.9) by Lemma 13.11, and Theorem 1.39 and Theorem 13.3 (2) hold. The 1-dimensional ℤG-chain complex ℤF ⟶^{r_{1−t}} ℤF is det-L^{2}-acyclic and has trivial L^{2}-torsion after tensoring with l^{2}(G) (see (3.24) and Theorem 3.14 (6)). Moreover, it is a subcomplex of the mapping cone of the chain endomorphism above. The weakly exact long L^{2}-homology sequence (see Theorem 1.21) and the sum formula (see Theorem 3.35 (1)) imply that the L^{2}-torsion of f, which is the L^{2}-torsion of this mapping cone after tensoring with l^{2}(G), is the L^{2}-torsion of the quotient ℤG-chain complex concentrated in dimension 2 and 1 after tensoring with l^{2}(G)

… → 0 → ⨁_{i=1}^{r} ℤG ⟶^{r_{I−tA}} ⨁_{i=1}^{r} ℤG → 0.

This implies

ρ^{(2)}(f) = − ln (det_{N (G)}(id_{l^{2}(G)} ⊗_{ℤG}r_{I−tA})) .

Since G belongs to the class G (see Definition 13.9) by Lemma 13.11, Theorem 13.3 (2) implies that ρ^{(2)}(f) is non-positive. The other claims follow from Theorem 3.172 (1) and (4). ⊓ ⊔

Suppose that the second Novikov-Shubin invariant α_{2}({ \atop B(F ⋊_{f} ℤ)}) is positive, what is conjectured to be true for any group (see Conjecture 2.82) and proved if f is induced by a surface homeomorphism (see Theorem 4.2 (1)). If α is a number with 0 < α < α_{2}(B(F ⋊_{f} ℤ)), then there is a real number C > 0 such that for all p

0 ≤ ρ^{(2)}(f) + 2r · ln(K) − ∑_{p=1}^{L} \frac{1}{p} · c(A, K)_{p} ≤ \frac{C}{α^{L}}

holds (see Theorem 3.172 (5)). In other words, the speed of convergence is exponential.

7.4.4 Generalizations

So far we have assumed that there is a finite model for BG. This implies that G is torsionfree. Similar to the notion of the virtual Euler characteristic (see Remark 6.81) of a group G, which possesses a subgroup G′_{0} ⊂ G of finite index with a finite CW -model for BG_{0}, one can extend the definition of ρ^{(2)}(f) to automorphisms f: G → G of such a group G. Namely, choose a subgroup G_{0} ⊂ G of finite index such that there is a finite CW -model for BG_{0} and f(G_{0}) = G_{0}. Then define

ρ_{virt}^{(2)}(f: G → G) = \frac{ρ^{(2)}(f|_{G_{0}} : G_{0} → G_{0})}{[G : G_{0}]}.  (7.30)

Given a subgroup G′_{0} of finite index in G with a finite CW -model for BG′_{0}, one obtains the desired subgroup G_{0} by ∩_{n∈ℤ}f^{n}(G′_{0}). We have to show that this is independent of the choice of G_{0}. Let G_{1} be another such choice. Then we obtain from Theorem 7.27 (5) if we put G_{2} = G_{0} ∩ G_{1}

\frac{ρ^{(2)}(f|_{G_{0}})}{[G : G_{0}]} = \frac{ρ^{(2)}(f|_{G_{2}})}{[G : G_{0}] · [G_{0} : G_{2}]} = \frac{ρ^{(2)}(f|_{G_{2}})}{[G : G_{1}] · [G_{1} : G_{2}]} = \frac{ρ^{(2)}(f|_{G_{1}})}{[G : G_{1}]}.

Another more general possibility is to use the classifying space E(G ⋊_{f} ℤ, FIN) of G ⋊_{f} ℤ for the family FIN of finite subgroups. Then one can define under the assumption that E(G, FIN) has a finite G-CW -model and G is of det ≥ 1-class

\overline{ρ}^{(2)}(f: G → G) = ρ^{(2)}(E(G ⋊_{f} ℤ; FIN); N (G ⋊_{f} ℤ)).  (7.31)

Here we use the fact that there is a finite G⋊_{f} ℤ-model for E(G⋊_{f} ℤ; FIN), provided that there is a finite G-CW -model for E(G, FIN). Namely, the to both ends infinite mapping telescope of the f: G → G-equivariant homotopy equivalence E(G, FIN) → E(G, FIN) induced by f is a model for E(G ⋊_{f} ℤ; FIN). Moreover, one has to use the fact that the definition of L^{2}-torsion carries over to finite proper G-CW -complexes and depends only on the Ghomotopy type provided that G is of det ≥ 1-class (see Section 3.8).

Any discrete cocompact subgroup of a connected Lie group and any wordhyperbolic group has a finite G-CW -model for E(G, FIN) (see [1, Corollary 4.14], [370]). For more information about E(G, FIN) we refer for instance to [336].

If G is torsionfree, then these three notions ρ^{(2)}(f), ρ_{virt}^{(2)}(f) and \overline{ρ}^{(2)}(f) are all defined if and only if one of them is defined, and in this case they coincide. Suppose that G possesses a torsionfree subgroup G_{0} of finite index. Then BG_{0} has a finite CW -model if there is a finite G-CW -model for E(G, FIN). If both ρ_{virt}^{(2)}(f) and \overline{ρ}^{(2)}(f) are defined, they coincide.

We mention without giving the proof that Theorem 7.27 carries directly over to \overline{ρ}^{(2)}(f: G → G), provided we assume in assertion (3) that G_{3} is torsionfree.

7.5 Miscellaneous

If one takes the third Novikov-Shubin invariant into account, one can improve Lemma 7.22 (1). Namely, if BG has finite 3-skeleton, b_{1}^{(2)}(K ×_{ϕ} EG; N (G)) = 0 and α_{3}(K ×_{ϕ} EG; N (G)) ≠ ∞^{+}, then

def(G) ≤ 0.  (7.32)

This can be seen as follows. It suffices to improve the inequality in Lemma 7.16 to a strict inequality

g−r < 1−b_{0}^{(2)}(K×_{ϕ}EG; N (K))+b_{1}^{(2)}(K×_{ϕ}EG; N (K))−b_{2}^{(2)}(K×_{ϕ}EG; N (K))

by the following modification of the proof of Lemma 7.16. Namely, since α_{3}(K ×_{ϕ} EG; N (G)) ≠ ∞^{+}, we have TH_{2}^{K}(K ×_{ϕ} EG; N (K)) ≠ 0. Notice that H_{2}^{K}(K ×_{ϕ} X̃; N (K)) → H_{2}^{K}(K ×_{ϕ} EG; N (K)) is surjective and H_{2}^{K}(K ×_{ϕ} X̃; N (K)) is a finitely generated projective N (K)-module since X is 2-dimensional and N (K) semihereditary by Theorem 6.7 (1). We conclude from Theorem 6.7 and Lemma 6.28 (3) that (7.18) becomes a strict inequality

b_{2}^{(2)}(K ×_{ϕ} X̃; N (K)) > b_{2}^{(2)}(K ×_{ϕ} EG; N (K))

and (7.32) follows.

We mention the following result of Lott [320, Theorem 2] which generalizes a result of Lubotzky [324]. The statement we present here is a slight improvement of Lott’s result due to Hillman [259].

Theorem 7.33. Let L be a connected Lie group. Let G be a lattice in L. If def(G) > 0, then one of the following assertions holds

(1) G is a lattice in P SL_{2}(ℂ);

(2) def(G) = 1. Moreover, either G is isomorphic to a torsionfree nonuniform lattice in ℝ × P SL_{2}(ℝ) or P SL_{2}(ℂ), or G is ℤ or ℤ^{2}.

More information about the class B_{1} can be found in [34]. For instance the first L^{2}-Betti number of a group having Kazhdan’s property (T) vanishes [34, Corollary 6].

The following result is due to Gaboriau [214, Theorem 6.3]. (An alternative proof using the dimension theory of Chapter 6 can be found in the Ph. D. thesis of Roman Sauer [456].) Two countable groups G_{0} and G_{1} are called measure equivalent if there exist commuting measure-preserving free actions of G_{0} and G_{1} on some infinite Lebesgue measure space (Ω, m) such that the actions of both G_{0} and G_{1} admit finite measure fundamental domains (see [237, 0.5E], [211] and [212]).

Theorem 7.34 (Measure equivalence and L^{2}-Betti numbers). Let G_{0} and G_{1} be two countable groups which are measure equivalent. Then there is a constant C > 0 such that for all p ≥ 0

b_{p}^{(2)}(G_{0}) = C · b_{p}^{(2)}(G_{1}).

Since any infinite amenable group is measure equivalent to ℤ [408], this gives another proof of the fact that the L^{2}-Betti numbers of infinite amenable countable groups vanish. Notice that ℤ^{m} and ℤ^{n} have different Novikov-Shubin invariants for m ≠ n (see Example 2.59) so that Novikov-Shubin invariants are not invariant under measure equivalence.

The notion of measure equivalence can be viewed as the measure theoretic analog of the metric notion of quasi-isometric groups. Namely, two finitely generated groups G_{0} and G_{1} are quasi-isometric if and only if there exist commuting proper (continuous) actions of G_{0} and G_{1} on some locally compact space such that each action has a compact fundamental domain [237, 0.2 C′_{2} on page 6].

If the finitely generated groups G_{0} and G_{1} are quasi-isometric and there exist finite models for BG_{0} and BG_{1} then b_{p}^{(2)}(G_{0}) = 0 ⇔ b_{p}^{(2)}(G_{1}) = 0 holds (see [237, page 224], [410]). But in general it is not true that there is a constant C > 0 such that b_{p}^{(2)}(G_{0}) = C · b_{p}^{(2)}(G_{1}) holds for all p ≥ 0 (cf. [215, page 7], [237, page 233], [523]). If F_{g} denotes the free group on g generators, then define G_{n} := (F_{3} × F_{3}) ∗ F_{n} for n ≥ 2. The groups G_{m} and G_{n} are quasi-isometric for m, n ≥ 2 (see [132, page 105 in IV-B.46], [523, Theorem 1.5]) and have finite models for their classifying spaces. One easily checks that b_{1}^{(2)}(G_{n}) = n and b_{2}^{(2)}(G_{n}) = 4 (see Example 1.38).

Gaboriau’s Theorem 7.34 implies that G_{n} and G_{m} are measure equivalent if and only if m = n holds. Hence there are finitely presented groups which are quasi-isometric but not measure equivalent. Another example is pointed out in [215, page 7]. There exist quasi-isometric finitely generated groups G_{0} and G_{1} such that G_{0} has Kazhdan’s property (T) and G_{1} does not, and Kazhdan’s property (T) is an invariant under measure equivalence [211, Theorem 8.2].

The converse is also true. The groups ℤ^{n} and ℤ^{m} are infinite amenable and hence measure equivalent. But they are not quasi-isometric for different m and n since n is the growth rate of ℤ^{n} and the growth rate is a quasi-isometry invariant.

Notice that Theorem 7.34 implies that the sign of the Euler characteristic of a group G is an invariant under measure equivalence, what is not true for quasi-isometry by the example above.

We mention the following not yet published result of Monod and Shalom that the non-vanishing of the second bounded cohomology H_{b}^{2}(G; l^{2}(G)) of a countable group G with coefficients in l^{2}(G) is an invariant of the measure equivalence class of G.

The following questions arose in discussions with R. Sauer in view of Theorem 3.113 and [343].

Question 7.35. (L^{2}-torsion of groups and quasi-isometry and measure equivalence). Let G_{i} for i = 0, 1 be a group such that there is a finite CW -model for BG_{i} and EG_{i} is det-L^{2}-acyclic. Suppose that G_{0} and G_{1} are measure equivalent or that G_{0} and G_{1} are quasi-isometric. Does then ρ^{(2)}(EG_{0}; N (G_{0})) = 0 ⇔ ρ^{(2)}(EG_{1}; N (G_{1})) = 0 hold?

Question 7.36. (Novikov-Shubin invariants and quasi-isometry). Let G_{i} for i = 0, 1 be a finitely generated group. Suppose that G_{0} and G_{1} are quasi-isometric. Does then α_{p}(EG_{0}; N (G_{0})) = α_{p}(EG_{1}; N (G_{1})) hold for p ≥ 1?

A lattice G in a locally compact second countable topological group T is a discrete subgroup such that the measure on T/G induced by a right invariant Haar measure on T has finite volume. (This implies that T is unimodular.) Since two lattices in the same locally compact second countable topological group T are measure equivalent [211, Example 1.2 on page 1059], Theorem 7.34 implies (see also Theorem 3.183)

Corollary 7.37. Let G_{1} and G_{2} be two lattices in the same locally compact second countable topological group T. Then for all p ≥ 0

\frac{b_{p}^{(2)}(G_{1})}{vol(T/G_{1})} = \frac{b_{p}^{(2)}(G_{2})}{vol(T/G_{2})}.

A survey article about orbit equivalent measure preserving actions and L^{2}-Betti numbers is written by Gaboriau [215].

For each Artin group A, Davis and Leary [129] compute the L^{2}-cohomology of the universal covering of its so called Salvetti complex. This is a finite CWcomplex which is conjectured to be a model for the classifying space BA. In the many cases when this conjecture is known to hold their calculation describes the reduced L^{2}-cohomology of EA, or, equivalently, the L^{2}-Betti numbers b_{p}^{(2)}(A).

Exercises

7.1. Let d be a non-negative integer. Suppose that the group G has a descending series of infinite subgroups G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ … ⊃ G_{d} such that BG_{n+1} has finite (d − n)-skeleton for each n = 0, 1, 2, …, d − 1, G_{n+1} is normal in G_{n} and the quotient G_{n}/G_{n+1} contains an element of infinite order or contains finite subgroups of arbitrary large order for n = 0, 1, …, d − 1. Prove G ∈ B_{d}.

7.2. Suppose that the group G has a descending series of infinite subgroups G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ … such that BG_{n+1} is of finite type, G_{n+1} is normal in G_{n} and the quotient G_{n}/G_{n+1} contains an element of infinite order or contains finite subgroups of arbitrary large order for n = 0, 1, . . .. Prove G ∈ B_{∞}.

7.3. Let G_{n} for n = 0, 1, 2, … be a sequence of non-trivial groups. Prove

b_{p}^{(2)}(∗_{n=0}^{∞}G_{n}) = ∑_{n=0}^{∞} b_{p}^{(2)}(G_{n}).

for p ≥ 2 and

b_{1}^{(2)}(∗_{n=0}^{∞}G_{n}) = ∞.

7.4. Let G_{1}, G_{2}, …, G_{n} be finitely many groups. Prove

b_{p}^{(2)}(∏_{k=1}^{n} G_{k}) = ∑_{\begin{matrix}0≤j_{1},j_{2},...,j_{n}, \\ ∑_{k=1}^{n} j_{k}=p \end{matrix}} ∏_{k=1}^{n} b_{j_{k}}^{(2)}(G_{k}),

where we use the convention that 0 · ∞ = 0, r · ∞ = ∞ for r ∈ (0, ∞] and r + ∞ = ∞ for r ∈ [0, ∞].

7.5. Let {G_{i} | i ∈ I} be a family of non-trivial groups for an infinite index set I. Put G = ∏_{i∈I} G_{i}. Prove G ∈ B_{∞}. Show by giving an example that it is not necessarily true that H_{p}^{(2)}(EG; N (G)) = 0 for all p ≥ 0.

7.6. Let G be an infinite locally finite group. Show that H_{p}^{G}(EG; N (G)) vanishes for p ≥ 1 and does not vanish for p = 0 and that b_{p}^{(2)}(G) = 0 holds for all p ≥ 0.

7.7. Let 1 → H → G → K → 1 be an extension of groups such that H or K respectively is finite. Show that K or H respectively belongs to B_{d}, BQ_{d} or BF_{d} if and only if G does.

7.8. Let d be a non-negative integer or d = ∞. Let F ⟶^{i} E ⟶^{p} B be a fibration of connected CW -complexes such that b_{p}^{(2)}(π_{1}(E) ×_{π_{1}(i)} F̃; N (π_{1}(E))) < ∞ for p ≤ d. Suppose that H_{p}(B̃; ℂ) is a finite dimensional complex vector space for p ≤ d and that π_{1}(B) ∈ BF_{d}. Show that then b_{p}^{(2)}(Ẽ) = 0 for p ≤ d.

7.9. Let G and H be non-trivial groups. Show that b_{1}^{(2)}(G ∗ H) = 0 if and only if G = H = ℤ/2.

7.10. Suppose there is a finite G-CW -model for E(G; FIN). Let d be the least common multiple of the order of finite subgroups of G. Show

d · χ^{(2)}(G) ∈ ℤ.

7.11. Prove χ^{(2)}(SL(2, ℤ)) = −\frac{1}{12}.

7.12. Let G be a finitely presented group. Show

def(G) ≤ b_{1}(G; ℤ/p) − b_{2}(G; ℤ/p),

where b_{p}(G; ℤ/p) is the p-th Betti number of EG with respect to singular homology with coefficients in ℤ/p.

7.13. Let f: G → G be a (not necessarily bijective) endomorphism of a group G which possesses a finite CW -model for BG and is of det ≥ 1-class. Let K be the colimit of the directed system … ⟶^{f} G ⟶^{f} G ⟶^{f} … indexed by ℤ. Let G_{f} be the semidirect product K ⋊ ℤ with respect to the shift automorphism of K. Show (1) There is a finite CW -model for BG_{f}, namely the mapping torus of

Bf: BG → BG;

(2) BG_{f} is det-L^{2}-acyclic and the L^{2}-torsion ρ^{(2)}({ \atop BG_{f}}) depends only on f.

Define

ρ^{(2)}(f) := ρ^{(2)}({ \atop BG_{f}});

(3) Show that this invariant reduces to the invariant ρ^{(2)}(f) of Definition 7.26 in the case where f is an automorphism;

(4) Show that Theorem 7.27 does hold also for group endomorphisms if one makes the following modifications or additional assumptions. Assertion (1), (2), (3) and (4) remain true. For assertion (5) one needs the condition that f induces a bijection G/G_{0} ⟶^{≅} G/G_{0}. In the formulation of assertion (6) one must replace the universal covering of BG by the covering associated to the canonical projection G → K. In assertion (7) the conditions must be required for K instead of G.

8. The Algebra of Affiliated Operators

Introduction

In this chapter we introduce and study the algebra U(G) of operators affiliated to N (G) for a group G. A G-operator f: dom(f) ⊂ V → W of Hilbert N (G)-modules is an operator whose domain dom(f) is a linear G-invariant subspace and which satisfies f(gx) = gf(x) for all x ∈ dom(f) and g ∈ G. The algebra U(G) consists of densely defined closed G-operators a: dom(a) ⊂ l^{2}(G) → l^{2}(G) and contains N (G) as a subalgebra. It is constructed in such a way that an element f: l^{2}(G) → l^{2}(G) in N (G) is a weak isomorphism if and only it is invertible in U(G). This is reflected algebraically by the fact that U(G) is the Ore localization of N (G) with respect to the set of non-zero divisors (see Theorem 8.22 (1)). It does not come with a natural topology anymore but has nice ring theoretic properties. Namely, U(G) is von Neumann regular, i.e. for any r ∈ U(G) there is s ∈ U(G) with rsr = r, or, equivalently, any finitely generated submodule of a projective U(G)-module is a direct summand (see Theorem 8.22 (3)). We have already mentioned in Example 6.12 that N (G) behaves in several ways like a principal ideal domain except that N (G) is not Noetherian and has zero-divisors if G is infinite. Any principal ideal domain R has a quotient field F, and in this analogy U(G) should be thought of as F. Recall that the (extended) dimension of an arbitrary R-module M is the same as the F -dimension of the F -vector space F ⊗_{R} M (see Example 6.12). In terms of this analogy it is not surprising that the extended dimension function dim_{N (G)} for arbitrary N (G)-modules comes from an extended U(G)-dimension function dim_{U(G)} over U(G) in the sense that dim_{N (G)}(M) = dim_{U(G)}(U(G) ⊗_{N (G)} M) holds for any N (G)-module M (see Theorem 8.29). The algebra U(G) will play a role in the proof of the Atiyah conjecture in Chapter 10. It will be the largest algebra attached to a given group which we will consider in this book. All other algebras attached to a given group will be contained in it.

8.1 The Algebra of Affiliated Operators

In this section we introduce the algebra U(G) of operators affiliated to the group von Neumann algebra N (G). It was originally introduced and studied in [396, Chapter XVI].

We have already studied the ℂ-category {fin. gen. Hilb. N (G)-mod.} with involution of finitely generated Hilbert N (G)-modules with bounded Goperators as morphisms (see Section 6.2). We have already seen that weak isomorphisms play an important role. Recall that an operator is a weak isomorphism if its kernel is trivial and its image is dense and that a weak isomorphism is in general not invertible. The basic idea for the following construction is to enlarge the set of morphisms in {fin. gen. Hilb. N (G)-mod.} to get a new ℂ-category with involution {fin. gen. Hilb. N (G)-mod.}_{U} with the same objects such that weak isomorphisms become invertible. To motivate it, let us consider a bounded G-operator f: V → W of finitely generated Hilbert N (G)-modules, which is a weak isomorphism, and check what is needed to find an “inverse”. The polar decomposition f = us consists of a unitary invertible G-operator u and a positive G-operator s which is a weak isomorphism. So it suffices to “invert” s. Since s is a weak isomorphism, its kernel is trivial and we obtain an unbounded densely defined closed G-operator ∫ λ^{−1}dE_{λ}, if {E_{λ} | λ ∈ [0, ∞)} is the spectral family of s (see Subsection 1.4.1). This should become the inverse of s. Hence we must enlarge the morphisms to include unbounded densely defined G-operators like ∫ λ^{−1}dE_{λ}. The difficulty will be that the composition and sum of unbounded densely defined closed operators (see Notation 1.69) is in general not again densely defined and closed. The main problem is that the intersection of two dense subspaces is not necessarily dense again. It will turn out that in the specific situation, which we are interested in, this problem can be solved. We need some preparation to handle this.

Definition 8.1 (Affiliated operator). An unbounded operator f: dom(f) ⊂ V → W of finitely generated Hilbert N (G)-modules is called affiliated (to N (G)) if f is densely defined with domain dom(f) ⊂ V, is closed and is a G-operator, i.e. dom(f) is a linear G-invariant subspace and f(gx) = gf(x) for all x ∈ dom(f) and g ∈ G.

Definition 8.2. Let V be a finitely generated Hilbert N (G)-module. A Ginvariant linear subspace L ⊂ V is called essentially dense if for any ϵ > 0 there is a finitely generated Hilbert N (G)-submodule P ⊂ V with P ⊂ L and dim_{N (G)}(V) − dim_{N (G)}(P) ≤ ϵ.

Notice that essentially dense implies dense.

Lemma 8.3. Let f, f′: V → W be affiliated operators of finitely generated Hilbert N (G)-modules. Then (1) A countable intersection of essentially dense linear G-subspaces of V is

again essentially dense;

(2) If L ⊂ W is essentially dense, then f^{−1}(L) ⊂ dom(f) is an essentially dense subspace of V. In particular dom(f) is essentially dense;

(3) If f′ ⊂ f, then f′ = f. In particular f = f′, if there is a G-invariant subset S of dom(f) ∩ dom(f′) which is dense in V and on which f and f′ agree;

(4) If f is bounded on its domain, i.e. there is C < 0 with |f(x)| ≤ C|x| for all x ∈ dom(f), then dom(f) = V.

Proof. (1) Let {L_{n} | n ≥ 0} be a countable set of essentially dense linear Gsubspaces of V. Given ϵ > 0, choose a Hilbert N (G)-submodule P_{n} ⊂ V with P_{n} ⊂ L_{n} and dim_{N (G)}(V)−dim_{N (G)}(P_{n}) ≤ 2^{−n−1}ϵ. Put P = ⋂_{n≥0} P_{n}. Then P is a Hilbert N (G)-submodule of V with P ⊂ ⋂_{n≥0} L_{n} and we conclude from Theorem 1.12 (2) and (4)

dim_{N (G)}(V) − dim_{N (G)}(P)

= dim_{N (G)}(V) − lim_{m→∞} dim_{N (G)} ( ⋂_{n=0}^{m} P_{n})

= lim_{m→∞} (dim_{N (G)}(V) − dim_{N (G)} ( ⋂_{n=0}^{m} P_{n}))

≤ lim_{m→∞} ∑_{n=0}^{m} (dim_{N (G)}(V) − dim_{N (G)}(P_{n}))

≤ lim_{m→∞} (∑_{n=0}^{m} 2^{−n−1}ϵ)
= ϵ.

(2) We can write f using polar decomposition as the composition

f: V ⟶^{pr} ker(f)^{⊥} ⟶^{u} \overline{im(f)} ⟶^{s} \overline{im(f)} ⟶^{i} W,

where pr and i are the canonical projection and inclusion, u is a unitary isomorphism and s is a densely defined unbounded closed G-operator which is positive and a weak isomorphism. Let {E_{λ} | λ ∈ [0, ∞)} be the spectral family of s. Since E_{λ} converges for λ → ∞ strongly to id, we get lim_{λ→∞} dim_{N (G)}(im(E_{λ})) = dim_{N (G)}(\overline{im(f)}). Fix ϵ > 0. Choose λ_{0} with dim_{N (G)}(\overline{im(f)}) − dim_{N (G)}(im(E_{λ_{0}})) ≤ ϵ/2. Since L is essentially dense in W, we can find a Hilbert N (G)-submodule P ⊂ W with P ⊂ L and dim_{N (G)}(W) − dim_{N (G)}(P) ≤ ϵ/2. We conclude from Theorem 1.12 (2)

dim_{N (G)}(\overline{im(f)}) − dim_{N (G)}(P ∩ im(E_{λ_{0}})) ≤ ϵ.  (8.4)

Notice that im(E_{λ_{0}}) ⊂ dom(s) follows from (1.66) since each y ∈ im(E_{λ_{0}}) lies in the domain of ∫_{−∞}^{∞} λdE_{λ}. Moreover, s induces a bounded G-operator s_{λ_{0}} : im(E_{λ_{0}}) → im(E_{λ_{0}}) which is a weak isomorphism, and s^{−1}(P ∩ im(E_{λ_{0}})) = s^{−1}(P ∩ im(E_{λ_{0}})) ∩ im(E_{λ_{0}}) = (s_{λ_{0}})^{−1}(P ∩ im(E_{λ_{0}})) is a closed G-invariant subspace of \overline{im(f)}. We conclude from Lemma 2.12

dim_{N (G)}(s^{−1}(P ∩ im(E_{λ_{0}}))) = dim_{N (G)}(P ∩ im(E_{λ_{0}})).  (8.5)

We conclude from Additivity (see Theorem 1.12 (2))

dim_{N (G)} (s^{−1}(P ∩ im(E_{λ_{0}}))) + dim_{N (G)}(ker(f))
= dim_{N (G)} ((s ∘ u ∘ pr)^{−1}(P ∩ im(E_{λ_{0}}))); (8.6)
dim_{N (G)}(V) = dim_{N (G)}(ker(f)) + dim_{N (G)}(\overline{im(f)}).  (8.7)

Hence (s ∘ u ∘ pr)^{−1}(P ∩ im(E_{λ_{0}})) is a Hilbert N (G)-submodule of V, is contained in f^{−1}(L) and because of (8.4), (8.5), (8.6) and (8.7)

dim_{N (G)}(V) − dim_{N (G)} ((s ∘ u ∘ pr)^{−1}(P ∩ im(E_{λ_{0}}))) ≤ ϵ.

Hence f^{−1}(L) is essentially dense.

(3) Let V be a finitely generated Hilbert N (G)-module. We first show that an affiliated operator h: V → V which is symmetric is already selfadjoint. We do this using the Cayley transform κ(s) of a symmetric densely defined operator s: dom(s) ⊂ H → H, which is the operator (s − i)(s + i)^{−1} with domain dom(κ(s)) = (s + i)(dom(s)). Let us summarize the basic properties of the Cayley transform (see [421, Section 5.2]). It is an isometry on its domain dom(κ(s)). The operator κ(s) − 1 with the same domain as κ(s) is injective and has range im(κ(s) − 1) = dom(s). If in addition s is closed, then the subspaces (s ± i)(dom(s)) are closed. The operator s is selfadjoint if and only if (s + i)(dom(s)) = (s − i)(dom(s)) = H.

We want to use the latter criterion to show that h is selfadjoint. Notice that κ(h) is a G-operator as h is a G-operator, and dom(κ(h)) = (h + i)(dom(h)) is closed as h is closed. Let p denote the projection onto the G-invariant closed subspace dom(κ(h)). Then κ(h) ∘ p − p is everywhere defined and im(κ(h) − 1)) = im(κ(h) ∘ p − p). Since

(κ(h)∘p−p)^{∗} = ((κ(h) ∘ p − p) ∘ p)^{∗} = p^{∗} ∘(κ(h)∘p−p)^{∗} = p∘(κ(h)∘p−p)^{∗},

we conclude using Additivity (see Theorem 1.12 (2))

dim_{N (G)}(dom(h)) = dim_{N (G)}(im(κ(h) − 1))
= dim_{N (G)}(im(κ(h) ∘ p − p))
= dim_{N (G)}(im((κ(h) ∘ p − p)^{∗}))
≤ dim_{N (G)}(im(p))
= dim_{N (G)}((h + i)(dom(h))).

This and assertion (2) imply

dim_{N (G)}((h + i)(dom(h))) = dim_{N (G)}(V).

Since (h + i)(dom(h)) ⊂ V is closed, we conclude (h + i)(dom(h)) = dom(κ(h)) = V from Theorem 1.12 (1) and (2). Since κ(h) is an isometry on its domain and a G-operator, we get dim_{N (G)}(im(κ(h)) = dim_{N (G)}(V). As (h − i)(dom(h)) = im(κ(h)) is closed, we conclude (h − i)(dom(h)) = V. This finishes the proof that h is selfadjoint.

Now we can give the proof of (3). Let f = uh be the polar decomposition of f. Put h′ = u^{∗}f′. Since u^{∗} is a bounded G-operator and f′ is affiliated, h′ is affiliated. We have h′ = u^{∗}f′ ⊂ u^{∗}f = h. We conclude h′ ⊂ h = h^{∗} ⊂ (h′)^{∗}, i.e. h′ is symmetric. Hence h′ is selfadjoint by the argument above. This implies h′ = h = h^{∗} = (h′)^{∗} and hence dom(f′) = dom(h′) = dom(h) = dom(f), i.e. f = f′.

(4) Any closed densely defined operator which is bounded on its domain is automatically everywhere defined. ⊓ ⊔

Now we are ready to define the desired ℂ-category with involution {fin. gen. Hilb. N (G)-mod.}_{U} as follows. Objects are finitely generated Hilbert N (G)-modules. Given two finitely generated Hilbert N (G)-modules V and W, the set of morphisms from V to W consists of all affiliated operators f: dom(f) ⊂ V → W. The identity element of V is given by the identity operator id: V → V. Given an affiliated operator f: dom(f) ⊂ V → W, its adjoint f^{∗}: dom(f^{∗}) ⊂ W → V is affiliated, since the adjoint of a closed densely defined operator is always closed and densely defined and the adjoint of a G-operator is always a G-operator. Thus we can define the involution by taking the adjoint.

Given morphisms f: dom(f) ⊂ U → V and g: dom(g) ⊂ V → W, define their composition g ∘ f in {fin. gen. Hilb. N (G)-mod.}_{U} by the minimal closure of the unbounded operator g ∘ f: U → W with domain f^{−1}(dom(g)) as defined in Notation 1.69. We have to check that this is well-defined. From Lemma 8.3 (2) we conclude that f^{−1}(dom(g)) is essentially dense and in particular dense in U. We know already that the adjoints g^{∗} and f^{∗} are affiliated again and hence f^{∗}∘g^{∗} with domain (g^{∗})^{−1}(dom(f^{∗})) is densely defined. Since the domain of the adjoint of g ∘ f contains (g^{∗})^{−1}(dom(f^{∗})), (g ∘ f)^{∗} is densely defined. This implies that g ∘ f is closable, namely, ((g ∘ f)^{∗})^{∗} is its minimal closure. Obviously g ∘ f is a G-operator. Hence its minimal closure is indeed affiliated. Analogously one defines the complex vector space structure on the set of morphisms from V to W by taking the minimal closure of the addition and scalar multiplication of unbounded operators defined in Notation 1.69. We leave it to the reader to check that the various axioms like associativity of the composition and so on are immediate consequences of Lemma 8.3 (3). Obviously {fin. gen. Hilb. N (G)-mod.} is a ℂsubcategory with involution of {fin. gen. Hilb. N (G)-mod.}_{U}. Next we show that {fin. gen. Hilb. N (G)-mod.}_{U} has the desired properties.

Lemma 8.8. (1) An affiliated operator f: dom(f) ⊂ V → W represents an isomorphism in {fin. gen. Hilb. N (G)-mod.}_{U} if and only if f is a weak isomorphism, i.e. f has trivial kernel and dense image f(dom(f));

(2) Let f: dom(f) ⊂ V → W be an affiliated operator. Then there are bounded (everywhere defined) G-operators a: V → W and b: V → V with the properties that b is a weak isomorphism and f = ab^{−1} holds in {fin. gen. Hilb. N (G)-mod.}_{U}.

Proof. (1) Suppose that f: dom(f) ⊂ V → W is an affiliated operator which is a weak isomorphism. Let f = us be its polar decomposition. Then u: V → W is a unitary bijective G-operator and s: dom(s) ⊂ V → V is an affiliated operator which is positive and a weak isomorphism. Let h(s): V → V be the operator given by the functional calculus applied to s for the Borel function h: ℝ → ℝ which sends λ ≠ 0 to λ^{−1} and 0 to, let us say, 0. Notice that h(s) is a densely defined selfadjoint G-operator and in particular affiliated. If {E_{λ} | λ ∈ [0, ∞)} is the spectral family of s, then im(E_{n}) ∩ im(E_{1/n}^{⊥}) is contained in dom(h(s)) for all integers n ≥ 1. Since s has trivial kernel, lim_{n→∞} dim_{N (G)}(im(E_{n}) ∩ im(E_{1/n}^{⊥})) = dim_{N (G)}(V). Since both s and h(s) map im(E_{n}) ∩ im(E_{1/n})^{⊥} to itself and define inverse operators there, Lemma 8.3 (3) implies that s and h(s) define inverse morphisms in {fin. gen. Hilb. N (G)-mod.}_{U}. Since u obviously defines an isomorphism in {fin. gen. Hilb. N (G)-mod.}_{U}, f is an isomorphism in {fin. gen. Hilb. N (G)-mod.}_{U}.

Now suppose that the affiliated operator f: dom(f) ⊂ V → W defines an isomorphism in {fin. gen. Hilb. N (G)-mod.}_{U}. Let h: dom(h) ⊂ W → V be an inverse. For x ∈ ker(f) we have x ∈ f^{−1}(dom(h)) and hence x = id(x) = h ∘ f(x) = 0. Hence f has trivial kernel. Since f ∘ h = id in U, the image of f contains the subspace h^{−1}(dom(f)) which is dense by Lemma 8.3 (2). Hence the image of f is dense. This shows that f is a weak isomorphism.

(2) Let f = us be the polar decomposition of f. Let p = 1 − u^{∗}u. This is the projection onto ker(u^{∗}u) = ker(us) = ker(s). We have f = u(s+p) and (s+p) is an affiliated positive operator dom(s + p) ⊂ V → V with trivial kernel. Define Borel functions h_{1}, h_{2} : ℝ → ℝ by h_{1}(λ) = λ and h_{2}(λ) = 1 for λ ≤ 1 and by h_{1}(λ) = 1 and h_{2}(λ) = 1/λ for λ ≥ 1. Then u∘h_{1}(s+p) and h_{2}(s+p) are bounded G-operators, h_{2}(s + p) represents an invertible morphism in {fin. gen. Hilb. N (G)-mod.}_{U} and f = (u ∘ h_{1}(s + p)) ∘ (h_{2}(s + p))^{−1} in {fin. gen. Hilb. N (G)-mod.}_{U}. ⊓ ⊔

Definition 8.9 (Algebra of affiliated operators). Define U(G) to be the ring with involution given by the endomorphisms of l^{2}(G) in the ℂ-category with involution {fin. gen. Hilb. N (G)-mod.}_{U}, i.e. the ring of affiliated operators f: dom(f) ⊂ l^{2}(G) → l^{2}(G).

Let {l^{2}(G)^{n}}_{U} ⊂ {fin. gen. Hilb. N (G)-mod.}_{U} be the full subcategory, whose objects are l^{2}(G)^{n} for n ≥ 0. Let {U(G)^{n}} be the category whose objects are the (left) U(G)-modules U(G)^{n} for n ≥ 0 and whose morphisms are U(G)-homomorphisms. Then we obtain an isomorphism of ℂ-categories with involution (compare with 6.22)

ν_{U}: {U(G)^{n}} → {l^{2}(G)^{n}}_{U}.  (8.10)

Example 8.11. Let G = ℤ^{n}. Using the identifications of Example 1.4 we get an identification of U(ℤ^{n}) with the complex algebra L(T^{n}) of equivalence classes of measurable functions f: T^{n} → ℂ. Recall that two such functions are equivalent if they differ on a subset of measure zero only.

Let i: H → G be an injective group homomorphism. We have associated to it a covariant functor of ℂ-categories with involution

i_{∗}: {fin. gen. Hilb. N (H)-mod.} → {fin. gen. Hilb. N (G)-mod.}

in Definition 1.23. The same construction induces a covariant functor of ℂcategories with involution

i_{∗}: {fin. gen. Hilb. N (H)-mod.}_{U} → {fin. gen. Hilb. N (G)-mod.}_{U}

because for an affiliated operator f: U → V of finitely generated Hilbert N (H)-modules the densely defined G-operator id ⊗_{ℂH}f: dom(id ⊗_{ℂH}f) = ℂG ⊗_{ℂH} dom(f) ⊂ i_{∗}U → i_{∗}V has a densely defined adjoint and therefore its minimal closure exists and is affiliated. Functoriality follows from Lemma 8.3 (3). In particular we get a ring homomorphism

i_{∗}: U(H) → U(G).  (8.12)

Notice that N (G) is a ∗-subring of U(G) and that U(G) does not carry a natural topology anymore. However, U(G) has nice properties as a ring what we will explain below.

8.2 Basic Properties of the Algebra of Affiliated Operators

In this section we explain and prove various ring-theoretic properties of U(G).

8.2.1 Survey on Ore Localization

Definition 8.13. Let R be a ring. Given a set S ⊂ R, a ring homomorphism f: R → R′ is called S-inverting if f(s) is invertible in R′ for all s ∈ S. An S-inverting ring homomorphism f: R → R_{S} is called universal S-inverting if for any S-inverting ring homomorphism g: R → R′ there is precisely one ring homomorphism \overline{g}: R_{S} → R′ satisfying \overline{g} ∘ f = g.

The universal property implies as usual that the universal S-inverting homomorphism f: R → R_{S} is unique up to unique isomorphism. One can construct such a universal S-inverting ring homomorphism by writing down a suitable ring with generators and relations. If \overline{S} is the set of elements s ∈ R for which f(s) ∈ R_{S} is a unit, then \overline{S} is multiplicatively closed, i.e s, t ∈ \overline{S} implies st ∈ \overline{S} and 1 ∈ \overline{S}, and the ring homomorphism ϕ: R_{S} → R\overline{S}, which is uniquely determined by f\overline{S} ∘ ϕ = f_{S}, is an isomorphism. Hence we can assume in the sequel without loss of generality that S is multiplicatively closed. If R is commutative and S ⊂ R multiplicatively closed, one can describe R_{S} in terms of fractions rs^{−1} with the obvious rules for addition and multiplications. Under a certain condition this nice approach works also for non-commutative rings.

Definition 8.14 (Ore localization). Let S ⊂ R be a multiplicatively closed subset of the ring R. The pair (R, S) satisfies the (right) Ore condition if i.) for (r, s) ∈ R × S there exists (r′, s′) ∈ R × S satisfying rs′ = sr′ and ii.) if for r ∈ R and s ∈ S with sr = 0 there is t ∈ S with rt = 0.

If (R, S) satisfies the Ore condition, define the (right) Ore localization to be the following ring RS^{−1}. Elements are represented by pairs (r, s) ∈ R × S, where two such pairs (r, s) and (r′, s′) are called equivalent if there are u, u′ ∈ R such that ru = r′u′ and su = s′u′ hold and su = s′u′ belongs to S. The addition and multiplication is given on representatives by (r, s) + (r′, s′) = (rc + r′d, t), where t = sc = s′d ∈ S and by (r, s) · (r′, s′) = (rc, s′t), where sc = r′t for t ∈ S. The unit element under addition is represented by (0, 1) and under multiplication by (1, 1). Let f: R → RS^{−1} be the ring homomorphism which sends r ∈ R to the class of (r, 1).

One may try to remember part i.) of the Ore condition by saying that for any left (= wrong way) fraction s^{−1}r there is a right fraction r′(s′)^{−1} with s^{−1}r = r′(s′)^{−1}. Notice that part ii.) of the Ore condition is automatically satisfied if S contains no zero-divisor.

Lemma 8.15. Suppose that (R, S) satisfies the Ore condition. Then

(1) The map f: R → RS^{−1} is the universal S-inverting ring homomorphism;

(2) The kernel of f: R → RS^{−1} is {r ∈ R | rs = 0 for some s ∈ S};

(3) The functor RS^{−1} ⊗_{R} − is exact;

(4) The pair (M_{n}(R), S · I_{n}) satisfies the Ore condition, where M_{n}(R) is the ring of (n, n)-matrices over R and I_{n} is the unit matrix. The canonical ring homomorphism induced by the universal property M_{n}(R)(S · I_{n})^{−1} ⟶^{≅} M_{n}(RS^{−1}) is an isomorphism.

Proof. [484, Proposition II.1.4 on page 51]. ⊓ ⊔ The next example is based on [489].

Example 8.16. Let G be a torsionfree amenable group. Suppose that the Kaplansky Conjecture 10.14 holds for G with coefficients in the field F, i.e.

F G has no non-trivial zero-divisors. We want to show that the set S of nonzero-divisor satisfies the left Ore condition. Since F G is a ring with involution the right Ore condition follows as well. Notice that then the Ore localization (F G)S^{−1} is a skewfield, in which F G embeds.

Consider elements β, γ in F G such that γ is a non-zero-divisor. Choose a finite subset S ⊂ G such that s ∈ S ⇒ s^{−1} ∈ S and β and γ can be written as β = ∑_{s∈S} b_{s} · s and γ = ∑_{s∈S} c_{s} · s. Since G satisfies the Følner condition (see Lemma 6.35), we can find a non-empty subset A ⊂ G satisfying |∂_{S}A|·|S| < |A|. We want to find elements δ = ∑_{a∈A} d_{a}·a and ϵ = ∑_{a∈A} e_{a}·a such that ϵ is a non-zero-divisor and ϵβ = δγ. Let AS ⊂ G be the subset {a · s | a ∈ A, s ∈ S}. Then the latter equation is equivalent to the following set of equations indexed by elements u ∈ AS

∑_{a∈A,s∈S,s·a=u} e_{a}b_{s} − d_{a}c_{s} = 0.

This is a system of |AS| homogeneous equations over F in 2|A| variables d_{a} and e_{a}. Suppose that |AS| < 2|A|. Then we can find a solution different from zero, in other words, we can find δ and ϵ such that ϵβ = δγ and at least one of the elements δ and ϵ is different from zero. Since γ is a non-zero-divisor, ϵ = 0 implies δ = 0. Therefore ϵ ≠ 0. By assumption ϵ must be a non-zero-divisor. It remains to prove |AS| < 2|A|. From

AS ⊂ A ∪ ⋃_{s∈S}{a ∈ A | a · s^{−1} ∉ A} · s ⊂ A ∪ ⋃_{s∈S} ∂_{S}A · s.

we conclude

|AS| ≤ |A| + |S| · |∂_{S}A| < 2|A|.

and the claim follows.

Let F_{g} be the free group with g ≥ 2 generators. Let S be the multiplicatively closed subset of ℂF_{g} consisting of all non-zero-divisors. We mention that ℂF_{g} has no non-trivial zero-divisors and that (ℂF_{g}, S) does not satisfy the Ore condition.

The lamplighter group L is amenable but the set S of non-zero-divisors does not satisfies the left Ore condition [311].

8.2.2 Survey on von Neumann Regular Rings

Definition 8.17 (Von Neumann regular). A ring R is called von Neumann regular if for any r ∈ R there is an s ∈ R with rsr = r.

This notion should not be confused with the notion of a regular ring, i.e. a Noetherian ring for which every R-module has a projective resolution of finite dimension. For instance a principal ideal domain R is always regular since submodules of free R-modules are free (see [15, Corollary 1.2 in chapter 10 on page 353]), but it is von Neumann regular if and only if it is a field. The definition above is appropriate to check whether a ring is von Neumann regular. However, for structural questions the following equivalent characterizations are more useful.

Lemma 8.18. The following assertions are equivalent for a ring R.

(1) R is von Neumann regular;

(2) Every principal (left or right) ideal in R is generated by an idempotent;

(3) Every finitely generated (left or right ) ideal in R is generated by an idempotent;

(4) Every finitely generated submodule of a finitely generated projective (left or right ) R-module is a direct summand;

(5) Any finitely presented (left or right ) R-module is projective;

(6) Every (left or right ) R-module is R-flat.

Proof. see [444, Lemma 4.15, Theorem 4.16 and Theorem 9.15], [518, Theorem 4.2.9 on page 98]. ⊓ ⊔

Several processes preserve the property of a ring to be von Neumann regular.

Lemma 8.19. (1) If R is von Neumann regular, then the matrix ring M_{n}(R)

is von Neumann regular;

(2) The center of a von Neumann regular ring is von Neumann regular;

(3) A directed union of von Neumann regular rings is von Neumann regular.

Proof. Assertions (1) and (2) follow from [224, Theorem 1.7 and Theorem 1.14], whereas assertion (3) follows directly from the Definition 8.17 of von Neumann regular. ⊓ ⊔

Recall that an R-module is semisimple if any submodule of a module is a direct summand. A ring R is called semisimple if any R-module is semisimple. A ring R is semisimple if and only if R considered as an R-module is semisimple.

Lemma 8.20. Let R be von Neumann regular. Then

(1) Any element in R is either a zero-divisor or a unit;

(2) If a von Neumann regular ring is Noetherian or Artinian, then it is already semisimple.

Proof. (1) This follows from Lemma 8.18 (2). Assertion (2) is proved in [224, page 21]. ⊓ ⊔

In our situation the rings are rings with involution ∗: R → R, sometimes also called ∗-rings. The involution is required to satisfy i.) ∗ ∘ ∗ = id; ii.) ∗(1) = 1, iii.) ∗(r + s) = ∗(r) + ∗(s) and iv.) ∗(r · s) = ∗(s) · ∗(r) for r, s ∈ R. We often write ∗(r) = r^{∗}.

Example 8.21. The complex group ring ℂG is von Neumann regular if and only if G is locally finite. If G is locally finite this follows from Lemma 8.19 (3) since G is the directed union of its finitely generated subgroups and the complex group ring of a finite group is semisimple and hence in particular von Neumann regular.

Suppose that ℂG is von Neumann regular. Let H ⊂ G be a finitely generated subgroup. Let s_{1}, s_{2}, …, s_{r} be a set of generators of H. There is an exact sequence of ℂH-modules ℂH^{n} ⟶^{i} ℂH ⟶^{ϵ} ℂ → 0, where ℂ comes with the trivial H-action, i is given by the (r, 1) matrix (s_{1} − 1, …, s_{r} − 1) and ϵ sends ∑_{h∈H} λ_{h}h to ∑_{h∈H} λ_{h}. This sequence stays exact after applying ℂG ⊗_{ℂH} −. From Lemma 8.18 (5) we conclude that ℂ[G/H] = ℂG ⊗_{ℂH} ℂ is a projective ℂG-module. This implies that H is finite. This shows that G is locally finite.

More information about von Neumann regular rings can be found for instance in [224].

8.2.3 Basic Properties of the Algebra of Affiliated Operators

The next result summarizes the main properties of the algebra of affiliated operators.

Theorem 8.22 (The algebra of affiliated operators).

(1) The set of non-zero-divisors S in N (G) satisfies the right Ore condition and the Ore localization N (G)S^{−1} is canonically isomorphic to U(G);

(2) U(G) is flat as N (G)-module;

(3) U(G) is von Neumann regular;

(4) Let M be a finitely presented N (G)-module. Then U(G) ⊗_{N (G)} M is a finitely generated projective U(G)-module. If additionally dim_{N (G)}(M) = 0, then U(G) ⊗_{N (G)} M = 0;

(5) A sequence P_{0} → P_{1} → P_{2} of finitely generated projective N (G)-modules is weakly exact at P_{1} if and only if the induced sequence U(G)⊗_{N (G)}P_{0} → U(G) ⊗_{N (G)} P_{1} → U(G) ⊗_{N (G)} P_{2} is exact at U(G) ⊗_{N (G)} P_{1};

(6) If q ∈ M_{n}(U(G)) is a projection, then q ∈ M_{n}(N (G)). If e ∈ M_{n}(U(G)) is an idempotent, then there is a projection p ∈ M_{n}(N (G)) such that pe = e and ep = p holds in M_{n}(U(G));

(7) Given a finitely generated projective U(G)-module Q, there is a finitely generated projective N (G)-module P such that U(G) ⊗_{N (G)} P and Q are U(G)-isomorphic;

(8) If P_{0} and P_{1} are two finitely generated projective N (G)-modules, then P_{0} ≅_{N (G)} P_{1} ⇔ U(G) ⊗_{N (G)} P_{0} ≅_{U(G)} U(G) ⊗_{N (G)} P_{1}.

Proof. (1) An element f ∈ N (G) = B(l^{2}(G))^{G} is not a zero-divisor if and only if f is a weak isomorphism (see Lemma 1.13). Hence the right Ore condition for the set S of non-zero-divisors of N (G) is satisfied by Lemma 8.8 (2). From Lemma 8.8 (2) and Lemma 8.15 (1) we obtain a canonical homomorphism of rings with involution N (G)S^{−1} → U(G) which is bijective.

(2) This follows from assertion (1) and Lemma 8.15 (3).

(3) Consider a ∈ U(G). Let a = us be its polar decomposition into a Gequivariant partial isometry u and a positive affiliated operator s. Let f: ℝ → ℝ be the function which sends λ to λ^{−1} if λ > 0 and to zero if λ ≤ 0. Then f(s) ∈ U and sf(s)s = s holds in U(G). If we put b = f(s)u^{∗}, we conclude

aba = usf(s)u^{∗}us = usf(s)s = us = a.

(4) Because of Theorem 6.5 and Theorem 6.7 (3) and (4e) it suffices to prove the claim that dim_{N (G)}(M) = 0 implies U(G) ⊗_{N (G)} M = 0. From Lemma 6.28 we get an exact sequence 0 → N (G)^{n} ⟶^{s} N (G)^{n} → M → 0 with a weak isomorphism s. By Theorem 6.24 (3) ν(s) is a weak isomorphism. We get from Lemma 8.8 (1) that U(G) ⊗_{N (G)} s is an isomorphism. Hence U(G) ⊗_{N (G)} M = 0.

(5) We firstly show for a finitely generated projective N (G)-module P

P = 0 ⇔ U(G) ⊗_{N (G)} P = 0.  (8.23)

Choose an idempotent e: N (G)^{n} → N (G)^{n} whose image is N (G)-isomorphic to P. Then U(G)⊗_{N (G)}e is an idempotent whose image is U(G)-isomorphic to U(G)⊗_{N (G)}P. The matrix in M_{n}(U(G)) describing U(G)⊗_{N (G)}e is the image of the matrix in M_{n}(N (G)) describing e under the injection M_{n}(N (G)) → M_{n}(U(G)) induced by the inclusion of rings N (G) ⊂ U(G). Hence e is zero if and only if U(G) ⊗_{N (G)} e = 0. Therefore (8.23) is true.

Consider a sequence of finitely generated projective N (G)-modules P_{0} ⟶^{f_{0}} P_{1} ⟶^{f_{1}} P_{2}. Since N (G) is semihereditary (see Theorem 6.5 and Theorem 6.7 (1)), the image of f_{1} ∘ f_{0} is a finitely generated projective N (G)- module. We conclude from assertion (2) and (8.23) that f_{1} ∘ f_{0} is zero if and only if U(G) ⊗_{N (G)} f_{1} ∘ U(G) ⊗_{N (G)} f_{0} is zero. Hence we can assume without loss of generality that im(f_{0}) ⊂ ker(f_{1}). Recall that the given sequence P_{0} ⟶^{f_{0}} P_{1} ⟶^{f_{1}} P_{2} is weakly exact if and only if the finitely generated projective N (G)-module P (ker(f_{1})/ im(f_{0})) is trivial and that ker(f_{1})/ im(f_{0}) = P (ker(f_{1})/ im(f_{0})) ⊕ T (ker(f_{1})/ im(f_{0})) (see Definition 6.1, Theorem 6.5 and Theorem 6.7 (3)). Now apply Theorem 6.7 (4e), assertion (2), assertion (4) and (8.23).

(6) Let q ∈ M_{n}(U(G)) be a projection. Then the induced affiliated operator l^{2}(G)^{n} → l^{2}(G)^{n} is bounded on its domain, namely by 1, and hence everywhere defined by Lemma 8.3 (4). Therefore q ∈ M_{n}(N (G)).

Let e ∈ M_{n}(U(G)) be an idempotent. Put z = 1 − (e^{∗} − e)^{2} = 1 + (e^{∗} − e)(e^{∗} − e)^{∗}. This element z is invertible in M_{n}(U(G)) by Lemma 1.13 and Lemma 8.8 (1) because the induced affiliated operator l^{2}(G) → l^{2}(G) is obviously injective. We have z = z^{∗}, ze = ez = ee^{∗}e, ze^{∗} = e^{∗}ee^{∗} = e^{∗}z. We also get z^{−1}e = ez^{−1} and z^{−1}e^{∗} = e^{∗}z^{−1}. Put p = ee^{∗}z^{−1}. Then one easily checks p^{∗} = p, p^{2} = p, pe = e and ep = p.

8.3 Dimension Theory and L^{2}-Betti Numbers over U(G) 329

(7) Let Q be a finitely generated projective U(G)-module. Let e ∈ M_{n}(U(G)) be an idempotent whose image is U(G)-isomorphic to Q. By assertion (6) there is a projection p ∈ M_{n}(N (G)) such that pe = p and ep = e holds in M_{n}(U(G)). Put P = im(p). Then P is a finitely generated projective N (G)- module with U(G) ⊗_{N (G)} P ≅_{U(G)} Q.

(8) Let P_{0} and P_{1} be two finitely generated projective N (G)-modules with U(G) ⊗_{N (G)} P_{0} ≅_{U(G)} U(G) ⊗_{N (G)} P_{0}. Because of assertion (1) there is an N (G)-map f: P_{0} → P_{1} such that U(G) ⊗_{N (G)} f is an isomorphism. By assertion (5) f: P_{0} → P_{1} is a weak isomorphism. Hence P_{0} and P_{1} are N (G)- isomorphic by Lemma 6.28 (1). ⊓ ⊔

8.3 Dimension Theory and L^{2}-Betti Numbers over the Algebra of Affiliated Operators

In this section we construct a dimension function dim_{U(G)} for arbitrary U(G)- modules such that Additivity, Cofinality and Continuity hold and for any N (G)-module M we recapture dim_{N (G)}(M) by dim_{U(G)}(U(G) ⊗_{N (G)} M). This allows us to define L^{2}-Betti numbers over U(G).

Notation 8.24. Denote by L^{dr}(R^{n}) the set of direct summands in R^{n} with the partial ordering ≤ induced by inclusion. Denote by L^{dr}(l^{2}(G)^{n}) the set of Hilbert N (G)-submodules of l^{2}(G)^{n} with the partial ordering ≤ induced by inclusion.

Define for a finitely generated projective U(G)-module Q

dim_{U(G)}(Q) := dim_{N (G)}(P) ∈ [0, ∞),  (8.25)

where P is any finitely generated projective N (G)-module P with U(G)⊗_{N (G)} P ≅_{U(G)} Q. This definition makes sense because of Lemma 8.22 (7) and (8)

Lemma 8.26. (1) Let j: L^{dr}(N (G)^{n}) → L^{dr}(U(G)^{n}) be the map which sends P ⊂ N (G)^{n} to the image of the composition

U(G) ⊗_{N (G)} P ⟶^{U(G)⊗_{N (G)}i} U(G) ⊗_{N (G)} N (G)^{n} ⟶^{l} U(G)^{n},

where i: P → N (G)^{n} is the inclusion and l the obvious isomorphism. Then j preserves the partial ordering and is bijective;

(2) The map k: L^{dr}(N (G)^{n}) → L^{dr}(l^{2}(G)^{n}) induced by the functor ν (see (6.22)) is bijective and preserves the ordering;

(3) For a subset S ⊂ L^{dr}(U(G)^{n}) there is a least upper bound sup(S) ∈ L^{dr}(U(G)^{n}). If S is directed under inclusion, then

dim_{U(G)}(sup(S)) = sup{dim_{U(G)}(P) | P ∈ S}.

The analogous statement holds for N (G) instead of U(G).

Proof. (1) Obviously j preserves the partial ordering. Bijectivity follows from Lemma 8.22 (6).

(2) This is obvious.

(3) Because of assertions (1) and (2) it suffices to show for any subset S ⊂ L^{dr}(l^{2}(G)^{n})) that a least upper bound sup(S) exists in L^{dr}(l^{2}(G)^{n}) and that dim_{N (G)}(sup(S)) = sup{dim_{N (G)}(V) | V ∈ S} holds if S is directed under ≤. We construct sup(S) as the intersection of all Hilbert N (G)-submodules V of l^{2}(G)^{n} for which each element of S is contained in V. Suppose that S is directed under inclusion. Then sup(S) is the same as the closure of the union of all elements in S. Now the second claim follows from Lemma 1.12 (3). This finishes the proof of Lemma 8.26. ⊓ ⊔

Lemma 8.27. The pair (U(G), dim_{U(G)}) satisfies Assumption 6.2.

Proof. We already know that (N (G), dim_{N (G)}) satisfies Assumption 6.2 (see Theorem 6.5). Hence (U(G), dim_{U(G)}) satisfies Assumption 6.2 (1). It remains to prove Assumption 6.2 (2). One easily checks that it suffices to do this for a submodule K ⊂ U(G)^{n} for some positive integer n. Let S = {P | P ⊂ K, P fin. gen. U(G)-module}. Notice that each P ∈ S belongs to L^{dr}(U(G)^{n}) since U(G) is von Neumann regular (see Lemma 8.18 (4) and Lemma 8.22 (3)). By assertion (3) of the previous Lemma 8.26 the supremum sup(S) ∈ L^{dr}(U(G)^{n}) exists and satisfies

dim_{U(G)}(sup(S)) = sup{dim_{U(G)}(P) | P ∈ S}.

Hence it remains to prove \overline{K} = sup(S). Consider f: U(G)^{n} → U(G) with K ⊂ ker(f). Then ker(f) ∈ L^{dr}(U(G)^{n}) and P ⊂ ker(f) for P ∈ S. This implies sup(S) ⊂ ker(f). Since any finitely generated submodule of K belongs to S and hence is contained in sup(S), we get K ⊂ sup(S) and hence \overline{K} ⊂ sup(S).

⊓ ⊔

Definition 8.28. Let M be a U(G)-module. Define its extended von Neumann dimension

dim_{U(G)}(M) ∈ [0, ∞]

by the extension constructed in Definition 6.6 of dim_{U(G)}(Q) which we have introduced for any finitely generated projective U(G)-module Q in (8.25).

Specializing Theorem 6.7 about dimension functions for arbitrary modules to this case and using Lemma 6.28 (3) and Theorem 8.22 (see also [435, Chapter 3]) we obtain

Theorem 8.29. [Dimension function for arbitrary U(G)-modules]. The dimension dim_{U(G)} satisfies Additivity, Cofinality and Continuity. Given any N (G)-module M, we get

dim_{N (G)}(M) = dim_{U(G)}(U(G) ⊗_{N (G)} M).

If P is a projective U(G)-module, then dim_{U(G)}(P) = 0 ⇔ P = 0.

Now we can define L^{2}-Betti numbers working with U(G) instead of N (G). This corresponds in the classical setting for Betti numbers of a CW -complex Y to define the p-th Betti number as the dimension of the rational vector space H_{p}(Y; ℚ) instead of the rank of the abelian group H_{p}(Y; ℤ).

Definition 8.30. Let X be a (left) G-space. Define the singular homology H_{p}^{G}(X; U(G)) of X with coefficients in U(G) to be the homology of the U(G)- chain complex U(G)⊗_{ℤG}C_{∗}^{sing}(X), where C_{∗}^{sing}(X) is the singular chain complex of X with the induced ℤG-structure. Define the p-th L^{2}-Betti number of X by

b_{p}^{(2)}(X; U(G)) := dim_{U(G)}(H_{p}^{G}(X; U(G))) ∈ [0, ∞],

where dim_{U(G)} is the extended dimension function of Definition 8.28.

We conclude from Lemma 8.22 (2) and the above Theorem 8.29

Theorem 8.31. Let X be a G-space with G-action. Then we get

b_{p}^{(2)}(X; N (G)) := b_{p}^{(2)}(X; U(G)).

8.4 Various Notions of Torsion Modules over a Group von Neumann Algebra

In this section we consider three different notions of torsion modules in the category of N (G)-modules and analyse their relationship. We have already introduced for an N (G)-module M its submodule TM in Definition 6.1 as the closure of {0} in M, or, equivalently, as the kernel of the canonical map i(M): M → (M^{∗})^{∗}. Next we define

Definition 8.32. Let M be an N (G)-module. Define N (G)-submodules

T_{dim}M := ⋃ {N | N is an N (G)-submodule of M with dim_{N (G)}(N) = 0};
T_{U}M := ker(j) for j: M → U(G) ⊗_{N (G)} M, m ↦ 1 ⊗ m.

Notice that T_{dim}(M) ⊂ M is indeed an N (G)-submodule by Additivity (see Theorem 6.7 (4b)) and can be characterized to be the largest N (G)- submodule of M of dimension zero. Because of Lemma 8.15 (2) or [484, Corollary II.3.3 on page 57] one can identify T_{U}M with the set of elements m ∈ M for which there is a non-zero-divisor r ∈ N (G) with rm = 0. Notice that

M = TM ⇔ M^{∗} = hom_{N (G)}(M, N (G)) = 0;
M = T_{dim}M ⇔ dim_{N (G)}(M) = 0;
M = T_{U}M ⇔ U(G) ⊗_{N (G)} M = 0.

Lemma 8.33. (1) If M is an N (G)-module, then T_{U}M ⊂ T_{dim}M ⊂ TM;

(2) If M is a finitely generated projective N (G)-module, then T_{U}M = T_{dim}M = TM = 0;

(3) If M is a finitely presented N (G)-module, then T_{U}M = T_{dim}M = TM;

(4) If M is a finitely generated N (G)-module, then T_{dim}M = TM.

Proof. (1) Since U(G) is flat over N (G), the U(G)-map U(G)⊗_{N (G)}(T_{U}M) → U(G) ⊗_{N (G)} M is injective. Since this map at the same time is the zero map, U(G) ⊗_{N (G)} (T_{U}M) is trivial. We get dim_{N (G)}(T_{U}M) = 0 from Theorem 8.29. This implies T_{U}M ⊂ T_{dim}M. Let f: M → N (G) be any N (G)- map. Then f(T_{dim}M) is an N (G)-submodule of dimension zero in N (G) by Additivity (Theorem 6.7 (4b)). By Theorem 6.7 (1) and (4b) each finitely generated N (G)-submodule of f(T_{dim}M) ⊂ N (G) is projective and has dimension zero. Any finitely generated projective N (G)-module of dimension zero is trivial (Lemma 6.28 (3)). Hence f(T_{dim}M) is trivial. This implies T_{dim}M ⊂ TM.

(2) Obviously TP is trivial for a finitely generated projective N (G)-module P. Now apply assertion (1).

(3) Let M be finitely presented. Then M = PM ⊕ TM with finitely generated projective PM by Theorem 6.7 (1). Because of assertions (1) and (2) it suffices to prove T_{U}(TM) = TM. This follows from Theorem 6.7 (4e) and Lemma 8.22 (4).

(4) This follows from Theorem 6.7 (4e). ⊓ ⊔

Example 8.34. We want to construct a finitely generated non-trivial N (G)- module M with the property that T_{U}(M) = 0 and T_{dim}M = M and in particular T_{U}(M) ≠ T_{dim}M. Let I_{1} ⊂ I_{2} ⊂ … ⊂ N (G) be a nested sequence of ideals which are direct summands in N (G) such that dim_{N (G)}(I_{n}) ≠ 1 and lim_{n→∞} dim_{N (G)}(I_{n}) = 1. Let I be the ideal ⋃_{n=1}^{∞} I_{n}. Put M = N (G)/I. Then dim_{N (G)}(M) = 0 by Theorem 6.7 (4b) and (4c). This implies T_{dim}M = M. By Lemma 8.33 (2) we get T_{U}N (G)/I_{n} = 0 for n ≥ 1. Lemma 8.36 (1) below implies that T_{U}M = colim_{n≥1} T_{U}N (G)/I_{n} = 0.

We get for instance for G = ℤ such a sequence of ideals (I_{n})_{n≥1} by the construction in Example 1.14.

Example 8.35. An example of an N (ℤ)-module M with M^{∗} = 0 and dim_{N (G)}(M) = 1 and hence with T_{dim}M ≠ TM is given in the exercises of Chapter 6.

The notion T_{U} has the best properties in comparison with T_{dim} and T.

Lemma 8.36. (1) The functor T_{U} is left exact and commutes with colimits

over directed systems;

(2) An N (G)-module M is cofinal measurable in the sense of [343, Definition 2.1], i.e. all its finitely generated submodules are quotients of finitely presented N (G)-modules of dimension zero, if and only if T_{U}M = M.

(1) Let {M_{i} | i ∈ I} be a directed system of N (G)-modules. Since the functor U(G) ⊗_{N (G)} − is compatible with arbitrary colimits and the functor colim_{i∈I} is exact for directed systems, the canonical map colim_{i∈I} T_{U}M_{i} → T_{U} (colim_{i∈I} M_{i}) is an isomorphism. Since U(G) is flat over N (G) by Theorem 8.22 (2), T_{U} is left exact.

(2) Obviously submodules of cofinal-measurable N (G)-modules are cofinalmeasurable again. Because of Lemma 8.33 (3) and assertion (1) it suffices to prove for a finitely generated N (G)-module M that U(G) ⊗_{N (G)} M = 0 is true if and only if M is the quotient of a finitely presented N (G)-module N with U(G) ⊗_{N (G)} N = 0.

The if-part is obvious, the only-if-part is proved as follows. Let M be a finitely generated U(G)-module with U(G) ⊗_{N (G)} M = 0. Choose an epimorphism f: N (G)^{n} → M. Since U(G) is flat over N (G) by Theorem 8.22 (2), the inclusion i: ker(f) → N (G)^{n} induces an isomorphism U(G) ⊗_{N (G)} ker(f) → U(G) ⊗_{N (G)} N (G)^{n}. Hence we can find an A-map g: N (G)^{n} → ker(f) such that U(G)⊗_{N (G)}g is an isomorphism. Hence coker(i∘g) is a finitely presented N (G)-module with U(G)⊗_{N (G)} coker(i∘g) = 0 which maps surjectively onto M. ⊓ ⊔

8.5 Miscellaneous

Let R be a principal ideal domain with quotient field F. We have already mentioned in Example 6.12 that R together with the usual notion of the rank of a finitely generated free R-module satisfies Assumption 6.2 and that the extended dimension function of Definition 6.6 is given by dim_{R}(M) = dim_{F} (F ⊗_{R} M). Notice that F is the Ore localization of R with respect to the set S of non-zero-divisors and that all the properties of U(G) as stated in Lemma 8.22 and Theorem 8.29 do also hold for F. Moreover, the condition M = T_{U}M translates to the classical notion of a torsion module, namely that F ⊗_{R} M = 0, or, equivalently, that for each element m ∈ M there is an element r ∈ R with r ≠ 0 and rm = 0.

More information about affiliated rings and their regularity properties can be found in [36], [37], [247]. A systematic study of U(G) and dimension functions is carried out in [435, chapters 2 and 3].

Exercises

8.1. A ring R with involution ∗ is called ∗-regular if it is von Neumann regular and for any r ∈ R we have r^{∗}r = 0 ⇔ r = 0. Show that U(G) is ∗-regular.

8.2. Construct a commutative diagram of ℂ-categories

{fin. gen. proj. N (G)-mod.} ⟶^{ν} {fin. gen. Hilb. N (G)-mod.}
↓  ↓
{fin. gen. proj. U(G)-mod.} ⟶^{ν_{U}} {fin. gen. Hilb. N (G)-mod.}_{U}

whose vertical arrows are the obvious inclusions and whose horizontal arrows are equivalences of ℂ-categories.

8.3. Let G be a group. Make sense out of the following chain of inclusions

G ⊂ ℤG ⊂ ℂG ⊂ l^{1}(G) ⊂ C_{r}^{∗}(G) ⊂ N (G) ⊂ l^{2}(G) ⊂ U(G),

where l^{1}(G) is the Banach algebra of formal sums ∑_{g∈G} λ_{g} · g which satisfy ∑_{g∈G} |λ_{g}| < ∞.

8.4. Let f: P → Q be a homomorphism of finitely generated projective U(G)- modules with dim_{U(G)}(P) = dim_{U(G)}(Q). Show that the following assertions are equivalent: i.) f is injective, ii.) f is surjective, iii.) f is bijective.

8.5. Show that the functor T_{dim} is left exact, but the functor T is not left exact.

8.6. Show that the functor T_{dim} does not commute with colimits over directed systems, unless all structure maps are injective.

8.7. Give an example of a non-trivial N (ℤ)-module M and a directed system {M_{i} | i ∈ I} of submodules (directed by inclusion) with the properties that M^{∗} = 0 and M_{i} ≅ N (ℤ) for i ∈ I. Show that this implies TM ≠ ⋃_{i∈I} TM_{i}.

8.8. Given an N (G)-module M, define T′_{U}M by the cokernel of i: M → U(G) ⊗_{N (G)} M, m ↦ 1 ⊗ m. Construct for an exact sequence of N (G)- modules 0 → L → M → N → 0 a natural exact sequence 0 → T_{U}L → T_{U}M → T_{U}N → T′_{U}L → T′_{U}M → T′_{U}N → 0.

8.9. Show for an infinite locally finite group G that

b_{p}^{(2)}(G; N (G)) = b_{p}^{(2)}(G; U(G)) = 0

for all p ≥ 0, but H_{0}^{G}(EG; U(G)) ≠ 0.

Show that we get an N (G)-module M such that

dim_{N (G)}(M) = dim_{U(G)}(U(G) ⊗_{U(G)} M) = 0,

but U(G) ⊗_{U(G)} M ≠ {0}.

9. Middle Algebraic K-Theory and L-Theory of von Neumann Algebras

Introduction

So far we have only dealt with the von Neumann algebra N (G) of a group G. We will introduce and study in Section 9.1 the general concept of a von Neumann algebra. We will explain the decomposition of a von Neumann algebra into different types. Any group von Neumann algebra is a finite von Neumann algebra. A lot of the material of the preceding chapters can be extended from group von Neumann algebras to finite von Neumann algebras as explained in Subsection 9.1.4. In Sections 9.2 and 9.3 we will compute K_{n}(A) and K_{n}(U) for n = 0, 1 in terms of the centers Z(A) and Z(U), where U is the algebra of operators which are affiliated to a finite von Neumann algebra A. The quadratic L-groups L_{n}^{ϵ}(A) and L_{n}^{ϵ}(U) for n ∈ ℤ and the decorations ϵ = p, h, s are determined in Section 9.4. The symmetric L-groups L_{ϵ}^{n}(A) and L_{ϵ}^{n}(U) turn out to be isomorphic to their quadratic counterparts.

In Section 9.5 we will apply the results above to detect elements in the Kand G-theory of the group ring. We will show for a finite normal subgroup H of an arbitrary (discrete) group G that the map Wh(H)^{G} → Wh(G) induced by the inclusion of H into G has finite kernel, where the action of G on Wh(H) comes from the conjugation action of G on H (see Theorem 9.38). We will present some computations of the Grothendieck group G_{0}(ℂG) of finitely generated (not necessarily projective) ℂG-modules. The main result, Theorem 9.65, says for an amenable group G that the rank of the abelian group G_{0}(ℂG) is greater or equal to the cardinality of the set con(G)_{f,cf} of conjugacy classes (g) of elements g ∈ G for which |g| < ∞ and |(g)| < ∞ hold. The map detecting elements in G_{0}(ℂG) is based on the center valued dimension which is related to the Hattori-Stallings rank. We will review the Hattori-Stallings rank, the Isomorphism Conjecture for K_{0}(ℂG) and the Bass Conjecture in Subsection 9.5.2. This chapter needs only a small input from Chaper 8 and is independent of the other chapters.

9.1 Survey on von Neumann Algebras

So far we have only considered the von Neumann algebra N (G) of a group G (see Definition 1.1). In this section we introduce and study the notion of a von Neumann algebra in general.

9.1.1 Definition of a von Neumann Algebra

Let H be a Hilbert space and B(H) be the C^{∗}-algebra of bounded (linear) operators from H to itself, where the norm is the operator norm. The norm topology is the topology on B(H) induced by the operator norm. The strong topology, ultra-weak topology or weak topology respectively is the unique topology such that a subset A ⊂ B(H) is closed if and only if for any net (x_{i})_{i∈I} of elements in A, which converges strongly, ultra-weakly or weakly respectively to an element x ∈ B(H), also x belongs to A (see Subsection 1.1.3 for the various notions of convergence). The norm topology contains both the strong topology and the ultra-weak topology, the strong topology contains the weak topology, the ultra-weak topology contains the weak topology. In general these inclusions are strict and there is no relation between the ultraweak and strong topology [144, I.3.1 and I.3.2]. A map of topological spaces h: X → Y is continuous if and only if for any net (x_{i})_{i∈I} converging to x the net (h(x_{i}))_{i∈I} converges to h(x). Notice that this characterization of continuity is valid for all topological spaces. Only if X satisfies the first countability axiom (i.e. any point has a countable neighborhood basis) one can use sequences (x_{n})_{n≥0} instead of nets, but this axiom will not be satisfied for some of the topologies on B(H) introduced above.

Definition 9.1 (Von Neumann algebra). A von Neumann algebra A is a sub-∗-algebra of B(H) which is closed in the weak topology and contains id: H → H.

The condition weakly closed can be rephrased in a more algebraic fashion as follows. Given a subset M ⊂ B(H), its commutant M′ is defined to be the subset {f ∈ B(H) | fm = mf for all m ∈ M}. If we apply this construction twice, we obtain the double commutant M′′. The proof of the following so called Double Commutant Theorem can be found in [282, Theorem 5.3.1. on page 326].

Theorem 9.2 (Double Commutant Theorem). Let M ⊂ B(H) be a sub-∗-algebra, i.e. M is closed under addition, scalar multiplication, multiplication and under the involution ∗ and contains 0 and 1. Then the following assertions are equivalent.

(1) M is closed in the weak topology;

(2) M is closed in the strong topology;

(3) M = M′′.

In particular the closure of M in the weak topology as well as in the strong topology is M′′.

9.1.2 Types and the Decomposition of von Neumann Algebras

Next we recall the various types of von Neumann algebras. A projection p in a von Neumann algebra A is an element satisfying p^{2} = p and p^{∗} = p. It is called abelian if pAp is a commutative algebra. Two projections p and q are called equivalent p ∼ q, if there is an element u ∈ A with p = uu^{∗} and q = u^{∗}u. We write p ≤ q if qp = p. A projection p is finite, if q ≤ p and q ∼ p together imply p = q, and infinite otherwise. A projection p is properly infinite if p is infinite and cp is either zero or infinite for all central projections c ∈ A. The central carrier c_{p} of a projection p is the smallest central projection c_{p} ∈ A satisfying p ≤ c_{p}.

A von Neumann algebra A is of type I if it has an abelian projection whose central carrier is the identity 1. If A has no non-zero abelian projection but possesses a finite projection with central carrier 1, then A is of type II. If A has no non-zero finite projection, it is of type III. We call A finite, infinite or properly infinite respectively if 1 is a projection, which is finite, infinite or properly infinite respectively. A von Neumann algebra A is of type I_{f} or of type II_{1} respectively if A is finite and of type I or of type II respectively. It is of type I_{∞} or of type II_{∞} respectively if A is properly infinite and of type I or of type II respectively. All von Neumann algebras of type III are properly infinite. A von Neumann algebra can only be of at most one of the types I_{f}, I_{∞}, II_{1}, II_{∞} and III. A von Neumann algebra A is called a factor if its center Z(A) := {a ∈ A | ab = ba for all b ∈ A} consists of {λ · 1 | λ ∈ ℂ}. A factor is of precisely one of the types I_{f}, I_{∞}, II_{1}, II_{∞} or III.

One has the following unique decomposition [283, Theorem 6.5.2 on page 422].

Theorem 9.3. Given a von Neumann algebra A, there is a natural unique decomposition

A = A_{I_{f}} × A_{I_{∞}} × A_{II_{1}} × A_{II_{∞}} × A_{III}

into von Neumann algebras of type I_{f}, I_{∞}, II_{1}, II_{∞} and III. In particular one obtains natural decompositions for the K-groups

K_{n}(A) = K_{n}(A_{I_{f}}) × K_{n}(A_{I_{∞}}) × K_{n}(A_{II_{1}}) × K_{n}(A_{II_{∞}}) × K_{n}(A_{III}).

Lemma 9.4. Let G be a discrete group. Let G_{f} be the normal subgroup of G consisting of elements g ∈ G, whose centralizer has finite index (or, equivalently, whose conjugacy class (g) consists of finitely many elements). Then

(1) The group von Neumann algebra N (G) is of type I if and only if G is virtually abelian; (2) The group von Neumann algebra N (G) is of type II if and only if the

index of G_{f} in G is infinite;

(3) Suppose that G is finitely generated. Then N (G) is of type I_{f} if G is virtually abelian, and of type II_{1} if G is not virtually abelian;

(4) The group von Neumann algebra N (G) is a factor if and only if G_{f} is the trivial group.

Proof. (1) This is proved in [285], [490].

(2) This is proved in [285],[366].

(3) This follows from (1) and (2) since for finitely generated G the group G_{f} has finite index in G if and only if G is virtually abelian.

(4) This follows from [144, Proposition 5 in III.7.6 on page 319]. ⊓ ⊔

9.1.3 Finite von Neumann Algebras and Traces

One of the basic properties of finite von Neumann algebras is the existence of the center valued trace which turns out to be universal.

A finite trace tr: A → ℂ on a von Neumann algebra is a ℂ-linear mapping satisfying tr(ab) = tr(ba) for a, b ∈ A and tr(a) ≥ 0 for a ≥ 0 (i.e. a = bb^{∗} for some b ∈ A). It is called faithful, if for a ∈ A with a ≥ 0 we have tr(a) = 0 ⇒ a = 0. It is called normal if for f ∈ A, which is the supremum with respect to the usual ordering ≤ of positive elements (see 1.7) of some monotone increasing net {f_{i} | i ∈ I} of positive elements in A, we get tr(f) = sup{tr(f_{i}) | i ∈ I}. The next result is taken from [283, Theorem 7.1.12 on page 462, Proposition 7.4.5 on page 483, Theorem 8.2.8 on page 517, Proposition 8.3.10 on page 525, Theorem 8.4.3 on page 532].

Theorem 9.5. Let A be a finite von Neumann algebra on H. There is a map

tr^{u} = tr_{A}^{u}: A → Z(A)

into the center Z(A) of A called the center valued trace or universal trace of A, which is uniquely determined by the following properties:

(1) tr^{u} is a trace with values in the center, i.e. tr^{u} is ℂ-linear, for a ∈ A with a ≥ 0 we have tr^{u}(a) ≥ 0 and tr^{u}(ab) = tr^{u}(ba) for all a, b ∈ A;

(2) tr^{u}(a) = a for all a ∈ Z(A).

The map tr^{u} has the following further properties:

(3) tr^{u} is faithful;

(4) tr^{u} is normal, or, equivalently, tr^{u} is continuous with respect to the ultraweak topology on A;

(5) || tr^{u}(a)|| ≤ ||a|| for a ∈ A;

(6) tr^{u}(ab) = a tr^{u}(b) for all a ∈ Z(A) and b ∈ A; (7) Let p and q be projections in A. Then p ∼ q, if and only if tr^{u}(p) = tr^{u}(q);

(8) Any linear functional f: A → ℂ which is continuous with respect to the norm topology on A and which is central, i.e. f(ab) = f(ba) for all a, b ∈ A, factorizes as

A ⟶^{tr^{u}} Z(A) ⟶^{f|_{Z(A)}} ℂ.

Example 9.6. Let X be a compact space together with a finite measure ν on its Borel-σ-algebra. Let L^{∞}(X, ν) be the Banach algebra of equivalence classes of essentially bounded measurable functions X → ℂ, where two such functions are called equivalent if they only differ on a set of measure zero. It becomes a Banach algebra with the norm

||f||_{∞} = inf{K > 0 | ν({x ∈ X | |f(x)| ≥ K}) = 0}

and the involution coming from complex conjugation. This turns out to be a commutative von Neumann algebra by the obvious embedding

L^{∞}(X, ν) → B(L^{2}(X, ν))

coming from pointwise multiplication. Any commutative von Neumann algebra is isomorphic to L^{∞}(X, ν) for appropriate X and ν [144, Theorem 1 and 2 in I.7.3 on page 132].

Example 9.7. Let G be a group. The right regular representation ρ_{r}: ℂG → B(l^{2}(G)) sends g ∈ G to the operator r_{g^{−1}} : l^{2}(G) → l^{2}(G), u ↦ ug, whereas the left regular representation ρ_{l}: ℂG → B(l^{2}(G)) sends g ∈ G to the operator l_{g}: l^{2}(G) → l^{2}(G), u ↦ gu. The left regular representation is a homomorphism of ℂ-algebras, whereas the right regular representation is an anti-homomorphism of ℂ-algebras, i.e. it respects the scalar multiplication and addition, but respects multiplication only up to changing the order. We get from [283, Theorem 6.7.2 on page 434]

im(ρ_{r})′′ = im(ρ_{l})′ = B(l^{2}(G))^{G}.

Hence N (G) as introduced in Definition 1.1 is the closure of ℂG, which we view as a ∗-subalgebra of B(l^{2}(G)) by the right regular representation ρ_{r}, in B(l^{2}(G)) with respect to the weak or strong topology. The closure of im(ρ_{r}) in B(l^{2}(G)) with respect to the norm topology is called the reduced C^{∗}-algebra C_{r}^{∗}(G) of G. In the special case G = ℤ^{n}, we get C_{r}^{∗}(ℤ^{n}) = C(T^{n}) (compare with Example 1.4), where C(T^{n}) denotes the space of continuous functions from T^{n} to ℂ.

We will later need for the computation of K_{1}(A) the following technical condition. It is always satisfied for a von Neumann algebra acting on a separable Hilbert space and in particular for the group von Neumann algebra N (G) of any countable group G.

Definition 9.8. A von Neumann algebra is countably composable if every orthogonal family of non-zero projections is countable.

9.1.4 Extending Results for Group von Neumann Algebras to Finite von Neumann Algebras

Let A be a finite von Neumann algebra with some faithful finite normal trace tr: A → ℂ. Define a pre-Hilbert structure on A by ⟨a, b⟩ = tr(ab^{∗}). Let l^{2}(A) be the Hilbert completion of A. Denote by ||a|| the induced norm on l^{2}(A). Given a ∈ A, we obtain a linear operator A → A sending b to ab. This operator is bounded with operator norm ||a||. Hence it extends uniquely to a bounded operator ρ_{l}(a): l^{2}(A) → l^{2}(A) satisfying ||ρ_{l}(a)||_{∞} = ||a||. This yields the left regular representation

ρ_{l}: A → B(l^{2}(A)).

Thus we obtain a left A-module structure on l^{2}(A). Analogously we get ρ_{r}(a): l^{2}(A) → l^{2}(A) induced by b ↦ ba. In particular we obtain the right regular representation ρ_{r}: A → B_{A}(l^{2}(A)) from A into the subalgebra B_{A}(l^{2}(A)) of linear bounded A-operators of B(l^{2}(A)). The following result is fundamental for the theory of Hilbert modules over a finite von Neumann algebra (see Dixmier [144, Theorem 1 in I.5.2 on page 80, Theorem 2 in I.6.2 on page 99]).

Theorem 9.9. Let A be a finite von Neumann algebra. Then the right regular representation

ρ_{r}: A → B_{A}(l^{2}(A))

is an isometric anti-homomorphism of ℂ-algebras.

In the special case where A is the group von Neumann algebra N (G) := B(l^{2}(G))^{G}, the inclusion N (G) → l^{2}(G) f → \overline{f^{∗}(1)} induces an isometric isomorphism l^{2}(N (G)) → l^{2}(G) and thus an identification of B_{N (G)}(l^{2}(N (G)) with B(l^{2}(G))^{G}. Under this identification the map ρ_{r} of Theorem 9.9 becomes the the anti-homomorphisms of ℂ-algebras N (G) → N (G) sending f to the operator i ∘ f^{∗} ∘ i, where i: l^{2}(G) → l^{2}(G) sends ∑_{g∈G} λ_{g} · g to ∑_{g∈G} \overline{λ_{g}} · g.

9.2 Middle K-Theory of a Neumann Algebra 341

Remark 9.10. In view of Example 9.7 and Theorem 9.9 it is clear that a lot of the material of the preceding chapters extends from group von Neumann algebras (with the standard trace) to finite von Neumann algebras with a given finite faithful normal trace. For instance, there is an obvious notion of a finitely generated Hilbert A-module and the corresponding category is equivalent to the category of finitely generated projective A-modules as ℂcategory with involution (cf. Theorem 6.24). It is clear how to define the von Neumann dimension for a finitely generated A-module and to extend for instance Theorem 6.5 and Theorem 6.7. Moreover, the definition of the L^{2}- Betti number b_{p}^{(2)}(X; V) of a G-space X (see Definition 6.50) can be extended to the case where an A-ℂG-bimodule V, which is finitely generated projective over A, is given. Then the definition of b_{p}^{(2)}(X; N (G)) is the special case V = N (G), where the N (G)-ℂG-bimodule structure comes from the inclusion of rings ℂG ⊂ N (G). Also the notion of the algebra U(G) of operators affiliated to N (G) extends to a finite von Neumann algebra A. All the nice properties of U(G) carry over to U = U(A), for instance U(A) is von Neumann regular and is the Ore localization of A with respect to the set of non-zero divisors.

For the rest of this chapter we will use some of the results, which we only have proved for group von Neumann algebras, also for finite von Neumann algebras.

9.2 Middle K-Theory of a Neumann Algebra

In this section we define and compute the K-groups K_{0}(A), K_{1}(A) and K_{1}^{inj}(A) for a von Neumann algebra A.

9.2.1 K_{0} of a von Neumann Algebra

Definition 9.11 (Projective class group K_{0}(R)). Let R be an (associative) ring (with unit). Define its projective class group K_{0}(R) to be the abelian group whose generators are isomorphism classes [P] of finitely generated projective R-modules P and whose relations are [P_{0}] + [P_{2}] = [P_{1}] for any exact sequence 0 → P_{0} → P_{1} → P_{2} → 0 of finitely generated projective R-modules. Define G_{0}(R) analogously but replacing finitely generated projective by finitely generated.

One should view K_{0}(R) together with the assignment sending a finitely generated projective R-module P to its class [P] in K_{0}(R) as the universal dimension for finitely generated projective R-modules. Namely, suppose we are given an abelian group and an assignment d which associates to a finitely generated projective R-module an element d(P) ∈ A such that d(P_{0}) + d(P_{2}) = d(P_{1}) holds for any exact sequence 0 → P_{0} → P_{1} → P_{2} → 0 of finitely generated projective R-modules. Then there is precisely one homomorphism ϕ of abelian groups from K_{0}(R) such that ϕ([P]) = d(P) holds for each finitely generated projective R-module. The analogous statement holds for G_{0}(R) if we consider finitely generated R-modules instead of finitely generated projective R-modules.

Definition 9.12 (Center valued trace). Let A be a finite von Neumann algebra and let tr^{u} be its center valued trace (see Theorem 9.5). For a finitely generated projective A-module P define its center valued von Neumann dimension by

dim^{u}(P) := tr^{u}(A) := ∑_{i=1}^{n} tr^{u}(a_{i,i}) ∈ Z(A)^{ℤ/2} = {a ∈ Z(A) | a = a^{∗}}

for any matrix A = (a_{i,j})_{i,j} ∈ M_{n}(A) with A^{2} = A such that im(R_{A}: A^{n} → A^{n}) induced by right multiplication with A is A-isomorphic to P.

The definition of dim^{u}(P) is independent of the choice of A (cf. (6.4)). The matrix A appearing in Definition 9.12 can be chosen to satisfy both A^{2} = A and A^{∗} = A. This follows from Theorem 6.24. Therefore dim^{u}(P) is an element in Z(A)^{ℤ/2} with respect to the ℤ/2-action coming from taking the adjoint.

The next result follows from [283, Theorem 8.4.3 on page 532, Theorem 8.4.4 on page 533].

Theorem 9.13 (K_{0} of finite von Neumann algebras). Let A be a finite von Neumann algebra.

(1) The following statements are equivalent for two finitely generated projective A-modules P and Q:

(a) P and Q are A-isomorphic;

(b) P and Q are stably A-isomorphic, i.e. P ⊕ V and Q ⊕ V are Aisomorphic for some finitely generated projective A-module V;

(c) dim^{u}(P) = dim^{u}(Q);

(d) [P] = [Q] in K_{0}(A);

(2) The center valued dimension induces an injection

dim^{u}: K_{0}(A) → Z(A)^{ℤ/2} = {a ∈ Z(A) | a = a^{∗}},

where the group structure on Z(A)^{ℤ/2} comes from the addition. If A is of type II_{1}, this map is an isomorphism. ⊓ ⊔

Example 9.14. Let A be an abelian von Neumann algebra. Then it is of the shape L^{∞}(X, ν) as explained in Example 9.6. Let L^{∞}(X, ν, ℤ) be the abelian subgroup of L^{∞}(X, ν) consisting of elements which can be represented by bounded measurable functions f: X → ℤ. We claim that the center-valued dimension induces an isomorphism

dim^{u}: K_{0}(L^{∞}(X, ν)) → L^{∞}(X, ν, ℤ).

This follows from Theorem 9.13 (2) and the following result taken from [331, Lemma 4.1]. Namely, for an abelian von Neumann algebra A and an Ahomomorphism t: A^{n} → A^{n}, which is normal, i.e. t and t^{∗} commute, there exists a unitary morphism u: A^{n} → A^{n} such that u^{∗} ∘ t ∘ u is diagonal.

Theorem 9.15. Let A be a properly infinite von Neumann algebra. Then K_{0}(A) = 0.

Proof. We firstly show that for a properly infinite projection in a von Neumann algebra A the class [im(p)] ∈ K_{0}(A) of the finitely generated projective A-module im(p) is zero. This follows essentially from [283, Lemma 6.3.3 on page 411] from which we get a projection q satisfying q ≤ p and q ∼ p − q ∼ p. Since equivalent projections have A-isomorphic images and im(p) is A-isomorphic to im(p − q) ⊕ im(q), we get in K_{0}(A)

[im(p)] = [im(q)] + [im(p − q)] = [im(p)] + [im(p)] ⇒ [im(p)] = 0.

Given any projective A-module P, choose a projection p ∈ M_{n}(A) such that im(p: A^{n} → A^{n}) is A-isomorphic to P. Notice that M_{n}(A) is again a von Neumann algebra.

Next we want to show that p ⊕ 1 ∈ M_{n+1}(A) is properly infinite. The center Z(M_{n+1}(A)) is {c · I_{n+1} | c ∈ Z(A)}, where I_{n+1} is the identity matrix in M_{n+1}(A). This follows for instance from Theorem 9.2. Hence any central projection c in M_{n+1}(A) is of the form c = q · I_{n+1} for some central projection q ∈ A. If c(p ⊕ 1) is not zero, then q ∈ A is a non-zero central projection. Since A is by assumption properly infinite, q is infinite. But then also c is infinite by [283, Theorem 6.3.8 on page 414]. Hence p ⊕ 1 is properly infinite.

The Morita isomorphism μ: K_{0}(M_{n+1}(A)) → K_{0}(A) is defined by μ([P]) = [A^{n+1} ⊗_{M_{n+1}(A)} P]. In particular it maps the class of im(p ⊕ 1) to the class of im(p)⊕im(1) = P ⊕A. Since we have already shown that properly infinite projections represent zero in K_{0}, we conclude [A] = 0 ∈ K_{0}(A) and [im(p ⊕ 1)] = 0 ∈ K_{0}(M_{n+1}(A)). Hence we get in K_{0}(A)

[P] = [im(p) ⊕ im(1)] − [A] = 0 − 0 = 0.⊓ ⊔

In view of Theorem 9.3 and Theorem 9.15 we get for any von Neumann algebra A that K_{0}(A) = K_{0}(A_{I_{f}}) ⊕ K_{0}(A_{II_{1}}) and hence the computation of K_{0}(A) follows from Theorem 9.13.

9.2.2 K_{1} of a von Neumann Algebra

Definition 9.16 (K-group K_{1}(R)). Let R be a ring. Define K_{1}(R) to be the abelian group whose generators are conjugacy classes [f] of automorphisms f: P → P of finitely generated projective R-modules with the following relations: i.) Given a commutative diagram of finitely generated projective R-modules

0 ⟶ P_{1} ⟶^{i} P_{2} ⟶^{p} P_{3} ⟶ 0
f_{1}↓  ^{f_{2}}↓  ^{f_{3}}↓
0 ⟶ P_{1} ⟶^{i} P_{2} ⟶^{p} P_{3} ⟶ 0

with exact rows and automorphisms as vertical arrows, we get [f_{1}]+[f_{3}] = [f_{2}]. ii.) Given automorphisms f, g: P → P of a finitely generated projective Rmodule P, we get [g ∘ f] = [f] + [g].

Define K_{1}^{inj}(R) analogously by replacing automorphisms by injective endomorphisms everywhere.

We leave it to the reader to check that the definition of K_{1}(R) by K_{1}(R) := GL(R)/[GL(R), GL(R)] in Subsection 3.1.1 coincides with the one of Definition 9.16.

Let Z(A)^{inv} be the abelian group of elements in the center of A which are invertible. The set of non-zero-divisors in Z(A) is an abelian monoid under multiplication and we denote by Z(A)^{w} the abelian group which is obtained from this abelian monoid by the Grothendieck construction. We can identify the abelian von Neumann algebra Z(A) with L^{∞}(X, ν) (see Example 9.6). Then Z(A)^{inv} becomes the space of equivalence classes of measurable functions f: X → ℂ which are essentially bounded from below and above, i.e. there are positive constants k and K such that {x ∈ X | |f(x)| ≤ k} and {x ∈ X | |f(x)| ≥ K} have measure zero. The space Z(A)^{w} becomes the space of equivalence classes of measurable functions from f: X → ℂ for which f^{−1}(0) has measure zero. Notice that Z(A)^{w} = U(Z(A))^{inv} = Z(U(A))^{inv}. We get from [344, Theorem 2.1]

Theorem 9.17 (K_{1} of von Neumann algebras of type I_{f}). Let A be a von Neumann algebra of type I_{f}. Then there is a so called normalized determinant

det_{norm} : M_{k}(A) → Z(A)

with the following properties:

(1) If A ∈ M_{k}(A) satisfies det_{norm}(A) = 1, then A is a product of two commutators in GL_{k}(A);

(2) The normalized determinant induces isomorphisms

det_{norm} : K_{1}(A) ⟶^{≅} Z(A)^{inv};
det_{norm} : K_{1}^{inj}(A) ⟶^{≅} Z(A)^{w},

which are compatible with the involutions.

We can use the center valued trace to define the center valued Fuglede-Kadison determinant

det_{F K}: GL_{k}(A) → Z(A)^{+,inv}, A ↦ exp (\frac{1}{2} · tr^{u}(ln(A^{∗}A))) ,

where Z(A)^{+,inv} is the (multiplicative) abelian group of elements in the center of A which are both positive and invertible. We get from [344, Theorem 3.3]

Theorem 9.18 (K_{1} of von Neumann algebras of type II_{1}). Let A be a countably composable von Neumann algebra of type II_{1}.

(1) If A ∈ GL_{k}(A) satisfies det_{F K}(A) = 1, then A is a product of nine commutators in GL_{k}(A);

(2) The center valued Fuglede-Kadison determinant induces an isomorphism

det_{F K}: K_{1}(A) ⟶^{≅} Z(A)^{+,inv};

9.3 Middle K-Theory of the Algebra of Affiliated Operators 345

(3) K_{1}^{inj}(A) = 0.

Theorem 9.18 (2) has also been proved in [179], provided that A is a factor. We get from [344, Theorem 4.2]

Theorem 9.19. Let A be a countably composable properly infinite von Neumann algebra. Then K_{1}(A) = K_{1}^{inj}(A) = 0.

In view of Theorem 9.3, Theorem 9.18 (3) and Theorem 9.19 we get for a countably composable von Neumann algebra A that K_{1}(A) = K_{1}(A_{I_{f}}) ⊕ K_{1}(A_{II_{1}}) and K_{1}^{inj}(A) = K_{1}^{inj}(A_{I_{f}}) and hence the computation of K_{1}(A) and K_{1}^{inj}(A) follows from Theorem 9.17 (2) and Theorem 9.18 (2).

The condition countably composable appearing in Theorem 9.18 and Theorem 9.19 is purely technical, it may be possible that it can be dropped.

9.3 Middle K-Theory of the Algebra of Affiliated Operators

In this section we compute the K-groups K_{0}(U) and K_{1}(U) of the algebra U of operators affiliated to a finite von Neumann algebra A and deal with a part of the localization sequence associated to A → U.

Theorem 9.20 (K-groups of U). Let A be a finite von Neumann algebra and U be the algebra of affiliated operators. Then

(1) The map i_{∗}: K_{0}(A) → K_{0}(U) induced by the inclusion i: A → U is an isomorphism;

(2) There is a natural isomorphism

j: K_{1}^{inj}(A) ⟶^{≅} K_{1}(U).

Proof. (1) This follows from Theorem 8.22 (7) and (8). (2 ) Since for an injective endomorphism f: P → P of a finitely generated projective A-module P the induced map U ⊗_{A} f: U ⊗_{A} P → U ⊗_{A} P is a U-isomorphism (see Lemma 6.28 (2) and Theorem 8.22 (5)), we obtain a natural map j: K_{1}^{inj}(A) → K_{1}(U). We define an inverse k: K_{1}(U) → K_{1}^{inj}(A) as follows. Let η ∈ K_{1}^{inj}(U) be an element for which there is a U-automorphism f: U^{n} → U^{n} with η = [f]. By Lemma 8.8 we can choose injective endomorphisms a, b: A^{n} → A^{n} such that U ⊗_{A} a = f ∘ U ⊗_{A} b (identifying U ⊗_{A} A^{n} and U^{n}). Define k(η) = [a] − [b]. We leave it to the reader to verify that k is well-defined and an inverse of j. ⊓ ⊔

Notice that Theorem 9.20 together with the results of Subsections 9.2.1 and 9.2.2 give the complete computation of K_{n}(U) for n = 0, 1, provided that A_{II} is countably composable.

Definition 9.21. Let K_{0}(A → U) be the abelian group whose generators are isomorphism classes of finitely presented A-modules M with U ⊗_{A} M = 0 and whose relations are [M_{0}] + [M_{2}] = [M_{1}] for any exact sequence 0 → M_{0} → M_{1} → M_{2} → 0 of such A-modules.

Recall that the following conditions for an A-module M are equivalent by Lemma 6.28 (4) and Lemma 8.33 (3): i.) M is finitely presented with U ⊗_{A} M = 0, ii.) M is finitely presented with dim_{A}(M) = 0, iii.) M has a resolution 0 → A^{n} → A^{n} → M → 0, iv.) M has a resolution 0 → P_{0} → P_{1} → M → 0 for finitely generated projective A-modules and U ⊗_{A} M = 0.

Let S ⊂ R be a multiplicatively closed subset of the ring R satisfying the Ore condition (see Definition 8.14). Provided that S contains no zero-divisors, there is an exact localization sequence associated to an Ore localization R → RS^{−1} [39]

K_{1}(R) ⟶^{i_{1}} K_{1}(RS^{−1}) ⟶^{j} K_{0}(R → RS^{−1}) ⟶^{k} K_{0}(R) ⟶^{i_{0}} K_{0}(RS^{−1}). (9.22)

Here i_{1} and i_{0} are induced by the canonical map i: R → RS^{−1} and K_{0}(R → RS^{−1}) is defined in terms of R-modules M which possess a resolution 0 → P_{1} → P_{0} → M → with finitely generated projective R-modules P_{0} and P_{1} and satisfy RS^{−1} ⊗_{R} M = 0. The map k sends the class of such an R-module M to [P_{0}] − [P_{1}]. The class of an automorphism f: (RS^{−1})^{n} → (RS^{−1})^{n} is sent by j to [coker(a)] − [coker(b)] for any R-endomorphisms a, b: R^{n} → R^{n}, for which RS^{−1} ⊗_{R} a and RS^{−1} ⊗_{R} b are bijective and satisfy f = RS^{−1} ⊗_{R} a ∘ (RS^{−1} ⊗_{R} b)^{−1}.

Lemma 9.23. Let A be a finite von Neumann algebra. Then

(1) The localization sequence (9.22) yields for A and its Ore localization U (see Theorem 8.22 (1)) the exact sequence

K_{1}(A) ⟶^{i_{1}} K_{1}(U) ⟶^{j} K_{0}(A → U) ⟶^{j} K_{0}(A) ⟶^{i_{1}} K_{0}(U);

It splits into an exact sequence

K_{1}(A) ⟶^{i_{1}} K_{1}(U) ⟶^{j} K_{0}(A → U) → 0  (9.24)

and an isomorphism

i_{0} : K_{0}(A) ⟶^{≅} K_{0}(U);

(2) The localization sequence is the direct product of the localization sequences of the summands A_{I_{f}} and A_{II_{1}} appearing in the decomposition A = A_{I_{f}} × A_{II_{1}} (see Theorem 9.3);

(3) If A is of type I_{f}, then the exact sequence (9.24) is isomorphic to the short exact sequence

0 → Z(A)^{inv} → Z(A)^{w} → Z(A)^{w}/Z(A)^{inv} → 0.

9.4 L-Theory of a von Neumann Algebra 347

In particular K_{0}(A → U) = Z(A)^{w}/Z(A)^{inv}; (4) If A is of type II_{1} and countably composable, then the exact sequence

(9.24) becomes

Z(A)^{+,inv} → 0 → 0 → 0.

In particular K_{0}(A → U) = 0.

Proof. (1) This follows from Theorem 9.20 (1).

(2) This is obvious.

(3) This follows from Theorem 9.17 and Theorem 9.20 (2).

(4) This follows from Theorem 9.18 and Theorem 9.20 (2). ⊓ ⊔

9.4 L-Theory of a von Neumann Algebra and the Algebra of Affiliated Operators

In this section we give the computation of the L-groups of a von Neumann algebra A and of the algebra U of affiliated operators. The definitions of the various decorated quadratic and symmetric L-groups L_{n}^{ϵ}(R) and L_{ϵ}^{n}(R) for a ring R with involution ∗: R → R are given for instance in [429], [430]. The decoration ϵ = p or ϵ = h respectively means that the underlying modules are finitely generated projective or finitely generated free respectively. If we write ϵ = s, we mean the L-groups with respect to the trivial subgroup in K̃_{1}.

Before we state the result, we need some preparation. A non-singular symmetric form a: P → P^{∗} on a finitely generated projective (left) R-module is an R-isomorphism P → P^{∗} such that the composition P ⟶^{i} (P^{∗})^{∗} ⟶^{a^{∗}} P^{∗} is equal to a, where i is the canonical isomorphism and the involution on R is used to transform the canonical right module structure on P^{∗} to a left module structure. Two symmetric non-singular forms a: P → P^{∗} and b: Q → Q^{∗} are isomorphic if there is an R-isomorphism f: P → Q satisfying f^{∗} ∘ b ∘ f = a. Given a finitely generated projective R-module P, the associated hyperbolic non-singular symmetric form h(P): P ⊗P^{∗} → (P ⊗P^{∗})^{∗} = P^{∗}⊕P is given by

the matrix ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). Two non-singular symmetric forms are called equivalent

if they become isomorphic after adding hyperbolic forms. The Witt group of equivalence classes of non-singular symmetric forms with the addition coming from the orthogonal sum can be identified with L^{0}(R) [429, Proposition 5.1 on page 160]. The analogous statement is true for L_{h}^{0}(R) or L_{s}^{0}(R) respectively if one considers only non-singular symmetric forms a: F → F^{∗} for finitely generated free R-modules or non-singular symmetric forms a: R^{n} → (R^{n})^{∗} respectively such that the element in K̃_{1}(R) given by the composition of a with the standard isomorphism

i: (R^{n})^{∗} → R^{n},  f ↦ (f(e_{1})^{∗}, f(e_{2})^{∗}, …, f(e_{n})^{∗})

vanishes. If A is a von Neumann algebra (actually it is enough to require A to be a C^{∗}-algebra), there are maps

sign^{(2)} : L_{p}^{0}(A) → K_{0}(A);  (9.25)
ι: K_{0}(A) → L_{p}^{0}(A),  (9.26)

which turn out to be inverse to one another [442, Theorem 1.6 on page 343]). The map ι of (9.26) above sends the class [P] ∈ K_{0}(A) of a finitely generated projective A-module P to the class of \overline{μ}: P → P^{∗} coming from some inner product μ on P (see Section 6.2). Such an inner product exists and the class of \overline{μ}: P → P^{∗} in L_{p}^{0}(A) is independent of the choice of the inner product by Lemma 6.23.

Next we define sign^{(2)}([a]) for the class [a] ∈ L^{0}(A) represented by a nonsingular symmetric form a: P → P^{∗}. Choose a finitely generated projective A-module Q together with an isomorphism u: A^{n} → P ⊕ Q. Let i: (A^{n})^{∗} → A^{n} be the standard isomorphism. Let \overline{a}: A^{n} → A^{n} be the endomorphism i ∘ u^{∗} ∘ (a + 0) ∘ u. We get by spectral theory projections χ_{(0,∞)}(\overline{a}): A^{n} → A^{n} and χ_{(−∞,0)}(\overline{a}): A^{n} → A^{n}. Define P_{+} and P_{−} to be image of χ_{(0,∞)}(\overline{a}) and χ_{(−∞,0)}(\overline{a}). Put sign^{(2)}([a]) = [P_{+}]−[P_{−}]. We leave it to the reader to check that this is well defined. The non-singular symmetric form a: P → P^{∗} is isomorphic to the orthogonal sum of a_{+} : P_{+} → P_{+}^{∗} and a_{−}: P_{−} → P_{−}^{∗}, where a_{+} and −a_{−} come from inner products. This implies that sign^{(2)} and ι are inverse to one another. One can define analogously isomorphisms, inverse to one another,

sign^{(2)} : L_{p}^{0}(U) → K_{0}(U);  (9.27)
ι: K_{0}(U) → L_{p}^{0}(U).  (9.28)

Let R be a ring with involution. The involution on R induces involutions on the reduced K-groups K̃_{i}(R) for i = 0, 1 which are defined as the cokernel of the homomorphisms K_{i}(ℤ) → K_{i}(R) induced by the obvious ring homomorphism ℤ → R. Denote by Ĥ^{n}(ℤ/2; K̃_{i}(R)) the Tate cohomology of the group ℤ/2 with coefficients in the ℤ[ℤ/2]-module K̃_{i}(R). For any ℤ[ℤ/2]- module M, Ĥ^{n}(ℤ/2; M) is 2-periodic with

Ĥ^{0}(ℤ/2; M) = ker(1 − t: M → M)/ im(1 + t: M → M);
Ĥ^{1}(ℤ/2; M) = ker(1 + t: M → M)/ im(1 − t: M → M),

where t ∈ ℤ/2 is the generator. We get long exact Rothenberg sequences [429, Proposition 9.1 on page 181]

… → Ĥ^{0}(ℤ/2; K̃_{0}(R)) → L_{h}^{1}(R) → L_{p}^{1}(R) → Ĥ^{1}(ℤ/2; K̃_{0}(R))
→ L_{h}^{0}(R) → L_{p}^{0}(R) ⟶^{j} Ĥ^{0}(ℤ/2; K̃_{0}(R)) → …,  (9.29)

where j sends the class of a symmetric non-singular form f: P → P^{∗} to the class of the element [P] ∈ K̃_{0}(R), and

… → Ĥ^{0}(ℤ/2; K̃_{1}(R)) → L_{s}^{1}(R) → L_{h}^{1}(R) → Ĥ^{1}(ℤ/2; K̃_{1}(R))
→ L_{s}^{0}(R) → L_{h}^{0}(R) ⟶^{k} Ĥ^{0}(ℤ/2; K̃_{1}(R)) → …,  (9.30)

where k sends the class of a symmetric non-singular form a: F → F^{∗} to the class of the element [i∘h^{∗} ∘a∘h] ∈ K̃_{1}(R) for any R-isomorphism h: R^{n} → F and i: (R^{n})^{∗} ⟶^{≅} R^{n} the standard isomorphism. There are also Rothenberg sequences for the quadratic L-groups, just replace the symmetric L-groups in (9.29) and (9.30) by the quadratic versions.

Theorem 9.31 (L-groups of von Neumann algebras). Let A be a von Neumann algebra. If A is finite, let U = U(A) be the algebra of affiliated operators. Then

(1) The symmetrization maps

L_{ϵ}^{n}(A) ⟶^{≅} L_{n}^{ϵ}(A);
L_{ϵ}^{n}(U) ⟶^{≅} L_{n}^{ϵ}(U)

are isomorphisms for n ∈ ℤ and ϵ = p, h, s;

(2) The quadratic L-groups are 2-periodic, i.e. there are natural isomorphisms

L_{n}^{ϵ}(A) ⟶^{≅} L_{n+2}^{ϵ}(A);
L_{n}^{ϵ}(U) ⟶^{≅} L_{n+2}^{ϵ}(U)

for n ∈ ℤ and ϵ = p, h, s and analogously for the symmetric L-groups;

(3) The L^{2}-signature maps sign^{(2)} defined in (9.25) and (9.27) and the maps ι defined in (9.26) and (9.28) are isomorphisms, inverse to one another, and yield a commutative square of isomorphisms

L_{p}^{0}(A) ⟶_{≅}^{sign^{(2)}} K_{0}(A)
↓≅  ↓≅
L_{p}^{0}(U) ⟶_{≅}^{sign^{(2)}} K_{0}(U)

(4) We have L_{p}^{1}(A) = 0 and L_{p}^{1}(U) = 0;

(5) If A is of type II_{1}, then for n ∈ ℤ the diagram of natural maps

0 ⟶ ℤ/2 ⟶ L_{h}^{0}(A) ⟶ L_{p}^{0}(A) ⟶ 0
↓^{id}  ↓≅  ↓≅
0 ⟶ ℤ/2 ⟶ L_{h}^{0}(U) ⟶ L_{p}^{0}(U) ⟶ 0

is commutative, the vertical maps are isomorphisms and the rows are exact. We have L_{h}^{1}(A) = L_{h}^{1}(U) = 0. If A is countably composable and of type II_{1}, then the natural maps appearing in the commutative square

L_{s}^{n}(A) ⟶^{≅} L_{h}^{n}(A)
↓≅  ↓≅
L_{s}^{n}(U) ⟶^{≅} L_{h}^{n}(U)

are isomorphisms for n ∈ ℤ;

(6) Suppose that A is of type I_{f}. Then the natural map L_{ϵ}^{n}(A) → L_{ϵ}^{n}(U) is bijective for n ∈ ℤ and ϵ = p, h, s. We have L_{h}^{1}(A) = 0. Let l: L_{h}^{0}(A) → K_{0}(A) be the composition of the natural map i: L_{h}^{0}(A) → L_{p}^{0}(A) with the isomorphism sign^{(2)} : L_{p}^{0}(A) ⟶^{≅} K_{0}(A). If [A] ∈ 2·K_{0}(A), then we get an exact sequence

0 → ℤ/2 → L_{h}^{0}(A) ⟶^{l_{1}} 2 · K_{0}(A) → 0,

where l_{1} is induced by l. If [A] ∉ 2 · K_{0}(A), then l is injective and its image is generated by 2 · K_{0}(A) and [A]. If K_{1}(ℤ) → K_{1}(A) is trivial, then we get an exact sequence

0 → L_{s}^{0}(A) → L_{h}^{0}(A) → {f ∈ Z(A)^{inv} | f^{2} = 1} → 0

and L_{1}^{s}(A) = 0. Suppose that K_{1}(ℤ) → K_{1}(A) is not trivial. Let κ be the subgroup of {f ∈ Z(A)^{inv} | f^{2} = 1} generated by the image of [− id: A → A] under the isomorphism det_{norm} : K_{1}(A) ⟶^{≅} Z(A)^{inv} of Theorem 9.17. Then we obtain an exact sequence

0 → L_{s}^{0}(A) → L_{h}^{0}(A) → {f ∈ Z(A)^{inv} | f^{2} = 1}/κ → 0

and L_{s}^{1}(A) ≅ ℤ/2;

(7) If A is properly infinite, then L_{p}^{n}(A) and L_{h}^{n}(A) vanish for n ∈ ℤ. If A is countably composable and properly infinite, then L_{s}^{n}(A) vanishes for n ∈ ℤ.

Proof. (1) This follows from the fact that 2 is invertible in A and U [429, Proposition 3.3 on page 139]. The proof is given there only for L^{p} but applies also to L^{h} and L^{s}.

(2) This follows from assertion (1) and from the fact that i = \sqrt{−1} belongs to Z(A) and Z(U) and is sent under the involution to −i (see [429, Proposition 4.3 on page 150]. The proof is given there only for L^{p} but applies also to L^{h} and L^{s}.

(3) This follows from the constructions and definitions and from Theorem 9.20 (1). The map sign^{(2)} : L_{p}^{0}(A) → K_{0}(A) is defined and bijective not only for a von Neumann algebra A, but also for a C^{∗}-algebra A (see [442, Theorem 1.6 on page 343]).

(4) The L-group L_{p}^{1}(A) is isomorphic to the topological K-group K_{top}^{1}(A) (for any C^{∗}-algebra A) [442, Theorem 1.8 on page 347]. For a von Neumann algebra K_{1}^{top}(A) is trivial. [51, Example 8.1.2 on page 67], [514, Example 7.1.11 on page 134].

Since U(G) is von Neumann regular (see Theorem 8.22 (3)), any finitely generated submodule of a finitely generated projective module is a direct summand (see Lemma 8.18 (4)). Hence the argument that L_{1}^{p}(R) vanishes for semisimple rings in [428] carries over to U. One could also argue by doing surgery on the inclusion of H_{0}(C_{∗}) → C_{∗} in the sense of [429, Section 4].

(5) We conclude from Theorem 9.13 (2) and Theorem 9.18 (2) that the involution on K_{i}(A) is trivial and 2 · id: K_{i}(A) ⟶^{≅} K_{i}(A) is bijective for i = 0, 1. Hence Ĥ^{n}(ℤ/2; K_{i}(A)) = 0 for i, n ∈ {0, 1}. The natural map K_{1}(A) → K̃_{1}(A) is bijective by Theorem 9.18 (2). We get a short exact sequence 0 → K_{0}(ℤ) = ℤ → K_{0}(A) → K̃_{0}(A) → 0 from Theorem 9.13 (2). It induces a long exact sequence of Tate cohomology groups. This implies that Ĥ^{1}(ℤ/2, K̃_{0}(A)) = ℤ/2, Ĥ^{0}(ℤ/2, K̃_{0}(A)) = 0 and Ĥ^{n}(ℤ/2, K̃_{1}(A)) = 0 for n = 0, 1. We conclude from Theorem 9.18 (3) and Theorem 9.20 that the natural map Ĥ^{n}(ℤ/2, K̃_{i}(A)) → Ĥ^{n}(ℤ/2, K̃_{i}(U)) is bijective for n, i ∈ {0, 1}. Now assertion (5) follows from the Rothenberg sequences (9.29) and (9.30).

(6) We have already proved in assertion (3) and (4) that i: L_{p}^{0}(A) → L_{p}^{0}(U) is bijective for all n.

The involution on K_{0}(A) is trivial and multiplication with 2 is injective by Theorem 9.13 (2). This implies

Ĥ^{0}(ℤ/2; K_{0}(A)) = K_{0}(A)/2 · K_{0}(A);
Ĥ^{1}(ℤ/2; K_{0}(A)) = 0.

The long exact Tate cohomology sequence associated to 0 → K_{0}(ℤ) = ℤ → K_{0}(A) → K̃_{0}(A) → 0 yields the exact sequence

0 → Ĥ^{1}(ℤ/2; K̃_{0}(A)) → ℤ/2 ⟶^{η} K_{0}(A)/2 · K_{0}(A) → Ĥ^{0}(ℤ/2; K̃_{0}(A)) → 0,

where η sends the generator of ℤ/2 to the class of [A]. The natural map Ĥ^{n}(ℤ/2; K̃_{0}(A)) → Ĥ^{n}(ℤ/2; K̃_{0}(U)) is bijective for n = 0, 1 by Theorem 9.20 (1). Hence the part of assertion (6) for L_{h}^{n} follows from assertions (3),

(4) and from the Rothenberg sequence (9.29). It remains to treat L_{s}^{n}. We conclude from Example 9.6 applied to Z(A), Theorem 9.17 (2) and

Theorem 9.20 (2) that det_{norm} induces isomorphisms

Ĥ^{0}(ℤ/2; K_{1}(A)) = {f ∈ Z(A) | f^{2} = 1};
Ĥ^{1}(ℤ/2; K_{1}(A)) = 0,

and that the natural map Ĥ^{n}(ℤ/2; K_{1}(A)) ⟶^{≅} Ĥ^{n}(ℤ/2; K_{1}(U)) is bijective for n = 0, 1. We begin with the case where K_{1}(ℤ) → K_{1}(A) is trivial. Then the natural maps K_{1}(A) → K̃_{1}(A) and K_{1}(U) → K̃_{1}(U) are bijective. The Rothenberg sequence (9.30) reduces to

0 → L_{s}^{0}(A) → L_{h}^{0}(A) ⟶^{k} {f ∈ Z(A) | f^{2} = 1} → L_{s}^{1}(A) → 0.

By inspecting the proof of Theorem 9.17 in [344, Theorem 2.1] one checks that any element in x ∈ K_{1}(A) with x^{∗} = x can be represented by an element a ∈ A^{inv} with a^{∗} = a. (Notice that this is not completely obvious since the composition of the canonical map Z(A)^{inv} → K_{1}(A) with det_{norm} : K_{1}(A) → Z(A)^{inv} is not the identity in general.) Hence the map k in the Rothenberg sequence above is surjective and assertion (6) follows.

It remains to treat the case where K_{1}(ℤ) → K_{1}(A) is not trivial. The exact Tate cohomology sequence associated to the exact sequence 0 → K_{1}(ℤ) = {±1} → K_{1}(A) → K̃_{1}(A) → 0 and the computations above yield a short exact sequence

0 → {f ∈ Z(A) | f^{2} = 1}/κ ⟶^{i} Ĥ^{0}(ℤ/2; K̃_{1}(A)) → {±1} → 0  (9.32)

and imply Ĥ^{1}(ℤ/2; K̃_{1}(A)) = 0 and that the natural map Ĥ^{n}(ℤ/2; K̃_{1}(A)) ⟶^{≅} Ĥ^{n}(ℤ/2; K̃_{1}(U)) is bijective for n = 0, 1. The Rothenberg sequence (9.30) reduces to

0 → L_{s}^{0}(A) → L_{h}^{0}(A) ⟶^{k} Ĥ^{0}(ℤ/2; K̃_{1}(A)) → L_{s}^{1}(A) → 0.

Since any element in x ∈ K_{1}(A) with x^{∗} = x can be represented by an element a ∈ A^{inv} with a^{∗} = a, the image of k contains the image of the map i appearing in the exact sequence (9.32). From the definition of k one concludes that im(k) ⊂ im(i). Namely, for a symmetric non-degenerate symmetric form f: P → P^{∗} or a: F → F^{∗} respectively the element [P] ∈ K̃_{0}(R) or [i∘h^{∗} ∘a∘ h] ∈ K̃_{1}(R) respectively, which appears in the definition of the image under k of the element [f] ∈ L_{p}^{0}(R) or [a] ∈ L_{h}^{0}(R) respectively, lifts to an element in the unreduced K-group, which is fixed under the involution. Hence we get im(k) = im(i). Now assertion (6) follows.

(7) We already know that K_{i}(A) = 0 for i = 0, 1 (see Theorem 9.15 and Theorem 9.19). Now the claim follows for ϵ = p from assertions (2), (3) and (4). The claim for the other decorations ϵ = h, s is a direct consequence of the Rothenberg sequences (9.29) and (9.30). This finishes the proof of Theorem 9.31. ⊓ ⊔

Remark 9.33. The canonical decomposition of A (see Theorem 9.3) induces an isomorphism

L_{p}^{n}(A) ⟶^{≅} L_{p}^{n}(A_{I_{f}}) × L_{p}^{n}(A_{I_{∞}}) × L_{p}^{n}(A_{II_{1}}) × L_{p}^{n}(A_{II_{∞}}) × L_{p}^{n}(A_{III}).

If A is finite, the canonical decomposition of A (see Theorem 9.3) induces a splitting

L_{p}^{n}(U(A)) ≅ L_{p}^{n}(U(A_{I_{f}})) × L_{p}^{n}(U(A_{II_{1}})).

This comes from a general splitting L_{p}^{n}(R × S) ⟶^{≅} L_{p}^{n}(R) × L_{p}^{n}(S) for rings with involution. This splitting is not available for the other decorations ϵ = h, s essentially because we only get a splitting K_{i}(R×S) ⟶^{≅} K_{i}(R)×K_{i}(S) for the unreduced K-groups, which does not carry over to the reduced K-groups.

Example 9.34. Let G be a finitely generated group which does not contain ℤ^{n} as subgroup of finite index. Then N (G) is of type II_{1} and countably composable by Lemma 9.4. We conclude from Theorem 9.13, Theorem 9.18, Theorem 9.20 and Theorem 9.31 that

K_{0}(N (G)) ≅ Z(N (G))^{ℤ/2};
K_{1}(N (G)) ≅ Z(N (G))^{+,inv};
K_{0}(U(G)) ≅ Z(N (G))^{ℤ/2};
K_{1}(U(G)) = 0;
L_{p}^{0}(N (G)) ≅ Z(N (G))^{ℤ/2};
L_{h}^{0}(N (G)) ≅ L_{s}^{0}(N (G));
L_{ϵ}^{1}(N (G)) ≅ 0  for ϵ = p, h, s;
L_{ϵ}^{n}(N (G)) ≅ L_{ϵ}^{n}(U(G))  for n ∈ ℤ, ϵ = p, h, s;
L_{n}^{ϵ}(N (G)) ≅ L_{ϵ}^{n}(N (G))  for n ∈ ℤ, ϵ = p, h, s;
L_{n}^{ϵ}(U(G)) ≅ L_{ϵ}^{n}(U(G))  for n ∈ ℤ, ϵ = p, h, s;

and that there is an exact sequence 0 → ℤ/2 → L_{h}^{0}(N (G)) → Z(N (G))^{ℤ/2} → 0. If G contains no element g ∈ G with g ≠ 1 and |(g)| < ∞, then we conclude from Lemma 9.4 (4)

Z(N (G))^{ℤ/2} ≅ ℝ;
Z(N (G))^{+,inv} ≅ ℝ^{>0}.

Remark 9.35. Analogously to the localization sequence in K-theory (9.22) there is a long exact localization sequence in L-theory [430, Section 3.2]. Let A be a finite von Neumann algebra and U the algebra of affiliated operators such that A_{II_{1}} is countably composable. Then the maps L_{ϵ}^{n}(A) → L_{ϵ}^{n}(U) and L_{n}^{ϵ}(A) → L_{n}^{ϵ}(U) are bijective for all n ∈ ℤ and ϵ = p, h, s by Theorem 9.31. Hence the relative terms L_{ϵ}^{n}(A → U) and L_{n}^{ϵ}(A → U) must vanish for n ∈ ℤ and ϵ = p, h, s. They are defined in terms of U-acyclic Poincaré Achain complexes. Equivalently, they can be defined in terms of linking forms and formations on finitely presented A-modules which are U-torsion [430, Proposition 3.4.1 on page 228, Proposition 3.4.7 on page 274, Proposition 3.5.2 on page 292, Proposition 3.5.5 on page 361]. See also [184], [186].

9.5 Application to Middle K- and G-Theory of Group Rings

9.5.1 Detecting Elements in K_{1} of a Complex Group Ring

We have introduced the Whitehead group Wh(G) of a group G in Subsection 3.1.1. Let i: H → G be the inclusion of a normal subgroup H ⊂ G. It induces a homomorphism i_{0} : Wh(H) → Wh(G). The conjugation action of G on H and on G induces a G-action on Wh(H) and on Wh(G) which turns out to be trivial on Wh(G). Hence i_{0} induces homomorphisms

i_{1} : ℤ ⊗_{ℤG} Wh(H) → Wh(G);  (9.36)
i_{2} : Wh(H)^{G} → Wh(G).  (9.37)

The main result of this subsection is the following theorem. We emphasize that it holds for all groups G.

Theorem 9.38 (Detecting elements in Wh(G)). Let i: H → G be the inclusion of a normal finite subgroup H into an arbitrary group G. Then the maps i_{1} and i_{2} defined in (9.36) and (9.37) have finite kernel.

Proof. Since H is finite, Wh(H) is a finitely generated abelian group. Hence it suffices to show for k = 1, 2 that i_{k} is rationally injective, i.e. id_{ℚ} ⊗_{ℤ}i_{k} is injective. The G-action on H by conjugation c: G → aut(H) factorizes as G ⟶^{pr} G/C_{G}H ⟶^{\overline{c}} aut(H), where C_{H} is the centralizer of H, i.e. the kernel of c, and \overline{c} is injective. Since H is finite, aut(H) and therefore G/C_{G}H are finite. This implies that the natural map

b: ℚ ⊗_{ℤ} Wh(H)^{G} = ℚ ⊗_{ℤ} Wh(H)^{G/C_{G}H} → ℚ ⊗_{ℤ[G/C_{G}H]} Wh(H) = ℚ ⊗_{ℤG} Wh(H)

sending q ⊗_{ℤ} x to q ⊗_{ℤG} x is an isomorphism. Its composition with ℚ ⊗_{ℤG} i_{1} is ℚ ⊗_{ℤG} i_{2}. Hence it suffices to show that i_{2} is rationally injective.

Next we construct the following commutative diagram

9.5 Application to Middle K- and G-Theory of Group Rings 355

(Z(ℂH)^{inv})^{G×ℤ/2} ⟶^{s∘i_{6}} Z(N (G))^{+,inv}
j_{3}↓  ^{det}_{F K} ↑
K_{1}(ℂH)^{G×ℤ/2} ⟶^{i_{5}} K_{1}(N (G))
j_{2}↑  ^{k_{2}}↑
K_{1}(ℤH)^{G×ℤ/2} ⟶^{i_{4}} K_{1}(ℤG)^{ℤ/2}
p↓  ^{q}↓
Wh(H)^{G×ℤ/2}  ⟶^{i_{3}} Wh(G)^{ℤ/2}
j_{1}↓  ^{k_{1}}↓
Wh(H)^{G}  ⟶^{i_{2}} Wh(G)

The G-actions on the various groups above come from the conjugation action of G on the normal subgroup H. The ℤ/2-actions are given by the involutions coming from the involutions on the rings. Notice that these two actions commute. The maps j_{1} and k_{1} are the obvious inclusions. The homomorphisms p and q are induced by the canonical projections K_{1}(ℤH) → Wh(H) and K_{1}(ℤG) → Wh(G). The maps j_{2} and k_{2} come from the obvious ring homomorphisms. Let j′_{3} : Z(ℂH)^{inv} → K_{1}(ℂH) be the homomorphism sending u ∈ Z(ℂH)^{inv} to the class in K_{1}(ℂH) represented by the ℂH-automorphism of ℂH given by multiplication with u. It induces the homomorphism j_{3}. We have indicated the definition of the center-valued Fuglede-Kadison determinant K_{1}(N (G)) → Z(N (G))^{+,inv} in Subsection 9.2.2. Notice that it is defined for any group G. The condition that N (G) is countably decomposable enters in the proof of Theorem 9.18, not in the construction of the Fuglede-Kadison determinant. The horizontal maps i_{2}, i_{3} and i_{4} are induced by the inclusion i: H → G in the obvious way. The map i_{5} comes from the inclusion of ℂH into N (G). Let i_{6} : (Z(ℂH)^{inv})^{G×ℤ/2} → (N (G)^{inv})^{G×ℤ/2} be the injection induced by the inclusion of rings ℂH → N (G). We have N (G)^{G} = Z(N (G)), since ℂG is dense in N (G) in the weak topology. We have N (G)^{inv} ∩ Z(N (G)) = Z(N (G))^{inv}. This implies (N (G)^{inv})^{G×ℤ/2} = (Z(N (G))^{inv})^{ℤ/2}.

Hence we can define a map s: (N (G)^{inv})^{G×ℤ/2} → Z(N (G))^{+,inv} by sending a to |a| = \sqrt{a^{∗}a}. For u ∈ Z(N (G))^{ℤ/2} the Fuglede-Kadison determinant of the N (G)-automorphism of N (G) given by multiplication with u is |u| = \sqrt{u^{∗}u}. Now one easily checks that the diagram above commutes. In order to show that i_{2} is rationally injective, it suffices to prove the following assertions:

(1) s is rationally injective;

(2) i_{6} is injective;

(3) j_{3} is an isomorphism;

(4) j_{2} is rationally injective; (5) p is rationally an isomorphism;

(6) j_{1} is rationally an isomorphism;

(7) k_{1} is injective;

(8) det_{F K} ∘k_{2}(ker(q)) = 1.

(1) This follows from s(a) = 1 ⇔ a^{2} = 1.

(2) This follows from the definition.

(3) Recall that j_{3} is induced by a homomorphism j′_{3} : Z(ℂH)^{inv} → K_{1}(ℂH) by taking the G × ℤ/2- fixed point set. Since the G × ℤ/2-action factorizes through the finite group G/C_{G}H × ℤ/2, j_{3} is a rational isomorphism if j′_{3} is a rational isomorphism. Since ℂH is semisimple, there is an isomorphism pr: ℂH → ∏_{a=1}^{b} M_{n_{a}}(ℂ). Thus we obtain a commutative diagram

Z(ℂH)^{inv} ⟶_{≅} ∏_{a=1}^{b} Z(M_{n_{a}}(ℂ))^{inv}  ∏_{a=1}^{b} d_{a} ; ⟵ ; ≅ ∏_{a=1}^{b} ℂ^{inv}
j′_{3}↓  ^{∏_{a=1}^{b} k′_{a}}↓  ^{∏_{a=1}^{b} m_{a}}↓
K_{1}(ℂH) ⟶_{≅} ∏_{a=1}^{b} K_{1}(M_{n_{a}}(ℂ)) ∏_{a=1} det_{ℂ} ∘μ_{a} ; ⟶ ; ≅ ∏_{a=1}^{b} ℂ^{inv}

Here the isomorphism d_{a}: ℂ^{inv} → Z(M_{n_{a}}(ℂ))^{inv} sends λ to the diagonal (n_{a}, n_{a})-matrix whose diagonal entries are all λ, the maps k′_{a} are defined analogously to j′_{3}, the map m_{a}: ℂ^{inv} → ℂ^{inv} sends λ to λ^{n_{a}}, the Morita isomorphism μ_{a}: K_{1}(M_{n_{a}}(ℂ)) → K_{1}(ℂ) is given by applying ℂ^{n_{a}} ⊗_{M_{na} (ℂ)} − and det_{ℂ}: K_{1}(ℂ) → ℂ^{inv} is the isomorphism induced by the determinant. Since all horizontal arrows and the maps m_{a} are rational isomorphisms, j′_{3} is rationally bijective.

(4) Wall [511] has shown for finite H that the kernel SK_{1}(ℤH) of the change of rings map K_{1}(ℤH) → K_{1}(ℚH) is finite and maps under the canonical projection K_{1}(ℤH) → Wh(H) bijectively onto the torsion subgroup tors(Wh(H)) of Wh(H) (see also [406, page 5 and page 180]). The change of rings map K_{1}(ℚH) → K_{1}(ℂH) is injective [406, page 5 and page 43]. Hence the change of rings map K_{1}(ℤH) → K_{1}(ℂH) is rationally injective. Since j_{2} is obtained by taking the fixed point set under the action of the finite group G/C_{G}H×ℤ/2 from it, j_{2} is rationally injective.

(5) The kernel of the projection K_{1}(ℤH) → Wh(H) is finite as H is finite. Hence p is rationally bijective since it is obtained from a rational isomorphism by taking the fixed point set under the operation of the finite group G/C_{G}H × ℤ/2.

(6) The involution on Wh′(H) := Wh(H)/ tors(Wh(H)) = Wh(H)/SK_{1}(ℤH) is trivial [511], [406, page 182]. This implies Wh′(H) = Wh′(H)^{ℤ/2}. Since SK_{1}(ℤH) is finite, we conclude that the inclusion Wh(H)^{ℤ/2} → Wh(H) is a rational isomorphism. This implies that j_{1} is rationally bijective.

(7) This is obvious.

(8) An element in the kernel of K_{1}(ℤG) → Wh(G) is represented by a ℤGautomorphism r_{±g}: ℤG → ℤG which is given by right multiplication with an element ±g for g ∈ G. Since for the induced N (G)-map r_{±g}: N (G) → N (G), we have (r_{±g})^{∗} ∘r_{±g} = 1, we get det_{F K} ∘k_{2}([r_{±g}]) = 1. This finishes the proof Theorem 9.38. ⊓ ⊔

9.5.2 Survey on the Isomorphism Conjecture for K_{0} of Complex Group Rings, the Bass Conjecture and the Hattori-Stallings Rank

In this section we explain the Isomorphism Conjecture for K_{0}(ℂG) and the Bass Conjecture and their relation. We introduce the Hattori-Stallings rank and study its relation to the center valued dimension of a group von Neumann algebra.

We begin with the Isomorphism Conjecture for K_{0}(ℂG). For more information about the Isomorphism Conjectures in algebraic K- and L-theory due to Farrell and Jones and the related Baum-Connes Conjecture for the topological K-theory of the reduced group C^{∗}-algebra we refer for instance to [27, Conjecture 3.15 on page 254], [128, section 6], [194], [256] [257], [339], [340], and [501].

The orbit category Or(G) has as objects homogeneous spaces G/H and as morphisms G-maps. Let Or(G, FIN) be the full subcategory of Or(G) consisting of objects G/H with finite H. We define a covariant functor

Or(G) → ABEL,  G/H ↦ K_{0}(ℂH)  (9.39)

as follows. It sends G/H to the projective class group K_{0}(ℂH). Given a morphism f: G/H → G/K there is an element g ∈ G with g^{−1}Hg ⊂ K such that f(g′H) = g′gK. Define f_{∗}: K_{0}(ℂH) → K_{0}(ℂK) as the map induced by induction with the group homomorphism c_{g}: H → K, h ↦ g^{−1}hg. This is independent of the choice of g since any inner automorphism of K induces the identity on K_{0}(ℂK).

Conjecture 9.40 (Isomorphism Conjecture). The Isomorphism Conjecture for K_{0}(ℂG) says that the map

a: colim_{Or(G,FIN)} K_{0}(ℂH) → K_{0}(ℂG)

induced by the various inclusions H ⊂ G is an isomorphism.

Notation 9.41. Let con(G) be the set of conjugacy classes (g) of elements g ∈ G. Let con(G)_{f} be the subset of con(G) of conjugacy classes (g) for which each representative g has finite order. Let con(G)_{cf} be the subset of con(G) of conjugacy classes (g) which contain only finitely many elements. Put con(G)_{f,cf} = con(G)_{f} ∩ con(G)_{cf}. We denote by class_{0}(G), class_{0}(G)_{f}, class_{0}(G)_{cf}, and class_{0}(G)_{f,cf} respectively the complex vector space with the set con(G), con(G)_{f}, con(G)_{cf} and con(G)_{f,cf} respectively as basis. We denote by class(G), class(G)_{f}, class(G)_{cf}, and class(G)_{f,cf} respectively the complex vector space of functions from the set con(G), con(G)_{f}, con(G)_{cf} and con(G)_{f,cf} respectively to the complex numbers ℂ.

Notice that class_{0}(G), class_{0}(G)_{f}, class_{0}(G)_{cf} and class_{0}(G)_{f,cf} can be identified with the complex vector space of functions with finite support from con(G), con(G)_{f}, con(G)_{cf} and con(G)_{f,cf} to ℂ. Recall that (g) is finite if and only if the centralizer C_{g} of g has finite index in G. We obtain an isomorphism of complex vector spaces

z: class_{0}(G)_{cf} ⟶^{≅} Z(ℂG), (g) ↦ ∑_{g′∈(g)} g′.  (9.42)

Define the universal ℂG-trace of ∑_{g∈G} λ_{g}g ∈ ℂG by

tr_{ℂG}^{u} (∑_{g∈G} λ_{g}g) := ∑_{g∈G} λ_{g} · (g).  ∈ class_{0}(G).  (9.43)

Under the obvious identification of class_{0}(G) with ℂG/[ℂG, ℂG], the trace tr_{ℂG}^{u} becomes the canonical projection. This extends to square matrices in the usual way

tr_{ℂG}^{u}: M_{n}(ℂG) → class_{0}(G),  A ↦ ∑_{i=1}^{n} tr_{ℂG}^{u}(a_{i,i}).  (9.44)

Let P be a finitely generated projective ℂG-module. Define its Hattori-Stallings rank by

HS(P) := tr_{ℂG}^{u}(A) ∈ class_{0}(G),  (9.45)

where A is any element in M_{n}(ℂG) with A^{2} = A such that the image of the map ℂG^{n} → ℂG^{n} given by right multiplication with A is ℂG-isomorphic to P. This definition is independent of the choice of A (cf. (6.4)). The Hattori-Stallings rank defines homomorphisms

HS: K_{0}(ℂG) → class_{0}(G),  [P] ↦ HS(P);  (9.46)
HS_{ℂ}: K_{0}(ℂG) ⊗_{ℤ} ℂ → class_{0}(G),  [P] ⊗ λ ↦ λ · HS(P). (9.47)

Conjecture 9.48 (Bass Conjecture). The strong Bass Conjecture for ℂG says that the image of the map HS_{ℂ}: K_{0}(ℂG) ⊗_{ℤ} ℂ → class_{0}(G) of (9.47) is class_{0}(G)_{f}.

The strong Bass Conjecture for ℤG says that for a finitely generated projective ℤG-module P

HS(ℂ ⊗_{ℤ} P)(g) = \{ \begin{matrix}0 & if g ≠ 1; \\ rk_{ℤ}(ℤ ⊗_{ℤG} P) & if g = 1. \end{matrix}

The weak Bass Conjecture for ℤG says that for a finitely generated projective ℤG-module P

HS(ℂ ⊗_{ℤ} P)(1) = ∑_{(g)∈con(G)} HS(P)(g) = rk_{ℤ}(ℤ ⊗_{ℤG} P).

Theorem 9.49. For any group G there is a commutative diagram whose left vertical arrow is an isomorphism

(colim_{Or(G,FIN)} K_{0}(ℂH)) ⊗_{ℤ} ℂ ⟶^{a⊗_{ℤ}id_{ℂ}} K_{0}(ℂG) ⊗_{ℤ} ℂ
h↓≅  ↓^{HS}_{ℂ}
class_{0}(G)_{f}  ⟶_{e} class_{0}(G)

Proof. Firstly we explain the maps in the square. We have introduced the map a in Conjecture 9.40 and the map HS_{ℂ} in (9.47). The map e is given extending a function con(G)_{f} → ℂ to a function con(G) → ℂ by putting it to be zero on elements in con(G) which do not belong to con(G)_{f}. Define for a group homomorphism ψ: G → G′ a map ψ_{∗}: con(G) → con(G′) by sending (h) to (ψ(h)). It induces a homomorphism ψ_{∗}: class_{0}(G) → class_{0}(G′). One easily checks that the following diagram commutes

K_{0}(ℂG) ⟶^{ψ_{∗}} K_{0}(ℂG′)
HS↓  ^{HS}↓
class_{0}(G) ⟶^{ψ_{∗}} class_{0}(G′)

(9.50)

There is a canonical isomorphism

f_{1} : (colim_{Or(G,FIN)} K_{0}(ℂH)) ⊗_{ℤ} ℂ ⟶^{≅}

colim_{Or(G,FIN)} (K_{0}(ℂH) ⊗_{ℤ} ℂ) . (9.51)

The Hattori-Stallings rank induces for each finite subgroup H of G an isomorphism K_{0}(ℂH) ⊗_{ℤ} ℂ → class(H) by elementary complex representation theory for finite groups [470, Theorem 6 in Chapter 2 on page 19]. Thus we get an isomorphism

f_{2} : colim_{Or(G,FIN)} (K_{0}(ℂH) ⊗_{ℤ} ℂ) ⟶^{≅} colim_{Or(G,FIN)} class(H). (9.52)

Let f′_{3} : colim_{Or(G,FIN)} con(H) → con(G)_{f} be the map induced by the inclusions of the finite subgroups H of G. Define a map f′_{4} : con(G)_{f} → colim_{Or(G,FIN)} con(H) by sending (g) ∈ con(G)_{f} to the image of (g) ∈ con(⟨g⟩) under the structure map from con(⟨g⟩) to colim_{Or(G,FIN)} con(H), where ⟨g⟩ is the cyclic subgroup generated by g. One easily checks that this is independent of the choice of the representative g of (g) and that f′_{3} and f′_{4} are inverse to one another. The bijection f′_{3} induces an isomorphism

f_{3} : colim_{Or(G,FIN)} class(H) ⟶^{≅} class_{0}(G)_{f},  (9.53)

because colimit and the functor sending a set to the complex vector space with this set as basis commute. Now the isomorphism h is defined as the composition of the isomorphisms f_{1} of (9.51), f_{2} of (9.52) and f_{3} of (9.53). It remains to check that the square in Theorem 9.49 commutes. This follows from the commutativity of (9.50). This finishes the proof of Theorem 9.49.

⊓ ⊔

A group G is poly-cyclic if there is a finite sequence of subgroups {1} = G_{0} ⊂ G_{1} ⊂ G_{2} ⊂ … G_{r} = G such that G_{i} is normal in G_{i+1} with cyclic quotient G_{i+1}/G_{i} for i = 0, 1, 2, …, r − 1. We call G virtually poly-cyclic if G contains a poly-cyclic subgroup of finite index.

Theorem 9.54. (1) The map a ⊗_{ℤ} id_{ℂ}: (colim_{Or(G,FIN)} K_{0}(ℂH)) ⊗_{ℤ} ℂ →

K_{0}(ℂG) ⊗_{ℤ} ℂ is injective;

(2) The Isomorphism Conjecture for K_{0}(ℂG) 9.40 implies that the map HS_{ℂ}: K_{0}(ℂG) ⊗_{ℤ} ℂ → class_{0}(G) of (9.47) is injective with image class_{0}(G)_{f} and hence implies the strong Bass Conjecture for K_{0}(ℂG) 9.48;

(3) Let P be a finitely generated projective ℂG-module and let g ∈ G be an element of infinite order with finite (g). Then HS(P)(g) = 0;

(4) Let P be a finitely generated projective ℤG-module. Then HS(ℂ ⊗_{ℤ} P)(g) = 0 for any element g ∈ G with g ≠ 1 for which |g| < ∞ or |(g)| < ∞;

(5) The strong Bass Conjecture for ℤG is true for residually finite groups and linear groups;

(6) The strong Bass Conjecture for ℂG is true for amenable groups;

(7) The strong Bass Conjecture for ℂG is true for a countable group G, if the Bost Conjecture holds for G, which is the version of the Baum-Connes Conjecture with C_{r}^{∗}(G) replaced by l^{1}(G). (A discussion of the Bost Conjecture and the class of groups for which it is known can be found in [38]. The class contains all countable groups which are a-T-menable);

(8) Suppose that G is virtually poly-cyclic. Then the map

a: colim_{Or(G,FIN)} K_{0}(ℂH) → K_{0}(ℂG)

is surjective and induces an isomorphism

a: (colim_{Or(G,FIN)} K_{0}(ℂH)) ⊗_{ℤ} ℂ ⟶^{≅} K_{0}(ℂG) ⊗_{ℤ} ℂ,

and the forgetful map

f: K_{0}(ℂG) → G_{0}(ℂG)

is an isomorphism;

(9) If P is a finitely generated projective ℂG-module, then HS(P)(1) ∈ ℚ.

Proof. (1) and (2) These follow from Theorem 9.49.

(3) This follows from [23, Theorem 8.1 on page 180].

(4) and (5) These follow from [307, Theorem 4.1 on page 96].

(6) This is proved for elementary amenable groups in [197, Theorem 1.6]. The proof for amenable groups can be found in [38].

(7) This is proved in [38].

(8) Moody has shown [384] that the obvious map ⨁_{H∈FIN} G_{0}(ℂH) → G_{0}(ℂG) given by induction is surjective. Since G is virtually poly-cyclic, the complex group ring ℂG is regular, i.e. Noetherian and any ℂG-module has a finite dimensional projective resolution [447, Theorem 8.2.2 and Theorem 8.2.20]. This implies that f: K_{0}(ℂG) → G_{0}(ℂG) is bijective. The same is true for any finite subgroup H ⊂ G. Now the claim follows using assertion (1).

(9) is proved in [529]. See also [78]. ⊓ ⊔ Next we investigate the relation between the center valued dimension

dim_{N (G)}^{u} (see Definition 9.12) and the Hattori-Stallings rank. Define a homomorphism

ϕ: Z(N (G)) → class(G)_{cf}  (9.55)

by assigning to u ∈ Z(N (G))

ϕ(u): con(G)_{cf} → ℂ, (h) ↦ tr_{N (G)} (u · \frac{1}{|(h)|} · ∑_{h̃∈(h)} h̃^{−1}) .

Equivalently, ϕ(u) can be described as follows. If we evaluate u ∈ Z(N (G)) ⊂ N (G) = B(l^{2}(G))^{G} at the unit element e ∈ G, we obtain an element u(e) = ∑_{g∈G} λ_{g} ·g ∈ l^{2}(G) with the property that λ_{g} depends only on the conjugacy class (g) of g ∈ G. This implies that λ_{g} = 0 if (g) is infinite and ϕ(u)(g) = λ_{g} for g ∈ G with finite (g).

Lemma 9.56. The map ϕ: Z(N (G)) → class(G)_{cf} is injective. The composition of k: class_{0}(G)_{cf} → Z(N (G)) with ϕ: Z(N (G)) → class(G)_{cf} is the canonical inclusion class_{0}(G)_{cf} → class(G)_{cf}, where k is given by

k ( ∑_{(g)∈con(G)} λ_{(g)} · (g)) = ∑_{(g)} λ_{(g)} · ( ∑_{g′∈(g)} g′) .

The following diagram commutes

K_{0}(ℂG) ⟶^{HS} class_{0}(G) ⟶^{r} class_{0}(G)_{cf}
i_{∗}↓  ^{k}↓
K_{0}(N (G)) ⟶^{dim_{N (G)}^{u}} Z(N (G))^{ℤ/2} ⟶^{j} Z(N (G)),

where r is given by restriction, i_{∗} is induced by the inclusion i: ℂG → N (G), j is the inclusion and the other maps have been defined in Definition 9.12 and (9.46).

Proof. Obviously ϕ is injective and ϕ ∘ k is the canonical inclusion. For the commutativity of the diagram above it is enough to show for an element A ∈ M_{n}(ℂG) and h ∈ G with finite (h)

tr_{ℂG}^{u}(A)(h) = (ϕ ∘ tr_{N (G)}^{u}(A))(h),  (9.57)

since for A ∈ M_{n}(ℂG) with A^{2} = A we have

ϕ ∘ k ∘ r ∘ HS(im(A))(h) = tr_{ℂG}^{u}(A)(h);
ϕ ∘ j ∘ dim_{N (G)}^{u} ∘i_{∗}([im(A)]) = (ϕ ∘ tr_{N (G)}^{u}(A))(h).

To prove (9.57) it suffices to show for g ∈ G and h ∈ G with finite (h)

tr_{ℂG}^{u}(g)(h) = (ϕ ∘ tr_{N (G)}^{u}(g))(h).  (9.58)

Obviously

tr_{ℂG}^{u}(g)(h) = tr_{N (G)} (g · ∑_{h̃∈(h)} h̃^{−1}) .  (9.59)

We conclude from the universal property of tr_{N (G)}^{u} for all x ∈ N (G) (see Theorem 9.5)

tr_{N (G)} (x · ∑_{h′∈(h)}(h′)^{−1}) = (ϕ ∘ tr_{N (G)}^{u}(x))(h).  (9.60)

Now (9.58) and hence (9.57) follow from (9.59) and (9.60). This finishes the proof of Lemma 9.56. ⊓ ⊔

We conclude from Theorem 9.54 (9) and Lemma 9.56

Corollary 9.61. Let P be a finitely generated projective ℂG-module. Then dim_{N (G)}(N (G) ⊗_{ℤG} P) = HS(P)(1) and this is a non-negative rational number.

The next result shows that one cannot detect elements in K̃_{0}(ℤG) by K̃_{0}(N (G)) in contrast to Wh(G) (cf. Theorem 9.38).

Theorem 9.62. The change of rings map K̃_{0}(ℤG) → K̃_{0}(N (G)) is trivial.

Proof. Because of Theorem 9.13 (2) and Lemma 9.56 it suffices to prove for a finitely generated projective ℤG-module P that HS(ℂ ⊗_{ℤ} P)(g) = 0 for g ∈ G with g ≠ 1 and |(g)| < ∞, and HS(ℂ ⊗_{ℤ} P)(1) ∈ ℤ. This follows from Theorem 9.54 (4) and the fact that P ≅_{ℤG} im(A) for some A ∈ M_{n}(ℤG) with A^{2} = A. ⊓ ⊔

Theorem 9.63. The image of the composition

K_{0}(ℂG) ⊗_{ℤ} ℂ ⟶^{HS_{ℂ}} class_{0}(G) ⟶^{r} class_{0}(G)_{cf}

is class_{0}(G)_{f,cf}.

Proof. Apply Theorem 9.49 and Theorem 9.54 (3). ⊓ ⊔

9.5.3 G-Theory of Complex Group Rings

In this subsection we investigate G_{0}(ℂG) (see Definition 9.11). We have defined TN (G) ⊗_{ℂG} M and PN (G) ⊗_{ℂG} M in Definition 6.1 (see also Section 8.4). Recall from Theorem 6.5 and Theorem 6.7 (3) that PN (G) ⊗_{ℂG} M is a finitely generated projective N (G)-module.

Theorem 9.64. If G is amenable, the map

l: G_{0}(ℂG) → K_{0}(N (G)), [M] ↦ [PN (G) ⊗_{ℂG} M]

is a well-defined homomorphism. If f: K_{0}(ℂG) → G_{0}(ℂG) is the forgetful map sending [P] to [P] and i_{∗}: K_{0}(ℂG) → K_{0}(N (G)) is induced by the inclusion i: ℂG → N (G), then the composition l ∘ f agrees with i_{∗}.

Proof. If 0 → M_{0} ⟶^{b} M_{1} ⟶^{p} M_{2} → 0 is an exact sequence of finitely generated ℂG-modules we have to check in K_{0}(N (G))

[PN (G) ⊗_{ℂG} M_{0}] − [PN (G) ⊗_{ℂG} M_{1}] + [PN (G) ⊗_{ℂG} M_{2}] = 0.

Consider the induced sequence PN (G) ⊗_{ℂG} M_{0} ⟶^{\overline{b}} PN (G) ⊗_{ℂG} M_{1} ⟶^{\overline{p}} PN (G) ⊗_{ℂG} M_{2}. Obviously \overline{p} is surjective as p is surjective. We conclude from Theorem 6.7 (1) that ker(\overline{b}) and ker(\overline{p}) are finitely generated projective N (G)-modules. Theorem 6.7 (4b) and (4e) and Theorem 6.37 imply

dim_{N (G)} (ker(\overline{b})) = 0;
dim_{N (G)} (PN (G) ⊗_{ℂG} M_{0}) = dim_{N (G)} (ker(\overline{p})) .

Lemma 6.28 shows that \overline{b} is injective and PN (G) ⊗_{ℂG} M_{0} ≅_{N (G)} ker(\overline{p}). Obviously ker(\overline{p})⊕PN (G)⊗_{ℂG}M_{2} and PN (G)⊗_{ℂG}M_{1} are N (G)-isomorphic.

⊓ ⊔

Now we can detect elements in G_{0}(ℂG) for amenable groups G.

Theorem 9.65. Suppose that G is amenable. Then the image of the composition

G_{0}(ℂG) ⊗_{ℤ} ℂ ⟶^{l⊗_{ℤ}ℂ} K_{0}(N (G)) ⊗_{ℤ} ℂ ⟶^{dim_{N (G)}^{u}} Z(N (G)) ⟶^{ϕ} class(G)_{cf}

contains the complex vector space class_{0}(G)_{f,cf}. In particular

rk_{ℤ}(G_{0}(ℂG)) ≥ | con(G)_{f,cf}|.

Proof. Apply Lemma 9.56, Theorem 9.63 and Theorem 9.64. ⊓ ⊔

Theorem 9.66. (1) If G is amenable, the class [ℂG] generates an infinite

cyclic subgroup in G_{0}(ℂG) and is in particular not zero;

(2) If G contains ℤ ∗ ℤ as subgroup, we get [ℂG] = 0 in G_{0}(ℂG).

Proof. (1) The map l: G_{0}(ℂG) → K_{0}(N (G)) of Theorem 9.64 sends [ℂG] to [N (G)] and [N (G)] ∈ K_{0}(N (G)) generates an infinite cyclic subgroup because of dim_{N (G)}(N (G)) = 1.

(2) We abbreviate F_{2} = ℤ ∗ ℤ. Induction with the inclusion F_{2} → G induces a homomorphism G_{0}(ℂF_{2}) → G_{0}(ℂG) which sends [ℂF_{2}] to [ℂG]. Hence it suffices to show [ℂF_{2}] = 0 in G_{0}(ℂF_{2}). The cellular chain complex of the universal covering of S^{1} ∨ S^{1} yields an exact sequence of ℂF_{2}-modules 0 → (ℂF_{2})^{2} → ℂF_{2} → ℂ → 0, where ℂ is equipped with the trivial F_{2}-action. This implies [ℂF_{2}] = −[ℂ] in G_{0}(ℂF_{2}). Hence it suffices to show [ℂ] = 0 in G_{0}(ℂF_{2}). Choose an epimorphism f: F_{2} → ℤ. Restriction with f defines a homomorphism G_{0}(ℂℤ) → G_{0}(ℂF_{2}). It sends ℂ viewed as trivial ℂℤ-module to ℂ viewed as trivial ℂF_{2}-module. Hence it remains to show [ℂ] = 0 in G_{0}(ℂℤ). This follows from the exact sequence 0 → ℂℤ ⟶^{s−1} ℂℤ → ℂ → 0 for s a generator of ℤ which comes from the cellular ℂℤ-chain complex of S̃^{1}. ⊓ ⊔

Theorem 9.66 gives evidence for

Conjecture 9.67. (Amenability and the regular representation in G-theory). A group G is amenable if and only if [ℂG] ≠ 0 in G_{0}(ℂG).

Example 9.68. Let D be the infinite dihedral group D = ℤ/2 ∗ ℤ/2. Then we compute

K̃_{0}(ℤD) = 0;
K_{0}(ℂD) ≅ ℤ^{3};
G_{0}(ℂD) ≅ K_{0}(ℂD).

as follows. Since K̃_{0}(ℤ[ℤ/2]) = 0 [125, Corollary 5.17] and the obvious map K̃_{0}(ℤ[ℤ/2])⊕K̃_{0}(ℤ[ℤ/2]) → K̃_{0}(ℤD) is bijective [508, Corollary 11.5 and the following remark], we conclude K̃_{0}(ℤD) = 0. If s_{1} and s_{2} are the generators of the subgroups ℤ/2 ∗ 1 and 1 ∗ ℤ/2, then the set con_{f}(G) consists of the three elements (1), (s_{1}) and (s_{2}). Let ℂ^{−} be the ℂ[ℤ/2]-module with ℂ as underlying complex vector space and the ℤ/2-action given by − id. Let P_{1} and P_{2} be the ℂD-modules given by induction with ℤ/2 = ⟨s_{1}⟩ ⊂ ℤ/2 ∗ ℤ/2 and ℤ/2 = ⟨s_{2}⟩ ⊂ ℤ/2 ∗ ℤ/2 applied to ℂ^{−}. One easily checks that HS(P_{i}) = 1/2·((e)−(s_{i})) for i = 1, 2 and e ∈ D the unit element. Notice that D contains an infinite cyclic subgroup of index 2, namely ⟨s_{1}s_{2}⟩. Because of Theorem 9.54 (1) and (8) the forgetful map f: K_{0}(ℂG) ⟶^{≅} G_{0}(ℂG) is bijective and we obtain isomorphisms u: ℤ^{3} ⟶^{≅} K_{0}(ℂD) and v: K_{0}(ℂD) ⟶^{≅} ℤ^{3}, which are inverse to one another and defined by

u(n_{0}, n_{1}, n_{2}) = n_{0} · [ℤD] + n_{1} · [P_{1}] + n_{2} · [P_{2}];
v([P]) = (HS(P)(1) + HS(P)(s_{1}) + HS(s_{2}), −2 · HS(P)(s_{1}), −2 · HS(s_{2})).

The composition

ℤ^{3} ⟶^{u} K_{0}(ℂD) ⟶^{i_{∗}} K_{0}(N (D)) ⟶^{dim_{N (D)}} ℝ

sends (n_{0}, n_{1}, n_{2}) to n_{0} + n_{1}/2 + n_{2}/2. In particular the map i_{∗}: K̃_{0}(ℂD) → K̃_{0}(N (D)) is not trivial (compare with Theorem 9.62.)

Remark 9.69. The knowledge about G_{0}(ℂG) is very poor. At least there is Moody’s result stated as Theorem 9.54 (8) and we will show in Example 10.13 that for the amenable locally finite group A = ⨁_{n∈ℤ} ℤ/2 the map K_{0}(ℂA) → G_{0}(ℂA) is not surjective. On the other hand we do not know a counterexample to the statement that for an amenable group with an upper bound on the orders of its finite subgroups both a: colim_{Or(G,FIN)} K_{0}(ℂH) → K_{0}(ℂG) and K_{0}(ℂG) → G_{0}(ℂG) are bijective. We also do not know a counterexample to the statement that for a group G, which is non amenable, G_{0}(ℂG) = {0}. To our knowledge the latter equality is not even known for G = ℤ ∗ ℤ.

9.6 Miscellaneous

We have emphasized the analogy between a finite von Neumann algebra A and a principal ideal domain R and their Ore localizations U and F with respect to the set of non-zero-divisors (see Example 6.12 and Section 8.5). Of course F is just the quotient field of R. Some of the results of this chapters are completely analogous for R and F. For instance K_{0}(R) → K_{0}(F) is bijective (compare Theorem 9.20 (1)). We have K_{1}(F) = F^{inv} and K_{0}(R → F) = F^{inv}/R^{inv}. We also get K_{1}(R) = R^{inv}, provided that R has a Euclidean algorithm, for instance if R is ℤ or K[x] for a field K (compare Theorem 9.17, Theorem 9.20 (2) and Theorem 9.23 (3)).

The topological K-theory K_{p}^{top}(A) of a von Neumann algebra A equals by definition K_{0}(A) for p = 0 and π_{0}(GL(A)) for p = 1. We have computed K_{0}^{top}(A) = K_{0}(A) in Subsection 9.2.1 and K_{1}^{top}(A) = 0 holds for any von Neumann algebra [51, Example 8.1.2 on page 67], [514, Example 7.1.11 on page 134].

More information about the Bass Conjecture can be found for instance in [23], [38], [159], [161], [165], [175], [176], [307], [458].

For an introduction to algebraic K-theory see for instance [378] and [441]. Computations of G_{0}(AG) for Noetherian rings A and finite nilpotent

groups G can be found in [246], [305] and [513]. In [246] a conjecture is stated which would give a computation for all finite groups G in terms of certain orders associated to the irreducible rational representations of G.

One can also consider instead of G_{0}(R) the version G_{0}^{F P_{∞}}(R) of Gtheory as suggested by Weibel. The group G_{0}^{F P_{∞}}(R) is defined in terms of R-modules which are of type F P_{∞}, i.e. possess a (not necessarily finite dimensional) finitely generated projective resolution (cf. Definition 9.11). This version is much closer to K-theory than G_{0}. For instance the forgetful map f: K_{0}(AG) → G_{0}^{F P_{∞}}(AG) is bijective for a commutative ring A if the trivial AG-module A has a finite dimensional (not necessarily finitely generated) projective AG-resolution. The latter condition is always satisfied for a ring A containing ℚ as subring if there is a finite dimensional G-CW -model for the classifying space E(G, FIN) for proper G-actions. This is the case for a word-hyperbolic group G and for a discrete subgroup G of a Lie group which has only finitely many path components (see [1, Corollary 4.14], [370]).

Exercises

9.1. Let A be a von Neumann algebra. Show that A is finite if and only if it possesses a finite normal faithful trace tr: A → ℂ.

9.2. Let M be a closed hyperbolic manifold. Show that N (π_{1}(M)) is of type II_{1}.

9.3. Let M be a compact connected orientable 3-manifold whose boundary does not contain S^{2} as a path component. Show that N (π_{1}(M)) is of type I_{f} if and only if M has a finite covering \overline{M} → M such that \overline{M} is homotopy equivalent to S^{3}, S^{1} × S^{2}, S^{1} × D^{2}, T^{2} × D^{1} or T^{3}, and that it is of type II_{1} otherwise.

9.4. Construct for any ring R and natural number m natural isomorphisms

K_{n}(M_{m}(R)) ⟶^{≅} K_{n}(R) for n = 0, 1.

9.5. Let A be an abelian von Neumann algebra. Show that [A] ∈ K_{0}(A) is not contained in 2 · K_{0}(A) and the canonical projection K_{1}(A) → K̃_{1}(A) is not bijective but that [M_{2}(A)] ∈ K_{0}(M_{2}(A)) is contained in 2 · K_{0}(M_{2}(A)) and K_{1}(M_{2}(A)) → K̃_{1}(M_{2}(A)) is bijective.

9.6. Show that the topological K-group K_{1}^{top}(A) := π_{0}(GL(A)) is trivial for any von Neumann algebra A.

9.7. Let G be a finite group. Let V and W be two finite dimensional unitary representations. Define the character of V by the function χ_{V} : G → ℂ which sends g ∈ G to the trace tr_{ℂ}(l_{g}: V → V) of the endomorphism l_{g} of the finite dimensional ℂ-vector space V given by left multiplication with g. Show that V and W are unitarily ℂG-isomorphic if and only if χ_{V} = χ_{W}.

9.8. Let R be a ring such that the set S of non-zero divisors satisfies the Ore condition. Let C_{∗} be a finite free R-chain complex such that each homology group H_{n}(C_{∗}) has a 1-dimensional finitely generated free R-resolution and is S-torsion, i.e. RS^{−1} ⊗_{R} H_{n}(C_{∗}) = 0. Define an element

ρ^{h}(C_{∗}) := ∑_{n∈ℤ}(−1)^{n} · [H_{n}(C_{∗})] ∈ K_{0}(R → RS^{−1}).

Choose an R-basis for C_{∗}. It induces an RS^{−1}-basis for RS^{−1} ⊗_{R} C_{∗}. Show that RS^{−1} ⊗_{R} C_{∗} is an acyclic based free RS^{−1}-chain complex. Define its torsion

ρ(RS^{−1} ⊗_{R} C_{∗}) ∈ K_{1}(RS^{−1})

as in (3.1). Show that the image of this element under the projection to the cokernel coker(i_{1}) of the map i_{1} : K_{1}(R) → K_{1}(RS^{−1}) appearing in the localization sequence (9.22) is independent of the choice of the Rbasis and is an R-chain homotopy invariant of C_{∗}. Show that the map j: K_{1}(RS^{−1}) → K_{0}(R → RS^{−1}) appearing in the localization sequence (9.22) maps ρ(RS^{−1} ⊗_{R} C_{∗}) to ρ^{h}(C_{∗}).

9.9. Compute explicitly for R = ℤ and S ⊂ ℤ the set of non-zero divisors that RS^{−1} = ℚ and that the localization sequence (9.22) becomes

{±1} → ℚ^{inv} ⟶^{p} ℚ^{+,inv} ⟶^{0} ℤ ⟶^{id} ℤ,

where p maps a rational number different from zero to its absolute value. Show for a finite free ℤ-chain complex C_{∗} that H_{n}(C_{∗}) has a 1-dimensional finitely generated free R-resolution and is S-torsion for all n ∈ ℤ, if and only if H_{n}(C_{∗}) is a finite abelian group for all n ∈ ℤ. Prove that in this case the invariant ρ^{h}(C_{∗}) introduced in the previous exercise becomes under these identifications

ρ^{h}(C_{∗}) = ∏_{n∈ℤ} |H_{n}(C_{∗})|^{(−1)^{n}} ∈ ℚ^{+,inv}.

9.10. Let G be a finite group. Show that ℂG is semisimple, i.e. any submodule of a finitely generated projective module is a direct summand. Conclude that ℂG is a product of matrix algebras M_{n}(ℂ). Compute K_{n}(ℂG) for n = 0, 1 and L_{n}^{ϵ}(ℂG) for ϵ = p, h, s and n ∈ ℤ as an abelian group.

9.11. Show that for a group G the following assertions are equivalent:

(1) N (G) is Noetherian;

(2) U(G) is Noetherian;

(3) U(G) is semisimple;

(4) K_{0}(U(G)) is finitely generated;

(5) K_{0}(N (G)) is finitely generated;

(6) G is finite.

9.12. For a ring R let Nil_{0}(R) be the abelian group whose generators [P, f] are conjugacy classes of nilpotent endomorphisms f: P → P of finitely generated projective R-modules and whose relations are given by [P_{0}, f_{1}] − [P_{1}, f_{1}] + [P_{2}, f_{2}] = 0 for any exact sequence 0 → (P_{0}, f_{0}) → (P_{1}, f_{1}) → (P_{2}, f_{2}) → 0 of such nilpotent endomorphisms. Nilpotent means that f^{n} = 0 for some natural number n. Show

(1) The natural map i: K_{0}(R) → Nil_{0}(R) sending [P] to [P, 0] is split injective;

(2) Let Ñil_{0}(R) be the cokernel of the natural map i: K_{0}(R) → Nil_{0}(R). If R is semihereditary, then Ñil_{0}(R) = 0;

(3) Show for a group G that Ñil_{0}(N (G)) = Ñil_{0}(U(G)) = 0.

9.13. Let G be a group such that Wh(G) vanishes. Show that any finite subgroup of the center of G is isomorphic to a product of finitely many copies of ℤ/2 and ℤ/3 or to a product of finitely many copies of ℤ/2 and ℤ/4.

9.14. Suppose that the group G satisfies the weak Bass Conjecture for ℤG. Then the change of rings homomorphism K_{0}(ℤG) → K_{0}(N (G)) maps the class [P] of a finitely generated projective ℤG-module P to rk_{ℤ}(ℤ ⊗_{ℤG} P) · [N (G)].

9.15. Show that G satisfies the strong Bass Conjecture for ℂG if and only if HS(P)(g) = 0 holds for any finitely generated projective ℂG-module P and any element g ∈ G with |g| = |(g)| = ∞.

9.16. Show that the group G satisfies con(G)_{f,cf} = {(1)} if and only if the image of the change of rings map K̃_{0}(ℂG) → K̃_{0}(N (G)) consists of torsion elements.

9.17. Let A be a finite von Neumann algebra. Construct homomorphisms d_{A}: G_{0}(A) → K_{0}(A) and d_{U}: G_{0}(U) → K_{0}(U) such that d_{A} ∘ f_{A} = id and d_{U} ∘ f_{U} = id holds for the forgetful maps f_{A}: K_{0}(A) → G_{0}(A) and f_{U}: K_{0}(U) → G_{0}(U).

9.18. Let F be a free group. Let V be a ℂF -module whose underlying complex vector space is finite dimensional and let H ⊂ F be a subgroup. Show that V ⊗_{ℂ} ℂ[F/H] with the diagonal F -action is a finitely generated ℂF -module whose class in G_{0}(ℂF) is zero.

9.19. Let G be finite. Show that the change of coefficients map j: G_{0}(ℤG) → K_{0}(ℚG) is well-defined and surjective.

10. The Atiyah Conjecture

Introduction

Atiyah [9, page 72] asked the question, whether the analytic L^{2}-Betti numbers b_{p}^{(2)}(M) of a cocompact free proper Riemannian G-manifold with G-invariant Riemannian metric and without boundary, which are defined in terms of the heat kernel (see (1.60)), are always rational numbers. This is implied by the following conjecture which we call the (strong) Atiyah Conjecture in view of Atiyah’s question above.

Given a group G, let FIN (G) be the set of finite subgroups of G. Denote by

\frac{1}{|FIN (G)|}ℤ ⊂ ℚ  (10.1)

the additive subgroup of ℝ generated by the set of rational numbers \{\frac{1}{|H|} | H ∈ FIN (G)}.

Conjecture 10.2 (Strong Atiyah Conjecture). A group G satisfies the strong Atiyah Conjecture if for any matrix A ∈ M(m, n, ℂG) the von Neumann dimension of the kernel of the induced bounded G-operator

r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}, x ↦ xA

satisfies

dim_{N (G)} (ker (r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n})) ∈ \frac{1}{|FIN (G)|}ℤ.

We will explain in Subsection 10.1.4 that there are counterexamples to the strong Atiyah Conjecture 10.2, but no counterexample is known at the time of writing if one replaces \frac{1}{|FIN (G)|}ℤ by ℚ or if one assumes that there is an upper bound for the orders of finite subgroups of G.

In Subsection 10.1.1 we will give various reformulations of the Atiyah Conjecture. In particular we will emphasize the module-theoretic point of view which seems to be the best one for proving the conjecture for certain classes of groups (see Lemma 10.7). This is rather surprising if one thinks about the original formulation of Atiyah’s question in terms of heat kernels. On the other hand it gives some explanation for the strong Atiyah Conjecture 10.2. Roughly speaking, the strong Atiyah Conjecture 10.2 predicts that all L^{2}-Betti numbers for G are induced by L^{2}-Betti numbers of finite subgroups H ⊂ G. This is the same philosophy as in the Isomorphism Conjecture 9.40 for K_{0}(ℂG) or in the Baum-Connes Conjecture for K_{∗}^{top}(C_{r}^{∗}(G)). We discuss the relation of the Atiyah Conjecture to other conjectures like the Kaplansky Conjecture in Subsection 10.1.2. We will give a survey on positive results about the Atiyah Conjecture in Subsection 10.1.3.

In Section 10.2 we will discuss first a general strategy how to prove the strong Atiyah Conjecture 10.2. Then we will explain the concrete form in which the strategy will appear to prove the strong Atiyah Conjecture 10.2 for Linnell’s class of groups C.

In Section 10.3 we give the details of the proof of Linnell’s Theorem 10.19. The proof is complicated, but along the way one can learn a lot of notions and techniques like transfinite induction, universal localization of rings, division closure and rational closure, crossed products and G- and Ktheory. Knowledge of these concepts is not required for any other parts of the book. To get a first impression of the proof, one may only consider the proof in the case of a free group presented in Subsection 10.3.1 which can be read without knowing any other material from this Chapter 10.

To understand this chapter, the reader is only required to be familiar with Sections 1.1, 1.2 and 6.1.

10.1 Survey on the Atiyah Conjecture

In this section we formulate various versions of the Atiyah Conjecture, discuss its consequences and a strategy for the proof.

10.1.1 Various Formulations of the Atiyah Conjecture

Conjecture 10.3 (Atiyah Conjecture). Let G be a group. Let F be a field ℚ ⊂ F ⊂ ℂ and let ℤ ⊂ Λ ⊂ ℝ be an additive subgroup of ℝ. We say that G satisfies the Atiyah Conjecture of order Λ with coefficients in F if for any matrix A ∈ M(m, n, F G) the von Neumann dimension (see Definition 1.10) of the kernel of the induced bounded G-operator r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}, x ↦ xA satisfies

dim_{N (G)} (ker (r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n})) ∈ Λ.

Notice that the strong Atiyah Conjecture 10.2 is the special case F = ℂ and Λ = \frac{1}{|FIN (G)|}ℤ, where the choices of F and Λ are the best possible ones. Sometimes we can only prove this weaker version for special choices of F and Λ but not the strong Atiyah Conjecture 10.2 itself.

Lemma 10.4. Let G be the directed union of the directed system {G_{i} | i ∈ I} of subgroups. Then G satisfies the Atiyah Conjecture 10.3 of order Λ with coefficients in F if and only if each G_{i} does. In particular a group G satisfies the Atiyah Conjecture 10.3 of order Λ with coefficients in F if and only if each finitely generated subgroup does.

Proof. Let A ∈ M(m, n, F G_{i}). Then we conclude from Lemma 1.24

dim_{N (G_{i})} (ker (r_{A}^{(2)} : l^{2}(G_{i})^{m} → l^{2}(G_{i})^{n}))
= dim_{N (G)} (ker (r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n})) .

For any A ∈ M(m, n, F G) there is G_{i} such that A already belongs to M(m, n, F G_{i}) because there are only finitely many elements in G which appear with non-trivial coefficient in one of the entries of A. A group G is the directed union of its finitely generated subgroups. ⊓ ⊔

We can reformulate the Atiyah Conjecture 10.3 for F = ℚ as follows.

Lemma 10.5. (Reformulation of the Atiyah Conjecture in terms of L^{2}-Betti numbers). Let G be a group. Let ℤ ⊂ Λ ⊂ ℝ be an additive subgroup of ℝ. Then the following statements are equivalent:

(1) For any cocompact free proper G-manifold M without boundary we have

b_{p}^{(2)}(M; N (G)) ∈ Λ;

(2) For any cocompact free proper G-CW -complex X we have

b_{p}^{(2)}(X; N (G)) ∈ Λ;

(3) The Atiyah Conjecture 10.3 of order Λ with coefficients in ℚ is true for G.

Proof. (3) ⇒ (2) This follows from Additivity (see Theorem 1.12 (2)).

(2) ⇒ (3) Because of Lemma 10.4 and the equality b_{p}^{(2)}(G ×_{H} Z; N (G)) = b_{p}^{(2)}(Z; N (H)) for p ≥ 0, H ⊂ G and Z an H-CW -complex (see Theorem 6.54 (7)), we can assume without loss of generality that G is finitely generated. Put Y = ⋁_{i=1}^{g} S^{1}. Choose an epimorphism f: π_{1}(Y) = ∗_{i=1}^{g}ℤ → G from the free group on g generators to G. Let p: \overline{Y} → Y be the G-covering associated to f. Let A be an (m,n)-matrix over ℚG and let d ≥ 3 be an integer. Choose k ∈ ℤ with k ≠ 0 such that k · A has entries in ℤG. By attaching n copies of G × D^{d−1} to \overline{Y} with attaching maps of the shape G × S^{d−2} → G → \overline{Y} and m d-cells G×D^{d} one can construct a pair of finite G-CW -complexes (X, \overline{Y}) such that the d-th differential of the cellular ℤG-chain complex C_{∗}(X) is given by k·A and the (d+1)-th differential is trivial. This follows from the observation that the composition of the Hurewicz map π_{d−1}(X_{d−1}) → H_{d−1}(X_{d−1}) with the boundary map H_{d−1}(X_{d−1}) → H_{d−1}(X_{d−1}, X_{d−2}) is surjective since each element of the cellular ℤG-basis for C_{d−1}(X) = H_{d−1}(X_{d−1}, X_{d−2}) obviously lies in the image of the composition above. Hence C_{∗}^{(2)}(X) is a finite Hilbert N (G)-chain complex such that H_{d}^{(2)}(C_{∗}^{(2)}(X)) = H_{d}^{(2)}(X; N (G)) is just the kernel of r_{k·A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}. Obviously r_{k·A}^{(2)} and r_{A}^{(2)} have the same kernel. Hence dim_{N (G)} (ker(r_{A}^{(2)})) = b_{d}^{(2)}(X; N (G)).

(1) ⇒ (2) Given a cocompact free proper G-CW -complex X of dimension n, we can find an embedding of G\X in ℝ^{2n+3} with regular neighborhood N with boundary ∂N [445, chapter 3]. Then there is an (n + 1)-connected map f: ∂N ⟶ G\X. Let \overline{f}: \overline{∂N} → X be obtained by the pullback of X → G\X with f. Then \overline{f} is (n + 1)-connected and \overline{∂N} is a cocompact free proper G-manifold without boundary. We get b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(\overline{∂N}; N (G)) for p ≤ n from Theorem 1.35 (1).

(2) ⇒ (1) This is obvious. This finishes the proof of Lemma 10.5. ⊓ ⊔ Atiyah’s question [9, page 72], whether the L^{2}-Betti numbers b_{p}^{(2)}(M; N (G))

of a cocompact free proper Riemannian G-manifold M with G-invariant Riemannian metric and without boundary are always rational numbers, has a positive answer if and only if the Atiyah Conjecture 10.3 of order Λ = ℚ with coefficients in F = ℚ is true for G. This follows from the L^{2}-Hodge-de Rham Theorem 1.59, the expression of the L^{2}-Betti numbers in terms of the heat kernel (see (1.60) and Theorem 3.136 (1)) and Lemma 10.5 above.

All examples of L^{2}-Betti numbers of G-coverings of closed manifolds or of finite CW -complexes, which have explicitly been computed in this book, are consistent with the strong Atiyah Conjecture 10.2, provided that there is an upper bound on the orders of finite subgroups of G. Without the latter condition counterexamples do exist as we will explain in Remark 10.24.

One can rephrase the Atiyah Conjecture also in a more module-theoretic fashion, where one emphasizes the possible values dim_{N (G)}(N (G) ⊗_{F G} M) for FG-modules M. This point of view turns out to be the best one for proofs of the Atiyah Conjecture. Recall that we have defined the von Neumann dimension of any N (G)-module in Definition 6.6 (using Theorem 6.5).

Notation 10.6. Let G be a group and let F be a field with ℚ ⊂ F ⊂ ℂ. Define Λ(G, F)_{fgp}, Λ(G, F)_{fp}, Λ(G, F)_{fg} or Λ(G, F)_{all} respectively to be the additive subgroup of ℝ given by differences

dim_{N (G)}(N (G) ⊗_{F G} M_{1}) − dim_{N (G)}(N (G) ⊗_{F G} M_{0}),

where M_{0} and M_{1} run through all finitely generated projective F G-modules, finitely presented F G-modules, finitely generated F G-modules or F G-modules with dim_{N (G)}(N (G) ⊗_{F G} M_{i}) < ∞ for i = 0, 1 respectively.

Lemma 10.7. (Reformulation of the Atiyah Conjecture in terms of modules). A group G satisfies the Atiyah Conjecture of order Λ with coefficients in F if and only if Λ(G, F)_{fp} ⊂ Λ, or equivalently, if and only if for any finitely presented F G-module M

dim_{N (G)}(N (G) ⊗_{F G} M) ∈ Λ.

Proof. Given a matrix A ∈ M(m, n, F G), let r_{A}: F G^{m} → F G^{n} and r_{A}^{N (G)} : N (G)^{m} → N (G)^{n} be the associated F G-homomorphism and N (G)- homomorphism given by right multiplication with A. Since the tensor product N (G)⊗_{F G} − is right exact, coker (r_{A}^{N (G)}) is N (G)-isomorphic to N (G)⊗_{F G} coker(r_{A}). We conclude from Additivity (see Theorem 6.7 (4b))

dim_{N (G)} (ker (r_{A}^{N (G)})) = m − n + dim_{N (G)} (N (G) ⊗_{F G} coker(r_{A})) . (10.8)

Since N (G) is semihereditary (see Theorem 6.5 and Theorem 6.7 (1)), ker (r_{A}^{N (G)}) is finitely generated projective. We get from Theorem 6.24

dim_{N (G)} (ker (r_{A}^{N (G)})) = dim_{N (G)} (ker (r_{A}^{(2)})) .  (10.9)

We conclude from (10.8) and (10.9) that dim_{N (G)} (ker (r_{A}^{(2)})) belongs to Λ if and only if dim_{N (G)} (N (G) ⊗_{F G} coker(r_{A})) belongs to Λ since m−n ∈ ℤ ⊂ Λ.

⊓ ⊔

Lemma 10.10. Let G be a group and let F be a field with ℚ ⊂ F ⊂ ℂ.

(1) We have \frac{1}{|FIN (G)|}ℤ ⊂ Λ(G, F)_{fgp}. If the Isomorphism Conjecture 9.40 for K_{0}(ℂG) is true, we have \frac{1}{|FIN (G)|}ℤ = Λ(G, F)_{fgp};

(2) Λ(G, F)_{fgp} ⊂ Λ(G, F)_{fp} ⊂ Λ(G, F)_{fg} ⊂ Λ(G, F)_{all};

(3) Λ(G, F)_{fg} ⊂ clos(Λ(G, F)_{fp}), where the closure is taken in ℝ;

(4) clos(Λ(G, F)_{fp}) = clos(Λ(G, F)_{fg}) = Λ(G, F)_{all}.

Proof. (1) The image of the following composition

colim_{Or(G,FIN)} K_{0}(F H) ⟶^{a} K_{0}(F G) ⟶^{i} K_{0}(N (G)) ⟶^{dim_{N (G)}} ℝ

is \frac{1}{|FIN (G)|}ℤ, where a is essentially given by the various inclusions of the finite subgroups H of G and is in the case F = ℂ the assembly map appearing in the Isomorphism Conjecture 9.40 for K_{0}(ℂG), and i is a change of rings homomorphism. This follows from the compatibility of the dimension with induction (see Theorem 6.29 (2)) and the fact that dim_{N (H)}(V) = \frac{1}{|H|} · dim_{ℂ}(V) holds for a finitely generated N (H)-module for finite H ⊂ G. The

image of the composition K_{0}(F G) ⟶^{i} K_{0}(N (G)) ⟶^{dim_{N (G)}} ℝ is by definition Λ(G, F)_{fgp}.

(2) This is obvious.

(3) Let M be a finitely generated F G-module. Choose an exact sequence ⨁_{i∈I} F G ⟶^{p} F G^{n} → M → 0 for an appropriate n ∈ ℕ and index set I. For a finite subset J ⊂ I let M_{J} be the cokernel of the restriction of p to ⨁_{i∈J} F G. We obtain a directed system of finitely presented F G-modules {M_{J} | J ⊂ I, |J| < ∞} such that colim_{J⊂I,|J|<∞} M_{J} = M. We conclude colim_{J⊂I,|J|<∞} N (G)⊗_{F G} M_{J} = N (G)⊗_{F G} M since N (G)⊗_{F G} − has a right adjoint and hence respects colimits. Each of the structure maps N (G) ⊗_{F G} M_{J_{1}} → N (G) ⊗_{F G} M_{J_{2}} for J_{1} ⊂ J_{2} ⊂ I with |J_{1}|, |J_{2}| < ∞ is surjective and each F G-module M_{J} is finitely presented. Hence the claim follows since the dimension is compatible with directed colimits (see Theorem 6.13 (2)).

(4) Let M be an F G-module. Let {M_{i} | i ∈ I} be the directed system of its finitely generated submodules. Then M is colim_{i∈I} M_{i}. Hence N (G) ⊗_{F G} M is colim_{i∈I} N (G) ⊗_{F G} M_{i}. Because of Theorem 6.13 (2) and assertion (3) it suffices to show that for any map f: M_{0} → M_{1} of finitely generated F Gmodules we have

dim_{N (G)} (im (id ⊗_{F G}f: N (G) ⊗_{F G} M_{0} → N (G) ⊗_{F G} M_{1})) ∈ Λ(G, F)_{fg}.

This follows from Additivity (see Theorem 6.7 (4b)) and the exact sequence

N (G) ⊗_{F G} M_{0} ⟶^{id ⊗_{F G}f} N (G) ⊗_{F G} M_{1} ⟶^{id ⊗_{F G} pr} N (G) ⊗_{F G} coker(f) → 0.⊓ ⊔

Remark 10.11. Let G be a group. Then \frac{1}{|FIN (G)|}ℤ is closed in ℝ if and only if there is an upper bound on the orders of finite subgroups. Suppose that there is such an upper bound. Then the least common multiple d of the orders of finite subgroups is defined and

\frac{1}{|FIN (G)|}ℤ = \{r ∈ ℝ | d · r ∈ ℤ\}

and by Lemma 10.7 and Lemma 10.10 the strong Atiyah Conjecture 10.2 is equivalent to the equality

{r ∈ ℝ | d · r ∈ ℤ} = Λ(G, ℂ)_{fgp} = Λ(G, ℂ)_{fp} = Λ(G, ℂ)_{fg} = Λ(G, ℂ)_{all}.

In particular the strong Atiyah Conjecture 10.2 for a torsionfree group G is equivalent to the equality

ℤ = Λ(G, ℂ)_{fgp} = Λ(G, ℂ)_{fp} = Λ(G, ℂ)_{fg} = Λ(G, ℂ)_{all}.

Remark 10.12. Assume that G is amenable and that there is an upper bound on the orders of finite subgroups of G. Then G satisfies the strong Atiyah Conjecture 10.2 if and only if the image of the map

dim_{N (G)} : G_{0}(ℂG) → ℝ, [M] ↦ dim_{N (G)}(N (G) ⊗_{ℂG} M)

(see Theorem 6.7 (4e) and Theorem 9.64) is contained in (and hence equal to) \frac{1}{|FIN (G)|}ℤ. Moreover, we conclude from Theorem 9.54 (8) and Lemma 10.10 (1) that the strong Atiyah Conjecture 10.2 holds for virtually poly-cyclic groups G.

Example 10.13. Let A = ⨁_{n∈ℤ} ℤ/2. This abelian group is locally finite and hence satisfies the Isomorphism Conjecture 9.40 for K_{0}(ℂG) and by Lemma 10.4 the strong Atiyah Conjecture 10.2 . This implies

ℤ[1/2] = \frac{1}{|FIN (A)|}ℤ = Λ(A, ℂ)_{fgp} = Λ(A, ℂ)_{fp}.

We want to show that each real number r ≥ 0 can be realized by r = dim_{N (A)}(N (A) ⊗_{ℚA} M) for a finitely generated ℚG-module M. This implies

Λ(A, ℂ)_{fg} = ℝ

or, equivalently, that the map

dim_{N (A)} : G_{0}(ℂA) → ℝ, [M] ↦ dim_{N (A)}(N (A) ⊗_{ℂA} M)

(see Theorem 6.7 (4e) and Theorem 9.64) is surjective. The image of its composition with the forgetful map f: K_{0}(ℂA) → G_{0}(ℂA) is ℤ[1/2]. In particular we conclude that f is not surjective and G_{0}(ℂA) is not countable.

Let A_{n} ⊂ A be the finite subgroup ⨁_{k=0}^{n−1} ℤ/2 of order 2^{n} for n ≥ 1. Denote by N_{n} ∈ ℚA the element 2^{−n} · ∑_{a∈A_{n}} a. Let I_{n} and (N_{n}) be the ideals of ℚA generated by the elements N_{n} − N_{n−1} and N_{n}. Since N_{m}N_{n} = N_{max{m,n}}, we get a direct sum decomposition (N_{n}) ⊕ I_{n} = (N_{n−1}) Since (N_{n}) is ℚG-isomorphic to ℚG ⊗_{ℚA_{n}} ℚ for ℚ with the trivial A_{n}-action and the dimension is compatible with induction (see Theorem 6.29 (2)), we get

dim_{N (A)}(N (A) ⊗_{ℚA} I_{n})
= dim_{N (A)}(N (A) ⊗_{ℚA} (N_{n−1})) − dim_{N (A)}(N (A) ⊗_{ℚA} (N_{n}))
= \frac{1}{|A_{n−1}|} − \frac{1}{|A_{n}|}
= 2^{−n}.

Fix any number r ∈ [0, 1). Because it has a 2-adic expansion, we can find a (finite or infinite) sequence of positive integers n_{1} < n_{2} < n_{3} < … such that

r = ∑_{i≥1} 2^{−n_{i}} = ∑_{i≥1} dim_{N (A)}(N (A) ⊗_{ℚA} I_{n_{i}}).

Let P_{k} ⊂ ℚA be the ℚA-submodule ∑_{i=1}^{k} I_{n_{i}}. By construction P_{k} = ⨁_{i=1}^{k} I_{n_{i}} and P_{k} is a direct summand in ℚA. This implies

dim_{N (A)}(N (A) ⊗_{ℚA} ℚA/P_{k}) = 1 − ∑_{i=1}^{k} 2^{−n_{i}}.

Let P ⊂ ℚA be the union of the submodules P_{k}. Then N (A) ⊗_{ℚA} ℚA/P = colim_{k→∞} N (A) ⊗_{ℚA} ℚA/P_{k}. We conclude from Theorem 6.13 (2)

dim_{N (A)}(N (A) ⊗_{ℚA} ℚA/P) = 1 − ∑_{i≥1} 2^{−n_{i}} = 1 − r.

Since M = ℚA^{n} ⊕ ℚA/P is a finitely generated ℚA-module and satisfies dim_{N (A)}(N (A) ⊗_{ℚA} M) = n + 1 − r, the claim follows.

10.1.2 Relations of the Atiyah Conjecture to Other Conjectures

In this subsection we relate the Atiyah Conjecture to other conjectures.

Conjecture 10.14 (Kaplansky Conjecture). The Kaplansky Conjecture for a torsionfree group G and a field F says that the group ring F G has no non-trivial zero-divisors.

If G contains an element g of finite order n > 1, then x = ∑_{i=1}^{n} g^{i} is a nontrivial zero-divisor, namely x(n − x) = 0. Hence the Kaplansky Conjecture is equivalent to the statement that, given a field F, a group G is torsionfree if and only if F G has no non-trivial zero-divisors.

If G is a right-ordered torsionfree group and F is a field, then the Kaplansky Conjecture is known to be true [302, Theorem 6.29 on page 101]), [310, Theorem 4.1]. Delzant [140] deals with group rings of word-hyperbolic groups and proves the Kaplansky Conjecture for certain word-hyperbolic groups. Given a torsionfree group G, a weaker version of the Kaplansky Conjecture predicts that F G has no non-trivial idempotents and a stronger version predicts that all units in F G are trivial, i.e. of the form λ·g for λ ∈ F, λ ≠ 0 and g ∈ G [302, (6.20) on page 95]). For more information about the Kaplansky Conjecture we refer for instance to [310, Section 4]. A proof of the next result can also be found in [172].

Lemma 10.15. Let F be a field with ℤ ⊂ F ⊂ ℂ and let G be a torsionfree group. Then the Kaplansky Conjecture holds for G and the field F if the Atiyah Conjecture 10.3 of order Λ = ℤ with coefficients in F is true for G.

Proof. Let x ∈ F G be a zero-divisor. Let r_{x}^{(2)} : l^{2}(G) → l^{2}(G) be given by right multiplication with x. We get 0 < dim_{N (G)} (ker(r_{x}^{(2)})) ≤ 1 from The-

orem 1.12 (1) and (2). Since by assumption dim_{N (G)} (ker(r_{x}^{(2)})) ∈ ℤ, we

conclude dim_{N (G)} (ker(r_{x}^{(2)})) = 1. Since ker(r_{x}^{(2)}) is closed in l^{2}(G), Theo-

rem 1.12 (1) and (2) imply ker(r_{x}^{(2)}) = l^{2}(G) and hence x = 0. ⊓ ⊔ Lemma 10.16. Let G be a torsionfree amenable group. Then the strong Atiyah Conjecture 10.2 for G is equivalent to the Kaplansky Conjecture 10.14 for G and the field ℂ.

Proof. Because of Lemma 10.15 it suffices to prove the strong Atiyah Conjecture 10.2, provided that the Kaplansky Conjecture 10.14 holds for G and ℂ. We first show for a ℂG-module M which admits an exact sequence

ℂG^{n} ⟶^{c} ℂG → M → 0 that dim_{N (G)}(N (G) ⊗_{ℂG} M) ∈ {0, 1}. Let C_{∗} be the

chain complex concentrated in dimension 1 and 0 with c as first differential. We conclude from Lemma 1.18, Theorem 6.24 and the dimension-flatness of N (G) over ℂG for amenable G (see Theorem 6.37)

dim_{N (G)}(N (G) ⊗_{ℂG} M) = dim_{N (G)}(coker(N (G) ⊗_{ℂG} c))
= dim_{N (G)}(H_{0}^{(2)}(l^{2}(G) ⊗_{ℂG} C_{∗}))
= dim_{N (G)}(ker(Δ_{0} : l^{2}(G) → l^{2}(G))
= dim_{N (G)}(ker(N (G) ⊗_{ℂG} c^{∗}c))
= dim_{N (G)}(N (G) ⊗_{ℂG} ker(c^{∗}c)).

Since c^{∗}c: ℂG → ℂG is ℂG-linear and hence given by right multiplication with an element in ℂG, its kernel is either trivial or ℂG. This implies dim_{N (G)}(N (G) ⊗_{ℂG} ker(c^{∗}c)) ∈ {0, 1} and thus dim_{N (G)}(N (G) ⊗_{ℂG} M) ∈ {0, 1}.

In order to prove the strong Atiyah Conjecture 10.2 it suffices to show for any finitely generated ℂG-module M that dim_{N (G)}(N (G) ⊗_{ℂG} M) is an integer. We proceed by induction over the number n of generators of M. In the induction beginning n = 1 we can assume the existence of an exact sequence ⨁_{i∈I} ℂG ⟶^{p} ℂG → M. Given a finite subset J ∈ I, let M_{J} be the cokernel of the restriction of p to ⨁_{i∈J} ℂG. Then dim_{N (G)}(N (G) ⊗_{ℂG} M_{J}) ∈ {0, 1} by the argument above. We conclude as in the proof of Lemma 10.10 (3) that dim_{N (G)}(N (G) ⊗_{ℂG} M) ∈ {0, 1}. It remains to prove the induction step from n to n + 1. Obviously we can find a short exact sequence of ℂG-modules 0 → L → M → N → 0 such that L is generated by one element and N by n elements. By induction hypothesis dim_{N (G)}(N (G)⊗_{ℂG}L) and dim_{N (G)}(N (G) ⊗_{ℂG} N) are integers. We conclude from Additivity (see Theorem 6.7 (4b)) and dimension-flatness of N (G) over ℂG for amenable G (see Theorem 6.37)

dim_{N (G)}(N (G)⊗_{ℂG}M) = dim_{N (G)}(N (G)⊗_{ℂG}L)+dim_{N (G)}(N (G)⊗_{ℂG}N).

Hence dim_{N (G)}(N (G) ⊗_{ℂG} M) is an integer. ⊓ ⊔ The result above follows also from [416, Theorem 2.2]. Suppose that there is an integer d such that the order of any finite sub-

group of G divides d and that the strong Atiyah Conjecture 10.2 holds for G. Suppose furthermore that h^{(2)}(G) < ∞ and hence χ^{(2)}(G) is defined (see Definition 6.79). Theorem 6.99 (2) says that d · b_{p}^{(2)}(G) ∈ ℤ holds for all p. This implies

d · χ^{(2)}(G) ∈ ℤ.

This is consistent with the result of Brown [69, Theorem IX.9.3 on page 257] that d · χ′(G) ∈ ℤ holds for a group G of finite homological type, i.e. a group G of finite virtual cohomological dimension such that for any ℤGmodule M, which is finitely generated as abelian group, H_{i}(G; M) is finitely generated for all i ≥ 0. Here χ′(G) is defined by

χ′(G) = \frac{1}{[G : G_{0}]} · ∑_{p≥0}(−1)^{p} rk_{ℤ}(H_{p}(G_{0}; ℤ))  (10.17)

for any torsionfree subgroup G_{0} ⊂ G of finite index [G : G_{0}]. Namely, if G contains a torsionfree subgroup G_{0} of finite index, which has a finite model for BG, then both χ^{(2)}(G) and χ′(G) are defined and agree. This follows from Remark 6.81, where χ^{(2)}(G) = χ_{virt}(BG) is shown, and the fact that χ_{virt}(BG) = χ′(G) [69, page 247].

We will state, discuss and give some evidence for the Singer Conjecture in Chapter 11. Because of the Euler-Poincaré formula (see Theorem 1.35 (2)) the Singer Conjecture implies for a closed aspherical manifold M that all the L^{2}-Betti numbers b_{p}^{(2)}(M̃) of its universal covering are integers, as predicted by the strong Atiyah Conjecture 10.2 in combination with Lemma 10.5.

A link between the Atiyah Conjecture and the Baum-Connes Conjecture will be discussed in Section 10.4. A connection between the Atiyah Conjecture and the Isomorphism Conjecture 9.40 for K_{0}(ℂG) has already been explained in Lemma 10.10 (1). Notice that the Atiyah Conjecture is harder than the Baum-Connes Conjecture and the Isomorphism Conjecture for K_{0}(ℂG) in the sense that it deals with finitely presented modules (see Lemma 10.7) and not only with finitely generated projective modules.

10.1.3 Survey on Positive Results about the Atiyah Conjecture

In this subsection we state some cases, where the Atiyah Conjecture is known to be true.

Definition 10.18 (Linnell’s class of groups C). Let C be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients.

We will extensively discuss the proof of the following theorem due to Linnell [309, Theorem 1.5] later in this chapter.

Theorem 10.19 (Linnell’s Theorem). Let G be a group such that there is an upper bound on the orders of finite subgroups and G belongs to C. Then the strong Atiyah Conjecture 10.2 holds for G.

The next result is a direct consequence of Theorem 13.3 (2), Theorem 13.31 (2) and Proposition 13.35 (1) which deal with the Approximation Conjecture 13.1.

Theorem 10.20. Let ℤ ⊂ Λ ⊂ ℝ be an additive subgroup of ℝ which is closed in ℝ. Let {G_{i} | i ∈ I} be a directed system of groups such that each G_{i} belongs to the class G (see Definition 13.9) and satisfies the Atiyah Conjecture 10.3 of order Λ with coefficients in ℚ. Then both its colimit (= direct limit) and its inverse limit satisfy the Atiyah Conjecture 10.3 of order Λ with coefficients in ℚ.

The next definition and the next theorem, which is essentially a consequence of the two Theorems 10.19 and 10.20 above by transfinite induction, are taken from [463].

Definition 10.21. Let D be the smallest non-empty class of groups such that

(1) If p: G → A is an epimorphism of a torsionfree group G onto an elementary amenable group A and if p^{−1}(B) ∈ D for every finite group B ⊂ A, then G ∈ D;

(2) D is closed under taking subgroups;

(3) D is closed under colimits and inverse limits over directed systems.

Theorem 10.22. (1) If the group G belongs to D, then G is torsionfree and the Atiyah Conjecture 10.3 of order Λ = ℤ with coefficients in F = ℚ is true for G;

(2) The class D is closed under direct sums, direct products and free products. Every residually torsionfree elementary amenable group belongs to D.

The fundamental group of compact 2-dimensional manifold M belongs to C since it maps onto the abelian group H_{1}(M) with a free group as kernel. If π_{1}(M) is torsionfree, it belongs also to D. This follows from the fact that a finitely generated free group is residually torsionfree nilpotent [354, §2]. The pure braid groups belong to D since they are residually torsionfree nilpotent [180, Theorem 2.6]. The Atiyah Conjecture 10.3 of order Λ = ℤ with coefficients in F = ℚ is also true for the braid group (see [313]). Positive one-relator groups, i.e. one relator groups whose relation can be written as a word in positive multiples of the generators, belong to D because they are residually torsionfree solvable.

10.1.4 A Counterexample to the Strong Atiyah Conjecture

The lamplighter group L is defined by the semidirect product

L := ⨁_{n∈ℤ} ℤ/2 ⋊ ℤ

with respect to the shift automorphism of ⨁_{n∈ℤ} ℤ/2, which sends (x_{n})_{n∈ℤ} to (x_{n−1})_{n∈ℤ}. Let e_{0} ∈ ⨁_{n∈ℤ} ℤ/2 be the element whose entries are all zero except the entry at 0. Denote by t ∈ ℤ the standard generator of ℤ which we will also view as element of L. Then {e_{0}t, t} is a set of generators for L. The associated Markov operator M: l^{2}(G) → l^{2}(G) is given by right multiplication with \frac{1}{4} · (e_{0}t + t + (e_{0}t)^{−1} + t^{−1}). It is related to the Laplace operator Δ_{0} : l^{2}(G) → l^{2}(G) of the Cayley graph of G by Δ_{0} = 4 · id −4 · M. The following result is a special case of the main result in the paper of Grigorchuk and Żuk [230, Theorem 1 and Corollary 3] (see also [229]). An elementary proof can be found in [143].

Theorem 10.23 (Counterexample to the strong Atiyah Conjecture). The von Neumann dimension of the kernel of the Markov operator M of the lamplighter group L associated to the set of generators {e_{0}t, t} is 1/3. In particular L does not satisfy the strong Atiyah Conjecture 10.2.

Notice that each finite subgroup of the lamplighter group L is a 2-group and for any power 2^{n} of 2 a subgroup of order 2^{n} exists. Hence \frac{1}{|FIN (L)|} ·ℤ = ℤ[1/2] and we obtain a counterexample to the strong Atiyah Conjecture 10.2. At the time of writing the author does not know of a counterexample to the strong Atiyah Conjecture in the case, where one replaces \frac{1}{|FIN (L)}ℤ by ℚ or where one assumes the existence of an upper bound on the orders of finite subgroups.

Remark 10.24. The lamplighter group L is a subgroup of a finitely presented group G such that any finite subgroup of G is a 2-group. By Theorem 10.23 above and a slight variation of the proof of Lemma 10.5 we conclude the existence of a closed Riemannian manifold M with G = π_{1}(M) such that not all the L^{2}-Betti numbers of its universal covering belong to \frac{1}{|FIN (π_{1}(M))|}ℤ = ℤ[1/2].

The group G is constructed as follows (see [229]). Let ϕ: L → L be the injective endomorphism sending t to t and e_{0} to e_{0}t^{−1}e_{0}t. Let G be the HHNextension associated to the subgroups L and im(ϕ) of L and the isomorphism ϕ: L → im(ϕ). This group G has the finite presentation

G = ⟨e_{0}, t, s | e_{0}^{2} = 1, [t^{−1}e_{0}t, e_{0}] = 1, [s, t] = 1, s^{−1}e_{0}s = e_{0}t^{−1}e_{0}t⟩.

The general theory of HHN-extensions implies that G contains L as a subgroup, namely, as the subgroup generated by e_{0} and t and each finite subgroup of G is isomorphic to a subgroup of L [350, Theorem 2.1 on page 182, Theorem 2.4 on page 185].

Remark 10.25. For the lamplighter group L the Isomorphism Conjecture 9.40 is true, i.e. the assembly map

a: colim_{L/H⊂Or(L,FIN)} K_{0}(ℂH) → K_{0}(ℂL)

is bijective.

We first prove surjectivity. A ring is called regular coherent, if any finitely presented module has a finite projective resolution. For a finite group H the rings ℂH and ℂ[H × ℤ] are regular coherent. Given a commutative ring R, any finitely presented R[⨁_{n∈ℤ} ℤ/2]-module or R[(⨁_{n∈ℤ} ℤ/2) × ℤ]-module respectively is the induction of a finitely presented RH-module or R[H × ℤ]- module respectively for some finite subgroup H ⊂ ⨁_{n∈ℤ} ℤ/2. Hence the rings ℂ[⨁_{n∈ℤ} ℤ/2] and ℂ[(⨁_{n∈ℤ} ℤ/2) × ℤ] are regular coherent. For a commutative ring R and automorphism α: R → R the change of rings homomorphism K_{0}(R) → K_{0}(R_{α}[t, t^{−1}]) is surjective, where R_{α}[t, t^{−1}] is the ring of α-twisted Laurent series over R, provided that R and R[ℤ] are regular coherent [508, Corollary 13.4 and the following sentence on page 221]. This implies that K_{0}(ℂ[⨁_{n∈ℤ} ℤ/2]) → K_{0}(ℂL) is surjective. Since ⨁_{n∈ℤ} ℤ/2 is locally finite, ⨁_{n∈ℤ} ℤ/2 satisfies the Isomorphism Conjecture 9.40. This proves the surjectivity of a: colim_{H⊂FIN (L)} K_{0}(ℂH) → K_{0}(ℂL).

Next we prove injectivity. For finite subgroups H ⊂ K of L the inclusion H → K is split injective, because both H and K are ℤ/2-vector spaces. Therefore K_{0}(ℂH) → K_{0}(ℂK) is a split injection of finitely generated free abelian groups. Hence colim_{H⊂FIN (L)} K_{0}(ℂH) is torsionfree. Since a ⊗_{ℤ} ℂ is injective by Theorem 9.54 (1), the assembly map a is injective.

We conclude for the lamplighter group L that Λ(G, ℂ)_{fgp} ≠ Λ(G, ℂ)_{fp}.

10.2 A Strategy for the Proof of the Atiyah Conjecture

In this section we discuss a strategy for the proof of the strong Atiyah Conjecture 10.2. We begin with a general strategy. Although it cannot work in full generality because of the counterexample presented in Subsection 10.1.4, it is nevertheless very illuminating and is the basic guideline for proofs in special cases. We will give some basic facts about localization in order to formulate the result which we will prove and which implies Linnell’s Theorem 10.19. After presenting some induction principles we will explain the concrete form in which the strategy will appear to prove the strong Atiyah Conjecture 10.2 for Linnell’s class of groups C. To get a first impression of the proof, one may only consider the proof in the case of a free group presented in Subsection 10.3.1 which can be read without knowing any other material from this Chapter 10.

Linnell proves his Theorem 10.19 in [309, Theorem 1.5]. Our presentation is based on the Ph. D. thesis of Reich [435].

10.2.1 The General Case

In this Subsection we discuss a general strategy for the proof of the strong Atiyah Conjecture 10.2. We have introduced and studied the notion of a von Neumann regular ring in Subsection 8.2.2 and of the algebra U(G) in Definition 8.9 and Subsection 8.2.3. It suffices to recall for the sequel the following facts. A ring is von Neumann regular if and only if any finitely presented Rmodule is finitely generated projective (see Lemma 8.18). The algebra U(G) is the Ore localization of N (G) with respect to the multiplicatively closed subset of non-zero-divisors in N (G) (see Theorem 8.22 (1)). The algebra U(G) is von Neumann regular (see Theorem 8.22 (3)). The explicit operator theoretic definition U(G) will not be needed. The dimension function dim_{N (G)} for arbitrary N (G)-modules extends to a dimension function dim_{U(G)} for arbitrary U(G)-modules (see Theorem 8.29).

Lemma 10.26 (General Strategy). Let G be a group. Suppose that there is a ring S(G) with ℂG ⊂ S(G) ⊂ U(G) which has the following properties:

(R’) The ring S(G) is von Neumann regular; (K’) The image of the composition

K_{0}(S(G)) ⟶^{i} K_{0}(U(G)) ⟶^{dim_{U(G)}} ℝ

is contained in \frac{1}{|FIN (G)|}ℤ, where i denotes the change of rings homomorphism.

Then G satisfies the strong Atiyah Conjecture 10.2.

Proof. Because of Lemma 10.7 it suffices to show for a finitely presented ℂG-module M that dim_{N (G)}(N (G) ⊗_{ℂG} M) belongs to \frac{1}{|FIN (G)|}ℤ . Since S(G) is von Neumann regular, the finitely presented S(G)-module S(G) ⊗_{ℂG} M is finitely generated projective (see Lemma 8.18) and hence defines a class in K_{0}(S(G)). The composition dim_{U(G)} ∘i: K_{0}(S(G)) → ℝ has image \frac{1}{|FIN (G)|}ℤ by assumption and sends the class of S(G) ⊗_{ℂG} M to dim_{N (G)}(N (G) ⊗_{ℂG} M) by Theorem 8.29. ⊓ ⊔

Remark 10.27. The role of the ring-theoretic condition (R’) is to reduce the problem from finitely presented modules to finitely generated projective modules. Thus one can use K-theory of finitely generated projective modules. The price to pay is that one has to enlarge ℂG to an appropriate ring S(G) which is nicer than ℂG, namely, which is von Neumann regular. This will be done by a localization process, namely, we will choose S(G) to be the division closure of ℂG in U(G). This will be a minimal choice for S(G). Notice that localization usually improves the properties of a ring and that in general ℂG does not have nice ring theoretic properties. For instance ℂG is von Neumann regular if and only if G is locally finite (see Example 8.21). No counterexample is known to the conjecture that ℂG is Noetherian if and only if G is virtually poly-cyclic. The implications G virtually poly-cyclic ⇒ ℂG regular ⇒ ℂG Noetherian are proved in [447, Theorem 8.2.2 and Theorem 8.2.20]. In order to be able to treat dimensions, the algebra U(G) comes in, which in contrast to N (G) does contain S(G). The passage from N (G) to U(G) does not cause any problems since the dimension theory of N (G) extends to a dimension theory for U(G) and the change of rings map K_{0}(N (G)) → K_{0}(U(G)) is bijective (see Theorem 8.29 and Theorem 9.20 (1)).

Lemma 10.28. Let S(G) be a ring with ℂG ⊂ S(G) ⊂ U(G).

(1) Suppose that S(G) is semisimple. Then there is an upper bound on the orders of finite subgroups of G;

(2) Suppose that S(G) satisfies the conditions (R’) and (K’) of Lemma 10.26 and that there is an upper bound on the orders of finite subgroups of G. Then S(G) is semisimple;

(3) Suppose that G is torsionfree. Then S(G) is a skew field if and only if S(G) satisfies the conditions (R’) and (K’) of Lemma 10.26. In this case we get for any ℂG-module M

dim_{N (G)}(N (G) ⊗_{ℂG} M) = dim_{U(G)}(U(G) ⊗_{ℂG} M)
= dim_{S(G)}(S(G) ⊗_{ℂG} M),

where dim_{S(G)}(V) for an S(G)-module V is defined to be n if V ≅_{S(G)} S(G)^{n} for a non-negative integer n and by ∞ otherwise.

Proof. (1) By Wedderburn’s Theorem the semisimple ring S(G) is a finite product of matrix rings over skewfields. Since K_{0} is compatible with products and K_{0}(D) ≅ ℤ for a skewfield D, the Morita isomorphism implies that K_{0}(S(G)) is a finitely generated free abelian group. Hence \frac{1}{|FIN (G)|}ℤ is finitely generated since it is contained in the image of the composition dim_{U(G)} ∘i: K_{0}(S(G)) → ℝ. If a_{1}, a_{2}, …, a_{r} are generators for \frac{1}{|FIN (G)|}ℤ, we can find l ∈ ℤ such that l·a_{i} ∈ ℤ for i = 1, 2 …, r. Hence l· \frac{1}{|FIN (G)|}ℤ ⊂ ℤ. Therefore l is an upper bound on the orders of finite subgroups of G.

(2) Let l be the least common multiple of the orders of finite subgroups of G. By condition (K’) the image of the composition

K_{0}(S(G)) → K_{0}(U(G)) ⟶^{dim_{U(G)}} ℝ

lies in \frac{1}{l} · ℤ = \{r ∈ ℝ | l · r ∈ ℤ\}. We want to show that for any chain of ideals {0} = I_{0} ⊂ I_{1} ⊂ I_{2} ⊂ … ⊂ I_{r} = S(G) of S(G) with I_{i} ≠ I_{i+1} we have r ≤ l. Then S(G) is Noetherian and hence is semisimple by Lemma 8.20 (2).

Choose x_{i} ∈ I_{i} with x_{i} ∉ I_{i−1} for 1 ≤ i ≤ r − 1. Let J_{i} be the ideal generated by x_{1}, x_{2}, …, x_{i} for 1 ≤ i ≤ r − 1. Then we obtain a sequence of finitely generated ideals of the same length {0} = J_{0} ⊂ J_{1} ⊂ J_{2} ⊂ … ⊂ J_{r} = S(G) of S(G) with J_{i} ≠ J_{i+1}. Since U(G) is von Neumann regular, J_{i−1} is a direct summand in J_{i} (see Lemma 8.18 (4)). Hence we get direct sum decompositions J_{i} = J_{i−1} ⊕ K_{i} for i = 1, 2, …, r for finitely generated projective nontrivial S(G)-modules K_{1}, K_{2}, …, K_{r}. Choose an idempotent p_{i} ∈ M_{n_{i}}(S(G)) representing K_{i}. Then p_{i} considered as an element M_{n_{i}}(U(G)) represents U(G) ⊗_{S(G)} K_{i} and is non-trivial. Hence U(G) ⊗_{S(G)} K_{i} is a non-trivial finitely generated projective U(G)-module. We conclude from Theorem 8.29 dim_{U(G)}(U(G) ⊗_{U(S)} K_{i}) > 0 for i = 1, 2, …, r and hence

0 < dim_{U(G)}(U(G) ⊗_{S(G)} J_{1}) < dim_{U(G)}(U(G) ⊗_{S(G)} J_{2})

< … < dim_{U(G)}(U(G) ⊗_{S(G)} J_{r−1}) < 1.

Since l · dim_{U(G)}(U(G) ⊗_{S(G)} J_{i}) is an integer for i = 1, 2, …, r − 1, we get r ≤ l.

(3) Suppose that S(G) satisfies conditions (R’) and (K’). By the argument above S(G) is semisimple and contains no non-trivial ideal. Hence it is a skew field. Recall that over a skew field S(G) any S(G)-module V is isomorphic to ⨁_{i∈I} S(G) for some index set I. Now apply Theorem 8.29. Any skewfield satisfies the conditions (R’) and (K’). This finishes the proof of Lemma 10.28.

⊓ ⊔

Example 10.29. Consider the free abelian group ℤ^{n} of rank n. Let ℂ[ℤ^{n}]_{(0)} be the quotient field of the commutative integral domain ℂ[ℤ^{n}]. By inspecting Example 8.11 we see that ℂ[ℤ^{n}] ⊂ ℂ[ℤ^{n}]_{(0)} ⊂ U(ℤ^{n}). By Lemma 10.28 (3) the quotient field ℂ[ℤ^{n}]_{(0)} satisfies the conditions of Lemma 10.26. This proves the strong Atiyah Conjecture 10.2 for ℤ^{n} and that for any ℂ[ℤ^{n}]-module M

dim_{N (ℤ^{n})}(N (ℤ^{n}) ⊗_{ℂ[ℤ^{n}]} M) = dim_{ℂ[ℤ^{n}]_{(0)}}(ℂ[ℤ^{n}]_{(0)} ⊗_{ℂ[ℤ^{n}]} M)

holds. The reader should compare this with the explicit proof of Lemma 1.34, which corresponds to the case of a finitely presented ℂ[ℤ^{n}]-module M.

Remark 10.30. Suppose that G is a torsionfree amenable group which satisfies the Kaplansky Conjecture 10.14. We have already shown in Example 8.16 that set S of non-zero-divisors of ℂG satisfies the Ore condition (see Definition 8.14) and that the Ore localization S^{−1}ℂG is a skew field. Recall from Theorem 8.22 (1) that U(G) is the Ore localization of the von Neumann algebra N (G) with respect to the set of all non-zero-divisors. A non-zero-divisor x of ℂG is still a non-zero-divisor when we regard it as an element in N (G) by the following argument. Denote by r_{x}^{N (G)} : N (G) → N (G) and r_{x}^{ℂG}: ℂG → ℂG the maps given by right multiplication with x. The kernel of r_{x}^{N (G)} is a finitely generated projective N (G)- module since N (G) is semihereditary (see Theorem 6.5 and Theorem 6.7 (1)). We conclude dim_{N (G)}(ker(r_{x}^{N (G)}) = dim_{N (G)}(N (G) ⊗_{ℂG} ker(r_{x}^{ℂG})) from the dimension-flatness of N (G) over ℂG for amenable G (see Theorem 6.37). Since ker(r_{x}) = {0}, Lemma 6.28 (3) implies that r_{x}^{N (G)} has trivial kernel and hence x is a non-zero-divisor in N (G). We conclude ℂG ⊂ S^{−1}ℂ[G] ⊂ U(G). Hence the strong Atiyah Conjecture 10.2 holds for G by Lemma 10.28 (3). Notice that this gives a different proof of Lemma 10.16.

Remark 10.31. We know already from Subsection 10.1.4 that the strong Atiyah Conjecture 10.2 does not hold for arbitrary groups G. We think that the extra condition that there is an upper bound on the order of finite subgroups of G is very important. Under this condition we see from Lemma 10.28 (2) that we can replace in Lemma 10.26 the condition (R’) that S(G) is von Neumann regular by the stronger condition (R) that S(G) is semisimple. Actually, the proof of Linnell’s Theorem does only work if we use (R) instead of (R’). ⊓ ⊔

10.2.2 Survey on Universal Localizations and Division and Rational Closure

We give some basic facts about universal localization and division and rational closure.

Let R be a ring and Σ be a set of homomorphisms between (left) Rmodules. A ring homomorphism f: R → S is called Σ-inverting if for every map α: M → N ∈ Σ the induced map S ⊗_{R} α: S ⊗_{R} M → S ⊗_{R} N is an isomorphism. A Σ-inverting ring homomorphism i: R → R_{Σ} is called universal Σ-inverting if for any Σ-inverting ring homomorphism f: R → S there is precisely one ring homomorphism f_{Σ}: R_{Σ} → S satisfying f_{Σ} ∘ i = f. This generalizes Definition 8.13. If f: R → R_{Σ} and f′: R → R′_{Σ} are two universal Σ-inverting homomorphisms, then by the universal property there is precisely one isomorphism g: R_{Σ} → R′_{Σ} with g ∘ f = f′. This shows the uniqueness of the universal Σ-inverting homomorphism. The universal Σ-inverting ring homomorphism exists if Σ is a set of homomorphisms of finitely generated projective modules [465, Section 4]. If Σ is a set of matrices, a model for R_{Σ} is given by considering the free R-ring generated by the set of symbols \{\overline{a_{i,j}} | A = (a_{i,j}) ∈ Σ\} and dividing out the relations given in matrix form by \overline{A}A = A\overline{A} = 1, where \overline{A} stands for (\overline{a_{i,j}}) for A = (a_{i,j}). The map i: R → R_{Σ} does not need to be injective and the functor R_{Σ} ⊗_{R} − does not need to be exact in general.

Notation 10.32. Let S be a ring and R ⊂ S be a subring. Denote by T (R ⊂ S) the set of all elements in R which become invertible in S. Denote by Σ(R ⊂ S) the set of all square matrices A over R which become invertible in S.

A subring R ⊂ S is called division closed if T (R ⊂ S) = R^{inv}, i.e. any element in R which is invertible in S is already invertible in R. It is called rationally closed if Σ(R ⊂ S) = GL(R), i.e. any square matrix over R which is invertible over S is already invertible over R. Notice that the intersection of division closed subrings of S is again division closed, and analogously for rationally closed subrings. Hence the following definition makes sense.

Definition 10.33 (Division and rational closure). Let S be a ring with subring R ⊂ S. The division closure D(R ⊂ S) or rational closure R(R ⊂ S) respectively is the smallest subring of S which contains R and is division closed or rationally closed respectively.

Obviously R ⊂ D(R ⊂ S) ⊂ R(R ⊂ S) ⊂ S. One easily checks D(D(R ⊂ S) ⊂ S) = D(R ⊂ S) and R(R(R ⊂ S) ⊂ S) = R(R ⊂ S). The rational closure R(R ⊂ S) of R in S is the set of elements s ∈ S for which there is a square matrix A over S and a matrix B over R such that AB = BA = 1 holds over S and s is an entry in A [118, section 7.1]. The easy proof of the next two results is left to the reader (or see [435, Proposition 13.17]).

Lemma 10.34. Let S be a ring with subring R ⊂ S.

(1) If R is von Neumann regular, then R is division closed and rationally closed in S;

(2) If D(R ⊂ S) is von Neumann regular, then D(R ⊂ S) = R(R ⊂ S).

Lemma 10.35. Let S be a ring with subring R ⊂ S.

(1) The map ϕ: R_{T (R⊂S)} → S given by the universal property factorizes as

ϕ: R_{T (R⊂S)} ⟶^{Φ} D(R ⊂ S) ⊂ S;

(2) Suppose that the pair (R, T (R ⊂ S)) satisfies the (right) Ore condition (see Definition 8.14). Then

R_{T (R⊂S)} = RT (R ⊂ S)^{−1} = D(R ⊂ S),

i.e. the map given by the universal property R_{T (R⊂S)} ⟶^{j} RT (R ⊂ S)^{−1} is an isomorphism, and Φ: R_{T (R⊂S)} → D(R ⊂ S) is an isomorphism;

(3) The map ψ: R_{Σ(R⊂S)} → S given by the universal property factorizes as

ψ: R_{Σ(R⊂S)} ⟶^{Ψ} R(R ⊂ S) ⊂ S. The map Ψ: R_{Σ(R⊂S)} → R(R ⊂ S) is

always surjective.

The map Ψ: R_{Σ(R⊂S)} → R(R ⊂ S) is not always injective. (An example will be given in the exercises.)

Lemma 10.36. Let R_{i} ⊂ S_{i} ⊂ S be subrings for i ∈ I and R = ⋃_{i∈I} R_{i} and S = ⋃_{i∈I} S_{i} directed unions of rings. Suppose that all rings S_{i} are von Neumann regular. Then S is von Neumann regular and we obtain directed unions

T (R ⊂ S) = ⋃_{i∈I} T (R_{i} ⊂ S_{i});
Σ(R ⊂ S) = ⋃_{i∈I} Σ(R_{i} ⊂ S_{i});
D(R ⊂ S) = ⋃_{i∈I} D(R_{i} ⊂ S_{i});
R(R ⊂ S) = ⋃_{i∈I} R(R_{i} ⊂ S_{i}).

Proof. One easily checks that S satisfies Definition 8.17 of von Neumann regular.

Suppose that x ∈ R becomes invertible in S. Choose i ∈ I with x ∈ R_{i}. Obviously x is not a zero-divisor in S_{i}. Since S_{i} is von Neumann regular, x_{i} is invertible in S_{i} by Lemma 8.20 (1). This shows T (R ⊂ S) = ⋃_{i∈I} T (R_{i} ⊂ S_{i}). The proof of Σ(R ⊂ S) = ⋃_{i∈I} Σ(R_{i} ⊂ S_{i}) is similar using the fact that a matrix ring over a von Neumann ring is again von Neumann regular by Lemma 8.19 (1).

Since S_{i} is von Neumann regular, Lemma 10.34 (1) implies D(R_{i} ⊂ S_{i}) = D(R_{i} ⊂ S) ⊂ D(R ⊂ S) and hence ⋃_{i∈I} D(R_{i} ⊂ S_{i}) ⊂ D(R ⊂ S). Since ⋃_{i∈I} D(R_{i} ⊂ S_{i}) = ⋃_{i∈I} D(R_{i} ⊂ S) is division closed in S, we get ⋃_{i∈I} D(R_{i} ⊂ S_{i}) = D(R ⊂ S). The proof for the rational closure is analogous. ⊓ ⊔

10.2.3 The Strategy for the Proof of Linnell’s Theorem

We will outline the strategy of proof for Linnell’s Theorem 10.19.

Notation 10.37. Let G be a group. Define D(G) to be the division closure D(ℂG ⊂ U(G)) and R(G) to be the rational closure R(ℂG ⊂ U(G)) of ℂG in U(G) (see Definition 10.33). We abbreviate T (G) = T (ℂG ⊂ U(G)) and Σ(G) = Σ(ℂG ⊂ U(G)) (see Notation 10.32).

The ring D(G) will be our candidate for S(G) in Lemma 10.26. We conclude from Lemma 10.34 that any von Neumann regular ring S(G) with ℂG ⊂ S(G) ⊂ U(G) satisfies D(G) ⊂ R(G) ⊂ S(G) and that D(G) = R(G) holds if D(G) is von Neumann regular. Hence D(G) is a minimal choice for S(G) in Lemma 10.26 and we should expect D(G) = R(G). In view of Remark 10.31 we should also expect that D(G) is semisimple. The upshot of this discussion is that, in order to prove Linnell’s Theorem 10.19, we will prove the following

Theorem 10.38 (Strong version of Linnell’s Theorem). Let G be a group in the class C (see Definition 10.18) such that there exists a bound on the orders of finite subgroups. Then

(R) The ring D(G) is semisimple; (K) The composition

colim_{G/H∈Or(G,FIN)} K_{0}(ℂH) ⟶^{a} K_{0}(ℂG) ⟶^{i} K_{0}(D(G))

is surjective, where a is the assembly map appearing in the Isomorphism Conjecture 9.40 for K_{0}(ℂG) and essentially given by the various inclusions of the finite subgroups of G, and i is a change of rings homomorphism.

Notice that we have replaced the condition (K’) appearing in Lemma 10.26 by the stronger condition (K) above. The stronger condition gives better insight in what is happening and also yields a connection to the Isomorphism Conjecture 9.40 for K_{0}(ℂG). Namely, the surjectivity of the change of rings map K_{0}(ℂG) → K_{0}(D(G)) is equivalent to the condition (K) provided that the Isomorphism Conjecture 9.40 for K_{0}(ℂG) holds.

We will prove in the strong version of Linnell’s Theorem 10.38 a statement which implies the strong Atiyah Conjecture 10.2 by Lemma 10.26 but which in general contains more information than the strong Atiyah Conjecture 10.2. At least for torsionfree groups G the statement turns out to be equivalent to the strong Atiyah Conjecture 10.2 by the next lemma.

Lemma 10.39. Let G be a torsionfree group. Then the strong Atiyah Conjecture 10.2 is true if and only if D(G) is a skewfield. In this case D(G) = R(G) and the composition

colim_{Or(G,FIN)} K_{0}(ℂH) ⟶^{a} K_{0}(ℂG) ⟶^{i} K_{0}(D(G))

is surjective.

Proof. Suppose that D(G) is a skewfield. Then D(G) = R(G) follows from Lemma 10.34 (2). Since any finitely generated module over a skewfield is finitely generated free, the composition colim_{Or(G,FIN)} K_{0}(ℂH) ⟶^{a}

K_{0}(ℂG) ⟶^{i} K_{0}(D(G)) is surjective. The strong Atiyah Conjecture 10.2 follows from Lemma 10.26.

Now suppose that the strong Atiyah Conjecture 10.2 holds. We want to show that R(G) is a skewfield. It suffices to show for any element x ∈ R(G) with x ≠ 0 that x is invertible in U(G), because then x is already invertible in R(G) since R(G) is rationally closed in U(G). By Lemma 10.35 (3) there is y ∈ ℂG_{Σ(G)} which is mapped to x under the canonical ring homomorphism Ψ: ℂG_{Σ(G)} → R(G). Two square matrices A and B over a ring R are called associated if there are invertible square matrices U and V and non-negative integers m and n satisfying

U ( \begin{matrix}A & 0 \\ 0 & I_{m} \end{matrix} ) V = ( \begin{matrix}B & 0 \\ 0 & I_{n} \end{matrix} ) .

For a set of matrices Σ over R, the universal Σ-inverting homomorphism f: R → R_{Σ} and s ∈ R_{Σ} we can find a square matrix A over R such that f(A) and s are associated in R_{Σ}. This follows from Cramer’s Rule (see [465, Theorem 4.3 on page 53]). If we apply this to y ∈ ℂG_{Σ(G)} regarded as (1, 1)-matrix and push forward the result to R(G) by Ψ, we obtain invertible matrices U, V over R(G) and a matrix A over ℂG satisfying

U ( \begin{matrix}x & 0 \\ 0 & I_{m−1} \end{matrix} ) V = A.

If we consider this over U(G), we conclude for the maps r_{x} and r_{A} induced by right multiplication with x and A

dim_{U}(im(r_{x}: U(G) → U(G))+m−1 = dim_{U(G)} (im(r_{A}: U(G)^{m} → U(G)^{m})) .

Since by assumption the strong Atiyah Conjecture 10.2 holds, Theorem 8.29 and Lemma 10.7 imply that dim_{U(G)} (im (r_{A}: U(G)^{m} → U(G)^{m})) must be an integer ≤ m and hence dim_{U}(ker(r_{x}: U(G) → U(G))) ∈ {0, 1}. Since x is by assumption non-trivial and U(G) is von Neumann regular by Theorem 8.22 (3), the kernel of r_{x} is trivial by Lemma 8.18 and Theorem 8.29. Hence x must be a unit by Lemma 8.20 (1). Hence R(G) is a skewfield. Since D(G) is division closed and contained in the skewfield R(G), D(G) is a skewfield. We conclude D(G) = R(G) from Lemma 10.34 (2). This finishes the proof of Lemma 10.39. ⊓ ⊔

The rest of this chapter is devoted to the proof of the strong version of Linnell’s Theorem 10.38. First of all we explain the underlying induction technique.

Lemma 10.40 (Induction principle). Suppose that (P) is a property for groups such that the following is true:

(1) (P) holds for any free group;

(2) Let 1 → H → G → Q → 1 be an extension of groups such that (P) holds for H and Q is virtually finitely generated abelian, then (P) holds for G.

(3) If (P) holds for any finitely generated subgroup of G, then (P) holds for G;

Then (P) holds for any group G in C.

If we replace condition (1) by the condition that (P) holds for the trivial group, then (P) holds for any elementary amenable group G.

Proof. This follows from the following description of the class C and the class EAM of elementary amenable groups respectively by transfinite induction in [300, Lemma 3.1] and [309, Lemma 4.9]. Define for any ordinal α the class of groups D_{α} as follows. Put D_{0} to be the class consisting of virtually free groups or of the trivial group respectively. If α is a successor ordinal, define D_{α} to be the class of groups G which fit into an exact sequence 1 → H → G → Q → 1 such that any finitely generated subgroup of H belongs to D_{α−1} and Q is either finite or infinite cyclic. If α is a limit ordinal, then D_{α} is the union of the D_{β} over all β with β < α. Then C or EAM respectively is ⋃_{α>0} D_{α}. ⊓ ⊔

Lemma 10.40 is the reason why it sometimes is easier to prove a property (P) for all elementary amenable groups than for all amenable groups. We conclude from Lemma 10.40.

Lemma 10.41 (Plan of proof). In order to prove the strong version of Linnell’s Theorem 10.38 and hence Linnell’s Theorem 10.19 it suffices to show that the following statements are true:

(1) Any free group G has the property (R), i.e. D(G) is semisimple;

(2) If 1 → H → G → Q → 1 is an extension of groups such that H has property (R) and Q is finite, then G has property (R);

(3) If 1 → H → G → Q → 1 is an extension of groups such that H has property (R) and Q is infinite cyclic, then G has property (R);

(4) Any virtually free group G has property (K), i.e. the composition

colim_{G/H∈Or(G,FIN)} K_{0}(ℂH) ⟶^{a} K_{0}(ℂG) ⟶^{i} K_{0}(D(G))

is surjective;

(5) Let 1 → H → G → Q → 1 be an extension of groups. Suppose that for any group H′, which contains H as subgroup of finite index, properties (K) and (R) are true. Suppose that Q is virtually finitely generated abelian. Then (K) holds for G;

(6) Suppose that G is the directed union of subgroups {G_{i} | i ∈ I} such that each G_{i} satisfies (K). Then G has property (K);

(7) Let G be a group such that there is an upper bound on the orders of finite subgroups. Suppose that G is the directed union of subgroups {G_{i} | i ∈ I} such that each G_{i} satisfies both (K) and (R). Then G satisfies (R).

Proof. We call two groups G and G′ commensurable if there exists subgroups G_{0} ⊂ G and G′_{0} ⊂ G′ such that [G : G_{0}] < ∞ and [G′ : G′_{0}] < ∞ holds and G_{0} and G′_{0} are isomorphic. In order to prove Lemma 10.41, it suffices to show for any G ∈ C the following property:

(P) If there is an upper bound on the orders of finite subgroups of G and G_{0} is commensurable to G, then G_{0} satisfies (R) and (K).

This is done by proving the conditions in Lemma 10.40.

Since any group G which is commensurable with a free group is virtually free, condition (1) of Lemma 10.40 follows from assumptions (1), (2) and (4) in Lemma 10.41.

Let 1 → H → G → Q → 1 be an extension such that Q is virtually finitely generated abelian, H has property (P) and there is an upper bound on the orders of finite subgroups of G. Let G_{0} be any group which is commensurable to G. Then one easily constructs an extension 1 → H_{0} → G_{0} → Q_{0} → 1 such that H_{0} is commensurable to H and Q_{0} is commensurable to Q and hence virtually finitely generated abelian. Then G_{0} satisfies (K) by assumption (5) of Lemma 10.41. Moreover one can find a filtration of G_{0} by a sequence of normal subgroups H_{0} = G_{n} ⊂ G_{n−1} ⊂ … ⊂ G_{1} ⊂ G_{0} such that G_{i}/G_{i−1} ≅ ℤ for 1 ≤ i ≤ n − 1 and G_{0}/G_{1} is finite. The group G_{0} has property (R) by assumptions (2) and (3) of Lemma 10.41. Hence G has property (P). This proves condition (2) of Lemma 10.40.

Finally we prove condition (3) of Lemma 10.40. Suppose that there is an upper bound on the orders of finite subgroups of G. Any finitely generated subgroup of G_{0} is commensurable to a finitely generated subgroup of G.

Hence G_{0} has properties (K) and (R) by assumptions (6) and (7) of Lemma 10.41. Now Lemma 10.41 follows from Lemma 10.40. ⊓ ⊔

Remark 10.42. Notice that in the formulation of condition (P) in the proof of Lemma 10.41 we have to build in the commensurable group G_{0} since we cannot prove in general that a group G has property (K) if a subgroup G_{0} ⊂ G of finite index has property (K).

In the statements (5) and (7) of Lemma 10.41 we need both (K) and (R) to get a conclusion for (K) or (R) alone. Therefore we cannot separate the proof of (K) from the one of (R) and vice versa.

If we would replace (R) by (R′) in Lemma 10.41, statement (7) remains true if we omit the condition that there is an upper bound on the orders of finite subgroups and that each G_{i} satisfies (K). We only need that each G_{i} satisfies (R′) to prove that G satisfies (R′). This looks promising. The problem is that we cannot prove statements (3) and (5) for (R′) instead of (R). Namely, the proofs rely heavily on the assumption that D(H) is Noetherian (see Goldie’s Theorem 10.61 and Moody’s Induction Theorem 10.67). Notice that a von Neumann regular ring is semisimple if and only if it is Noetherian (see Lemma 8.20 (2)). The counterexample to the strong Atiyah Conjecture given by the lamplighter group L (see Subsection 10.1.4) actually shows that at least one of the statements (3) and (5) applied to 1 → ⨁_{n∈ℤ} ℤ/2 → L → ℤ → 0 is wrong, if we replace (R) by (R′).

We can replace condition (K) by (K′) in Lemma 10.41 getting a weaker conclusion in the strong version of Linnell’s Theorem 10.38. This still gives the strong Atiyah Conjecture and the proof of statement (4) simplifies considerably. But we prefer the stronger condition (K) because we get a stronger result in strong version of Linnell’s Theorem 10.38.

10.3 The Proof of Linnell’s Theorem

This section is devoted to the proof that the conditions appearing in Lemma 10.41 are satisfied. Recall that then the strong version of Linnell’s Theorem 10.38 and hence Linnell’s Theorem 10.19 follow.

10.3.1 The Proof of Atiyah’s Conjecture for Free Groups

In this subsection we prove

Lemma 10.43. The strong Atiyah Conjecture 10.2 holds for any free group G, i.e. for any matrix A ∈ M(m, n, ℂG) the von Neumann dimension of the kernel of the induced bounded G-operator

r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}, x ↦ xA

satisfies

dim_{N (G)} (ker (r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n})) ∈ ℤ.

The proof uses the notion of a Fredholm module and is designed along the new conceptual proof of the Kadison Conjecture for the free group F_{2} on two generators due to Connes [120, IV.5] (see also [166]). The Kadison Conjecture says that there are no non-trivial idempotents in C_{r}^{∗}(G) for a torsionfree group G. We begin with introducing Fredholm modules.

Let H be a Hilbert space. Given a Hilbert basis {b_{i} | i ∈ I} and f ∈ B(H), define tr(f) ∈ ℂ to be ∑_{i∈I}⟨f(b_{i}), b_{i}⟩ if this sum converges. We say that f is of trace class if tr(|f|) < ∞ holds for one (and then automatically for all) Hilbert basis, where |f| is the positive part in the polar decomposition of f. (Notice that this agrees with Definition 1.8 applied to |f| in the case G = {1}.) If f is of trace class, then ∑_{i∈I}⟨f(b_{i}), b_{i}⟩ converges and defines a number tr(f) ∈ ℂ which is independent of the choice of Hilbert basis. We call f ∈ B(H) compact if the closure of the image of the unit disk {x ∈ H | |x| ≤ 1} is compact. We call f ∈ B(H) an operator of finite rank if the image of f is a finite dimensional vector space. We denote by L^{0}(H), L^{1}(H), and K(H) the (twosided) ideal in B(H) of operators of finite rank, of operators of trace class and of compact operators. Denote by L^{p}(H) = {f ∈ B(H) | |f|^{p} ∈ L^{1}(H)} the Schatten ideal in B(H) for p ∈ [1, ∞). Notice that L^{0}(H) ⊂ L^{1}(H) ⊂ L^{q}(H) ⊂ L^{q′}(H) ⊂ K(H) holds for 1 ≤ q ≤ q′. A ∗-algebra or algebra with involution B is an algebra over ℂ with an involution of rings ∗: B → B, which is compatible with the ℂ-multiplication in the sense ∗(λ·b) = \overline{λ}·∗(b) for λ ∈ ℂ and b ∈ B. We call a ∗-homomorphism ρ: B → B(H) a B-representation. Here we do not require that ρ sends 1 ∈ B to id_{H} ∈ B(H).

Definition 10.44. Let B ⊂ A be an arbitrary ∗-closed subalgebra of the C^{∗}-algebra A. For p ∈ {0} ∪ [1, ∞) a p-summable (A, B)-Fredholm module consists of two A-representations ρ_{±}: A → B(H) such that ρ_{+}(a) − ρ_{−}(a) ∈ L^{p}(H) holds for each a ∈ B.

We will later construct a 0-summable Fredholm (N (G), ℂG)-module. The next lemma follows using the equation

ρ_{+}(ab) − ρ_{−}(ab) = ρ_{+}(a)(ρ_{+}(b) − ρ_{−}(b)) + (ρ_{+}(a) − ρ_{−}(a))ρ_{−}(b).

Lemma 10.45. Let ρ_{±}: A → B(H) be a p-summable (A, B)-Fredholm module. Then

(1) For q ∈ {0} ∪ [1, ∞) the set A^{q} = {a ∈ A | ρ_{+}(a) − ρ_{−}(a) ∈ L^{q}(H)} is a ∗-subalgebra of A. For q ≤ q′ we have the inclusions A^{0} ⊂ A^{1} ⊂ A^{q} ⊂ A^{q′} ⊂ A;

(2) The map

τ: A^{1} → ℂ, a ↦ tr(ρ_{+}(a)) − tr(ρ_{−}(a))

is linear and has the trace property τ(ab) = τ(ba); (3) Tensoring ρ_{±} with the standard representation ρ_{st} of M_{n}(ℂ) on ℂ^{n} yields

a p-summable (M_{n}(A), M_{n}(B))-Fredholm module. We have M_{n}(A^{p}) ⊂ M_{n}(A)^{p}.

To show that certain numbers are integers, the key ingredient will be the following lemma.

Lemma 10.46. Let p, q ∈ B(H) be two projections with p−q ∈ L^{1}(H). Then tr(p − q) is an integer.

Proof. The operator (p − q)^{2} is a selfadjoint compact operator. We get a decomposition of H into the finite dimensional eigenspaces of (p − q)^{2} [434, Theorem IV.16 on page 203]

H = ker((p − q)^{2}) ⊕ ⨁_{λ≠0} E_{λ}.

This decomposition is respected by both p and q since both p and q commute with (p−q)^{2}. Notice that ker(p−q) = ker((p−q)^{2}). Next we compute the trace of (p − q) with respect to a Hilbert basis which respects this decomposition

tr(p − q) = ∑_{λ≠0} tr((p − q)|_{E_{λ}}) = ∑_{λ≠0} tr(p|_{E_{λ}}) − tr(q|_{E_{λ}}).

Each difference tr(p|_{E_{λ}}) − tr(q|_{E_{λ}}) is an integer, namely, the difference of the dimensions of finite dimensional vector spaces dim_{ℂ}(p(E_{λ})) − dim_{ℂ}(q(E_{λ})). Since the sum of these integers converges to the real number tr(p − q), the claim follows. ⊓ ⊔

Lemma 10.47. Let a, b ∈ B(H) be elements with a − b ∈ L^{0}(H). Denote by p_{ker(a)} and p_{ker(b)} the orthogonal projections onto their kernels. Then p_{ker(a)}− p_{ker(b)} ∈ L^{0}(H).

Proof. One easily checks that p_{ker(a)} and p_{ker(b)} agree on ker(a − b). Since the orthogonal complement of ker(a − b) is finite dimensional because of a − b ∈ L^{0}(H), we conclude p_{ker(a)} − p_{ker(b)} ∈ L^{0}(H). ⊓ ⊔

In order to construct the relevant Fredholm module, we need some basic properties of the free group F_{2} in two generators s_{1} and s_{2}. The associated Cayley graph is a tree with obvious F_{2}-action. The action is free and transitive on the set V of vertices. It is free on the set E of edges such that F_{2}\E maps bijectively to the set of generators {s_{1}, s_{2}}. Hence we can choose bijections of F_{2}-sets V ≅ F_{2} and E ≅ F_{2} ∐ F_{2}. We equip the Cayley graph with the word-length metric. Then any to distinct points x and y on the Cayley graph can be joined by a unique geodesic which will be denoted by x → y. On x → y there is a unique initial edge init(x → y) on x → y starting at x. Fix a base point x_{0} ∈ V. Define a map

f: V → E ∐{∗}  (10.48)

by sending x to init(x → x_{0}) if x ≠ x_{0} and to ∗ otherwise. Notice for x ≠ x_{0} that g init(x → x_{0}) = init(gx → gx_{0}) is different from init(gx → x_{0}) if and only if gx lies on gx_{0} → x_{0}. This implies

Lemma 10.49. The map f is bijective and almost equivariant in the following sense: For a fixed g ∈ F_{2} there is only a finite number of vertices x ≠ x_{0} with gf(x) ≠ f(gx). The number of exceptions is equal to the distance from x_{0} to gx_{0}.

It is shown in [141] that free groups are the only groups which admit maps with properties similar to the one of f.

Let l^{2}(V) and l^{2}(E) be the Hilbert spaces with the sets V and E as Hilbert basis. We obtain isomorphisms of ℂ[F_{2}]-representations l^{2}(V) ≅ l^{2}(F_{2}) and l^{2}(V) ≅ l^{2}(F_{2}) ⊕ l^{2}(F_{2}) from the bijections V ≅ F_{2} and E ≅ F_{2} ∐ F_{2} above. Thus we can view l^{2}(V) and l^{2}(E) as N (F_{2})-representations. Denote by ℂ the trivial N (F_{2})-representation for which ax = 0 for all a ∈ N (F_{2}) and x ∈ ℂ. Denote by ρ_{+} : N (F_{2}) → B(l^{2}(V)) and by ρ: N (F_{2}) → B(l^{2}(E) ⊕ ℂ) the corresponding representations. The map f from (10.48) induces an isometric isomorphism of Hilbert spaces F: l^{2}(V) ⟶^{≅} l^{2}(E) ⊕ ℂ. Let ρ_{−}: N (G) → B(l^{2}(V)) be defined by ρ_{−}(a) = F^{−1} ∘ ρ(a) ∘ F.

Lemma 10.50. (1) We have for a ∈ N (F_{2})

tr_{N (F_{2})}(a) = ∑_{x∈V}⟨(ρ_{+}(a) − ρ_{−}(a))(x), x⟩;

(2) The representations ρ_{+} and ρ_{−} define a 0-summable (N (F_{2}), ℂ[F_{2}])- Fredholm module;

(3) We get tr_{N (F_{2})}(A) = τ(A) for A ∈ M_{n}(N (F_{2}))^{1}, if τ: M_{n}(N (F_{2}))^{1} → ℝ is the map defined in Lemma 10.45 for the Fredholm module ρ_{±} ⊗ ρ_{st}.

Proof. (1) We get for g ∈ F_{2} and x ∈ V with x ≠ x_{0}

⟨(ρ_{+}(g) − ρ_{−}(g))(x_{0}), x_{0}⟩ = tr_{N (F_{2})}(g);
⟨(ρ_{+}(g) − ρ_{−}(g))(x), x⟩ = 0.

By linearity we get the equations above for any a ∈ ℂ[F_{2}]. Recall that N (F_{2}) is the weak closure of ℂ[F_{2}] in B(l^{2}(F_{2})). One easily checks that tr_{N (F_{2})}(a) and ⟨ρ_{+}(a) − ρ_{−}(a))(x), x⟩ are weakly continuous functionals on a ∈ N (G) for each x ∈ E. This implies for a ∈ N (F_{2}) and x ∈ V with x ≠ x_{0}

⟨(ρ_{+}(a) − ρ_{−}(a))(x_{0}), x_{0}⟩ = tr_{N (F_{2})}(a);
⟨(ρ_{+}(a) − ρ_{−}(a))(x), x⟩ = 0.

Now assertion (1) follows by summing over x ∈ E.

(2) By Lemma 10.49 we have

ρ_{+}(g)(x) = gx = f^{−1}(gf(x)) = ρ_{−}(g)(x)

for all x ∈ V with a finite number of exceptions. Hence ρ_{+}(g) − ρ_{−}(g) has finite rank for g ∈ F_{2}. By linearity ρ_{+}(a)−ρ_{−}(a) has finite rank for a ∈ ℂ[F_{2}].

(3) This follows from (1) and Lemma 10.45 (3). ⊓ ⊔ Now we are ready to prove Lemma 10.43.

Proof. Since any finitely generated free group occurs as a subgroup in F_{2} and any finitely generated subgroup of a free group is a finitely generated free group, it suffices to prove the strong Atiyah Conjecture only for F_{2} by Lemma 10.4. Since for any matrix B ∈ M(m, n, ℂ[F_{2}]) the matrix BB^{∗} is a square matrix over ℂ[F_{2}] and ker (r_{BB^{∗}}^{(2)}) = ker (r_{B}^{(2)}) holds, it suffices to prove for each square matrix A ∈ M_{n}(ℂ[F_{2}])

dim_{N (F_{2})} (ker (r_{A}^{(2)} : l^{2}(F_{2})^{n} → l^{2}(F_{2})^{n})) ∈ ℤ.

Consider the 0-summable (N (F_{2}), ℂF_{2})-Fredholm module of Lemma 10.50 (2). By 0-summability we know ℂ[F_{2}] ⊂ N (F_{2})^{0}. From Lemma 10.45 (3) we conclude M_{n}(ℂ[F_{2}]) ⊂ M_{n}(N (F_{2})^{0}) ⊂ M_{n}(N (F_{2}))^{0}. Hence ρ_{+} ⊗ ρ_{st}(A) − ρ_{−} ⊗ ρ_{st}(A) is a finite rank operator. We conclude from Lemma 10.47 that p_{ker(ρ_{+}⊗ρ_{st}(A))}−p_{ker(ρ_{−}⊗ρ_{st}(A))} is of finite rank. Since up to the summand ℂ we

are dealing with sums of regular representations, we get ρ_{+}⊗ρ_{st} (p_{ker(r_{A}^{(2)})}) =

p_{ker(ρ_{+}⊗ρ_{st}(A))} and ρ_{−} ⊗ ρ_{st} (p_{ker(r_{A}^{(2)})}) ⊕ p_{ℂ^{n}} = p_{ker(ρ_{−}⊗ρ_{st}(A))}, where p_{ℂ^{n}}

is the projection onto the summand ℂ^{n}. Hence p_{ker(r_{A}^{(2)})} ∈ M_{n}(N (F_{2}))^{0} ⊂ M_{n}(N (F_{2}))^{1}. Lemma 10.50 (3) implies

dim_{N (F_{2})} (ker (r_{A}^{(2)})) = tr_{N (F_{2})} (p_{ker(r_{A}^{(2)})}) = τ (p_{ker(r_{A}^{(2)})}) .

We conclude from Lemma 10.46 that τ (p_{ker(r_{A}^{(2)})}) is an integer. This finishes the proof of Lemma 10.43. ⊓ ⊔

Lemma 10.39 and Lemma 10.43 imply

Lemma 10.51. Let G be a free group. Then D(G) is a skewfield and in particular statement (1) in Lemma 10.41 holds.

We will give a different proof for the Atiyah Conjecture 10.3 of order ℤ with coefficients in ℚ for any free group F in Example 13.5, which is based on approximation techniques.

Remark 10.52. The strong Atiyah Conjecture 10.2 for a virtually free group G follows if one knows it for free groups by the following argument. It suffices to treat the case where G is finitely generated by Lemma 10.4. If d is the least common multiple of the orders of finite subgroups of G, then one can find a finitely generated free subgroup F ⊂ G of index d. For a proof of this well-known fact we refer for instance to [24, Theorem 8.3 and Theorem 8.4], [461, Theorem 5 in Section 7 on page 747]. For any matrix A ∈ M(m, n, ℂG) we get from Theorem 1.12 (6)

dim_{N (G)}(ker(r_{A}^{(2)})) = \frac{1}{d} · dim_{N (F)}(ker(r_{A}^{(2)})).

10.3.2 Survey on Crossed Products

In this subsection we will introduce the concept of crossed product which we will need later.

Let R be a ring and let G be a group. Suppose that we are given maps of sets c: G → aut(R), g ↦ c_{g} and τ: G × G → R^{inv} satisfying

c_{τ(g,g′)} ∘ c_{gg′} = c_{g} ∘ c_{g′};
τ(g, g′) · τ(gg′, g′′) = c_{g}(τ(g′, g′′)) · τ(g, g′g′′)

for g, g′, g′′ ∈ G, where c_{τ(g,g′)} : R → R is conjugation with τ(g, g′), i.e. it sends r to τ(g, g′)rτ(g, g′)^{−1}. Let R ∗G = R ∗_{c,τ} G be the free R-module with the set G as basis. It becomes a ring with the following multiplication

(∑_{g∈G} λ_{g}g) · (∑_{h∈G} μ_{h}h) = ∑_{g∈G} ( ∑_{\begin{matrix}g′,g′′∈G, \\ g′g′′=g \end{matrix}} λ_{g′}c_{g′}(μ_{g′′})τ(g′, g′′)) g.

This multiplication is uniquely determined by the properties g · r = c_{g}(r) · g and g · g′ = τ(g, g′) · (gg′). The two conditions above relating c and τ are equivalent to the condition that this multiplication is associative. We call R ∗ G = R ∗_{c,τ} G the crossed product of R and G with respect to c and τ.

Example 10.53. Let 1 → H ⟶^{i} G ⟶^{p} Q → 1 be an extension of groups. Let s: Q → G be a map satisfying p ∘ s = id. We do not require s to be a group homomorphism. Let R be a commutative ring. Define c: Q → aut(RH) by c_{q}(∑_{h∈H} λ_{h}h) = ∑_{h∈H} λ_{h}s(q)hs(q)^{−1}. Define τ: Q × Q → (RH)^{inv} by τ(q, q′) = s(q)s(q′)s(qq′)^{−1}. Then we obtain a ring isomorphism RH ∗Q → RG by sending ∑_{q∈Q} λ_{q}q to ∑_{q∈Q} i(λ_{q})s(q), where i: RH → RG is the ring homomorphism induced by i: H → G. Notice that s is a group homomorphism if and only if τ is constant with value 1 ∈ R.

Example 10.54. Let ϕ: R → R be a ring automorphism. Fix a generator t ∈ ℤ. Define \overline{c}: ℤ → aut(R) by \overline{c}(t^{n}) = ϕ^{n} and let \overline{τ}: ℤ × ℤ → R^{inv} be the constant map with value 1. Then R ∗_{\overline{c},\overline{τ}} ℤ is the ϕ-twisted Laurent ring and sometimes denoted by R[t, t^{−1}]_{ϕ}. Given any other data c: ℤ → aut(R) and τ: ℤ × ℤ → R^{inv} for a crossed product structure, there is an isomorphism of rings

R[t, t^{−1}]_{ϕ} ⟶^{≅} R ∗_{c,τ} ℤ

which sends ∑_{n∈ℤ} λ_{n}t^{n} to ∑_{n∈ℤ} (λ_{n}r_{n}) t^{n}, where

r_{n} = τ(t, t)τ(t^{2}, t) … τ(t^{n−1}, t)  for n ≥ 2;
r_{n} = 1  for n = 0, 1;
r_{n} = τ(t^{−1}, t)c_{t^{−1}}(τ(t^{−1}, t))τ(t^{−1}, t^{−1}) …
c_{t^{n+1}}(τ(t^{−1}, t))τ(t^{n+1}, t^{−1}) for n ≤ −2.

The crossed product R ∗_{c,τ} G has the following universal property. Let S be a ring together with a map ν: G → S^{inv}. Denote by τ_{ν}: G × G → S^{inv} the map which sends (g, g′) to ν(g)ν(g′)ν(gg′)^{−1}. Suppose that we are given a ring homomorphism f: R → S satisfying c_{ν(g)} ∘ f = f ∘ c_{g} and τ_{ν} = f ∘ τ, where c_{ν(g)} : S → S sends s to ν(g)sν(g)^{−1}. Then there is exactly one ring homomorphism called crossed product homomorphism F = f ∗_{ν}id: R∗_{c,τ} G → S with the properties that F (r) = f(r) for r ∈ R and F (g) = ν(g). It sends ∑_{g∈G} λ_{g}g to ∑_{g∈G} f(λ_{g})ν(g).

If \overline{R}∗_{\overline{c},\overline{τ}} G is another crossed product and f: R → \overline{R} a ring homomorphism satisfying \overline{c}_{g} ∘ f = f ∘ c_{g} and \overline{τ} = f ∘ τ, then by the universal property above there is exactly one ring homomorphism, also called crossed product homomorphism, F = f ∗id: R∗_{c,τ} G → \overline{R}∗_{\overline{c},\overline{τ}} G which in uniquely determined by the properties that F (r) = f(r) for r ∈ R and F (g) = g for g ∈ G.

Next we collect the basic properties of crossed products which we will need later. Recall that a ring R is (left) Artinian if any descending sequence of finitely generated (left) R-modules M_{1} ⊃ M_{2} ⊃ M_{2} ⊃ … becomes stationary, i.e. there is an integer n with M_{m} = M_{m+1} for m ≥ n. A ring R is (left) Noetherian if any ascending sequence of finitely generated (left) R-modules becomes stationary. It suffices to check these condition for ideals. A ring R is (left) Noetherian if and only if any submodule of a finitely generated (left) Rmodule is again finitely generated. A ring R is semiprime if for any two-sided ideal I ⊂ R the implication I^{2} = {0} ⇒ I = {0} holds. For more information about these notions we refer for instance to [448].

Lemma 10.55. Let R ∗ G be a crossed product. Then

(1) If R is Artinian and G is finite, then R ∗ G is Artinian;

(2) If R is Noetherian and G is virtually poly-cyclic, then R∗G is Noetherian;

(3) If R is semisimple of characteristic 0, then R ∗ G is semiprime;

(4) If R is semisimple of characteristic 0 and G is finite, then R ∗ G is semisimple.

Proof. (1) An ascending chain of finitely generated R ∗ G-modules can be viewed as an ascending chain of finitely generated R-modules since G is by assumption finite. Hence it is stationary since R is Artinian by assumption.

(2) The same argument as in the proof of assertion (1) shows that R ∗ G is Noetherian if R is Noetherian and G is finite. A virtually poly-cyclic group can be obtained by iterated extensions with infinite cyclic groups or finite groups as quotient. If G is an extension 1 → H → G → Q → 1, one gets R ∗ G ≅ (R ∗ H) ∗ Q. Hence it remains to treat the case where Q = ℤ. By Example 10.54 it suffices to show that the twisted Laurent ring R[t, t^{−1}]_{ϕ} is Noetherian for an automorphism ϕ: R → R of a Noetherian ring R. This follows from [446, Proposition 3.1.13 on page 354, Proposition 3.5.2 on page 395].

(3) This follows from [417, Theorem I].

(4) A ring is semiprime and Artinian if and only if it is semisimple [448, Theorem 2.3.10 on page 139]. Now apply assertions (2) and (3). ⊓ ⊔

Lemma 10.56. Let R ⊂ S be an inclusion of rings and let R ∗ G → S ∗ G be an inclusion of crossed products.

(1) The subring of S ∗ G generated by D(R ⊂ S) and G in S ∗ G is a crossed product D(R ⊂ S) ∗ G such that R ∗ G ⊂ D(R ⊂ S) ∗ G is an inclusion of crossed products. The analogous statement holds for R(R ⊂ S);

(2) Let Σ be a set of matrices of R which is invariant under the automorphisms c_{g}: R → R for all g ∈ G. Let i: R → R_{Σ} be universal Σinverting. Then c: G × G → aut(R) induces c^{Σ}: G → aut(R_{Σ}) such that i ∘ c_{g} = c_{g}^{Σ} ∘ i holds for all g ∈ G and the map of crossed products i ∗ G: R ∗ G → R_{Σ} ∗ G is universal Σ-inverting.

Proof. (1) Since R ∗ G ⊂ S ∗ G is by assumption an inclusion of crossed products, we get for the crossed product structure maps c^{R}: G → aut(R), τ^{R}: G × G → R^{inv}, c^{S}: G → aut(S) and τ^{S}: G × G → S^{inv} that c_{g}^{S}: S → S induces c_{g}^{R}: R → R and τ^{R} = τ^{S}. By the universal property of the division closure c_{g}^{S} induces also an automorphism c_{g}^{D(R⊂S)} : D(R ⊂ S) → D(R ⊂ S). Thus we can form the crossed product D(R ⊂ S) ∗ G with respect to c^{D(R⊂S)} : G × G → aut(D(R ⊂ S)) and τ^{D(R⊂S)} : G × G → D(R ⊂ S)^{inv}, where τ^{D(R⊂S)} is given by τ^{R}. There is an obvious map from D(R ⊂ S) ∗ G to S ∗ G whose image is the subring generated by D(R ⊂ S) and G. One easily checks that it is injective since S ∗ G is the free S-module with G as basis.

(2) The extension c^{Σ} exists by the universal property of i: R → R_{Σ} since c_{g}(Σ) = Σ holds for g ∈ G by assumption. From the universal property of the crossed product we deduce that i ∗ G is universal Σ-inverting, since i is universal Σ-inverting by assumption. ⊓ ⊔

We conclude from Lemma 10.56 (using Notation 10.37)

Lemma 10.57. Let 1 → H → G → Q → 1 be an extension of groups. Let s: Q → G be a set-theoretic section.

(1) The subring of U(G) generated by D(H) and s(Q) is a crossed product D(H) ∗ Q such that

ℂG = ℂH ∗ Q ⊂ D(H) ∗ Q ⊂ U(H) ∗ Q ⊂ U(G)

is given by inclusions of rings or crossed products. The analogous statement holds for R(H);

(2) If ℂH → R(H) is universal Σ(H)-inverting, then ℂG = ℂH ∗ Q → R(H) ∗ Q is universal Σ(H)-inverting;

Lemma 10.58. Let 1 → H → G → Q → 1 be an extension of groups. Then

D(D(H) ∗ Q ⊂ U(G)) = D(G);
R(R(H) ∗ Q ⊂ U(G)) = R(G);

Proof. Since U(H) is von Neumann regular (see Theorem 8.22 (3)) and hence division closed and rationally closed by Lemma 10.34 (1), we get an inclusion

D(H) = D(ℂH ⊂ U(H)) = D(ℂH ⊂ U(G)) ⊂ D(ℂG ⊂ U(G)) = D(G).

Fix a set-theoretic section s: Q → G. Since D(H) ∗ Q can be viewed as the subring of U(G) generated by D(H) and s(Q) by Lemma 10.57 (1) and s(Q) lies in D(G), we get D(H) ∗ Q ⊂ D(G). This implies D(D(H) ∗ Q ⊂ U(G)) ⊂ D(D(G) ⊂ U(G)) = D(G). Since D(G) = D(ℂG ⊂ U(G)) = D(ℂH ∗ Q ⊂ U(G)) ⊂ D(D(H) ∗ Q ⊂ U(G)) obviously holds, Lemma 10.58 follows for D(G). The proof for R(G) is analogous. ⊓ ⊔

10.3.3 Property (R) Ascends to Finite Extensions

Lemma 10.59. Statement (2) in Lemma 10.41 is true, i.e. if 1 → H → G → Q → 1 is an extension of groups such that D(H) is semisimple and Q is finite, then D(G) is semisimple. Moreover, D(G) agrees with D(H) ∗ Q.

Proof. We conclude from Lemma 10.55 (4) that D(H) ∗ Q is semisimple. In particular D(H) ∗ Q is von Neumann regular. From Lemma 10.34 (1) and Lemma 10.58 we conclude D(G) = D(H) ∗ Q. ⊓ ⊔

10.3.4 Property (R) Ascends to Extensions by Infinite Cyclic Groups

This subsection is devoted to the proof of

Lemma 10.60. Statement (3) in Lemma 10.41 is true, i.e. if 1 → H → G → ℤ → 1 is an extension of groups such that D(H) is semisimple, then D(G) is semisimple. Moreover, NZD(D(H) ∗ ℤ) = T (D(H) ∗ ℤ ⊂ U(G)), the Ore localization (D(H) ∗ ℤ) NZD(D(H) ∗ ℤ)^{−1} exists and is isomorphic to D(G).

Here and elsewhere NZD(R) denotes the set of non-zero-divisors of a ring R. Its proof needs the following ingredients. We will fix throughout this subsection an extension 1 → H → G → ℤ → 1 together with a section s: ℤ → G which is a group homomorphism.

Theorem 10.61 (Goldie’s Theorem). Let R be a Noetherian semiprime ring. Then (R, NZD(R)) satisfies the Ore condition and the Ore localization R NZD(R)^{−1} is semisimple.

Proof. [119, Section 9.4]. ⊓ ⊔ We will need the following consequence

Lemma 10.62. Let R be a semisimple ring. Let R ∗ ℤ be a crossed product which is contained in the ring S. Suppose that NZD(R ∗ ℤ) = T (R ∗ ℤ ⊂ S). Then the Ore localization (R ∗ ℤ) NZD(R ∗ ℤ)^{−1} exists, is a semisimple ring and embeds into S as the division closure D(R ∗ ℤ ⊂ S).

Proof. Since R∗ℤ is Noetherian and semiprime by Lemma 10.55, we conclude from Goldie’s Theorem 10.61 that the Ore localization (R ∗ ℤ) NZD(R ∗ ℤ)^{−1} exists and is semisimple. By Lemma 10.35 (2) (R ∗ ℤ) NZD(R ∗ ℤ)^{−1} is isomorphic to D(R ∗ ℤ ⊂ S). ⊓ ⊔

Theorem 10.63. Suppose that the first non-vanishing coefficient of the Laurent polynomial f(z) ∈ N (H) ∗ ℤ is a non-zero-divisor in N (H). Then f is a non-zero-divisor in N (H) ∗ ℤ.

Proof. This is a special case of [308, Theorem 4]. ⊓ ⊔ We will need the following conclusion of Theorem 10.63 above.

Lemma 10.64. Suppose that f(z) ∈ U(H) ∗ ℤ is of the form 1 + a_{1}z + … + a_{n}z^{n}. Then f is invertible in U(G).

Proof. Recall from Theorem 8.22 (1) that U(H) is the Ore localization of N (H) with respect to the set of non-zero-divisors in N (H). Hence we can find a non-zero-divisor s ∈ N (H) and g(z) ∈ N (H) ∗ ℤ such that f(z) = s^{−1}g(z). The first non-vanishing coefficient of g(z) is s. By Theorem 10.63 above g(z) is a non-zero-divisor in N (G) and hence a unit in U(G). Therefore f(z) = s^{−1}g(z) is a unit in U(G). ⊓ ⊔

Finally we will need

Lemma 10.65. Let R be a semisimple ring. Let R ∗ ℤ be a crossed product which is contained in the ring S. Suppose that polynomials of the form f(z) = 1 + a_{1}z + … + a_{n}z^{n} in R ∗ ℤ become invertible in S. Then NZD(R ∗ ℤ) = T (R ∗ ℤ ⊂ S).

Before we give the proof of Lemma 10.65, we explain how the main result of this subsection Lemma 10.60 follows from the results above. By assumption D(H) is semisimple. Lemma 10.64 ensures that the assumptions of Lemma 10.65 are satisfied for R = D(H) and S = U(G). Hence NZD(D(H) ∗ ℤ) = T (D(H)∗ℤ ⊂ U(G)). Lemma 10.62 implies that the Ore localization (D(H)∗ ℤ) NZD(D(H) ∗ ℤ)^{−1} exists, is semisimple and agrees with D(D(H) ∗ ℤ ⊂ U(G)). Since D(D(H) ∗ ℤ ⊂ U(G)) is D(G) by Lemma 10.58, Lemma 10.60 follows.

It remains to prove Lemma 10.65. Recall that by Wedderburn’s Theorem [302, Theorem 3.5 in Chapter I on page 35] any semisimple ring R can be written as a finite product ∏_{i=1}^{r} M_{n_{i}}(D_{i}) of matrix rings over skewfields D_{i}.

We begin with the case where R itself is a skewfield. Then any non-zero element in R ∗ ℤ can be written up to multiplication with a unit in R and an element of the form z^{n} in the form f(z) = 1 + a_{1}z + … + a_{n}z^{n}. Hence it is invertible by assumption in S. This proves NZD(R ∗ ℤ) = T (R ∗ ℤ ⊂ S) provided that R is a skewfield.

Next we treat the case where R looks like M_{n}(D) for a skewfield D. We first change the given crossed product structure (c, τ) to a better one by a construction similar to the one in Example 10.54. We can write c_{t}: M_{n}(D) → M_{n}(D) as a composition c_{u} ∘ M_{n}(θ) for some u ∈ M_{n}(D) and automorphism θ: D → D, where c_{u} is conjugation with u and M_{n}(θ) is given by applying θ to each entry of a matrix [277, Theorem 8 on page 237]. Define a new crossed product structure \overline{c}: ℤ → aut(M_{n}(D)) and \overline{τ}: ℤ × ℤ → M_{n}(D) by \overline{c}_{t^{n}} = (u^{−1} ∘ c_{t})^{n} and \overline{τ}(t^{a}, t^{b}) = 1. Notice that M_{n}(D) ∗_{c,τ} ℤ and M_{n}(D) ∗_{\overline{c},\overline{τ}} ℤ are isomorphic. In the sequel we will use the new crossed product structure given by \overline{c} and \overline{τ}. Notice that the assumption about polynomials is invariant under this change of crossed product structure since 1 + a_{1}t + … + a_{n}t^{n} becomes with respect to the new crossed product structure r_{0} + a_{1}r_{1}t + … + a_{n}r_{n}t^{n} for appropriate elements r_{0}, r_{1}, …, r_{n} in M_{n}(D)^{inv}.

In the sequel we consider D ⊂ M_{n}(D) by the diagonal embedding which sends d ∈ D to the diagonal matrix whose diagonal entries are all d. The crossed product structure on M_{n}(D) induces one on D and we obtain an inclusion of crossed products D ∗ ℤ ⊂ M_{n}(D) ∗ ℤ. In particular D ∗ ℤ becomes a subring of S because by assumption M_{n}(D) ∗ ℤ ⊂ S. Since D is a skewfield, we know already NZD(D ∗ ℤ) = T (D ∗ ℤ ⊂ S). By Lemma 10.62 the Ore localization (D ∗ ℤ) NZD(D ∗ ℤ)^{−1} exists, is semisimple and embeds into S. In the sequel we identify D ∗ ℤ as a subring of M_{n}(D ∗ ℤ) by the diagonal embedding. Since NZD(D ∗ ℤ) ⊂ D ∗ ℤ satisfies the Ore condition, one easily checks that NZD(D ∗ ℤ) ⊂ M_{n}(D ∗ ℤ) satisfies the Ore condition and the inclusion M_{n}(D ∗ ℤ) ⊂ S is NZD(D ∗ ℤ)-inverting and hence induces a homomorphism M_{n}(D ∗ ℤ) NZD(D ∗ ℤ)^{−1} → S. It is injective since M_{n}(D ∗ ℤ) ⊂ S and (D ∗ ℤ) NZD(D ∗ ℤ)^{−1} ⊂ S are injective. Hence we get an embedding M_{n}(D ∗ ℤ) NZD(D ∗ ℤ)^{−1} ⊂ S. Consider x ∈ NZD(M_{n}(D) ∗ ℤ). It is a non-zero-divisor when considered in the Ore localization M_{n}(D ∗ ℤ) NZD(D ∗ ℤ)^{−1}. Since M_{n}(D ∗ ℤ) NZD(D ∗ ℤ)^{−1} = M_{n} ((D ∗ ℤ) NZD(D ∗ ℤ)^{−1}) is von Neumann regular by Lemma 8.19 (1), any non-zero-divisor in M_{n}(D ∗ ℤ) NZD(D ∗ ℤ)^{−1} is invertible by Lemma 8.20 (1). Hence x becomes invertible in M_{n}(D ∗ ℤ) NZD(D ∗ ℤ)^{−1} and in particular in S. This shows NZD(M_{n}(D) ∗ ℤ) = T (M_{n}(D) ∗ ℤ ⊂ S).

Finally it remains to treat the general case R = ∏_{i=1}^{r} M_{n_{i}}(D_{i}). We will equip R ∗ ℤ with the crossed product structure constructed in Example 10.54 for an appropriate automorphism ϕ: R → R. An automorphism of R permutes the factors M_{n_{i}}(R). Hence we can find a positive integer m such that ϕ^{m} leaves the various factors invariant and can be written as ∏_{i=1}^{r} ϕ_{i} for appropriate automorphisms ϕ_{i}: M_{n_{i}}(D_{i}) → M_{n_{i}}(D_{i}). This induces decompositions

R ∗ mℤ = ∏_{i=1}^{r} (M_{n_{i}}(D_{i}) ∗ ℤ) ;

NZD(R ∗ mℤ) = ∏_{i=1}^{r} NZD (M_{n_{i}}(D_{i}) ∗ ℤ) ;

T (R ∗ mℤ ⊂ S) = ∏_{i=1}^{r} T (M_{n_{i}}(D_{i}) ∗ ℤ ⊂ S) ;

We have already proved NZD (M_{n_{i}}(D_{i}) ∗ ℤ) = T (M_{n_{i}}(D_{i}) ∗ ℤ ⊂ S) for i = 1, 2 … r. This implies NZD(R ∗ mℤ) = T (R ∗ mℤ ⊂ S). We conclude from Lemma 10.62 that the Ore localization (R ∗ mℤ) NZD(R ∗ mℤ)^{−1} exists, is semisimple and embeds as the division closure D(R ∗ mℤ ⊂ S) into S. One easily checks that ((R ∗ mℤ) NZD(R ∗ mℤ)^{−1}) ∗ ℤ/m is isomorphic to ((R ∗ mℤ) ∗ ℤ/m) NZD(R ∗ mℤ)^{−1}. We conclude from Lemma 10.55 (4) that ((R ∗ mℤ) ∗ ℤ/m) NZD(R ∗ mℤ)^{−1} is semisimple. Since R ∗ ℤ agrees with (R ∗ mℤ) ∗ ℤ/m and R ∗ ℤ ⊂ S by assumption, we get inclusions R ∗ ℤ ⊂ (R∗ℤ) NZD(R∗mℤ)^{−1} ⊂ S and (R∗ℤ) NZD(R∗mℤ)^{−1} is semisimple. Since any non-zero-divisor in R ∗ ℤ is a non-zero-divisor in the Ore localization (R ∗ ℤ) NZD(R ∗ mℤ)^{−1} and any non-zero-divisor in a semisimple ring is a unit, we get NZD(R ∗ Z) = T (R ∗ ℤ ⊂ S). This finishes the proof of Lemma 10.65 and hence of Lemma 10.60. ⊓ ⊔

10.3.5 Property (K) and Extensions by Virtually Finitely Generated Abelian Groups

The main result of this section will be

Lemma 10.66. The statement (5) in Lemma 10.41 is true.

Again the proof will need some preparation. The main ingredient will be the next result. For its proof we refer to [383], [384], or to [112], [188] and [418, Chapter 8]. We have introduced the Grothendieck group of finitely generated R-modules G_{0}(R) in Definition 9.11.

Theorem 10.67 (Moody’s Induction Theorem). Let R be a Noetherian ring and Q be a virtually finitely generated abelian group. Then the induction map

colim_{Q/K∈Or(Q,FIN)} G_{0}(R ∗ K) → G_{0}(R ∗ Q)

is surjective.

However, more input is needed. We begin with the following general result about iterated localization.

Lemma 10.68. Let R ⊂ S be a ring extension and T ⊂ T (R ⊂ S) be a multiplicatively closed subset. Assume that the pairs (R, T) and (RT^{−1}, T (RT^{−1} ⊂ S)) satisfy the Ore condition. Then also (R, T (R ⊂ S)) satisfies the Ore condition and RT (R ⊂ S)^{−1} = (RT^{−1})T (RT^{−1} ⊂ S)^{−1}. We get inclusions

R ⊂ (RT^{−1})T (RT^{−1} ⊂ S)^{−1} = RT (R ⊂ S)^{−1} ⊂ S.

If we assume in addition that T (RT^{−1} ⊂ S) = NZD(RT^{−1}), then T (R ⊂ S) = NZD(R).

Proof. We begin with verifying the Ore condition for (R, T (R ⊂ S)). Given r_{1} ∈ R and t_{1} ∈ T, we must find r_{2} ∈ R and t_{2} ∈ T with t_{1}r_{2} = r_{1}t_{2}. Since (RT^{−1}, T (RT^{−1} ⊂ S)) satisfies the Ore condition, we can find r_{3}, r_{4} ∈ R and t_{3}, t_{4} ∈ T satisfying t_{1}r_{3}t_{3}^{−1} = r_{1}r_{4}t_{4}^{−1} in RT^{−1} and r_{4}t_{4}^{−1} ∈ T (RT^{−1} ⊂ S). Since (R, T) satisfies the Ore condition, we can find r_{5} ∈ R and t_{5} ∈ T satisfying t_{3}t_{5} = t_{4}r_{5}. This implies t_{1}r_{3}t_{5} = r_{1}r_{4}r_{5} and r_{5} ∈ T (R ⊂ S). Put r_{2} = r_{3}t_{5} and t_{2} = r_{4}r_{5}. Since r_{4}t_{4}^{−1} ∈ T (RT^{−1} ⊂ S) and r_{5} ∈ T (R ⊂ S) holds, we get t_{2} ∈ T (R ⊂ S).

One easily checks that the composition R → RT^{−1} → (RT^{−1})T (RT^{−1} ⊂ S)^{−1} is universally T (R ⊂ S)-inverting. Since the same holds for R → RT (R ⊂ S)^{−1}, we conclude RT (R ⊂ S)^{−1} = (RT^{−1})T (RT^{−1} ⊂ S)^{−1}. Since a non-zero-divisor in R is a non-zero-divisor in RT^{−1}, we conclude from the assumption T (RT^{−1} ⊂ S) = NZD(RT^{−1}) that T (R ⊂ S) = NZD(R) holds.

⊓ ⊔

Lemma 10.69. Let 1 → H → G → Q → 1 be an extension of groups such that Q is virtually finitely generated abelian. Suppose that D(H) is semisimple. Then NZD(D(H) ∗ Q) = T (D(H) ∗ Q ⊂ U(G)), and the Ore localization (D(H) ∗ Q) NZD(D(H) ∗ Q)^{−1} exists, is semisimple and agrees with D(G).

Proof. We begin with the special case where Q is finite. Then D(G) = D(H)∗ Q and is semisimple by Lemma 10.59. Since in a semisimple ring any nonzero-divisor is a unit, we conclude

NZD(D(H) ∗ Q) = (D(H) ∗ Q)^{inv} = T (D(H) ∗ Q ⊂ U(G)).

In particular D(H) ∗ Q = (D(H) ∗ Q) NZD(D(H) ∗ Q)^{−1}. So the claim holds for finite Q.

If Q is infinite cyclic, the claim follows from Lemma 10.60.

Any extension by a virtually finitely generated abelian group can be replaced by an iterated extension by infinite cyclic groups and by finite groups. Hence we must ensure that we can iterate. But this follows from Lemma 10.68. ⊓ ⊔

Lemma 10.70. (1) Let T ⊂ R be a set of non-zero-divisors. Suppose that (R, T) satisfies the Ore condition. Then the inclusion R → RT^{−1} induces an epimorphism G_{0}(R) → G_{0}(RT^{−1});

(2) If R is semisimple, then the forgetful map K_{0}(R) → G_{0}(R) is bijective.

Proof. (1) Let M be a finitely generated RT^{−1}-module. Choose an epimorphism f: (RT^{−1})^{n} → M. Let i: R^{n} → (RT^{−1})^{n} be the obvious injection. Now N = f ∘ i(R^{n}) is a finitely generated R-submodule of M. The inclusion j: N → M induces an injection id ⊗_{R}j: RT^{−1} ⊗_{R} N → RT^{−1} ⊗_{R} M since RT^{−1} ⊗_{R} − is exact by Lemma 8.15 (3). Since f is surjective, id ⊗_{R}j is bijective. Since for any RT^{−1}-module L we have RT^{−1} ⊗_{R} L ≅_{RT^{−1}} L, the RT^{−1}-modules M and RT^{−1} ⊗_{R} N are isomorphic.

(2) Any finitely generated module over a semisimple ring is finitely generated projective. ⊓ ⊔

Lemma 10.71. Let 1 → H → G ⟶^{p} Q → 1 be an extension of groups such that Q is virtually finitely generated abelian and D(H) is semisimple. Then the map

G_{0}(D(H) ∗ Q) → G_{0}(D(G))

is surjective.

Proof. This follows from Lemma 10.69 and Lemma 10.70 (1). ⊓ ⊔

Now we are ready to prove the main result of this subsection Lemma 10.66. We have to show for an extension of groups 1 → H → G → Q → 1 with virtually finitely generated abelian Q that the canonical map

colim_{G/G_{0}∈Or(G,FIN)} K_{0}(ℂG_{0}) → K_{0}(D(G))

is surjective, provided that for any group K with H ⊂ K ⊂ G and [K : H] < ∞ the canonical map

colim_{K/K_{0}∈Or(K,FIN)} K_{0}(ℂK_{0}) → K_{0}(D(K))

is surjective and D(K) is semisimple. We will show that the map of interest can be written as the following composition of maps which are surjective or even bijective: colim_{G/G_{0}∈Or(G,FIN)} K_{0}(ℂG_{0})

⟶^{f_{1}} colim_{Q/Q_{0}∈Or(Q,FIN)} colim_{p^{−1}(Q_{0})/L∈Or(p^{−1}(Q_{0}),FIN)} K_{0}(ℂL)
⟶^{f_{2}} colim_{Q/Q_{0}∈Or(Q,FIN)} K_{0}(D(p^{−1}(Q_{0})))
⟶^{f_{3}} colim_{Q/Q_{0}∈Or(Q,FIN)} G_{0}(D(p^{−1}(Q_{0})))
⟶^{f_{4}} colim_{Q/Q_{0}∈Or(Q,FIN)} G_{0}(D(H) ∗ Q_{0})
⟶^{f_{5}} G_{0}(D(H) ∗ Q)
⟶^{f_{6}} G_{0}(D(G))
⟶^{f_{7}} K_{0}(D(G))

The isomorphism f_{1} is due to the fact that any finite subgroup G_{0} ⊂ G occurs as finite subgroup in p^{−1}(p(G_{0})) and p(G_{0}) is finite. The map f_{2} is bijective by assumption. The map f_{3} is bijective by Lemma 10.70 (2) since D(p^{−1}(Q_{0})) is semisimple by assumption. The map f_{4} is an isomorphism since D(H) ∗ Q_{0} = D(p^{−1}(Q_{0})) holds by Lemma 10.59. The map f_{5} is surjective by Moody’s Induction Theorem 10.67. The map f_{6} is surjective by Lemma 10.71. The map f_{7} is an isomorphism by Lemma 10.70 (2) since D(G) is semisimple by Lemma 10.59 and Lemma 10.60. This finishes the proof of Lemma 10.66. ⊓ ⊔

10.3.6 Property (K) holds for Virtually Free Groups

The main goal of this subsection is to prove

Lemma 10.72. Statement (4) in Lemma 10.41 is true, i.e. for any virtually free group G the composition

colim_{Or(G,FIN)} K_{0}(ℂH) ⟶^{a} K_{0}(ℂG) ⟶^{i} K_{0}(D(G))

is surjective.

The main ingredients in the proof of Lemma 10.72 will be the following two results.

Theorem 10.73. For a virtually free group the Isomorphism Conjecture 9.40 for K_{0}(ℂG) is true, i.e. the assembly map

a: colim_{G/H∈Or(G,FIN)} K_{0}(ℂH) → K_{0}(ℂG)

is surjective.

Proof. [309, Lemma 4.8]. ⊓ ⊔ Theorem 10.74. Let R be a hereditary ring with faithful projective rank function ρ: K_{0}(R) → ℝ and let Σ be a set of full matrices with respect to ρ. Then the universal Σ-inverting homomorphism R → R_{Σ} induces an epimorphism K_{0}(R) → K_{0}(R_{Σ}).

Some explanations about Theorem 10.74 are in order. Recall that a ring is called hereditary, if any submodule of a projective module is again projective. A projective rank function for R is a function ρ: K_{0}(R) → ℝ such that for any finitely generated projective R-module P we have ρ([P]) ≥ 0 and ρ([R]) = 1. It is called faithful if ρ([P]) = 0 implies P = 0. Our main example of a faithful projective rank function will be (see Theorem 8.29)

K_{0}(ℂG) → ℝ, [P] ↦ dim_{U(G)}(U(G) ⊗_{ℂG} P).

Given an R-map α: P → Q of finitely generated projective R-modules, define ρ(α) to be the infimum over all numbers ρ(P′), where P′ runs through all finitely generated projective R-modules P′ for which there exists a factorization α: P → P′ → Q. We call α left full or right full respectively if ρ(α) = ρ(P) or ρ(α) = ρ(Q) respectively holds. It is called full if it is both left full and right full. Theorem 10.74 is now a special case of [465, Theorem 5.2], since by [465, Lemma 1.1 and Theorem 1.11] a projective rank function is in particular a Sylvester projective function and by [465, Theorem 1.16] there are enough left and right full maps.

We want to apply Theorem 10.74 to the case R = ℂH for a virtually finitely generated free group H, the rank function ρ given by dim_{U(H)} and Σ = Σ(H) := Σ(ℂH ⊂ U(H)). The trivial ℂH-module ℂ has a 1dimensional projective resolution P_{∗} since this is obviously true for a finitely generated free subgroup F ⊂ H of finite index [H : F] and [H : F] is invertible in ℂ. Hence any ℂH-module M has a 1-dimensional projective resolution, namely P_{∗} ⊗_{ℂ} M with the diagonal H-action. This implies that ℂH is hereditary.

Next we check that Σ(H) consists of full maps. Suppose that α: P → Q is a homomorphism of ℂH-modules such that id ⊗_{ℂH}α: U(H) ⊗_{ℂH} P → U(H) ⊗_{ℂH} Q is an isomorphism. If α factorizes over the finitely generated projective ℂH-module P′, then the isomorphism id ⊗_{ℂH}α factorizes over U(H) ⊗_{ℂH} P′. Theorem 8.29 implies

ρ(α) ≤ ρ(P) = dim_{U(H)}(U(H) ⊗_{ℂH} P) ≤ dim_{U(H)}(U(H) ⊗_{ℂH} P′).

Hence ρ(α) = ρ(P) = ρ(Q), i.e. α is full.

Now Theorem 10.74 implies that the map K_{0}(ℂH) → K_{0}(ℂH_{Σ(H)}) is surjective for virtually finitely generated free group H. The third ingredient will be the next lemma

Lemma 10.75. Let G be a virtually free group. Then the inclusion ℂG → D(G) is universally Σ(G)-inverting.

Before we give its proof, we explain how we can derive Lemma 10.72 from it. Namely, we conclude that for any virtually finitely generated free group H the map K_{0}(ℂH) → K_{0}(D(H)) is surjective. Let G be a virtually free group G. It is the directed union G = {G_{i} | i ∈ I} of its finitely generated subgroups G_{i}. Each G_{i} is virtually finitely generated free. The ring ℂG is the directed union of the subrings ℂG_{i} and the ring D(G) is the directed union of the subrings D(G_{i}) (see Lemma 10.83). Since K_{0} commutes with colimits, we get

K_{0}(ℂG) = colim_{i∈I} K_{0}(ℂG_{i});
K_{0}(D(G)) = colim_{i∈I} K_{0}(D(G_{i})).

Since a directed colimit of epimorphisms is again an epimorphism, we get

Lemma 10.76. If G is a virtually free group, then K_{0}(ℂG) → K_{0}(D(G)) is surjective.

Theorem 10.73 and Lemma 10.76 together imply Lemma 10.72. It remains to give the proof of Lemma 10.75. This needs some preparation.

We will need the following notions. A ring homomorphism f: R → K from a ring into a skewfield is called an R-field. It is called epic R-field if K is generated as a skewfield by the image of f, or, equivalently, for any skewfield K′ with im(f) ⊂ K′ ⊂ K we have K = K′. An epic R-field is called field of fractions for R if f is injective. Unfortunately, this notion is not unique, a ring can have several non-isomorphic field of fractions. We have to develop this notion further in order to achieve uniqueness.

A local homomorphism from the R-field f: R → K to the R-field g: R → L is an R-algebra homomorphism u: K_{0} → L, where K_{0} ⊂ K is an R-subalgebra of K and K_{0} − ker(u) ⊂ K_{0}^{inv}. The ring K_{0} is a local ring with maximal ideal ker(u). Two local homomorphisms are equivalent if they restrict to a common homomorphism, which is again local. An equivalence class is called specialization. An initial object f: R → K in the category of epic R-fields is called a universal R-field. If f is a universal R-field with injective f, then it is called universal R-field of fractions. The universal Rfield is unique up to isomorphism, but even if there exists a field of fractions for R, there need not exist a universal R-field. Moreover, if a universal Rfield f: R → K exists, the map f is not necessarily injective, i.e. it is not necessarily a field of fractions for R.

The ring R = ℤ has the following epic ℤ-fields f_{0} : ℤ → ℚ and f_{p}: ℤ → 𝔽_{p}, where 𝔽_{p} is the finite field of prime order p. Notice that for any R-field f: R → K there is an obvious factorization f: R → R_{Σ(f: R→K)} ⟶^{Ψ} K. In the case f_{p} it is given by ℤ → ℤ_{(p)} → 𝔽_{p}. The next lemma will give a criterion when Ψ is bijective. A ring R is called semifir if every finitely generated submodule of a free module is free and for a free module any two basis have the same cardinality.

Lemma 10.77. Let R be a semifir. Then there exists a universal field of fractions f: R → K and the canonical map Ψ: R_{Σ(f: R→K)} → K is an isomorphism.

Proof. By [118, Corollary 5.11 in Chapter 7 on page 417] there is a universal field of fractions f: R → K such that every full homomorphism P → Q of finitely generated projective R-modules becomes an isomorphism over K. Full is meant with respect to the faithful projective rank function given by the isomorphism K_{0}(R) ⟶^{≅} ℤ ⊂ ℝ. Hence the set of full matrices over R agrees with Σ(R ⊂ K). Hence Ψ is bijective by [118, Proposition 5.7 in Chapter 7 on page 415]. ⊓ ⊔

Given a free group F, there is a universal field of fractions f: ℂF → K by Lemma 10.77 since ℂF is a semifir [117, Corollary on page 68], [142].

The next result is shown in [306, Proposition 6]. The notion of Hughes-free is explained below in the proof of Lemma 10.81.

Lemma 10.78. The universal field of fractions of ℂF is Hughes-free.

A special case of the main theorem in [267] says

Lemma 10.79. Let F be a free group. Any two ℂF -fields of fractions which are Hughes-free are isomorphic as ℂF -fields.

We conclude from Lemma 10.77, Lemma 10.78 and Lemma 10.79.

Lemma 10.80. Let F be a free group. Let ℂF → K be a Hughes-free ℂF - field of fractions. Then it is a universal ℂF -field of fractions and it is universal Σ(ℂF → K)-inverting.

Now we are ready to show

Lemma 10.81. Let F be a free group. Then D(F) is a skewfield and the inclusion ℂF → D(F) is universally Σ(F)-inverting.

Proof. We know already that D(F) is a skewfield by Lemma 10.51. We have Σ(F) = Σ(ℂF → D(F)). Because of Lemma 10.80 it suffices to show that ℂF → D(F) is Hughes-free. This means the following. Given any finitely generated subgroup H ⊂ F and an element t ∈ H together with a homomorphism p_{t}: H → ℤ which maps t to a generator, the set {t^{i} | i ∈ ℤ} is D(ℂ[ker(p_{t})] ⊂ D(F))-linear independent in D(F). Since U(ker(p_{t})) is von Neumann regular by Theorem 8.22 (3), Lemma 10.34 (1) implies

D(ℂ[ker(p_{t})] ⊂ D(F)) = D(ℂ[ker(p_{t})] ⊂ U(F)) = D(ℂ[ker(p_{t})] ⊂ U(ker(p_{t}))) = D(ker(p_{t})).

From the split exact exact sequence 1 → ker(p_{t}) → H ⟶^{p_{t}} ℤ → 1 and Lemma 10.57 (1) we conclude that the subring of U(ker(p_{t})) ∗ ℤ generated by D(ker(p_{t})) and ℤ is itself a crossed product D(ker(p_{t})) ∗ ℤ. In particular it is a free D(ker(p_{t}))-module with basis {t^{i} | i ∈ ℤ}. ⊓ ⊔

Now we are ready to prove Lemma 10.75. Choose an extension 1 → F → G → Q → 1 for a free group F and a finite group Q. We conclude from Lemma 10.51 and Lemma 10.59 that D(G) is semisimple and D(G) = D(F)∗Q. Since ℂF → D(F) is universally Σ(F)-inverting by Lemma 10.81, the inclusion ℂG = ℂF ∗ Q ⊂ D(F) ∗ Q = D(G) is universally Σ(F)-inverting by Lemma 10.57 (2). Notice that D(G) = R(G) by Lemma 10.34 (2). Hence ℂG → D(G) is both Σ(G)-inverting and universally Σ(F)-inverting. One easily checks that then it is automatically universally Σ(G)-inverting. This finishes the proof of Lemma 10.75 and hence of the main result of this subsection, Lemma 10.72.

10.3.7 The Induction Step for Directed Unions

In this subsection we want to show

Lemma 10.82. The statements and (6) and (7) in Lemma 10.41 are true.

We begin with the following result

Lemma 10.83. Let the group G be the directed union of subgroups G = ⋃_{i∈I} G_{i}. Then

D(G) = ⋃_{i∈I} D(G_{i});
R(G) = ⋃_{i∈I} R(G_{i}).

If each D(G_{i}) or R(G_{i}) respectively is von Neumann regular, then D(G) or R(G) respectively is von Neumann regular.

Proof. We have ℂG = ⋃_{i∈I} ℂG_{i}. We conclude from Lemma 10.36 that

D (ℂG ⊂ ⋃_{i∈I} U(G_{i})) = ⋃_{i∈I} D(G_{i}).

Since each U(G_{i}) is von Neumann regular (see Theorem 8.22 (3)), ⋃_{i∈I} U(G_{i}) is von Neumann regular. Lemma 10.34 (1) implies

D(G) = D (ℂG ⊂ ⋃_{i∈I} U(G_{i}) .

The proof for the rational closure is similar. ⊓ ⊔

Now we can give the proof of Lemma 10.82. We begin with statement (6) We have to show the surjectivity of the map

colim_{G/H∈Or(G,FIN)} K_{0}(ℂH) → K_{0}(D(G)),

provided that G is a directed union G = ⋃_{i∈I} G_{i} and each of the maps colim_{G_{i}/K∈Or(G,FIN)} K_{0}(ℂK) → K_{0}(D(G_{i}) is surjective. This follows from ℂG = ⋃_{i∈I} ℂG_{i} and D(G) = ⋃_{i∈I} D(G_{i}) (see Lemma 10.83) and the facts that K_{0} commutes with colimits over directed systems and the colimit over a directed system of epimorphisms is again an epimorphism.

Next we prove statement (7). We have to show for G the directed union of subgroups G = ⋃_{i∈I} G_{i} that D(G) is semisimple provided that there is an upper bound on the orders of finite subgroups of G, each D(G_{i}) is semisimple and each map colim_{G_{i}/K∈Or(G_{i},FIN)} K_{0}(ℂK) → K_{0}(D(G_{i}) is surjective. We have already shown that the latter condition implies that colim_{G/H∈Or(G,FIN)} K_{0}(ℂH) → K_{0}(D(G) is surjective. Because of Lemma 10.28 (2) it suffices to prove that D(G) is von Neumann regular. This follows from Lemma 10.83, since each D(G_{i}) is by assumption semisimple and hence von Neumann regular. This finishes the proof of Lemma 10.82. ⊓ ⊔

Notice that we have proved the conditions appearing in Lemma 10.41, namely, statement (1) in Lemma 10.51, statement (2) in Lemma 10.59, statement (3) in Lemma 10.60, statement (5) in Lemma 10.66, statement (4) in Lemma 10.72 and statements (6) and (7) in Lemma 10.82. This proves the strong version of Linnell’s Theorem 10.38 and hence Linnell’s Theorem 10.19.

⊓ ⊔

10.4 Miscellaneous

No counterexamples to the Atiyah Conjecture 10.3 of order Λ = ℚ with coefficients in F = ℂ are known to the author at the time of writing. Recall that the Atiyah Conjecture 10.3 of order Λ = ℚ with coefficients in F = ℚ is equivalent to a positive answer to Atiyah’s original question whether the L^{2}- Betti numbers of a G-covering of a closed Riemannian manifold are rational numbers. In [143, Example 5.4] the real number

κ(1/2, 1/2) = ∑_{k≥2} \frac{ϕ(k)}{(2^{k} − 1)^{2}} = 0, 1659457149 …

is studied, where ϕ(k) denotes the number of primitive k-th roots of unity. It is shown that this number occurs as an L^{2}-Betti number of the universal covering of a closed manifold. Some evidence is given that this number may not be rational but no proof of the irrationality of this number is known to the author at the time of writing. So there are some doubts about this version of the Atiyah Conjecture. There is some hope that the strong Atiyah Conjecture 10.2 is true under the additional assumption that there is an upper bound on the orders of finite subgroups of G.

At the time of writing there is no proof known to the author that the strong Atiyah Conjecture 10.2 holds for a group G if it is true for a subgroup H ⊂ G of finite index. Some special results in this direction can be found in [313].

Some of the results of this Chapter 10 have been extended from ℚ as coefficients to the field of algebraic numbers \overline{ℚ} ⊂ ℂ. Namely, Theorem 10.20 and Theorem 10.22 have also been proved in the case where one replaces ℚ by \overline{ℚ} in [148, Theorem 1.6 and Proposition 1.9]. (There only torsionfree groups and Λ = ℤ is considered, but the same proof applies to the case where Λ is assumed to be closed in ℝ.) This is interesting because of Lemma 10.15 and the fact that a torsionfree group G satisfies the Kaplansky Conjecture for the field \overline{ℚ} if and only if is satisfies the Kaplansky Conjecture for the complex numbers ℂ. The well-known proof of this fact can be found for instance in [148, Proposition 4.1].

A group G has the algebraic eigenvalue property if for every matrix A ∈ M_{d}(\overline{ℚ}G) the eigenvalues of the operator r_{A}: l^{2}(G)^{d} → l^{2}(G)^{d} are algebraic. There is the conjecture that every group has this property [148, Conjecture 6.8]. It is based on the following results. If G is a free group, then the series

f(z) = ∑_{n=0}^{∞} tr_{\overline{ℚ}G}(A^{n}) · z^{−n−1}

is an algebraic function over \overline{ℚ}(z) and hence each eigenvalue of r_{A} is an algebraic number since it is a pole of f(z) [456]. Hence a free group has the algebraic eigenvalue property. More generally any group in Linnell’s class C (see Definition 10.18) has the algebraic eigenvalue property [148, Corollary 6.7].

So far we have studied the passage from modules over ℂG to N (G). Let C_{r}^{∗}(G) be the reduced C^{∗}-algebra of G. We have ℂG ⊂ C_{r}^{∗}(G) ⊂ N (G). We have mentioned in Lemma 10.10 that \frac{1}{|FIN (G)|}ℤ ⊂ Λ(G, ℂ)_{fgp} and that the Isomorphism Conjecture 9.40 for K_{0}(ℂG) implies \frac{1}{|FIN (G)|}ℤ = Λ(G, ℂ)_{fgp}, where Λ(G, ℂ)_{fgp} is the image of the composition

K_{0}(ℂG) ⟶^{i} K_{0}(N (G)) ⟶^{dim_{N (G)}} ℝ.

Let Λ(G, C_{r}^{∗})_{fgp} be the image of the composition

K_{0}(C_{r}^{∗}(G)) ⟶^{i} K_{0}(N (G)) ⟶^{dim_{N (G)}} ℝ.

Obviously

Λ(G, ℂ)_{fgp} ⊂ Λ(G, C_{r}^{∗})_{fgp}.

If equality would hold, this would suggest to study modules over C_{r}^{∗}(G) in order to attack the strong Atiyah Conjecture 10.2. This equality was conjectured in [26, page 21]. The question whether this equality holds is related to the Baum-Connes Conjecture (see [26], [27] [257]), which is the analog of the Isomorphism Conjecture 9.40 for K_{0}(ℂG) and known for a much larger class of groups. However, a counterexample to this equality has meanwhile been given by Roy [449]. There a group G is constructed which fits into an exact sequence 1 → π_{1}(M) → G → ℤ/3 × ℤ/3 → 1 for a closed aspherical 4-dimensional manifold M and whose non-trivial finite subgroups are isomorphic to ℤ/3, but for which −\frac{1105}{9} belongs to Λ(G, C_{r}^{∗})_{fgp}. It is not known to the author whether the Isomorphism Conjecture 9.40 for K_{0}(ℂG), the strong Atiyah Conjecture 10.2 or the Baum-Connes Conjecture is true or false for the group G above. The counterexample of Roy does not imply that one of these conjectures is false for this group G.

Let ℤ ⊂ ℤ [\frac{1}{|FIN (G)|}] ⊂ ℚ be the subring of ℚ obtained from ℤ by inverting all the orders of finite subgroups of G. One should not confuse the abelian group \frac{1}{|FIN (G)|}ℤ defined in (10.1) with the larger ring ℤ [\frac{1}{|FIN (G)|}]. Provided that G satisfies the Baum-Connes-Conjecture,

Λ(G, C_{r}^{∗})_{fgp} ⊂ ℤ [\frac{1}{|FIN (G)|}]

is shown in [339]. Notice that the lamplighter group L satisfies the Baum-Connes Conjecture by [257]) since it it amenable. Hence Λ(L, C_{r}^{∗})_{fgp} ⊂ ℤ [\frac{1}{2}] but Λ(L, ℂ)_{fp} contains 1/3 (see Theorem 10.23) and hence is not contained in ℤ [\frac{1}{2}] or Λ(L, C_{r}^{∗})_{fgp}.

The next result is taken from [435, Theorem 9.1].

Theorem 10.84. Let G be a group such that there is an upper bound on the orders of finite subgroups. Then

(1) If G belongs to C, then for any ℂG-module M

Tor_{p}^{ℂG}(M; D(G)) = 0 for all p ≥ 2;

(2) If G is elementary amenable, then for any ℂG-module M

Tor_{p}^{ℂG}(M; D(G)) = 0 for all p ≥ 1.

In other words, the functor − ⊗_{ℂG} D(G) is exact.

Notice that −⊗_{ℤG} ℂG is always exact and −⊗_{D(G)} U(G) is exact provided that D(G) is semisimple. Consider a group G, which belongs to C and has an upper bound on the orders of its finite subgroups, and a G-space X. We conclude from the strong version of Linnell’s Theorem 10.38, Theorem 10.84 and the universal coefficients spectral sequence that there is an exact sequence

0 → H_{n}(X; ℤ)⊗_{ℤG}U(G) → H_{n}^{G}(X; U(G)) → Tor_{1}^{ℤG}(H_{n−1}(X; ℤ), U(G)) → 0.

In particular we get in the special case X = EG

H_{1}^{G}(EG; U(G)) = Tor_{1}^{ℤG}(ℤ, U(G));  (10.85)
H_{n}^{G}(EG; U(G)) = 0  for n ≥ 2.  (10.86)

Notice that for an infinite group G we have b_{0}^{(2)}(G; N (G)) = 0 (see Theorem 6.54) (8b)) and hence dim_{U(G)}(H_{0}^{G}(EG; U(G))) = 0 by Theorem 8.31. We conclude for an infinite group G, which belongs to C and has an upper bound on the orders of its finite subgroups, that χ^{(2)}(G) is defined if and only if dim_{U(G)}(Tor_{1}^{ℤG}(ℤ, U(G))) < ∞, and in this case

χ^{(2)}(G) = − dim_{U(G)}(Tor_{1}^{ℤG}(ℤ, U(G))) ≤ 0.

Let G be a group in the class C such that there exists a bound on the orders of finite subgroups. Then ℂG → D(G) is universally Σ(G)-inverting (see [435, Theorem 8.3]). The proof of this statement is along the lines of the one of the strong version of Linnell’s Theorem 10.38. This is also true for the following statement (see [435, Theorem 8.4]).

Let G be an elementary amenable group such that there exists a bound on the orders of finite subgroups. Then the set of non-zero-divisors of ℂG equals T (G), i.e. any non-zero-divisor in ℂG becomes invertible in U(G). The pair (ℂG, T (G)) satisfies the Ore condition and the Ore localization is D(G). In particular ℂG → D(G) is both universally Σ(G)-inverting and universally T (G)-inverting.

Notice, however, that the lamplighter group L is amenable and neither D(L) nor U(L) is flat over ℂL, and that ℂL does not have a classical ring of quotients, i.e. the set S of non-zero-divisors does not satisfies the left Ore condition [311].

Lemma 10.87. Suppose for the group G that D(G) is semisimple. Then the change of rings map

K_{0}(D(G)) → K_{0}(U(G))

is injective.

Proof. We have a decomposition D(G) = ∏_{i=1}^{r} M_{n_{i}}(D_{i}) for skewfields D_{1}, D_{2}, …, D_{r} by Wedderburn’s Theorem. Let e_{i} be the projection onto M_{n_{i}}(D_{i}). Then each e_{i} is central in D(G) and e_{i}e_{j} = 0 for i ≠ j and K_{0}(D(G)) is the free abelian group with \{\frac{1}{n_{i}} · [im(e_{i})] | i = 1, 2 …, r\} as basis. By Theorem 8.22 (6) we can choose a projection p_{i} in N (G) such that p_{i}e_{i} = e_{i} and e_{i}p_{i} = p_{i} holds in U(G).

The element e_{i} ∈ U(G) commutes with any element f ∈ N (G) by the following argument. Because of the Double Commutant Theorem 9.2 we can find a net {f_{j} | j ∈ J} of elements in ℂG ⊂ N (G) = B(l^{2}(G))^{G} which converges to f in the strong topology. Since e_{i} commutes with any element in D(G), it commutes with any element f_{j}, i .e. we have the equality e_{i} ∘ f_{j} = f_{j} ∘ e_{i} of affiliated unbounded operators from l^{2}(G) to l^{2}(G). Obviously dom(e_{i}) ⊂ dom(f_{j} ∘ e_{i}) and dom(e_{i}) ⊂ dom(f ∘ e_{i}) since each f_{j} and f are bounded. By definition dom(f_{j} ∘ e_{i}) = dom(e_{i} ∘ f_{j}) and dom(f ∘ e_{i}) = dom(e_{i} ∘ f). For x ∈ dom(e_{i}) we have

lim_{j∈J} f_{j}(x) = f(x);
lim_{j∈J} e_{i} ∘ f_{j}(x) = lim_{j∈J} f_{j} ∘ e_{i}(x) = f ∘ e_{i}(x).

Hence f(x) ∈ dom(e_{i}) and e_{i} ∘ f(x) = f ∘ e_{i}(x). Lemma 8.3 (3) implies that f ∘ e_{i} = e_{i} ∘ f holds in U(G) for each f ∈ N (G).

Since U(G) is the Ore localization of N (G) with respect to all non-zero-divisors (see Theorem 8.22 (1)), e_{i} ∈ U(G) is central in U(G). This implies e_{i} = p_{i} in U(G) and p_{i} ∈ Z(N (G))^{ℤ/2}. The composition

K_{0}(D(G)) → K_{0}(U(G)) ⟵^{≅} K_{0}(N (G)) ⟶^{dim_{N (G)}} Z(A)^{ℤ/2}

(see Theorem 9.13 (2) and Theorem 9.20 (1)) sends [im(e_{i})] to p_{i}. Since p_{i}p_{j} = 0 for i ≠ j and p_{i}^{2} = p_{i}, the elements p_{1}, p_{2}, …, p_{r} are linearly independent over ℝ. Hence the composition above is injective. This finishes the proof of Lemma 10.87. ⊓ ⊔

Exercises

10.1. Let G be a finitely presented group. Let ℤ ⊂ Λ ⊂ ℝ be an additive subgroup of ℝ. Show that the following statements are equivalent:

(1) For any closed manifold M with G = π_{1}(M) we have b_{p}^{(2)}(M̃) ∈ Λ;

(2) For any finite CW -complex X with G = π_{1}(X) we have b_{p}^{(2)}(X̃) ∈ Λ;

(3) The Atiyah Conjecture 10.3 of order Λ with coefficients in ℚ is true for G.

10.2. Suppose that there exists a cocompact free proper G-CW -complex X which has finitely many path components. Show that then G is finitely generated. On the other hand show for a finitely generated group G that there is a cocompact connected free proper G-manifold M without boundary.

10.3. Let Λ ⊂ ℝ be an additive subgroup different from {0}. Let d(Λ) be the infimum of the subset {λ ∈ Λ | λ > 0} of ℝ. Show that Λ ⊂ ℝ is closed if and only if

Λ = {r ∈ ℝ | d(Λ) · r ∈ ℤ}.

10.4. Let G be a discrete group, let F be a field with ℚ ⊂ F ⊂ ℂ and let ℤ ⊂ Λ ⊂ ℝ be an additive subgroup of ℝ. Suppose that Λ ⊂ ℝ is closed and that G satisfies the Atiyah Conjecture 10.3 of order Λ with coefficients in F. Prove that then for any F G-chain complex C_{∗}

dim_{N (G)}(H_{p}(N (G) ⊗_{F G} C_{∗})) ∈ Λ ∐{∞}.

10.5. Let X be a G-CW -complex such that h^{(2)}(X; N (G)) < ∞ and hence χ^{(2)}(X; N (G)) is defined (see Definition 6.79). Suppose that there is an integer d such that the order of any finite subgroup of G divides d. Assume that the strong Atiyah Conjecture 10.2 holds for G. Then show

d · χ^{(2)}(X; N (G)) ∈ ℤ.

10.6. Let G be a group. Suppose that there is a chain of finite subgroups 1 ⊂ H_{1} ⊂ H_{2} ⊂ … and a prime number p such that the orders of the p-Sylow subgroups of H_{n} become arbitrary large. Show that there exists a chain of subgroups 1 ⊂ K_{1} ⊂ K_{2} ⊂ … with |K_{n}| = p^{n} and Λ(A, ℂ)_{fg} = ℝ holds.

10.7. Prove for the class C:

(1) C is closed under taking subgroups;

(2) If {G_{i} | i ∈ I} is a collection of groups in C, then ∗_{i∈I}G_{i} belongs to C;

(3) Let 1 → H → G → K → 1 be an extension of groups such that H is finite or infinite cyclic and K belongs to C, then G belongs to C;

(4) If G belongs to C and H ⊂ G is an elementary amenable normal subgroup of G, then G/H belongs to C.

10.8. Let (G_{n})_{n≥1} be a sequence of groups and (d_{n})_{n≥1} be a sequence of positive integers such that G_{n} belongs C and d_{n} is an upper bound on the orders of finite subgroups of G_{n}. Show that G = ∗_{n≥1}G_{n} satisfies the strong Atiyah Conjecture 10.2. Give an example of a group G such that G satisfies the strong Atiyah Conjecture 10.2, G contains only 2-groups as subgroups and has no upper bound on the orders of finite subgroups.

10.9. Show for any 2-dimensional manifold that π_{1}(M) belongs to C and satisfies the strong Atiyah Conjecture 10.2.

10.10. Let M be a closed 3-manifold such that there is a finite covering \overline{M} → M and a fibration F → \overline{M} → S^{1}, where F is a closed connected orientable manifold. Show that π_{1}(M) belongs to C and satisfies the strong Atiyah Conjecture 10.2.

10.11. Let G be a torsionfree group which satisfies the Atiyah Conjecture 10.3 of order Λ = ℤ with coefficients in F = ℚ. Suppose that there is a model for BG of finite type. Show for any p ≥ 0 that the Hilbert N (G)-module H_{p}^{(2)}(EG) is isomorphic to l^{2}(G)^{n} for an appropriate integer n ≥ 0.

10.12. Let M be a closed hyperbolic manifold of dimension 2n. Show that π_{1}(M) does belong to C if and only if n = 1.

10.13. Find an extension of groups 1 → H → G → K → 1 such that H and K satisfy the strong Atiyah Conjecture 10.2 but G does not.

10.14. Let G be the directed union of the directed system of subgroups {G_{i} | i ∈ I}. Suppose that for any i ∈ I there is a ring S(G_{i}) with ℂG_{i} ⊂ S(G_{i}) ⊂ U(G_{i}) which satisfies condition (R’) or (K’) respectively of Lemma 10.26. Suppose that S(G_{i}) ⊂ S(G_{j}) holds for i ≤ j. Put S(G) = ⋃_{i∈I} S(G_{i}). Show that ℂG ⊂ S(G) ⊂ U(G) and that S(G) satisfies conditions (R’) or (K’) respectively of Lemma 10.26. Give an example to show that the assertion above is not true in general for (R).

10.15. Show that the map τ defined in Lemma 10.45 (2) and (3) induces a map

τ: K_{0}(A^{1}) → ℝ

by sending an element in K_{0}(A^{1}) represented by an idempotent p to τ(p). Show that the image of this map is contained in ℤ.

10.16. Let H ⊂ G be a subgroup of finite index d. Suppose that D(H) is semisimple. Show that the following diagram is well-defined and commutes

K_{0}(D(G)) ⟶ K_{0}(U(G)) ⟶^{dim_{U(G)}} ℝ
res↓  ^{res}↓  ^{d·id}↓
K_{0}(D(H)) ⟶ K_{0}(U(H)) ⟶^{dim_{U(H)}} ℝ

Show that for all virtually free groups G the image of the composition

K_{0}(D(G)) ⟶^{i} K_{0}(U(G)) ⟶^{dim_{U(G)}} ℝ

is contained in \frac{1}{|FIN (G)|}ℤ.

10.17. Let ℂ⟨x, y, z⟩ and ℂ⟨x, u⟩ be non-commutative polynomial rings. It is known that non-commutative polynomial rings are semifirs [118, Section 10.9]. Hence there are universal fields of fractions ℂ⟨x, y, z⟩ → K and ℂ⟨x, u⟩ → L by Lemma 10.77. Define a map f: ℂ⟨x, y, z⟩ → ℂ⟨x, u⟩ by f(x) = x, f(y) = xu and f(z) = xu^{2}. Show that this map is injective. Let D be the division closure of the image of f in L. Show that f yields a field of fractions ℂ⟨x, y, z⟩ → D. It factorizes by the universal property

as the composition ℂ⟨x, y, z⟩ → ℂ⟨x, y, z⟩_{Σ(ℂ⟨x,y,z⟩→D)} ⟶^{Ψ} D. Show that Ψ

maps y^{−1}z − x^{−1}y to zero and deduce that Ψ in not injective. Prove that ℂ⟨x, y, z⟩ → D is not the universal field of fractions of ℂ⟨x, y, z⟩ and that there are two non-isomorphic fields of fractions for ℂ⟨x, y, z⟩.

10.18. Let G be the directed union of subgroups ⋃_{i∈I} G_{i}. Give an example to show that in general neither N (G) = ⋃_{i∈I} N (G_{i}) nor U(G) = ⋃_{i∈I} U(G_{i}) is true.

10.19. Let G be a group in the class C, which has an upper bound on the orders of finite subgroups. Show that the kernel of K_{0}(ℂG) → K_{0}(D(G)) and of K_{0}(ℂG) → K_{0}(N (G)) agree.

11. The Singer Conjecture

Introduction

This chapter is devoted to the following conjecture (see [477] and also [146, Conjecture 2]).

Conjecture 11.1 (Singer Conjecture). If M is an aspherical closed manifold, then

b_{p}^{(2)}(M̃) = 0 if 2p ≠ dim(M).

If M is a closed connected Riemannian manifold with negative sectional curvature, then

b_{p}^{(2)}(M̃) \{ \begin{matrix}= 0 & if 2p ≠ dim(M); \\ > 0 & if 2p = dim(M). \end{matrix}

Because of the Euler-Poincaré formula χ(M) = ∑_{p≥0}(−1)^{p} · b_{p}^{(2)}(M̃) (see Theorem 1.35 (2)) the Singer Conjecture 11.1 implies the following conjecture in the cases where M is aspherical or has negative sectional curvature.

Conjecture 11.2 (Hopf Conjecture). If M is an aspherical closed manifold of even dimension, then

(−1)^{dim(M)/2} · χ(M) ≥ 0.

If M is a closed Riemannian manifold of even dimension with sectional curvature sec(M), then

(−1)^{dim(M)/2} · χ(M) > 0 if sec(M) < 0;
(−1)^{dim(M)/2} · χ(M) ≥ 0 if sec(M) ≤ 0;
χ(M) = 0 if sec(M) = 0;
χ(M) ≥ 0 if sec(M) ≥ 0;
χ(M) > 0 if sec(M) > 0.

In original versions of the Singer Conjecture 11.1 and the Hopf Conjecture 11.2 the statements for aspherical manifolds did not appear. Notice that any Riemannian manifold with non-positive sectional curvature is aspherical by Hadamard’s Theorem. We will also discuss Conjecture 11.3 (L^{2}-torsion for aspherical manifolds). If M is an aspherical closed manifold of odd dimension, then M̃ is det-L^{2}-acyclic and

(−1)^{\frac{dim(M)−1}{2}} · ρ^{(2)}(M̃) ≥ 0.

If M is a closed connected Riemannian manifold of odd dimension with negative sectional curvature, then M̃ is det-L^{2}-acyclic and

(−1)^{\frac{dim(M)−1}{2}} · ρ^{(2)}(M̃) > 0.

If M is an aspherical closed manifold whose fundamental group contains an amenable infinite normal subgroup, then M̃ is det-L^{2}-acyclic and

ρ^{(2)}(M̃) = 0.

Notice that for aspherical closed manifolds of odd dimension Conjecture 11.3 is stronger than the Singer Conjecture 11.1. Recall that the Euler characteristic of a closed manifold of odd dimension vanishes and that for a closed Riemannian manifold M of even dimension ρ^{(2)}(M̃) = 0 holds. Therefore the Hopf Conjecture or Conjecture 11.3 respectively are not interesting for manifolds whose dimension is odd or even respectively.

We will discuss special cases for which these conjectures are known in Section 11.1. We give the proof of Gromov’s result on Kähler manifolds in Section 11.2.

To understand this chapter, it is only required to be familiar with Sections 1.1 and 1.2 and to have some basic knowledge of Section 3.4. For Subsections 11.2.3 and 11.2.4 small input from Chapter 2 is needed.

11.1 Survey on Positive Results about the Singer Conjecture

In this section we discuss results which give some evidence for the Singer Conjecture 11.1, the Hopf Conjecture 11.2 and Conjecture 11.3. Notice that it suffices to consider closed connected manifolds which are orientable. If M is not orientable, one can pass to the orientation covering \overline{M} → M and use the fact that b_{p}^{(2)}(M̃; N (π_{1}(M))) = 2 · b_{p}^{(2)}(˜\overline{M}; N (π_{1}(\overline{M}))) and ρ^{(2)}(M̃; N (π_{1}(M))) = 2 · ρ^{(2)}(˜\overline{M}; N (π_{1}(\overline{M}))) holds (see Theorem 1.35 (9) and Theorem 3.96 (5)).

11.1.1 Low-Dimensional Manifolds

We begin with low-dimensional manifolds. Let F_{g} be the closed orientable surface of genus g. It is aspherical if and only if g ≥ 1. It carries a hyperbolic structure if and only if g ≥ 2, a flat Riemannian metric if and only if g = 1 and a Riemannian metric with positive constant sectional curvature if and only if g = 0. If F_{g} carries a Riemannian metric with sectional curvature sec(M) < 0, sec(M) ≤ 0, sec(M) = 0, sec(M) ≥ 0 or sec(M) > 0 respectively, the Gauss-Bonnet Theorem [218, 3.20 on page 107 and 3.111 on page 147] implies χ(M) < 0, χ(M) ≤ 0, χ(M) = 0. χ(M) ≥ 0 or χ(M) > 0 respectively. If g ≥ 1, then b_{p}^{(2)}(F̃_{g}) = 0 for p ≠ 1 and b_{1}^{(2)}(F̃_{g}) = 2g − 2 = −χ(M) holds by Example 1.36. Hence the Singer Conjecture 11.1 and the Hopf Conjecture 11.2 are true in dimension 2.

Next we consider dimension 3 and Conjecture 11.3. We have already explained in Section 4.1 that the Sphere Theorem [252, Theorem 4.3] implies that a compact connected orientable irreducible 3-manifold is aspherical if and only if it is a 3-disk or has infinite fundamental group. This implies that a closed connected orientable 3-manifold is aspherical if and only if its prime decomposition contains at most one factor which is not a homotopy sphere and this factor must be irreducible with infinite fundamental group. Hence a closed orientable 3-manifold M is aspherical if and only if it is diffeomorphic to N#Σ for a closed connected orientable irreducible 3-manifold N with infinite fundamental group and a homotopy sphere Σ. Choose a homotopy equivalence g: Σ → S^{3}. We have briefly introduced the notion of Whitehead torsion and simple homotopy equivalence in Section 3.1.2. Since the Whitehead group of the trivial group is trivial and S^{3} has trivial fundamental group, the Whitehad torsion of g is trivial. By homotopy invariance and the sum formula for Whitehead torsion we conclude that id #g: N#Σ → N#S^{3} has trivial Whitehead torsion. This shows that there is a simple homotopy equivalence f: M → N. Hence a closed orientable 3-manifold M is aspherical if and only if it is simply homotopy equivalent to a closed connected orientable irreducible 3-manifold with infinite fundamental group. We conclude from Theorem 3.96 (1) that M is det-L^{2}-acyclic if and only if N is det-L^{2}-acyclic, and in this case ρ^{(2)}(M̃) = ρ^{(2)}(Ñ).

Suppose from now on that N satisfies Thurston’s Geometrization Conjecture. Then we conclude from Theorem 4.3 that Ñ is det-L^{2}-acyclic and ρ^{(2)}(Ñ) ≤ 0. Suppose that N carries a Riemannian metric of negative sectional curvature. Then its fundamental group is word-hyperbolic and hence cannot contain ℤ ⊕ ℤ as subgroup [65, Corollary 3.10 on page 462], Then N is already hyperbolic and ρ^{(2)}(Ñ) < 0. This is consistent with Conjecture 11.3. In other words, Conjecture 11.3 is true for 3-dimensional manifolds if Thurston’s Geometrization Conjecture holds. Notice that there exist closed graph manifolds, i.e. closed irreducible 3-manifolds M whose decomposition along a family of disjoint, pairwise-nonisotopic incompressible tori in M contains only Seifert pieces, which do not admit a Riemannian metric of non-positive sectional curvature but are aspherical. [304, Example 4.2]. So it makes a difference whether one makes the assumption aspherical instead of the stronger condition that there is a Riemannian metric with non-positive sectional curvature. Also, weakening the assumption in the conjectures, gives them a different flavour because aspherical is a purely homotopy-theoretic condition.

The Hopf Conjecture 11.2 is known for a closed 4-dimensional Riemannian manifold with non-positive or non-negative curvature [59], [108], where the result is attributed to Milnor. Actually it is shown that the integrand appearing in the Gauss-Bonnet formula is non-negative. It follows from the example of Geroch [219] that one cannot deduce in higher even dimension the positivity of the Gauss-Bonnet integrand at a point x purely algebraically from the fact that the sectional curvature is positive. The Singer Conjecture 11.1 for a closed orientable 4-manifold M is equivalent to the statement that b_{1}^{(2)}(M̃) = 0 because of Poincaré duality and the fact that infinite π_{1}(M) implies b_{0}^{(2)}(M̃) = 0 (see Theorem 1.35 (3) and (8)). We have given a lot of sufficient conditions on π_{1}(M) to ensure b_{1}^{(2)}(M̃) = 0 in Subsection 7.1.1. To the authors knowledge at the time of writing the Singer Conjecture 11.1 is not known for all closed 4-dimensional manifolds.

11.1.2 Pinched Curvature

The following two results are taken from the paper by Jost and Xin [280, Theorem 2.1 and Theorem 2.3].

Theorem 11.4. Let M be a closed connected Riemannian manifold of dimension dim(M) ≥ 3. Suppose that there are real numbers a > 0 and b > 0 such that the sectional curvature satisfies −a^{2} ≤ sec(M) ≤ 0 and the Ricci curvature is bounded from above by −b^{2}. If the non-negative integer p satisfies 2p ≠ dim(M) and 2pa ≤ b, then

b_{p}^{(2)}(M̃) = 0.

Theorem 11.5. Let M be a closed connected Riemannian manifold of dimension dim(M) ≥ 4. Suppose that there are real numbers a > 0 and b > 0 such that the sectional curvature satisfies −a^{2} ≤ sec(M) ≤ −b^{2}. If the nonnegative integer p satisfies 2p ≠ dim(M) and (2p − 1) · a ≤ (dim(M) − 2) · b, then

b_{p}^{(2)}(M̃) = 0.

The next result is a consequence of a result of Ballmann and Brüning [18, Theorem B on page 594].

Theorem 11.6. Let M be a closed connected Riemannian manifold. Suppose that there are real numbers a > 0 and b > 0 such that the sectional curvature satisfies −a^{2} ≤ sec(M) ≤ −b^{2}. If the non-negative integer p satisfies 2p < dim(M) − 1 and p · a < (dim(M) − 1 − p) · b, then

b_{p}^{(2)}(M̃) = 0;
α_{p}^{Δ}(M) = ∞^{+}.

Notice that Theorem 11.5 and Theorem 11.6 are improvements of the older results by Donnelly and Xavier [155], where b_{p}(M̃) = 0 for p ∉ \{\frac{dim(M)−1}{2}, dim(M), \frac{dim(M)+1}{2}\} is shown provided that there is a real number ϵ satisfying −1 ≤ sec(M) ≤ −1 + ϵ and 0 ≤ ϵ < 1 − \frac{(dim(M)−2)^{2}}{(dim(M)−1)^{2}} . The proof of Theorem 11.6 follows along the lines of [155], the decisive improvement is [18, Theorem 5.3 on page 619]. The Hopf Conjecture for pinched sectional curvature is also treated in [58].

11.1.3 Aspherical Manifolds and Locally Symmetric Spaces

Suppose that M is an aspherical closed Riemannian manifold. Then Corollary 5.16 shows that M satisfies the Singer Conjecture 11.1, the Hopf Conjecture 11.2 and Conjecture 11.3 provided that M carries the structure of a locally symmetric space. Suppose that M is a closed locally symmetric Riemannian manifold whose sectional curvature is negative. Then M satisfies the Singer Conjecture 11.1, the Hopf Conjecture 11.2 and Conjecture 11.3 by Corollary 5.16.

If the aspherical closed manifold M carries a non-trivial S^{1}-action, then it satisfies the Singer Conjecture 11.1, the Hopf Conjecture 11.2 and Conjecture 11.3 by Theorem 3.111.

If the aspherical closed manifold M appears in a fibration F → M → S^{1} such that F is a connected CW -complex of finite type, then the Singer Conjecture 11.1 and the Hopf Conjecture 11.2 hold for M by Theorem 1.39. If additionally F̃ is det-L^{2}-acyclic, M satisfies also Conjecture 11.3 by Theorem 3.106.

We have proved in Theorem 7.2 (1) and (2) that the Singer Conjecture 11.1 and the Hopf Conjecture 11.2 hold for an aspherical closed manifold M whose fundamental group contains an amenable infinite normal subgroup. We have proved in Theorem 3.113 that Conjecture 11.3 is true in the special case, where M is a closed aspherical manifold, whose fundamental group is of det ≥ 1-class and contains an elementary amenable infinite normal subgroup.

We will deal with the Singer Conjecture 11.1 for Kähler manifolds in Corollary 11.17.

11.2 The Singer Conjecture for Kähler Manifolds

Let M be a complex manifold (without boundary). Let h denote a Hermitian metric. Thus we have for any x ∈ M a Hilbert space structure

h_{x}: T_{x}M × T_{x}M → ℂ  (11.7)

on the complex vector space T_{x}M. It induces a Riemannian metric g by

g_{x}: T_{x}M × T_{x}M → ℝ,  (v, w) ↦ ℜ(h_{x}(v, w)).  (11.8)

The associated fundamental form is the 2-form ω defined by

ω_{x}: T_{x}M × T_{x}M → ℝ,  (v, w) ↦ − \frac{1}{2} · ℑ(h_{x}(v, w)). (11.9)

Obviously h contains the same information as g and ω together.

Definition 11.10 (Kähler manifold). Let M be a complex manifold without boundary. A Hermitian metric h is called Kähler metric if (M, g) is a complete Riemannian manifold and the fundamental form ω ∈ Ω^{2}(M) is closed, i.e. dω = 0. A Kähler manifold M = (M, h) is a connected complex manifold M without boundary together with a Kähler metric h.

Definition 11.11. Let (M, g) be a connected Riemannian manifold. A (p − 1)-form η ∈ Ω^{p−1}(M) is bounded if ||η||_{∞} := sup{||η||_{x} | x ∈ M} < ∞ holds, where ||η||_{x} is the norm on Alt^{p−1}(T_{x}M) induced by g_{x}. A p-form ω ∈ Ω^{p}(M) is called d(bounded) if ω = d(η) holds for some bounded (p − 1)- form η ∈ Ω^{p−1}(M). A p-form ω ∈ Ω^{p}(M) is called d̃(bounded) if its lift ω̃ ∈ Ω^{p}(M̃) to the universal covering M̃ is d(bounded).

The next definition is taken from [236, 0.3 on page 265].

Definition 11.12 (Kähler hyperbolic manifold). A Kähler hyperbolic manifold is a closed connected Kähler manifold (M, h) whose fundamental form ω is d̃(bounded).

Example 11.13. The following list of examples of Kähler hyperbolic manifolds is taken from [236, Example 0.3]:

(1) M is a closed Kähler manifold which is homotopy equivalent to a Riemannian manifold with negative sectional curvature;

(2) M is a closed Kähler manifold such that π_{1}(M) is word-hyperbolic in the sense of [234] and π_{2}(M) = 0;

(3) M̃ is a symmetric Hermitian space of non-compact type;

(4) M is a complex submanifold of a Kähler hyperbolic manifold;

(5) M is a product of two Kähler hyperbolic manifolds.

The complex projective space ℂℙ^{n} equipped with the Fubini-Study metric, which is up to scaling with a constant the only U(n + 1)-invariant Riemannian metric, is a closed Kähler manifold but is not Kähler hyperbolic. Namely, it is simply connected and its fundamental form is not of the form d(η) since in H^{2}(ℂℙ^{n}; ℤ) it represents the non-trivial class c_{1}(γ_{1}|_{ℂℙ^{n}}) given by the first Chern class of the restriction of the universal complex line bundle γ_{1}, which lives over ℂℙ^{∞}, to ℂℙ^{n} (see [520, Example 2.3 in VI.2 on page 224]).

The following result is due to Gromov [234, Theorem 1.2.B and Theorem 1.4.A on page 274].

Theorem 11.14. (L^{2}-Betti numbers and Novikov-Shubin invariants of Kähler hyperbolic manifolds). Let M be a closed Kähler hyperbolic manifold of complex dimension m and real dimension n = 2m. Then

b_{p}^{(2)}(M̃) = 0  if p ≠ m;
b_{m}^{(2)}(M̃) > 0;
(−1)^{m} · χ(M) > 0;
α_{p}(M̃) = ∞^{+}  for p ≥ 1.

This has interesting consequences in algebraic geometry.

Theorem 11.15. Let M be a closed Kähler hyperbolic manifold of complex dimension m and real dimension n = 2m.

(1) The canonical line bundle L = Λ^{m}T^{∗}M is quasi-ample, i.e. its Kodaira dimension is m;

(2) M satisfies all of the following four assertions (which are equivalent for closed Kähler manifolds):

(a) M is Moishezon, i.e. the transcendental degree of the field M(M) of meromorphic functions is equal to m;

(b) M is Hodge, i.e. the Kähler form represents a class in H^{2}(M; ℂ) which lies in the image of H^{2}(M; ℤ) → H^{2}(M; ℂ);

(c) M can be holomorphically embedded into ℂℙ^{N} for some N;

(d) M is a projective algebraic variety;

(3) The fundamental group is an infinite non-amenable group of deficiency ≤ 1. It cannot be a non-trivial free product.

Proof. (1) and (2a) We will see in Corollary 11.36 that there is a holomorphic L^{2}-integrable m-form on the universal covering M̃ of any closed Kähler hyperbolic manifold. This is used by Gromov [236, Section 3] to construct meromorphic functions on M and leads to the result that the canonical line bundle L is quasi-ample and M is Moishezon. The equivalence of the assertions (2a), (2b), (2c) and (2d) for closed Kähler manifolds is due to results of Chow, Kodaira and Moishezon (see [249, Appendix B4 on page 445], [294], [382], [520, Remark on page 11, Theorem 4.1 in VI.4 on page 234]).

We conclude α_{1}(M̃) = ∞^{+} and b_{0}^{(2)}(M̃) = 0 from Theorem 11.14. Theorem 1.35 (8) and Theorem 2.55 (5b) imply that π_{1}(M) is an infinite non-amenable group. If m = 1, then π_{1}(M) is the fundamental group of a closed orientable surface with infinite fundamental group and hence has deficiency ≤ 1 and cannot be a non-trivial free product. Suppose m ≥ 2. Then b_{1}^{(2)}(M̃) = 0 by Theorem 11.14. This shows b_{1}^{(2)}(π_{1}(M)) = 0. Lemma 7.22 implies that π_{1}(M) has deficiency ≤ 1. We conclude from Theorem 1.35 (5) that the only finitely presented group G, which satisfies b_{1}^{(2)}(G) = 0 and is a non-trivial free product, is G = ℤ/2 ∗ ℤ/2. Since this group is amenable, π_{1}(M) cannot be a non-trivial free product. ⊓ ⊔

Gromov’s notion of Kähler hyperbolic reflects the negative curvature case. The following notion due to Cao and Xavier [86] and Jost and Zuo [281] is weaker and modelled upon non-positive sectional curvature. A Kähler manifold M is called Kähler non-elliptic manifold if the lift ω̃ of the fundamental form ω to the universal covering M̃ can be written as d(η) for some 1-form η which satisfies for some point x_{0} ∈ M̃ and constants C_{1}, C_{2} > 0, which depend on η but not on x ∈ M̃,

||η||_{x} ≤ C_{1} · d(x, x_{0}) + C_{2}.

Here d(x, x_{0}) is the distance of x and x_{0} with respect to the metric on M̃ associated to its Riemannian metric. (Actually Jost and Zuo consider more generally arbitrary coverings and local systems.) Of course a Kähler hyperbolic manifold is in particular Kähler non-elliptic. If a closed Kähler manifold carries a Riemannian metric with non-positive sectional curvature, then it is Kähler non-elliptic. The following result is due to Cao and Xavier [86, Theorem 2 on page 485] and Jost and Xin [281, Theorem 3], where also applications to algebraic geometry are given.

Theorem 11.16. (L^{2}-Betti numbers of Kähler non-elliptic manifolds). Let M be a closed Kähler non-elliptic manifold of complex dimension m and real dimension n = 2m. Then

b_{p}^{(2)}(M̃) = 0  if p ≠ m;
(−1)^{m} · χ(M) ≥ 0.

Theorem 11.14 and Theorem 11.16 imply

Corollary 11.17. Let M be a closed Riemannian manifold whose sectional curvature is negative or non-positive respectively. Then M satisfies the Singer Conjecture 11.1 if M carries some Kähler structure. An aspherical closed Kähler manifold satisfies the Singer Conjecture 11.1 if its fundamental group is word-hyperbolic in the sense of [234].

The remainder of this section is devoted to the proof of Theorem 11.14 and Theorem 11.16.

11.2.1 Hodge Theory on Kähler manifolds

We give a brief introduction to Hodge theory on a complex manifold M (without boundary). Let m be its complex and n = 2m be its real dimension.

Recall that each tangent space T_{x}M is a complex vector space. We introduce the following notation for a complex vector space V

ℂ ⊗_{ℝ} V := ℂ ⊗_{ℝ} res_{ℂ}^{ℝ} V;
Alt^{p}(V ⊗_{ℝ} ℂ) := Alt_{ℝ}^{p} (res_{ℂ}^{ℝ} V, res_{ℂ}^{ℝ} ℂ) = Alt_{ℂ}^{p} (ℂ ⊗_{ℝ} V, ℂ) .

Denote by J: V → V multiplication with i. It induces an involution of complex vector spaces ℂ ⊗_{ℝ} J: ℂ ⊗_{ℝ} V → ℂ ⊗_{ℝ} V. Denote by (ℂ ⊗_{ℝ} V)^{+} and (ℂ ⊗_{ℝ} V)^{−} the eigenspace for +i and −i. We get canonical identifications of complex vector spaces

ℂ ⊗_{ℝ} V = (ℂ ⊗_{ℝ} V)^{+} ⊕ (ℂ ⊗_{ℝ} V)^{−};
V = (ℂ ⊗_{ℝ} V)^{+};
Alt^{r}(ℂ ⊗_{ℝ} V) = ⨁_{p+q=r} Alt^{p}((ℂ ⊗_{ℝ} V)^{+}) ⊗_{ℂ} Alt^{q}((ℂ ⊗_{ℝ} V)^{−}).

We have already defined the complex vector space Ω^{p}(M) of smooth p-forms on M in Section 1.3. We define the space of (p, q)-forms Ω^{p,q}(M) by

Alt^{p,q}(ℂ ⊗_{ℝ} T_{x}M) := Alt^{p}((ℂ ⊗_{ℝ} T_{x}M)^{+}) ⊗_{ℂ} Alt^{q}((ℂ ⊗_{ℝ} T_{x}M)^{−}),
Ω^{p,q}(M) := C^{∞} (Alt^{p,q}(ℂ ⊗_{ℝ} T M)) .  (11.18)

We obtain a canonical decomposition

Ω^{r}(M) = ⨁_{p+q=r} Ω^{p,q}(M).  (11.19)

We have introduced the exterior differential d in (1.46), the Hodge-star operator ∗ in (1.48) and adjoint of the exterior differential δ in (1.50). The Hodge-star operator ∗^{r}: Ω^{r}(M) → Ω^{n−r}(M) decomposes as

∗^{p,q}: Ω^{p,q}(M) → Ω^{m−p,m−q}(M).

Define

∂_{p,q}: Ω^{p,q}(M) → Ω^{p+1,q}(M);  (11.20)
\overline{∂}_{p,q}: Ω^{p,q}(M) → Ω^{p,q+1}(M)  (11.21)

by the composition

Ω^{p,q}(M) ⟶^{i} Ω^{p+q}(M) ⟶^{d^{p+q}} Ω^{p+q+1}(M) ⟶^{pr} \{ \begin{matrix}Ω^{p+1,q}(M) \\ Ω^{p,q+1}(M) \end{matrix}

where i is the inclusion and pr the projection. Define

Δ_{r} = d^{r−1}δ^{r} + δ^{r+1}d^{r}: Ω^{r}(M) → Ω^{r}(M); (11.22)
□_{p,q} := ∂^{p−1,q}(∂^{p−1,q})^{∗} + (∂^{p,q})^{∗}∂^{p,q}: Ω^{p,q}(M) → Ω^{p,q}(M); (11.23)
\overline{□}_{p,q} = \overline{∂}^{p,q−1}(\overline{∂}^{p,q−1})^{∗} + (\overline{∂}^{p,q})^{∗}\overline{∂}^{p,q}: Ω^{p,q}(M) → Ω^{p,q}(M). (11.24)

The proof of the following lemma can be found in [520, Theorem 3.7 in Chapter I on page 34 and Theorem 4.7 in Chapter V on page 191].

Lemma 11.25. If M is a Kähler manifold, then

∂ ∘ ∂ = 0;
\overline{∂} ∘ \overline{∂} = 0;
d = ∂ + \overline{∂};
Δ = 2 · □ = 2 · \overline{□}.

Definition 11.26. Let M be a Kähler manifold. Define the space of harmonic L^{2}-integrable (p, q)-forms and harmonic r-forms

H_{(2)}^{p,q}(M) := \{ω ∈ Ω^{p,q}(M) | \overline{□}(ω) = 0, ∫_{M} ω ∧ ∗ω < ∞\};
H_{(2)}^{r}(M) := {ω ∈ Ω^{r}(M) | Δ(ω) = 0, ∫_{M} ω ∧ ∗ω < ∞}.

We get the following extension of Theorem 1.57.

Theorem 11.27 (L^{2}-Hodge decomposition for Kähler manifolds). If M is a Kähler manifold, then we get orthogonal decompositions

L^{2}Ω^{r}(M) = H^{r}(M) ⊕ clos(d^{r−1}(Ω_{c}^{r−1}(M))) ⊕ clos(δ^{r+1}(Ω_{c}^{r+1}(M)));
H_{(2)}^{r}(M) = ⨁_{p+q=r} H_{(2)}^{p,q}(M),

where Ω_{c}^{p}(M) ⊂ Ω^{p}(M) is the subspace of p-forms with compact support.

11.2.2 The L^{2}-Lefschetz Theorem

Theorem 11.28 (L^{2}-Lefschetz Theorem). Let M be a (not necessarily compact) Kähler manifold of complex dimension m and real dimension n = 2m. Let ω be its fundamental form. Then the linear map

L^{k}: Ω^{r}(M) → Ω^{r+2k}(M), ϕ ↦ ϕ ∧ ω^{k}

satisfies

(1) L commutes with Δ and d;

(2) L^{k} induces bounded operators denoted in the same way

L^{k}: L^{2}Ω^{r}(M) → L^{2}Ω^{r+2k}(M);
L^{k}: H_{(2)}^{r}(M) → H_{(2)}^{r+2k}(M);

(3) The operators L^{k} of assertion (2) are quasi-isometries, i.e. there is a positive constant C such that C^{−1} · ||ϕ|| ≤ ||L^{k}(ϕ)|| ≤ C · ||ϕ|| holds, and in particular are injective, provided that k satisfies 2r + 2k ≤ n. These operators are surjective, if k satisfies 2r + 2k ≥ n.

Proof. (1) [520, Theorem 4.7 in Chapter V on page 191 and Theorem 4.8 in Chaper V on page 192].

(2) Notice that L^{k} is defined fiberwise, for each x we have a linear map L_{x}^{k}: Alt^{r}(T_{x}M ⊗_{ℝ} ℂ) → Alt^{r+2k}(T_{x}M ⊗_{ℝ} ℂ). The Kähler condition implies that ω_{x}^{m} ≠ 0 for all x ∈ M (see [520, (1.12) on page 158]). The fundamental form ω is parallel with respect to the Levi-Civita connection since the fiber transport t_{w}: T_{x}M → T_{y}M for any path w in M from x to y with respect to the Levi-Civita connection is an isometric ℂ-linear isomorphism. The following diagram commutes

Alt^{r}(T_{x}M ⊗_{ℝ} ℂ) ⟶^{L_{x}^{k}} Alt^{r+2k}(T_{x}M ⊗_{ℝ} ℂ)
t_{w}↓  ^{t_{w}}↓
Alt^{r}(T_{y}M ⊗_{ℝ} ℂ) ⟶^{L_{y}^{k}} Alt^{r+2k}(T_{y}M ⊗_{ℝ} ℂ)

This implies that L^{k} is bounded.

(3) Elementary linear algebra shows L_{x}^{k}: Alt^{r}(ℂ⊗_{ℝ}T_{x}M) → Alt^{r+2k}(ℂ⊗_{ℝ} T_{x}M) is injective for 2r + 2k = n [520, Theorem 3.12 (c) in V.3 on page 182]. Since the source and target are complex vector spaces of the same finite dimension, it is bijective for 2r+2k = n. Because of the identity L_{x}^{k} ∘L_{x}^{l} = L_{x}^{k+l} we conclude that L_{x}^{k} is injective, if k satisfies 2r + 2k ≤ n, and surjective, if k satisfies 2r + 2k ≥ n. This implies that L^{k} is a quasi-isometry and injective if k satisfies 2r + 2k ≤ n. It remains to prove the claim about surjectivity. Because of the factorization

L^{l+k}: Ω^{r−2l}(M) ⟶^{L^{l}} Ω^{r}(M) ⟶^{L^{k}} Ω^{r+2k}(M)

it suffices to prove surjectivity for L^{k} in the case 2r + 2k = n.

Consider the adjoint K_{x}: Alt^{r+2k}(ℂ ⊗_{ℝ} T_{x}M) → Alt^{r}(ℂ ⊗_{ℝ} T_{x}M) of L_{x}^{k}: Alt^{r}(ℂ ⊗_{ℝ} T_{x}M) → Alt^{r+2k}(ℂ ⊗_{ℝ} T_{x}M). Since L_{x}^{k} is surjective, K_{x}^{k} is injective. We get a quasi-isometry K: L^{2}Ω^{r+2k}(M) → L^{2}Ω^{r}(M) which is the adjoint of L^{k}. Since K is injective, L^{k} has dense image. Since L^{k} is a quasi-isometry, L^{k} has closed image and hence is surjective. The proof for H_{(2)}^{r}(M) is analogous. This finishes the proof of Theorem 11.28. ⊓ ⊔

Corollary 11.29. Let M be a closed Kähler manifold of complex dimension m and real dimension n = 2m. Let b_{r}(M) = dim_{ℂ} (H_{(2)}^{r}(M)) = dim_{ℂ}(H_{r}(M; ℂ)) be the ordinary r-th Betti number and define h_{p,q}(M) := dim_{ℂ} (H_{(2)}^{p,q}(M)). Then

(1) b_{r}(M) = ∑_{p+q=r} h_{p,q}(M) for r ≥ 0;

(2) b_{r}(M) = b_{n−r}(M) and h_{p,q}(M) = h_{m−p,m−q}(M) for r, p, q ≥ 0;

(3) h_{p,q}(M) = h_{q,p}(M) for p, q ≥ 0;

(4) b_{r}(M) is even for r odd; (5) h_{1,0}(M) = \frac{b_{1}(M)}{2} and depends only on π_{1}(M);

(6) b_{r}(M) ≤ b_{r+2}(M) for r < m.

The corresponding statements except for (4) hold for the L^{2}-versions b_{r}^{(2)}(M̃) and h^{p,q}(M̃).

Now we can give the proof of Theorem 11.16.

Proof. Because of Poincaré duality (see Theorem 1.35 (3)) it suffices to show H_{(2)}^{r}(M̃) = 0 for r < m. Let ω̃ be the lift of the fundamental form on M to

the universal covering M̃. Then

L^{1} : H_{(2)}^{r}(M̃) → H_{(2)}^{r+2}(M̃), ϕ ↦ ϕ ∧ ω̃

is injective by Theorem 11.28 (3). Consider ϕ ∈ H_{(2)}^{r}(M̃). By assumption

there is η ∈ Ω^{1}(M̃) which satisfies

ω̃ = d(η);
||η||_{x} ≤ C_{1} · d(x, x_{0}) + C_{2}

for positive constants C_{1}, C_{2} > 0 and some point x_{0} ∈ M̃. By the Hodge Decomposition Theorem 11.27 it suffices to show ϕ∧ω̃ ∈ clos (d^{r+1}(Ω_{c}^{r+1}(M̃)))

because then L^{1}(ϕ) = ϕ ∧ ω̃ ∈ H^{r+2}(M̃) implies L^{1}(ϕ) = 0 and thus ϕ = 0.

Let B_{r} be the closed ball of radius r around x_{0}. Choose a smooth function a_{r}: M̃ → [0, 1] together with a constant C_{3} > 0 such that a_{r}(x) = 1 for x ∈ B_{r}, a_{r}(x) = 0 for x ∉ B_{2r} and ||da_{r}||_{x} ≤ \frac{C_{3}}{d(x,x_{0})} for x ∈ B_{2r} − B_{r}. Since

ϕ ∧ a_{r}η lies in Ω_{c}^{r+1}(M̃), it suffices to prove that (−1)^{r} · d(ϕ ∧ a_{r}η) converges to ϕ ∧ ω̃ in the L^{2}-norm.

We have

||ϕ ∧ ω̃ − ϕ ∧ a_{r}ω̃||_{L^{2}}^{2} = ∫M̃ ||ϕ ∧ ω̃ − ϕ ∧ a_{r}ω̃||_{x}^{2} dvol
= ∫M̃(1 − a_{r}(x))^{2} · ||ϕ ∧ ω̃||_{x}^{2} dvol
≤ ∫_{M̃−B_{r}} ||ϕ ∧ ω̃||_{x}^{2} dvol .

Since ϕ ∧ ω̃ is L^{2}-integrable, we get lim_{r→∞} ∫_{M̃−B_{r}} ||ϕ ∧ ω̃||_{x}^{2} dvol = 0 from Lebesgue’s Theorem of majorized convergence. This shows that ϕ ∧ a_{r}ω̃ converges to ϕ ∧ ω̃ in the L^{2}-norm. Since

(−1)^{r} · d(ϕ ∧ a_{r}η) = ϕ ∧ da_{r} ∧ η + ϕ ∧ a_{r}ω̃,

it remains to show that ϕ ∧ da_{r} ∧ η converges to zero in the L^{2}-norm. We get for appropriate constants C_{4}, C_{5} > 0 and r ≥ 1 ∫M̃ ||ϕ ∧ da_{r} ∧ η||_{x}^{2} dvol

≤ ∫_{B_{2r}−B_{r}} ||ϕ ∧ da_{r} ∧ η||_{x}^{2} dvol
≤ ∫_{B_{2r}−B_{r}} C_{4} · ||ϕ||_{x}^{2} · ||da_{r}||_{x}^{2} · ||η||_{x}^{2} dvol
≤ ∫_{B_{2r}−B_{r}} C_{4} · ||ϕ||_{x}^{2} · \frac{C_{3}^{2}}{d(x, x_{0})^{2}} · (C_{1} · d(x, x_{0}) + C_{2})^{2} dvol
≤ C_{5} · ∫_{B_{2r}−B_{r}} ||ϕ||_{x}^{2} dvol .

Since ϕ is L^{2}-integrable, lim_{r→∞} ∫_{B_{2r}−B_{r}} ||ϕ||_{x}^{2} dvol = 0. Hence ϕ ∧ da_{r} ∧ η converges to zero in the L^{2}-norm. This finishes the proof of Theorem 11.16.

⊓ ⊔

11.2.3 Novikov-Shubin Invariants for Kähler Hyperbolic Manifolds

In this subsection we will prove

Theorem 11.30. Let M be a closed Kähler hyperbolic manifold. Then

α_{p}(M̃) = ∞^{+} for p ≥ 1.

Proof. Because of Poincaré duality (see Theorem 2.55 (2)) it suffices to treat the case p > m, if the real dimension of M is 2m.

We will use the analytic definition of the Novikov-Shubin invariants in terms of the analytic spectral density function F_{p}(M̃) (see Definition 2.64). Recall that the analytic and cellular versions agree by Theorem 2.68. We have to find ϵ > 0 such that F_{p}(M̃)(λ) = F_{p}(M̃)(0) holds for 0 ≤ λ ≤ ϵ. We conclude from the Hodge Decomposition Theorem 1.57 that the orthogonal complement of the kernel of d_{min}^{p}: dom(d_{min}^{p}) ⊂ L^{2}Ω^{p}(M̃) → L^{2}Ω^{p+1}(M̃) is clos (δ^{p+1}(Ω_{c}^{p+1}(M̃))). Because of Definition 2.1 and Lemma 2.2 (1) is suffices to find ϵ > 0 such that for all elements ψ in dom(d_{min}^{p}) ∩ clos (δ^{p+1}(Ω_{c}^{p+1}(M̃))) with d(ψ) := d_{min}^{p}(ψ) ∈ dom(δ^{p+1}) we have

||ψ||_{L^{2}} ≤ ϵ · ||d(ψ)||_{L^{2}}.  (11.31)

Put k = p − m. In the sequel C_{1}, C_{2}, … are positive constants which depend only on M (but not on ψ). Let ω̃ ∈ Ω^{2}(M̃) be the lift of the fundamental form ω of M to the universal covering M̃. The map L^{k}: L^{2}Ω^{p−2k}(M̃) → L^{2}Ω^{p}(M̃) sending ϕ to ϕ ∧ ω^{k} is bijective, commutes with Δ and satisfies

C_{1} · ||ϕ|| ≤ ||L^{k}(ϕ)||_{L^{2}} ≤ \frac{1}{C_{1}} · ||ϕ||

by Theorem 11.28. Consider ψ in dom(d_{min}^{p}) ∩ clos (δ^{p+1}(Ω_{c}^{p+1}(M̃))) with

d(ψ) ∈ dom(δ^{p+1}). We can choose ϕ with

ϕ ∈ dom((Δ_{p−2k})_{min});
ψ = L^{k}(ϕ).

Notice that some complication in the proof comes from the fact that d(ϕ) (in contrast to δ ∘ d(ϕ) = Δ(ϕ)) does not lie in the range, where we know that L^{k} is a quasi-isometry. Since L^{k} commutes with Δ and hence with Δ^{1/2} we conclude using partial integration (see Lemma 1.56)

||d(ψ)||_{L^{2}}^{2} = ⟨d(ψ), d(ψ)⟩ = ⟨δ∘d(ψ), ψ⟩ = ⟨Δ(ψ), ψ⟩ = ⟨Δ^{1/2}(ψ), Δ^{1/2}(ψ)⟩

= ⟨Δ^{1/2} ∘ L^{k}(ϕ), Δ^{1/2} ∘ L^{k}(ϕ)⟩ = ⟨L^{k} ∘ Δ^{1/2}(ϕ), L^{k} ∘ Δ^{1/2}(ϕ)⟩ =

||L^{k} ∘ Δ^{1/2}(ϕ)||_{L^{2}}^{2} ≥ C_{1}^{2} · ||Δ^{1/2}(ϕ)||_{L^{2}}^{2} = C_{1}^{2} · ⟨Δ^{1/2}(ϕ), Δ^{1/2}(ϕ)⟩ =

C_{1}^{2} · ⟨Δ(ϕ), ϕ⟩ = C_{1}^{2} · (⟨δ ∘ d(ϕ), ϕ⟩ + ⟨d ∘ δ(ϕ), ϕ⟩)

= C_{1}^{2} · (⟨d(ϕ), d(ϕ)⟩ + ⟨δ(ϕ), δ(ϕ)⟩) = C_{1}^{2} · (||d(ϕ)||_{L^{2}}^{2} + ||δ(ϕ)||_{L^{2}}^{2})

≥ C_{1}^{2} · ||d(ϕ)||_{L^{2}}^{2}.

This shows

||d(ϕ)||_{L^{2}} ≤ \frac{1}{C_{1}} · ||d(ψ)||_{L^{2}}.  (11.32)

Since M is Kähler hyperbolic by assumption, we can find η ∈ Ω^{1}(M̃) with ||η||_{∞} < ∞ and ω̃ = d(η). We have ||ω̃||_{L^{2}} < ∞. Put

θ := (−1)^{p} · ϕ ∧ ω^{k−1} ∧ η;
μ := (−1)^{p} · dϕ ∧ ω^{k−1} ∧ η.

Both forms θ and μ are L^{2}-integrable and

||μ||_{L^{2}} ≤ C_{2} · ||dϕ||_{L^{2}} · ||ω^{k−1}||_{∞} · ||η||_{∞};  (11.33)
ψ = dθ − μ.

We conclude using partial integration (see Lemma 1.56)

||ψ||_{L^{2}}^{2} = ⟨ψ, ψ⟩_{L^{2}} = ⟨ψ, dθ − μ⟩_{L^{2}} = ⟨ψ, dθ⟩_{L^{2}} − ⟨ψ, μ⟩_{L^{2}}
= ⟨δψ, θ⟩_{L^{2}} − ⟨ψ, μ⟩_{L^{2}} = − ⟨ψ, μ⟩_{L^{2}} ≤ ||ψ||_{L^{2}} · ||μ||_{L^{2}}.

This implies

||ψ||_{L^{2}} ≤ ||μ||_{L^{2}}.  (11.34)

If we put ϵ = \frac{C_{2}}{C_{1}} · ||ω^{k−1}||_{∞} · ||η||_{∞}, we get the desired inequality (11.31) from the inequalities (11.32), (11.33) and (11.34). This finishes the proof of Theorem 11.30. ⊓ ⊔ 11.2.4 Non-Vanishing of the Middle L^{2}-Betti Number for Kähler Hyperbolic Manifolds

In this subsection we sketch the proof of the following result. The proof is due to Gromov [236, Section 2] and inspired by [499]. Notice that Theorem 11.14 then follows from Theorem 11.16, Theorem 11.30 and Theorem 11.35 below.

Theorem 11.35. Let M be a closed Kähler hyperbolic manifold of complex dimension m and real dimension n = 2m. Then

H_{(2)}^{p,q}(M̃) ≠ 0 for p + q = m.

Proof. Fix p with 0 ≤ p ≤ m. Consider the following elliptic cochain complex of differential operators of order 1

… ⟶^{\overline{∂}_{p,q−2}} Ω^{p,q−1}(M̃) ⟶^{\overline{∂}_{p,q−1}} Ω^{p,q}(M̃) ⟶^{\overline{∂}_{p,q}} …

We define its (reduced) L^{2}-cohomology as in Subsection 1.4.2 by

Z^{p,q}(M̃) := ker ((\overline{∂}^{p,q}: Ω_{c}^{p,q}(M̃) → L^{2}Ω^{p,q+1}(M̃))_{min});
B^{p,q}(M̃) := im ((\overline{∂}^{p,q−1} : Ω_{c}^{p,q−1}(M̃) → L^{2}Ω^{p,q}(M̃))_{min});
H_{(2)}^{p,q}(M̃) := Z^{p,q}(M̃)/ clos(B^{p,q}(M̃)).

The Hilbert spaces H_{(2)}^{p,q}(M̃) are finitely generated Hilbert N (π_{1}(M))-modules

and isomorphic to H_{(2)}^{p,q}(M̃) since the Laplacian associated to the elliptic com-

plex (Ω^{p,∗}(M̃), \overline{∂}_{p,∗}) is by definition \overline{□}_{p,q} introduced in (11.24). The L^{2}-index of the elliptic complex (Ω^{p,∗}(M̃), \overline{∂}_{p,∗}) is by definition

index^{(2)} (Ω^{p,∗}(M̃), \overline{∂}_{p,∗}) = ∑_{q≥0}(−1)^{q} · dim_{N (π_{1}(M))} (H_{(2)}^{p,q}(M̃)) .

Using the lift ω̃ of the fundamental form ω of M to M̃, Gromov [236, Section 2] constructs a family of elliptic complexes (Ω^{p,∗}(M̃), \overline{∂}_{p,∗}[t]) with t ∈ ℝ, which depends continuously on t and agrees with (Ω^{p,∗}(M̃), \overline{∂}_{p,∗}) for t = 0. Moreover, he shows using a twisted L^{2}-index theorem (see also [361, Theorem 3.6 on page 223]) that for some non-constant polynomial p(t)

index^{(2)} (Ω^{p,∗}(M̃), \overline{∂}_{p,∗}[t]) = p(t).

This implies that there exist arbitrary small t > 0 with the property that at least one of the Hilbert modules H_{(2)}^{p,q} (Ω^{p,∗}(M̃), \overline{∂}_{p,∗}[t]) = ker ((\overline{□}_{p,q}[t])_{min}) is different from zero.

We want to use this fact to show that the assumption H_{(2)}^{p,m−p}(M̃) =

0 leads to a contradiction. We already know that H_{(2)}^{p,q}(M̃) = 0 for q ≠ m − p from Theorem 11.16 and Theorem 11.27. In Theorem 11.30 we have already proved that α_{r}(M̃) = ∞^{+} for all r ≥ 1. Lemma 2.66 (2) implies that α((Δ_{r})_{min} : dom((Δ_{r})_{min}) ⊂ L^{2}Ω^{r}(M̃) → L^{2}Ω^{r}(M̃)) = ∞^{+} for all r ≥ 0. We conclude from Lemma 11.25 that

α ((\overline{□}_{p,q})_{min} : dom((\overline{□}_{p,q})_{min}) ⊂ L^{2}Ω^{p,q}(M̃) → L^{2}Ω^{p,q}(M̃)) = ∞^{+}

for q ≥ 0. Hence there is ϵ > 0 such that for the spectral density function F ((\overline{□}_{p,q})_{min}) (λ) = 0 holds for all q ≥ 0 and λ ≤ ϵ. Since by construction \overline{□}_{p,q}[t] − \overline{□}_{p,q} is a bounded operator with operator norm ≤ C · t for some constant C, this contradicts the existence of arbitrary small t > 0 such that ker ((\overline{□}_{p,q}[t])_{min}) ≠ 0 for some q. This finishes the proof of Theorem 11.35.

⊓ ⊔

For applications in algebraic geometry the following consequence is crucial.

Corollary 11.36. Let M be a closed Kähler hyperbolic manifold of complex dimension m and real dimension n = 2m. Then there exists a non-trivial holomorphic L^{2}-integrable m-form on M̃.

Proof. Choose a non-trivial element ϕ in H^{0,m}(M̃) whose existence follows from Theorem 11.35. Obviously \overline{□}_{0,m}(ϕ) = 0 implies \overline{∂}_{0,m}(ϕ), i.e. ϕ is holomorphic. ⊓ ⊔

11.3 Miscellaneous

One may consider the following more general version of the Singer Conjecture.

Conjecture 11.37. (Singer Conjecture for contractible proper cocompact Poincaré G-CW -complexes). Let X be a finite proper G-CW -complex. Suppose that X is contractible (after forgetting the group action). Suppose that there is an element [X] ∈ H_{dim(X)}^{G}(X; ℚ) for ℚ with the trivial action such that the ℚG-chain map ∩[X]: C^{dim(X)−∗}(X) → C_{∗}(X), which is uniquely defined up to ℚG-chain homotopy, is a ℚG-chain homotopy equivalence. Then

b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(G) = 0 if 2p ≠ dim(X).

This version seems to cut down the assumptions to the absolutely necessary and decisive properties, namely contractibility, Poincaré duality, properness and cocompactness. If M is an aspherical closed orientable manifold, then M̃ with the canonical π_{1}(M)-action satisfies the assumptions of Conjecture 11.37. Hence Conjecture 11.37 implies the Singer Conjecture 11.1 for aspherical closed orientable manifolds. Notice that in the situation of Conjecture 11.37 the equality

b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(G)

follows from Theorem 6.54 (3). If a group G acts properly on a smooth contractible manifold M by orientation preserving diffeomorphisms, the assumption of Conjecture 11.37 are satisfied because there exist smooth equivariant triangulations [273]. Poincaré duality is explained for instance in [345, page 245].

Anderson [5], [6] has constructed simply connected complete Riemannian manifolds with negative sectional curvature such that the space of harmonic forms H_{(2)}^{p}(M) is non-trivial for some p with 2p ≠ dim(M). This manifold M does not admit a cocompact free proper action of a group G by isometries. Hence it yields a counterexample neither to the Singer Conjecture 11.1 nor to Conjecture 11.37. But this example shows the necessity of the condition of cocompactness for the Singer Conjecture 11.1 to be true.

The author does not know a counterexample to the conjecture that for a closed even-dimensional Riemannian manifold M with negative sectional curvature α_{p}(M̃) = ∞^{+} holds for p ≥ 1. This is true for locally symmetric spaces (see Corollary 5.16 (3)), Kähler manifolds (see Theorem 11.14) and for pinched curvature −1 ≤ sec(M) < −(1 − \frac{2}{dim(M)})^{2} (see Theorem 11.6).

Next we mention the work of Davis and Okun [130]. A simplicial complex L is called a flag complex if each finite non-empty set of vertices, which pairwise are connected by edges, spans a simplex of L. To such a flag complex they associate a right-angled Coxeter group W_{L} defined by the following presentation [130, Definition 5.1]. Generators are the vertices v of L. Each generator v satisfies v^{2} = 1. If two vertices v and w span an edge, there is the relation (vw)^{2} = 1. Given a finite flag complex L, Davis and Okun associate to it a finite proper W_{L}-CW -complex Σ_{L}, which turns out to be a model for the classifying space of the family of finite subgroups E(G, FIN) [130, 6.1, 6.1.1 and 6.1.2]. Equipped with a specific metric, Σ_{L} turns out to be non-positive curved in a combinatorial sense, namely, it is a CAT(0)-space [130, 6.5.3]. If L is a generalized rational homology (n − 1)-sphere, i.e. a homology (n − 1)- manifold with the same rational homology as S^{n−1}, then Σ_{L} is a polyhedral homology n-manifold with rational coefficients [130, 7.4]. So Σ_{L} is a reminiscence of the universal covering of a closed n-dimensional manifold with non-positive sectional curvature and fundamental group G. In view of the Singer Conjecture 11.1 the conjecture makes sense that b_{p}^{(2)}(Σ_{L}; N (W_{L})) = 0 for 2p ≠ n provided that the underlying topological space of L is S^{n−1} (or, more generally, that it is a homology (n−1)-sphere) [130, Conjecture 0.4 and 8.1]. Davis and Okun show that the conjecture is true in dimension n ≤ 4 and that it is true in dimension (n + 1) if it holds in dimension n and n is odd [130, Theorem 9.3.1 and Theorem 10.4.1].

A group G is called Kähler group if it is the fundamental group of a closed Kähler manifold. A lot of information about Kähler groups can be found in [3]. For instance if G is a Kähler group with b_{1}^{(2)}(G) ≠ 0, then it contains a subgroup G_{0} of finite index for which there is an epimorphism G_{0} → π_{1}(F_{g}) with finite kernel for some closed orientable surface F_{g} of genus g ≥ 2 [235, 3.1 on page 69]. Thus a Kähler group has b_{1}^{(2)}(G) = 0 or looks like a surface group of genus ≥ 2.

Exercises

11.1. Construct a closed connected Riemannian manifold M of even dimension, whose sectional curvature is non-positive but not identically zero, such that b_{p}^{(2)}(M̃) = 0 holds for all p ≥ 0.

11.2. Show for a flat closed connected Riemannian manifold M that it is det-L^{2}-acyclic and ρ^{(2)}(M̃) = 0.

11.3. Let M be a closed connected 4-manifold. Suppose that there is a fibration F → M → B for closed connected manifolds F and B of dimension 1, 2 or 3. Prove:

(1) b_{0}^{(2)}(M̃) ≠ 0 if and only if F and B have finite fundamental groups. In this case

b_{p}^{(2)}(M̃) = \frac{1}{|π_{1}(M)|}  for p = 0, 4;
b_{p}^{(2)}(M̃) = 0  for p ≠ 0, 2, 4;
b_{2}^{(2)}(M̃) = χ(F) · χ(B) − \frac{2}{|π_{1}(M)|};
χ(M) > 0;

(2) Suppose that π_{1}(F) is finite and π_{1}(B) is infinite or that π_{1}(F) is infinite and π_{1}(B) is finite. Then

b_{p}^{(2)}(M̃) = \frac{χ(F)·χ(B)}{2} for p = 1, 3;
b_{p}^{(2)}(M̃) = 0  for p ≠ 1, 3;
χ(M) ≤ 0;

(3) Suppose that F and B have infinite fundamental groups. Then

b_{p}^{(2)}(M̃) = 0  for p ≠ 2;
b_{p}^{(2)}(M̃) = χ(F) · χ(B) for p = 2;
χ(M) ≥ 0;

(4) Suppose that M is aspherical. Show that χ(M) = χ(B) · χ(F) ≥ 0 and that we have χ(M) > 0 if and only if both F and B carry a hyperbolic structure.

11.4. Let M be a closed connected Riemannian manifold of even dimension such that its sectional curvature satisfies −1 ≤ sec(M) < −(1 − \frac{2}{dim(M)})^{2}.

Show that b_{p}^{(2)}(M̃) = 0 for 2p ≠ dim(M) and α_{p}(M̃) = ∞^{+} for p ≥ 1 holds.

11.5. Let M be a Kähler manifold and N ⊂ M be a complex submanifold. Show that N inherits the structure of a Kähler manifold. Show the analogous statement for Kähler hyperbolic instead of Kähler.

11.6. Let M be a Kähler manifold of complex dimension m. Let ω be its fundamental form and dvol be its volume form. Show

m! · dvol = ω^{m},

where ω^{m} denotes the m-fold wedge product of ω with itself.

11.7. Let M be a Kähler hyperbolic manifold. Show that its universal covering does not contain a closed complex submanifold of positive dimension.

11.8. Show that the torus T^{2n} carries the structure of a Kähler manifold but not of a Kähler hyperbolic manifold.

11.9. Let M be a Kähler manifold. Show that there is an isometric ℂisomorphism from H^{p,q}(M̃) to the complex dual of H^{q,p}(M̃).

11.10. Suppose that the group G contains a subgroup G_{0} of finite index for which there is an epimorphism G_{0} → π_{1}(F_{g}) with finite kernel for some closed connected orientable surface F_{g} of genus g ≥ 2. Show that b_{1}^{(2)}(G) > 0.

11.11. A symplectic form ω on a smooth manifold of even dimension n = 2m is a 2-form ω ∈ Ω^{2}(M) such that ω_{x}^{m} ≠ 0 for all x ∈ M and d(ω) = 0. Suppose that M comes with a complex structure. We call ω compatible with the complex structure if ω(Jv, Jw) = ω(v, w) holds, where J is multiplication with i. Show that a Kähler structure on a complex manifold is the same as a symplectic structure which is compatible with the complex structure.

12. The Zero-in-the-Spectrum Conjecture

Introduction

In this section we deal with the zero-in-the-spectrum Conjecture which to the author’s knowledge appears for the first time in Gromov’s article [233, page 120].

Conjecture 12.1 (Zero-in-the-spectrum Conjecture). Let M̃ be a complete Riemannian manifold. Suppose that M̃ is the universal covering of an aspherical closed Riemannian manifold M (with the Riemannian metric coming from M). Then for some p ≥ 0 zero is in the spectrum of the minimal closure

(Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(M̃) → L^{2}Ω^{p}(M̃)

of the Laplacian acting on smooth p-forms on M̃.

Remark 12.2. Lott [318, page 347] gives five versions of this conjecture, stated as a question, where the conditions on M̃ are varied. Namely,

(1) M̃ is a complete Riemannian manifold;

(2) M̃ is a complete Riemannian manifold with bounded geometry, i.e. the injectivity radius is positive and the sectional curvature is pinched between −1 and 1;

(3) M̃ is a complete Riemannian manifold, which is uniformly contractible, i.e. for all r > 0 there is an R(r) > 0 such that for all m ∈ M the open ball B_{r}(m) around m of radius r is contractible within B_{R(r)}(m);

(4) M̃ is the universal covering of a closed Riemannian manifold M (with the Riemannian metric coming from M);

(5) M̃ is the universal covering of an aspherical closed Riemannian manifold M (with the Riemannian metric coming from M).

Notice that version (5) agrees with Conjecture 12.1. Meanwhile there are counterexamples to version (4) due to Farber and Weinberger [187] which also yield counterexamples to versions (1) and (2). To the authors knowledge there are no counterexamples known to versions (3) and (5) (at the time of writing).

We will give a purely algebraic reformulation of the zero-in-the-spectrum Conjecture 12.1 in Section 12.1. Some evidence for the zero-in-the-spectrum Conjecture 12.1 is presented in Section 12.2. Most of the positive results are based on direct computations or on appropriate index theorems applied to pertubations of invertible operators. We deal with the counterexamples of Farber and Weinberger in the non-aspherical case in Section 12.3.

To understand this chapter some knowledge about Chapters 1 and 2 is required.

12.1 An Algebraic Formulation of the Zero-in-the-Spectrum Conjecture

Next we reformulate the zero-in-the-spectrum Conjecture 12.1 into a purely algebraic statement, namely, that for an aspherical closed manifold M with fundamental group π = π_{1}(M) at least one of the homology groups H_{p}^{π}(M̃; N (π)) is different from zero. Recall that for a G-CW -complex X the N (G)-module H_{p}^{G}(X; N (G)) is defined by H_{p}(N (G) ⊗_{ℤG} C_{∗}(X)), where N (G) is just viewed as a ring, no topology enters. Analogously define H_{p}^{G}(X; C_{r}^{∗}(G)), where C_{r}^{∗}(G) is the reduced C^{∗}-algebra of G.

Lemma 12.3. Let X be a free G-CW -complex of finite type. Let n be a nonnegative integer or ∞. Then the following assertions are equivalent, where in assertion (5) we assume that X is a cocompact free proper G-manifold with G-invariant Riemannian metric.

(1) We have

b_{p}^{(2)}(X; N (G)) = 0  for p ≤ n;
α_{p}(X; N (G)) = ∞^{+}  for p ≤ n + 1;

(2) H_{p}^{G}(X; N (G)) vanishes for p ≤ n;

(3) H_{p}^{G}(X; C_{r}^{∗}(G)) vanishes for p ≤ n;

(4) For each p ≤ n zero is not in the spectrum of the combinatorial Laplacian Δ_{p}: C_{p}^{(2)}(X) → C_{p}^{(2)}(X), or, equivalently, Δ_{p} is invertible;

(5) For each p ≤ n zero is not in the spectrum of the minimal closure

(Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(X) → L^{2}Ω^{p}(X)

of the Laplacian acting on smooth p-forms on X.

Proof. (1) ⇒ (5) We conclude from Lemma 2.66 (2) and Theorem 2.68 that assertion (1) is equivalent to the condition that for each p ≤ n there exists ϵ_{p} > 0 such that the spectral density function F_{p}^{Δ} of (Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(X) → L^{2}Ω^{p}(X) satisfies F_{p}^{Δ}(λ) = 0 for λ ≤ ϵ_{p}. This is equivalent to the existence of ϵ_{p} > 0 such that

ϵ_{p} · ||ω||_{L^{2}} ≤ ||(Δ_{p})_{min}(ω)||_{L^{2}} for ω ∈ dom ((Δ_{p})_{min}) .

We conclude that (Δ_{p})_{min} is injective and has closed image. Since (Δ_{p})_{min} is selfadjoint (see Lemma 1.70 (1)), we have im ((Δ_{p})_{min})^{⊥} = ker ((Δ_{p})_{min}) = 0. Hence (Δ_{p})_{min} : dom ((Δ_{p})_{min}) → L^{2}Ω^{p}(X) is bijective. One easily checks that the inverse is a bounded operator. Hence zero is not in the spectrum of (Δ_{p})_{min} for p ≤ n.

(5) ⇒ (1) By assumption we can find for each p ≤ n a bounded operator S_{p}: L^{2}Ω^{p}(X) → L^{2}Ω^{p}(X) with im(S_{p}) = dom ((Δ_{p})_{min}) such that S_{p} ∘ (Δ_{p})_{min} = id_{dom((Δ_{p})_{min})} and (Δ_{p})_{min} ∘ S_{p} = id_{L^{2}Ω^{p}(X)}. Consider ω ∈ dom ((Δ_{p})_{min}). Choose ϕ ∈ L^{2}Ω^{p}(X) with S_{p}(ϕ) = ω. Denote by ||S_{p}||_{∞} the operator norm of S_{p}. We get

||(Δ_{p})_{min}(ω)||_{L^{2}} = ||(Δ_{p})_{min} ∘ S_{p}(ϕ)||_{L^{2}} = ||ϕ||_{L^{2}}
≥ ||S_{p}||_{∞}^{−1} · ||S_{p}(ϕ)||_{L^{2}} = ||S_{p}||_{∞}^{−1} · ||ω||_{L^{2}}.

This implies F_{p}^{Δ}(λ) = 0 for λ < ||S_{p}||_{L^{2}}^{−1}. Hence (1) is true.

(1) ⇔ (4) Its proof is analogous but simpler because the combinatorial Laplacian is a bounded everywhere defined operator.

(1) ⇔ (2) This follows from Lemma 6.98.

(4) ⇒ (3) Let f: ℂG^{m} → ℂG^{n} be a ℂG-linear map. Let A be the (m, n)- matrix describing f. Let A^{∗} be obtained from A by transposing and applying to each entry the standard involution ℂG → ℂG which sends ∑_{g∈G} λ_{g} · g to ∑_{g∈G} \overline{λ_{g}} · g^{−1}. Define the adjoint f^{∗}: ℂG^{n} → ℂG^{m} of f to be the ℂGlinear map given by A^{∗}. Recall that C_{∗}(X) is the cellular ℤG-chain complex and the finitely generated Hilbert N (G)-chain complex C_{∗}^{(2)}(X) is defined by l^{2}(G) ⊗_{ℤG} C_{∗}(X). Define Δ_{p}^{c}: C_{p}(X) → C_{p}(X) to be the ℂG-linear map c_{p+1}c_{p+1}^{∗} + c_{p}^{∗}c_{p}, where the adjoints of c_{p} and c_{p+1} are taken with respect to a cellular basis. (The cellular basis is not quite unique but the adjoints are). Then the combinatorial Laplacian Δ_{p}: C_{p}^{(2)}(X) → C_{p}^{(2)}(X) is id_{l^{2}(G)} ⊗_{ℤG} Δ_{p}^{c}. We conclude from (6.22) that Δ_{p} is invertible if and only if id_{N (G)} ⊗_{ℤG} Δ_{p}^{c}: N (G) ⊗_{ℤG} C_{p}(X) → N (G) ⊗_{ℤG} C_{p}(X) is bijective. We will need the following standard result about C^{∗}-algebras. If A ⊂ B is an inclusion of C^{∗}-algebras and a ∈ A is invertible in B, then a is already invertible in A [282, Proposition 4.1.5 on page 241]. If we apply this to A = M_{l}(C_{r}^{∗}(G)) and B = M_{l}(N (G)), we get that id_{C_{r}^{∗}(G)} ⊗_{ℤG} Δ_{p}^{c}: C_{r}^{∗}(G) ⊗_{ℤG} C_{p}(X) → C_{r}^{∗}(G) ⊗_{ℤ} C_{p}(X) is invertible for p ≤ n. One easily checks c_{p} ∘ Δ_{p}^{c} = Δ_{p−1}^{c} ∘ c_{p} for all p. This implies that id_{C_{r}^{∗}(G)} ⊗_{ℤG} c_{p} ∘ (id_{C_{r}^{∗}(G)} ⊗_{ℤG} Δ_{p}^{c})^{−1} = (id_{C_{r}^{∗}(G)} ⊗_{ℤG} Δ_{p−1}^{c})^{−1} ∘ id_{C_{r}^{∗}(G)} ⊗_{ℤG} c_{p} holds for p ≤ n. For p ≤ n consider x ∈ ker (id_{C_{r}^{∗}(G)} ⊗_{ℤG} c_{p}). Then x lies in the image of id_{C_{r}^{∗}(G)} ⊗_{ℤG} c_{p+1} because of the easily verified equation

id_{C_{r}^{∗}(G)} ⊗_{ℤG} c_{p+1} ∘ id_{C_{r}^{∗}(G)} ⊗_{ℤG} c_{p+1}^{∗} ∘ (id_{C_{r}^{∗}(G)} ⊗_{ℤG} Δ_{p}^{c})^{−1} (x) = x.

This proves H_{p}^{G}(X; C_{r}^{∗}(G)) = 0 for p ≤ n.

(3) ⇒ (2) Since C_{r}^{∗}(G)⊗_{ℤG}C_{∗}(X) is a projective C_{r}^{∗}(G)-chain complex whose modules vanish in negative dimensions, (3) implies that it is C_{r}^{∗}(G)-chain homotopy equivalent to a projective C_{r}^{∗}(G)-chain complex P_{∗} with P_{p} = 0 for p ≤ n. Since N (G) ⊗_{ℤG} C_{∗}(X) is N (G)-chain homotopy equivalent to N (G)⊗_{C_{r}^{∗}(G)} P_{∗}, assertion (2) follows. This finishes the proof of Lemma 12.3.

⊓ ⊔

Remark 12.4. Let M be an aspherical closed Riemannian manifold with fundamental group π. We conclude from Lemma 12.3 that M̃ satisfies the zero-in-the-spectrum Conjecture 12.1 if and only if H_{p}^{π}(Eπ; N (π)) ≠ 0 for at least one p ≥ 0. In particular the answer to the problem, whether the universal covering of a closed connected Riemannian manifold M satisfies the zero-in-the-spectrum Conjecture 12.1, depends only on the homotopy type of M and not on its Riemannian metric.

So one may ask the question for which groups G, whose classifying space BG possesses a finite CW -model, finite dimensional CW -model or CW - model of finite type, H_{p}^{π}(Eπ; N (π)) = 0 holds for p ≥ 0. In comparison with the zero-in-the-spectrum Conjecture 12.1, the condition Poincaré duality is dropped but a finiteness condition and the contractibility of Eπ remain. Without the finiteness assumption such groups G exists. For example the group G = ∏_{n=1}^{∞}(ℤ ∗ ℤ) satisfies H_{p}^{G}(EG; N (G)) = 0 for all p ≥ 0 (see Lemma 12.11 (5))).

12.2 Survey on Positive Results about the Zero-in-the-Spectrum Conjecture

In this section we present some positive results on the zero-in-the-spectrum Conjecture 12.1. Notice that it suffices to consider closed connected manifolds which are orientable. If M is not orientable, one can pass to the orientation covering \overline{M} → M and use the fact that b_{p}^{(2)}(M̃; N (π_{1}(M))) = 2 · b_{p}^{(2)}(˜\overline{M}; N (π_{1}(\overline{M}))) and α(M̃; N (π_{1}(M))) = α(˜\overline{M}; N (π_{1}(\overline{M}))) holds (see Theorem 1.35 (9) and Theorem 2.55 (6)).

12.2.1 The Zero-in-the-Spectrum Conjecture for Low-Dimensional Manifolds

The zero-in-the-spectrum Conjecture 12.1 is true for universal coverings of closed 2-dimensional manifolds by Example 1.36, actually zero is contained in the spectrum of the Laplacian in dimension 1.

We have already explained in Section 4.1 that the Sphere Theorem [252, Theorem 4.3] implies that a closed connected orientable 3-manifold is aspherical if and only if it is homotopy equivalent to a closed connected orientable irreducible 3-manifold M with infinite fundamental group. If M is non-exceptional, then M̃ satisfies the zero-in-the-spectrum Conjecture 12.1 by Theorem 4.2 (2) and (4). More precisely, zero is contained in the spectrum of the Laplacian in dimension 1 and 2. In other words, if Thurston’s Geometrization Conjecture is true, then the zero-in-the-spectrum Conjecture 12.1 holds for universal coverings of aspherical closed 3-manifolds.

Lemma 12.5. Let M be a closed connected 4-dimensional manifold with fundamental group π. Then zero is not the spectrum of (Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(M̃) → L^{2}Ω^{p}(M̃) for all p ≥ 0 if and only if H_{p}^{π}(Eπ; N (π)) = 0 for p = 0, 1 and χ(M) = 0.

Proof. We conclude from Lemma 12.3 that zero is not the spectrum of (Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(M̃) → L^{2}Ω^{p}(M̃) for all p ≥ 0 if and only if b_{p}^{(2)}(M̃) = 0 and α_{p+1}(M̃) = ∞^{+} for p ≥ 0. Since the classifying map f: M → Bπ for π = π_{1}(M) is 2-connected, we have b_{p}^{(2)}(M̃) = b_{p}^{(2)}(π) for p = 0, 1 and α_{p}(M̃) = α_{p}(π) for p ≤ 2 (see Theorem 1.35 (1) and Theorem 2.55 (1)). Now the claim follows from Lemma 12.3, Poincaré duality (see Theorem 1.35 (3) and Theorem 2.55 (2)) and the Euler-Poincaré formula (see Theorem 1.35 (2)). ⊓ ⊔

The possible geometries on 4-manifolds have been classified by Wall [512]. If a closed connected 4-manifold possesses a geometry, one can compute the L^{2}-Betti numbers and get enough information about their Novikov-Shubin invariants to conclude that M satisfies the zero-in-the-spectrum Conjecture 12.1 [329, Theorem 4.3] and [318, Proposition 18]. Let M be a 4-manifold with a complex structure. Then its universal covering satisfies the zero-in-the-spectrum Conjecture 12.1. Otherwise we would get a contradiction. Namely, a closed 4-manifold with complex structure, which satisfies b_{p}^{(2)}(M̃) = 0 for p ≥ 0 and hence χ(M) = sign(M) = 0 by Theorem 1.35 (2) and Lemma 7.22, possesses a geometry [512, page 148-149].

12.2.2 The Zero-in-the-Spectrum Conjecture for Locally Symmetric Spaces

The zero-in-the-spectrum Conjecture 12.1 holds for the universal covering M̃ of aspherical closed locally symmetric spaces M by Corollary 5.16 (2). Actually zero is in the spectrum of the Laplacian in dimension p, if \frac{dim(M)−1}{2} ≤ p ≤ \frac{dim(M)+1}{2}.

12.2.3 The Zero-in-the-Spectrum Conjecture for Kähler Hyperbolic Manifolds

If M is a closed Kähler hyperbolic manifold (see Definition 11.12), then zero is in the spectrum of the Laplacian on the universal covering in dimension dim(M)/2 and in particular M satisfies the zero-in-the-spectrum Conjecture 12.1 by Theorem 11.14.

12.2.4 The Zero-in-the-Spectrum Conjecture for HyperEuclidean Manifolds

A complete Riemannian manifold N is called hyperEuclidean if there is a proper distance non-increasing map F: N → ℝ^{n} of nonzero degree for n = dim(N). Proper means that the preimage of a compact set is again compact. The degree of a proper map f: N → P of oriented manifolds of the same dimension n is the integer deg(f) for which the map induced by f on the n-th homology with compact support sends the fundamental class [N] to the fundamental class [P] multiplied by deg(f). Suppose that f: N → P is smooth and proper and y is a regular value of f, i.e. T_{x}f: T_{x}N → T_{y}P is bijective for all x ∈ f^{−1}(y). Then f^{−1}(y) consists of finitely many points. Define ϵ(x) for x ∈ f^{−1}(y) to be 1 if T_{x}f is orientation preserving and to be −1 otherwise. Then deg(f) = ∑_{x∈f^{−1}(y)} ϵ(x).

Any simply connected Riemannian manifold with non-positive sectional curvature is hyperEuclidean, namely, take f to be the inverse of the exponential map exp_{y}: T_{y}M → M for some point y ∈ N.

Theorem 12.6. (1) Let N be a complete Riemannian manifold which is hyperEuclidean. Then zero is in the spectrum of (Δ_{p})_{min} if \frac{dim(N)−1}{2} ≤ p ≤ \frac{dim(N)+1}{2};

(2) Let M be a closed Riemannian manifold with non-positive sectional curvature. Then the universal covering M̃ satisfies the zero-in-the-spectrum Conjecture 12.1.

Proof. (1) We only sketch the proof, more details can be found in [233] and [318, Proposition 7]. Suppose that n = dim(N) is even and (1) is not true. This leads to a contradiction by the following argument. Take a non-trivial complex vector bundle over ℝ^{n} with fixed trivialization at infinity and pull it back via the composition of f with ϵ · id: ℝ^{n} → ℝ^{n}. Then the pullback F_{ϵ} carries an ϵ-flat connection which is trivial at infinity. The signature operator is invertible, since we assume that zero is not in the spectrum of (Δ_{p})_{min} for all p. If we twist the signature operator with this ϵ-flat connection, then for small enough ϵ this pertubation is still invertible and hence its index is zero. On the other hand a relative index theorem [239, Proposition 4.13 on page 121] shows that the index of the pertubated operator is non-zero for small ϵ. Here one needs the fact that the vector bundle F is non-trivial and f has degree different from zero. The case, where n is odd, is reduced to the case, where n is even, by considering N × ℝ.

(2) This follows from (1) since M̃ is hyperEuclidean. ⊓ ⊔

12.2.5 The Zero-in-the-Spectrum Conjecture and the Strong Novikov Conjecture

The strong Novikov Conjecture says for a group G that the so called assembly map a: K_{p}(BG) → K_{p}^{top}(C_{r}^{∗}(G)) from the topological K-homology of BG to the topological K-theory of the reduced group C^{∗}-algebra C_{r}^{∗}(G) of G is rationally injective for all p ∈ ℤ. For more information about this conjecture we refer for instance to [199]. Notice that the Baum-Connes Conjecture implies the strong Novikov Conjecture.

Theorem 12.7. Suppose that M is an aspherical closed Riemannian manifold. If its fundamental group π = π_{1}(M) satisfies the strong Novikov Conjecture, then M̃ satisfies the zero-in-the-spectrum Conjectur 12.1. Actually zero is in the spectrum of the Laplacian on M̃ in dimension p if \frac{dim(M)−1}{2} ≤ p ≤ \frac{dim(M)+1}{2}.

Proof. We give a sketch of the proof. More details can be found in [318, Corollary 4]. We only explain that the assumption that in every dimension zero is not in the spectrum of the Laplacian on M̃, yields a contradiction in the case that n = dim(M) is even. Namely, this assumption implies that the index of the signature operator twisted with the flat bundle M̃×_{π}C_{r}^{∗}(π) → M in K_{0}(C_{r}^{∗}(π)) is zero. This index is the image of the class [D] defined by the signature operator in K_{0}(Bπ) under the assembly a: K_{0}(Bπ) → K_{0}(C_{r}^{∗}(π)). Since by assumption the assembly map is injective, this implies [D] = 0 in K_{0}(Bπ). Notice that M is aspherical by assumption and hence M = Bπ. The homological Chern character defines an isomorphism

ℚ ⊗_{ℤπ} K_{0}(Bπ) = ℚ ⊗_{ℤπ} K_{0}(M) → H_{ev}(M; ℚ) = ⨁_{p≥0} H_{2p}(M; ℚ),

which sends D to the Poincaré dual L(M) ∩ [M] of the Hirzebruch L-class L(M) ∈ H^{ev}(M; ℚ). This implies that L(M)∩[M] = 0 and hence L(M) = 0. This contradicts the fact that the component of L(M) in H^{0}(M; ℚ) is 1. ⊓ ⊔

12.2.6 The Zero-in-the-Spectrum Conjecture and Finite Asymptotic Dimension

A metric space X has finite asymptotic dimension if there exists an integer n such that for any r > 0 there is a covering X = ⋃_{i∈I} C_{i} together with a constant D such that the diameter of the subset C_{i} ⊂ X is bounded by D for all i ∈ I and a ball of radius r intersects at most (n + 1) of the sets C_{i} nontrivially. The smallest such n is called the asymptotic dimension. This is the coarse geometric analog of the notion of finite covering dimenson in topology. The following theorem is due to Yu [528, Theorem 1.1 and Corollary 7.4]. Its proof is based on the fact that the coarse Baum-Connes Conjecture holds for proper metric spaces with finite asymptotic dimension [528, Corollary 7.1] and coarse index theory.

Theorem 12.8. (1) Let M be a uniformly contractible Riemannian manifold without boundary which has finite asymptotic dimension. Then zero is in the spectrum of (Δ_{p})_{min} for at least one p ≥ 0;

(2) Let G be a group such that there is a finite model for BG and G regarded as a metric space with respect to the word-length metric has finite asymptotic dimension. Then G satisfies the strong Novikov Conjecture;

(3) Let M be an aspherical closed manifold. Suppose that π_{1}(M) regarded as a metric space with respect to the word-length metric has finite asymptotic dimension. Then the universal covering M̃ satisfies the zero-in-the-spectrum Conjecture 12.1.

If G is a finitely generated group which is word-hyperbolic in the sense of [234], then G regarded as a metric space with respect to the word-length metric has finite asymptotic dimension [237, Remark on page 31]. A torsionfree word-hyperbolic group has a finite model for BG given by the Rips complex [65, Corollary 3.26 in III. Γ on page 470] and hence satisfies the assumptions of Theorem 12.8 (2).

If the finitely generated group G regarded as a metric space with respect to the word-length metric has finite asymptotic dimension, then any finitely generated subgroup H ⊂ G has the same property [528, Proposition 6.2].

Meanwhile Gromov has announced that he can construct groups with finite models for BG such that G as a metric space with respect to the wordlength metric does not have finite asymptotic dimension.

12.3 Counterexamples to the Zero-in-the-Spectrum Conjecture in the Non-Aspherical Case

For some time it was not known whether the zero-in-the-spectrum Conjecture 12.1 is true if one drops the condition aspherical. Farber and Weinberger [187] gave the first example of a closed Riemannian manifold for which zero is not in the spectrum of the minimal closure (Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(M̃) → L^{2}Ω^{p}(M̃) of the Laplacian acting on smooth p-forms on M̃ for each p ≥ 0. We will present a short proof of a more general result following ideas of Higson, Roe and Schick [258] which also were obtained independently by the author. We want to emphasize that the key idea is due to Farber and Weinberger [187]. Namely, they considered H_{p}^{G}(X̃; C_{r}^{∗}(G; ℝ)) instead of H_{p}^{G}(X̃; N (G)), where C_{r}^{∗}(G; ℝ) is the reduced real group C^{∗}-algebra. This is possible in the context of the zero-in-the-spectrum Conjecture 12.1 since M_{l}(C_{r}^{∗}(G; ℝ)) is division closed in M_{l}(N (G; ℝ)) (see [282, Proposition 4.1.5 on page 241]). The crucial observation is that for any invertible element in M_{l}(C_{r}^{∗}(G; ℝ)) we can find an element B ∈ M_{l}(ℚG), which is arbitrary close to A in the C^{∗}- norm and hence invertible in M_{l}(C_{r}^{∗}(G; ℝ)). (This is not true for N (G; ℝ) with its C^{∗}-norm because C_{r}^{∗}(G; ℝ) ≠ N (G; ℝ) in general). We have used this fact at several places before when we applied index theorems to small pertubations of invertible operators of Hilbert modules over C^{∗}-algebras.

Theorem 12.9. Let G be a finitely presented group. Then the following assertions are equivalent:

(1) We have H_{p}^{G}(EG; N (G)) = 0 for p = 0, 1, 2;

(2) We have H_{p}^{G}(EG; C_{r}^{∗}(G)) = 0 for p = 0, 1, 2;

(3) There is a finite 3-dimensional CW -complex X with fundamental group π_{1}(X) = G such that H_{p}^{G}(X̃; C_{r}^{∗}(G)) = 0 for p ≥ 0;

(4) There is a finite 3-dimensional CW -complex X with fundamental group π_{1}(X) = G such that H_{p}^{G}(X̃; N (G)) = 0 for p ≥ 0;

(5) For any integer n ≥ 6 there is a connected closed orientable Riemannian manifold M with fundamental group π_{1}(M) = G of dimension n such that H_{p}^{G}(M̃; N (G)) = 0 for p ≥ 0;

(6) For any integer n ≥ 6 there is a connected closed orientable Riemannian manifold M of dimension n with fundamental group π_{1}(M) = G such that for each p ≥ 0 zero is not in the spectrum of (Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(M̃) → L^{2}Ω^{p}(M̃).

Proof. (1) ⇒ (2) This follows from Lemma 12.3.

(2) ⇒ (3) Since G is finitely presented, we can find a CW -model for BG whose 2-skeleton is finite. By assumption H_{p}^{G}(EG; C_{r}^{∗}(G)) = 0 for p = 0, 1, 2. Let C_{r}^{∗}(G; ℝ) be the reduced real group C_{r}^{∗}-algebra of G. Since C_{r}^{∗}(G) = ℂ ⊗_{ℝ} C_{r}^{∗}(G; ℝ), we conclude H_{p}^{G}(EG; C_{r}^{∗}(G; ℝ)) = 0 for p = 0, 1, 2. Let Z be the 2-skeleton of BG. Then we obtain an exact sequence of C_{r}^{∗}(G; ℝ)-modules 0 → H_{2}^{G}(Z̃; C_{r}^{∗}(G; ℝ)) → C_{r}^{∗}(G; ℝ) ⊗_{ℤG} C_{2}(Z̃) → C_{r}^{∗}(G; ℝ) ⊗_{ℤG} C_{1}(Z̃) → C_{r}^{∗}(G; ℝ)⊗_{ℤG} C_{0}(Z̃) → 0. Since the C_{r}^{∗}(G; ℝ)-module C_{r}^{∗}(G; ℝ)⊗_{ℤG} C_{p}(Z̃) is finitely generated free for p = 0, 1, 2, the C_{r}^{∗}(G; ℝ)-module H_{2}^{G}(Z̃; C_{r}^{∗}(G; ℝ)) becomes finitely generated free after taking the direct sum with C_{r}^{∗}(G; ℝ)^{l} for an appropriate integer l ≥ 0. Put Y = Z ∨ ⋁_{i=1}^{l} S^{2}. Then Y is a finite 2dimensional CW -complex with π_{1}(Y) = G such that H_{p}^{G}(Ỹ; C_{r}^{∗}(G)) vanishes for p = 0, 1 and is a finitely generated free C_{r}^{∗}(G)-module for p = 2.

The universal coefficient spectral sequence converges to H_{p+q}^{G}(Ỹ; C_{r}^{∗}(G; ℝ)) and its E^{2}-term is E_{p,q}^{2} = Tor_{p}^{ℤG}(H_{q}(Ỹ); C_{r}^{∗}(G; ℝ)). Since H_{1}(Ỹ) = 0, it gives an exact sequence of C_{r}^{∗}(G; ℝ)-modules

C_{r}^{∗}(G; ℝ) ⊗_{ℤG} H_{2}(Ỹ) ⟶^{ψ} H_{2}^{G}(Ỹ; C_{r}^{∗}(G; ℝ)) → H_{2}^{G}(EG; C_{r}^{∗}(G; ℝ)) → 0,

where ψ is the obvious map. Since H_{2}^{G}(EG; C_{r}^{∗}(G; ℝ)) = 0 by assumption, ψ: C_{r}^{∗}(G; ℝ) ⊗_{ℤG} H_{2}(Ỹ) → H_{2}^{G}(Ỹ; C_{r}^{∗}(G; ℝ)) is surjective.

Let ϕ: H_{2}(Ỹ) → C_{r}^{∗}(G; ℝ)⊗_{ℤG} H_{2}(Ỹ) be the ℤG-map sending u to 1⊗_{ℤG} u. Next we show that the image of ψ∘ϕ: H_{2}(Ỹ) → H_{2}^{G}(Ỹ; C_{r}^{∗}(G; ℝ)) contains a C_{r}^{∗}(G; ℝ)-basis. Fix a C_{r}^{∗}(G; ℝ)-basis {b_{1}, b_{2}, …, b_{l}} of H_{2}^{G}(Ỹ; C_{r}^{∗}(G; ℝ)). Since ψ is surjective, we can find elements c_{i,j} ∈ C_{r}^{∗}(G; ℝ) and h_{i,j} ∈ H_{2}(Ỹ) such that

b_{i} = ∑_{j_{i}=1}^{s_{i}} c_{i,j_{i}} · ψ ∘ ϕ(h_{i,j_{i}})

holds for i = 1, 2, …, l.

Let μ: ⨁_{i=1}^{l} C_{r}^{∗}(G; ℝ)^{s_{i}} → M_{l}(C_{r}^{∗}(G; ℝ)) be the C_{r}^{∗}(G; ℝ)-linear map which is uniquely determined by the property

∑_{k=1}^{l} (μ(x_{1}, …, x_{l}))_{i,k} · b_{k} = ∑_{j_{i}=1}^{s_{i}} x_{i,j_{i}} · ψ ∘ ϕ(h_{i,j_{i}}) for i = 1, 2, …, l.

We equip C_{r}^{∗}(G; ℝ)^{m} with the norm ||(x_{1}, …, x_{m})||_{1} = ∑_{i=1}^{m} ||x_{i}||_{C_{r}^{∗}(G;ℝ)} and M_{l}(C_{r}^{∗}(G; ℝ)) with the standard C^{∗}-norm. Since μ is C_{r}^{∗}(G; ℝ)-linear, it is continuous with respect to these norms by the following calculation. Here e_{i,j_{i}} is the obvious element of the standard basis for the source of μ and C := max{||μ(e_{i,j_{i}})||_{M_{l}(C_{r}^{∗}(G;ℝ))} | 1 ≤ j_{i} ≤ s_{i}, 1 ≤ i ≤ l}

||μ(x)||_{M_{l}(C_{r}^{∗}(G;ℝ))} ≤ ||∑_{i=1}^{l} ∑_{j_{i}=1}^{s_{i}} x_{i,j_{i}} · μ(e_{i,j_{i}})||_{M_{l}(C_{r}^{∗}(G;ℝ))}

≤ ∑_{i=1}^{l} ∑_{j_{i}=1}^{s_{i}} ||x_{i,j_{i}} · μ(e_{i,j_{i}})||_{M_{l}(C_{r}^{∗}(G;ℝ))}

≤ ∑_{i=1}^{l} ∑_{j_{i}=1}^{s_{i}} ||x_{i,j_{i}}||_{C_{r}^{∗}(G;ℝ)} · ||μ(e_{i,j_{i}})||_{M_{l}(C_{r}^{∗}(G;ℝ))}

≤ ∑_{i=1}^{l} ∑_{j_{i}=1}^{s_{i}} ||x_{i,j_{i}}||_{C_{r}^{∗}(G;ℝ)} · C
= C · ||x||_{1}.

The set of invertible matrices Gl_{l}(C_{r}^{∗}(G; ℝ)) ⊂ M_{l}^{∗}(C_{r}^{∗}(G; ℝ)) is open. This standard fact for C^{∗}-algebras follows from the observation that for an element A in a C^{∗}-algebra with norm ||A|| < 1 the sequence ∑_{n≥0} A^{n} converges to an inverse of 1 − A. With respect to the C^{∗}-norm ℚG ⊂ C_{r}^{∗}(G; ℝ) is dense. Since μ sends ((c_{1,j_{1}})_{j_{1}=1}^{s_{1}}, …, (c_{l,j_{l}})_{j_{l}=1}^{s_{l}}) to the identity matrix, we can find q := ((q_{1,j_{1}})_{j_{1}=1}^{s_{1}}, …, (q_{l,j_{l}})_{j_{l}=1}^{s_{l}}) in ⨁_{i=1}^{l} ℚG^{s_{i}} such that μ(q) lies in GL_{l}(C_{r}^{∗}(G; ℝ)). Choose an integer N such that each number n_{i,j_{i}} := N · q_{i,j_{i}} is an integer. Obviously μ(N · q) = N · μ(q) lies in GL_{l}(C_{r}^{∗}(G; ℝ)). Define u_{i} ∈ H_{2}(Ỹ) for i = 1, 2, …, l by u_{i} = ∑_{j_{i}=1}^{s_{i}} n_{i,j} · h_{i,j}. Then {ψ ∘ ϕ(u_{i}) | i = 1, 2, …, l} is a C_{r}^{∗}(G; ℝ)-basis for H_{2}^{G}(Ỹ; C_{r}^{∗}(G; ℝ)).

Since Ỹ is simply connected, the Hurewicz homomorphism h: π_{2}(Ỹ) → H_{2}(Ỹ) is surjective [521, Corollary 7.8 in Chapter IV.7 on page 180]. Hence we can find maps a_{i}: S^{2} → Ỹ for i = 1, 2, …, l such that the composition ψ ∘ ϕ ∘ h: π_{2}(Ỹ) → H_{2}^{G}(Ỹ, C_{r}^{∗}(G; ℝ)) sends {[a_{1}], …, [a_{l}]} to a C_{r}^{∗}(G; ℝ)- basis. Now attach to Z a 3-cell with attaching map p ∘ a_{i}: S^{2} → Y for each i = 1, 2, …, l, where p: Ỹ → Y is the projection. We obtain a CW - complex X such that the inclusion Y → X is 2-connected. In particular π_{1}(X) = π_{1}(Y) = G and H_{p}^{G}(X̃; C_{r}^{∗}(G; ℝ)) = H_{p}^{G}(Ỹ; C_{r}^{∗}(G; ℝ)) for p = 0, 1. Moreover, there is an exact sequence of C_{r}^{∗}(G; ℝ)-modules

0 → H_{3}^{G}(X̃; C_{r}^{∗}(G; ℝ)) → C_{r}^{∗}(G; ℝ) ⊗_{ℤG} C_{3}(X̃) ⟶^{ν} H_{2}^{G}(Ỹ; C_{r}^{∗}(G; ℝ))

→ H_{2}^{G}(X̃; C_{r}^{∗}(G; ℝ)) → 0,

where ν sends 1⊗v_{i} for v_{i} the i-th element of the cellular basis of C_{3}(X̃) = ℤG^{l} to ψ∘ϕ∘h([a_{i}]). Hence ν is an isomorphism. This implies H_{p}^{G}(X̃, C_{r}^{∗}(G; ℝ)) = 0 for p = 2, 3 and hence H_{p}^{G}(X̃, C_{r}^{∗}(G)) = 0 for p ≥ 0.

(3) ⇔ (4) This follows from Lemma 12.3.

(4) ⇒ (5) We can find an embedding of X in ℝ^{n+1} for n ≥ 6 with regular neighborhood N with boundary ∂N [445, chapter 3]. Then X and N are homotopy equivalent and the inclusion ∂N → N is 2-connected. In particular we get identifications G = π_{1}(X) = π_{1}(N) = π_{1}(∂N) and H_{p}^{G}(Ñ; N (G)) = H_{p}^{G}(X̃; N (G)) = 0 for p ≥ 0. By Poincaré duality H_{p}^{G}(Ñ, ˜∂N; N (G)) is N (G)-isomorphic to H^{n+1−p}(Ñ; N (G)), where the latter is defined by the cohomology of hom_{ℤG}(C_{∗}(Ñ), N (G)). There is a canonical N (G)-isomorphism hom_{ℤG}(C_{∗}(Ñ), N (G)) ≅ hom_{N (G)}(N (G) ⊗_{ℤG} C_{∗}(Ñ), N (G)). Since N (G) ⊗_{ℤG} C_{∗}(Ñ) is a free N (G)-chain complex with trivial homology, it is contractible. This implies that hom_{N (G)}(N (G) ⊗_{ℤG} C_{∗}(Ñ), N (G)) is contractible. Hence H_{p}^{G}(Ñ, ˜∂N; N (G)) vanishes for p ≥ 0. The long exact homology sequence

… → H_{p+1}^{G}(Ñ, ˜∂N; N (G)) → H_{p}^{G}(˜∂N; N (G)) → H_{p}^{G}(Ñ; N (G))

→ H_{p}^{G}(Ñ, ˜∂N; N (G)) → …

implies that ∂N is a closed connected orientable Riemannian manifold with G = π_{1}(∂N) and H_{p}^{G}(˜∂N; N (G)) = 0 for p ≥ 0.

(5) ⇒ (1) This follows from the fact that the classifying map M → BG is 2connected and hence induces epimorphisms H_{p}^{G}(M̃; N (G)) → H_{p}^{G}(EG; N (G)) for p = 0, 1, 2.

(5) ⇔ (6) This follows from Lemma 12.3. This finishes the proof of Theorem 12.9. ⊓ ⊔

It remains to investigate the class of groups for which Theorem 12.9 applies.

Definition 12.10. Let d be a non-negative integer or d = ∞. Define Z_{d} to be the class of groups for which H_{p}^{G}(EG; N (G)) = 0 holds for p ≤ d.

Lemma 12.11. Let d be a non-negative integer or d = ∞. Then

(1) Let G be the directed union ⋃_{i∈} G_{i} of subgroups G_{i} ⊂ G. Suppose that G_{i} ∈ Z_{d} for each i ∈ I. Then G ∈ Z_{d};

(2) If G contains a normal subgroup H ⊂ G with H ∈ Z_{d}, then G ∈ Z_{d};

(3) If G ∈ Z_{d} and H ∈ Z_{e}, then G × H ∈ Z_{d+e+1};

(4) Z_{0} is the class of non-amenable groups;

(5) Let G be a non-amenable group, for instance a non-abelian free group. Then ∏_{i=0}^{d} G lies in Z_{d}.

Proof. (1) Inspecting for instance the bar-resolution or the infinite join model for EG, one sees that EG is the colimit of a directed system of G-CW - subcomplexes of the form G ×_{G_{i}} EG_{i} directed by I. Hence

H_{p}^{G}(EG; N (G)) = colim_{i∈I} H_{p}^{G}(G ×_{G_{i}} EG_{i}; N (G)).

We get H_{p}^{G}(G ×_{G_{i}} EG_{i}; N (G)) = N (G) ⊗_{N (G_{i})} H_{p}^{G_{i}}(EG_{i}, N (G_{i})) = 0 from Theorem 6.29 (1).

(2) The Serre spectral sequence applied to BH → BG → B(G/H) with coefficients in the ℤG-module N (G) converges to H_{p+q}^{G}(EG; N (G)) and has as E^{2}-term E_{p,q}^{2} = H_{p}^{G/H}(E(G/H); H_{q}^{H}(EH; N (G))). Theorem 6.29 (1) implies H_{q}^{H}(EH; N (G)) = N (G) ⊗_{N (H)} H_{q}^{H}(EH; N (H)) = 0 for q ≤ d. Hence H_{n}^{G}(EG; N (G)) = 0 for n ≤ d.

(3) The condition G ∈ Z_{d} implies that N (G) ⊗_{ℤG} C_{∗}(EG) is N (G)-chain homotopy equivalent to an N (G)-chain complex P_{∗} such that P_{p} = 0 holds for p ≤ d, and analogously for the condition H ∈ Z_{e}. Notice that C_{∗}(E(G × H)) is ℤ[G × H]-isomorphic to C_{∗}(EG) ⊗_{ℤ} C_{∗}(EH). Hence N (G × H) ⊗_{ℤ[G×H]} C_{∗}(E(G × H)) is N (G × H)-chain homotopy equivalent to the N (G × H)-chain complex N (G × H) ⊗_{N (G)⊗_{ℂ}N (H)} (P_{∗} ⊗_{ℂ} Q_{∗}), whose chain modules in dimension n ≤ d + e + 1 are zero. Hence G × H ∈ Z_{d+e+1}.

(4) This follows from Lemma 6.36.

(5) This follows from assertions (3) and (4) for d < ∞. The case d = ∞ follows from (2). This finishes the proof of Lemma 12.10. ⊓ ⊔

12.4 Miscellaneous

The Singer Conjecture 11.1 and the results presented in Section 12.2 suggest to consider the following stronger version of the zero-in-the-spectrum Conjecture 12.1. Let M be an aspherical closed manifold. Then zero is in the spectrum of (Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(M̃) → L^{2}Ω^{p}(M̃) for p with \frac{dim(M)−1}{2} ≤ p ≤ \frac{dim(M)+1}{2}. Roughly speaking, zero is always in the spectrum of the Laplacian in the middle dimension.

If M is a 2-dimensional complete Riemannian manifold, then zero is in the spectrum of (Δ_{p})_{min} for some p ≥ 0 by an argument due to Dodziuk and Lott [318, Proposition 10].

Let M be a complete Riemannian manifold. We have introduced its reduced L^{2}-cohomology H_{(2)}^{p}(M) and unreduced L^{2}-cohomology H_{(2),unr}^{p}(M) in Definition 1.71. The unreduced cohomology H_{(2),unr}^{p}(M) vanishes if and only if zero is not in the spectrum of (Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(M) → L^{2}Ω^{p}(M). The question whether the unreduced and reduced L^{2}-cohomology are quasi-isometry invariants is investigated in [237, page 291] and [410].

We have seen for an integer n and a free G-CW -complex X of finite type in Lemma 12.3 that H_{p}^{G}(X; N (G)) = 0 for p ≤ n is equivalent to H_{p}^{G}(X; C_{r}^{∗}(G))) = 0 for p ≤ n. The proof was essentially based on the fact that C_{r}^{∗}(G) is rationally closed in N (G), i.e. a matrix in M_{l}(C_{r}^{∗}(G)) which is invertible in M_{l}(N (G)) is already invertible in M_{l}(C_{r}^{∗}(G)). This does not imply that N (G) is flat over C_{r}^{∗}(G) as the following elementary example illustrates.

Example 12.12. We want to show that N (ℤ) is not flat over C_{r}^{∗}(ℤ). Choose a numeration by the positive integers of ℚ ∩ [0, 1] = {q_{n} | n = 1, 2, …}. Fix a sequence of positive real numbers (ϵ_{n})_{n≥1} with ∑_{n≥1} ϵ_{n} < 1/2. Let A ⊂ S^{1} be the complement of the open subset ⋃_{n≥0}{exp(2πix) | x ∈ (q_{n}−ϵ_{n}, q_{n}+ϵ_{n})} of S^{1}. Obviously A is closed and its volume is greater than zero. Choose a continuous function f: S^{1} → ℂ such that f^{−1}(0) = A. (Actually one can choose f to be smooth [67, Satz 14.1 on page 153].) Then multiplication with f ∈ C_{r}^{∗}(ℤ) = C(S^{1}) induces an injective C_{r}^{∗}(ℤ)-homomorphism r_{f}: C(S^{1}) → C(S^{1}) since the complement of A contains the dense subset {exp(2πiq) | q ∈ ℚ ∩ [0, 1]}. Multiplication with f ∈ N (ℤ) = L^{∞}(S^{1}) (see Example 1.4) induces a N (ℤ)-homomorphism L^{∞}(S^{1}) → L^{∞}(S^{1}) which is not injective since its kernel contains the non-zero element χ_{A} given by the characteristic function of A. We mention that both C_{r}^{∗}(ℤ) and N (ℤ) are flat over ℂℤ (see Conjecture 6.49).

A result similar to Theorem 12.9 has been proved by Kervaire [289].

Theorem 12.13. Let G be a finitely presented group. Suppose H_{p}(BG; ℤ) = 0 for p = 1, 2. Then there is for each n ≥ 5 a closed n-dimensional manifold M with H_{p}(M; ℤ) ≅ H_{p}(S^{n}; ℤ) for p ≥ 0 and π_{1}(M) ≅ G.

More information about the zero-in-the-spectrum Conjecture 12.1 can be found in [318] and in [237, section 8.A5].

Exercises

12.1. Let T: dom(T) ⊂ H → H be a densely defined closed operator. Suppose that T is selfadjoint. Prove

resolv(T) = {λ ∈ ℂ | (T − λ id)(dom(T)) = H}.

12.2. Let M be a compact connected orientable 3-manifold such that each prime factor is non-exceptional. Show that then H_{p}^{π}(Eπ; N (π)) is non-zero for p = 0 or p = 1.

12.3. Let M be an aspherical closed locally symmetric space with fundamental group π. Then H_{p}^{π}(Eπ; N (π)) = 0 for p < \frac{dim(M_{ncp})−f-rk(M_{ncp})}{2} and H_{p}^{π}(Eπ; N (π)) ≠ 0 for p = \frac{dim(M_{ncp})−f-rk(M_{ncp})}{2}.

12.4. Show that the universal covering of an aspherical closed Riemannian manifold is uniformly contractible.

12.5. Let f: M → N be a map of closed connected orientable manifolds of the same dimension. Suppose that f has non-zero degree and induces an isomorphism on the fundamental groups. Assume that zero is not in the spectrum of (Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(M̃) → L^{2}Ω^{p}(M̃) for p ≥ 0. Show that then zero is not in the spectrum of (Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(Ñ) → L^{2}Ω^{p}(Ñ) for p ≥ 0.

12.6. Let F → E → B be a fibration of closed connected manifolds such that π_{1}(F) → π_{1}(E) is injective. Suppose that zero is not in the spectrum of (Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(F̃) → L^{2}Ω^{p}(F̃) for p ≥ 0. Show that then zero is not in the spectrum of (Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(Ẽ) → L^{2}Ω^{p}(Ẽ) for p ≥ 0.

12.7. Let X be a finite 2-dimensional CW -complex with fundamental group π. Suppose that π is not amenable, H_{1}^{π}(Eπ; N (π)) = 0 and χ(X) = 0. Show that H_{p}^{π}(X; N (π)) = 0 for p ≥ 0 and X is aspherical.

12.8. Let G be a finitely presented group G such that H_{p}^{G}(EG; N (G)) = 0 for p ≤ 1. Show

def(G) ≤ 1.

Prove that def(G) = 1 implies the existence of an aspherical finite CW - complex X of dimension 2 such that G = π_{1}(X) and H_{p}^{G}(X̃; N (G)) = 0 for p ≥ 0.

12.9. Let G be a finitely presented group G such that H_{p}^{G}(EG; N (G)) = 0 for p ≤ 1. Define q(G) to be the infimum over the Euler characteristics χ(M) of all closed connected orientable 4-manifolds with π_{1}(M) ≅ G. Show

q(G) ≥ 0.

Suppose q(G) = 0. Then there exists a closed connected orientable 4manifold M with G ≅ π_{1}(M) such that zero is not in the spectrum of (Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊂ L^{2}Ω^{p}(M̃) → L^{2}Ω^{p}(M̃) for p ≥ 0.

12.10. Let G and H be groups with models for BG and BH of finite type. Let d and e be integers or be ∞. Suppose that

H_{p}^{G}(EG; N (G)) = 0  p ≤ d;
H_{p}^{G}(EG; N (G)) ≠ 0  p = d + 1;
H_{p}^{H}(EH; N (H)) = 0  q ≤ e;
H_{p}^{H}(EH; N (H)) ≠ 0  q = e + 1.

Then H_{n}^{G×H}(E(G × H); N (G × H)) is zero for n ≤ d + e + 1 and is not zero for p = d + e + 2.

12.11. Determine for the following groups G the supremum s over the set of integers n for which H_{p}^{G}(EG; N (G)) = 0 for p ≤ n holds: G = ℤ/2, ℤ^{k}, ∗_{i=1}^{r}ℤ, ∏_{i=1}^{r} ℤ ∗ ℤ, ℤ × (ℤ ∗ ℤ), ∏_{i=1}^{∞}(ℤ/2 ∗ ℤ/2), ∏_{i=1}^{∞}(ℤ/2 ∗ ℤ/3).

12.12. Let N be a compact connected oriented manifold of dimension n = 4k for some integer k. Let M be a complete Riemannian manifold which is diffeomorphic to the interior N − ∂N. Show

(1) The image of H^{2k}(N; ∂N; ℂ) → H^{2k}(N; ℂ) is isomorphic to the image of H_{c}^{2k}(M; ℂ) → H^{2k}(M; ℂ), where H_{c}^{2k}(M; ℂ) denotes cohomology with compact support;

(2) Suppose that the signature of N is different from zero. Conclude that then the image of H_{c}^{2k}(M; ℂ) → H^{2k}(M; ℂ) is non-trivial and zero is in the spectrum of (Δ_{2k})_{min} : dom ((Δ_{2k})_{min}) ⊂ L^{2}Ω^{2k}(M) → L^{2}Ω^{2k}(M).

13. The Approximation Conjecture and the Determinant Conjecture

The Approximation Conjecture and the Determinant Conjecture

Introduction

This chapter is devoted to the following two conjectures.

Conjecture 13.1 (Approximation Conjecture). A group G satisfies the Approximation Conjecture if the following holds:

Let {G_{i} | i ∈ I} be an inverse system of normal subgroups of G directed by inclusion over the directed set I. Suppose that ⋂_{i∈I} G_{i} = {1}. Let X be a G-CW -complex of finite type. Then G_{i}\X is a G/G_{i}-CW -complex of finite type and

b_{p}^{(2)}(X; N (G)) = lim_{i∈I} b_{p}^{(2)}(G_{i}\X; N (G/G_{i})).

Let us consider the special case where the inverse system {G_{i} | i ∈ I} is given by a nested sequence of normal subgroups of finite index

G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ G_{3} ⊃ ….

Notice that then b_{p}^{(2)}(G_{i}\X; N (G/G_{i})) = \frac{b_{p}(G_{i}\X)}{[G:G_{i}]}, where b_{p}(G_{i}\X) is the ordinary p-th Betti number of the finite CW - complex G_{i}\X (see Example 1.32). The inequality

lim sup_{i→∞} \frac{b_{p}(G_{i}\X)}{[G : G_{i}]} ≤ b_{p}^{(2)}(X; N (G))

is discussed by Gromov [237, pages 20, 231] and is essentially due to Kazhdan [284]. In this special case Conjecture 13.1 was formulated by Gromov [237, pages 20, 231] and proved in [328, Theorem 0.1]. Thus we get an asymptotic relation between the L^{2}-Betti numbers and Betti numbers, namely

lim_{i→∞} \frac{b_{p}(G_{i}\X)}{[G : G_{i}]} = b_{p}^{(2)}(X; N (G)),

although the L^{2}-Betti numbers of the universal covering Ỹ and the Betti numbers of a finite CW -complex Y have nothing in common except the fact that their alternating sum gives χ(Y) (see Example 1.38).

The Approximation Conjecture 13.1 in its present form for I = {0, 1, 2, …} appears in [462, Conjecture 1.10].

The second conjecture is

Conjecture 13.2 (Determinant Conjecture). Every group G is of det ≥ 1-class (see Definition 3.112), i.e. for any A ∈ M(m, n, ℤG) the Fuglede-Kadison determinant (see Definition 3.11) of the morphism r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n} given by right multiplication with A satisfies

det(r_{A}^{(2)}) ≥ 1.

We will see that the Determinant Conjecture 13.2 implies both the Approximation Conjecture 13.1 and the Conjecture 3.94 about the homotopy invariance of L^{2}-torsion (see Lemma 13.6 and Theorem 13.3 (1)).

In Section 13.1 we will explain the relevance of the two conjectures above and formulate the main Theorem 13.3 of this chapter, which says that the two conjectures above are true for a certain class G of groups. This theorem is proved in Section 13.2. Variations of the approximation results are presented in Section 13.3.

To understand this chapter, it is only required to be familiar with Sections 1.1, 1.2 and to have some basic knowledge of Section 3.2.

13.1 Survey on the Approximation Conjecture and Determinant Conjecture

In this section we discuss the Approximation Conjecture 13.1 and the Determinant Conjecture 13.2 and introduce the class of groups for which they are known.

13.1.1 Survey on Positive Results about the Approximation Conjecture and the Determinant Conjecture

The following theorem is the main result of this chapter. It is proved in [462, Theorem 1.14, Theorem 1.19 and Theorem 1.21] for a class of groups which is slightly smaller than our class G which will be introduced in Definition 13.9 and studied in Subsection 13.1.3. The general strategy is based on [328], where the case that each quotient G/G_{i} is finite is carried out. See also [82, Theorem 5.1 on page 71] for a treatment of determinant class in this special case. The treatment of amenable exhaustions uses ideas of [149]. We will present the proof of Theorem 13.3 in Section 13.2.

Theorem 13.3. (1) The Approximation Conjecture 13.1 for G and the in-

verse system {G_{i} | i ∈ I} is true if each group G_{i} is of det ≥ 1-class;

(2) If the group G belongs G, then G is of det ≥ 1-class and satisfies the Approximation Conjecture 13.1; (3) The class of groups, which are of det ≥ 1-class, is closed under the oper-

ations (1), (2), (3), (4) and (5) appearing in Definition 13.9;

(4) The class of groups, for which assertion (1) of Conjecture 3.94 is true, is closed under the operations (2), (3) and (4) appearing in Definition 13.9.

13.1.2 Relations to Other Conjectures

We leave the proof of the next lemma, which is similar to the one of Lemma 10.5, to the reader.

Lemma 13.4. Let G be a group. Let {G_{i} | i ∈ I} be an inverse system of normal subgroups directed by inclusion over the directed set I. Suppose that ⋂_{i∈I} G_{i} = {1}. Then the following assertions are equivalent:

(1) The group G satisfies the Approximation Conjecture 13.1 for the inverse system {G_{i} | i ∈ I};

(2) The group G satisfies the Approximation Conjecture 13.1 for the inverse system {G_{i} | i ∈ I} for all cocompact free proper G-manifolds X without boundary;

(3) Let A ∈ M(m, n, ℚG) be a matrix. Let A_{i} ∈ M(m, n, ℚ[G/G_{i}]) be the matrix obtained from A by applying elementwise the ring homomorphism ℚG → ℚ[G/G_{i}] induced by the projection G → G/G_{i}. Then

dim_{N (G)} (ker (r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}))
= lim_{i→∞} dim_{N (G/G_{i})} (ker (r_{A_{i}}^{(2)} : l^{2}(G/G_{i})^{m} → l^{2}(G/G_{i})^{n})) ,

where r_{A}^{(2)} and r_{A_{i}}^{(2)} are given by right multiplication.

The Approximation Conjecture is interesting in connection with the Atiyah Conjecture. The main application has already been stated in Theorem 10.20.

Example 13.5. We can give another proof that the Atiyah Conjecture 10.3 of order ℤ with coefficients in ℚ holds for any free group F. Namely, we can assume that F is finitely generated free by Lemma 10.4. The descending central series of a finitely generated free group F yields a sequence of in F normal subgroups F = F_{0} ⊃ F_{1} ⊃ F_{2} ⊃ … such that ⋂_{i≥1} F_{i} = {1} and each quotient F/F_{i} is torsionfree and nilpotent [354, §2]. Since finitely generated nilpotent groups are poly-cyclic groups, each quotient F/F_{i} satisfies the Atiyah Conjecture of order ℤ with coefficients in ℚ by Moody’s induction theorem (see Remark 10.12). Theorem 13.3 (2) and Theorem 10.20 imply that lim_{i∈I} F/F_{i} satisfies the Atiyah Conjecture of order ℤ with coefficients in ℚ. Since F is a subgroup of lim_{i∈I} F/F_{i}, Lemma 10.4 shows that F satisfies Atiyah Conjecture of order ℤ with coefficients in ℚ. Recall that we have already proved the strong Atiyah Conjecture 10.2 for a free group F_{2} of rank two and thus for all free groups using Fredholm modules in Subsection 10.3.1.

The Determinant Conjecture is important in connection with L^{2}-torsion because of the next result.

Lemma 13.6. If the group G satisfies the Determinant Conjecture 13.2 (for instance if G belongs to the class G introduced in Subsection 13.1.3), then G satisfies Conjecture 3.94.

Proof. Suppose that G satisfies the Determinant Conjecture 13.2. Then assertion (3) of Conjecture 3.94 follows directly from the definitions. Assertion (1) of Conjecture 3.94 follows from the following calculation based on Theorem 3.14 (1)

1 ≤ det(r_{A}^{(2)}) = \frac{1}{det(r_{A^{−1}}^{(2)})} ≤ 1.

Theorem 3.93 (1) implies that assertion (1) of Conjecture 3.94 implies assertion (2) of Conjecture 3.94. ⊓ ⊔

We conclude from Theorem 2.68 and Theorem 3.28

Theorem 13.7. (Logarithmic estimate for spectral density functions). If G satisfies Conjecture 3.94 (for instance if G belongs to the class G introduced in Subsection 13.1.3), then we get for any cocompact free proper G-manifold Mwith G-invariant Riemannian metric and p ≥ 1 that its p-th analytic spectral density function F_{p} (see Definition 2.64) satisfies for appropriate ϵ > 0 and C > 0

F_{p}(λ) ≤ \frac{C}{− ln(λ)} for 0 < λ < ϵ.

13.1.3 A Class of Groups

Now we define the class G of groups for which we can prove the Approximation Conjecture 13.1 and the Determinant Conjecture 13.2.

Definition 13.8. Let H ⊂ G be a subgroup. The discrete homogeneous space G/H is called amenable if we can find a G-invariant metric d: G/H × G/H → ℝ such that any set with finite diameter consists of finitely many elements and for all K > 0 and ϵ > 0 there is a finite non-empty subset A ⊂ G/H with

|B_{K}(A) ∩ B_{K}(G/H − A)| ≤ ϵ · |A|.

Here and elsewhere B_{K}(A) denotes for a subset A of a metric space (X, d) the set {x ∈ X | d(x, A) < K}. If H ⊂ G is a normal subgroup with amenable quotient, then G/H is an amenable discrete homogeneous space by the following argument. Consider K > 0 and ϵ > 0. Equip G/H with the word length metric. Because of Lemma 6.35 we can choose a set B ⊂ G/H such that for S = B_{2K}(eH) we get |∂_{S}B| ≤ \frac{ϵ}{1+ϵ} ·|B|. Put A = B −(B ∩B_{K}(G/H −B)). Then B_{K}(A) ∩ B_{K}(G/H − A) ⊂ ∂_{S}B. This implies

|B_{K}(A) ∩ B_{K}(G/H − A)| ≤ \frac{ϵ}{1 + ϵ} · |B|.

Since

(1 − \frac{ϵ}{1 + ϵ}) · |B| ≤ |B| − |∂_{S}B| = |B| − |B ∩ B_{2K}(G/H − B)|

≤ |B| − |B ∩ B_{K}(G/H − B)| ≤ |A|,

we get

|B_{K}(A) ∩ B_{K}(G/H − A)| ≤ ϵ · |A|.

Definition 13.9. Let G be the smallest class of groups which contains the trivial group and is closed under the following operations:

(1) Amenable quotient Let H ⊂ G be a (not necessarily normal) subgroup. Suppose that H ∈ G and the quotient G/H is an amenable discrete homogeneous space. Then G ∈ G (In particular G ∈ G if G contains a normal amenable subgroup H ⊂ G with G/H ∈ G) ;

(2) Colimits If G = colim_{i∈I} G_{i} is the colimit of the directed system {G_{i} | i ∈ I} of groups indexed by the directed set I and each G_{i} belongs to G, then G belongs to G;

(3) Inverse limits If G = lim_{i∈I} G_{i} is the limit of the inverse system {G_{i} | i ∈ I} of groups indexed by the directed set I and each G_{i} belongs to G, then G belongs to G;

(4) Subgroups If H is isomorphic to a subgroup of the group G with G ∈ G, then H ∈ G;

(5) Quotients with finite kernel Let 1 → K → G → Q → 1 be an exact sequence of groups. If K is finite and G belongs to G, then Q belongs to G.

Next we provide some information about the class G. Notice that Schick defines in [462, Definition 1.12] a smaller class, also denoted by G, by requiring that his class contains the trivial subgroup and is closed under operations (1), (2), (3) and (4), but not necessarily under operation (5).

Let (P) be a property of groups, e.g. being finite or being amenable. Then a group is residually (P) if for any g ∈ G with g ≠ 1 there is an epimorphism p: G → G′ to a group having property (P) such that p(g) ≠ 1. If G is countable and (P) is closed under taking subgroups and extensions, this is equivalent to the condition that there is a nested sequence of subgroups G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ … such that each G_{i} is normal in G and each quotient G/G_{i} has property (P). In particular we get the notions of a residually finite and residually amenable group. For the reader’s convenience we record some basic properties of residually finite groups. For more information we refer to the survey article [355].

Theorem 13.10 (Survey on residually finite groups).

(1) The free product of two residually finite groups is again residually finite [241], [114, page 27];

(2) A finitely generated residually finite group has a solvable word problem [389];

(3) The automorphism group of a finitely generated residually finite group is residually finite [29];

(4) A finitely generated residually finite group is Hopfian, i.e, any surjective endomorphism is an automorphism [356], [399, Corollary 41.44];

(5) Let G be a finitely generated group possessing a faithful representation into GL(n, F) for F a field. Then G is residually finite [356], [517, Theorem 4.2];

(6) Let G be a finitely generated group. Let G^{rf} be the quotient of G by the normal subgroup which is the intersection of all normal subgroups of G of finite index. The group G^{rf} is residually finite and any finite dimensional representation of G over a field factorizes over the canonical projection G → G^{rf};

(7) The fundamental group of a compact 3-manifold whose prime decomposition consists of non-exceptional manifolds (i.e., which are finitely covered by a manifold which is homotopy equivalent to a Haken, Seifert or hyperbolic manifold) is residually finite (see [253, page 380]);

(8) There is an infinite group with four generators and four relations which has no finite quotient except the trivial one [255].

Lemma 13.11. (1) A group G belongs to G if and only if each finitely gen-

erated subgroup of G belongs to G;

(2) The class G is residually closed, i.e. if there is a nested sequence of subgroups G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ … such that ⋂_{i≥0} G_{i} = {1} and each G_{i} belongs to G, then G belongs to G;

(3) Any residually amenable and in particular any residually finite group belongs to G;

(4) Suppose that G belongs to G and f: G → G is an endomorphism. Define the “mapping torus group” G_{f} to be the quotient of G ∗ ℤ obtained by introducing the relations t^{−1}gt = f(g) for g ∈ G and t ∈ ℤ a fixed generator. Then G_{f} belongs to G;

(5) Let {G_{j} | j ∈ J} be a set of groups with G_{j} ∈ G. Then the direct sum ⨁_{j∈J} G_{j} and the direct product ∏_{j∈J} G_{j} belong to G.

Proof. (1) Any group G is the directed union of its finitely generated subgroups and a directed union is a special case of a colimit of a directed system.

(2) There is an obvious group homomorphism G → lim_{i∈I} G/G_{i}. It is injective because of ⋂_{i≥0} G_{i} = {1}.

(3) This follows from (2).

(4) There is an exact sequence 1 → K → G_{f} → ℤ → 1, where K is the

colimit of the directed system … → G ⟶^{f} G ⟶^{f} G → … indexed by the integers.

(5) We first show that G × H belongs to G, provided that G and H belong to G. Let G_{G}^{×} be the class of groups {H | G × H ∈ G}. It contains the trivial group because G belongs to G by assumption. It suffices to show that G_{G}^{×} is closed under the operation appearing in Definition 13.9 because then G ⊂ G_{G}^{×} holds by definition of G. This is obvious for operations (1), (4) and (5). For operations (2) and (3) one uses the fact that for a directed system or inverse system {G_{i} | i ∈ I} over the directed set I the canonical maps induce isomorphisms

colim_{i∈I}(G × G_{i}) ≅ G × colim_{i∈I} G_{i};
G × lim_{i∈I} G_{i} ≅ lim_{i∈I}(G × G_{i}).

Since ∏_{j∈J} G_{j} is the limit of the inverse system {∏_{j∈I} G_{j} | I ⊂ J, I finite} and ⨁_{i∈I} G_{i} is by definition a subgroup of ∏_{j∈J} G_{i}, assertion (5) follows. This finishes the proof of Lemma 13.11. ⊓ ⊔

13.2 The Proof of the Approximation Conjecture and the Determinant Conjecture in Special Cases

In this section we give the proof of Theorem 13.3. As a warm-up we prove

Lemma 13.12. The trivial group is of det ≥ 1-class.

Proof. Let A ∈ M_{n}(ℤ) be a matrix. Let λ_{1}, λ_{2}, …, λ_{r} be the eigenvalues of AA^{∗} (listed with multiplicity), which are different from zero. Then we get from Example 3.12

det(r_{A}^{(2)}) = \sqrt{∏_{i=1}^{r} λ_{i}}.

Let p(t) = det_{ℂ}(t − AA^{∗}) be the characteristic polynomial of AA^{∗}. It can be written as p(t) = t^{a} · q(t) for some polynomial q(t) with integer coefficients and q(0) ≠ 0. One easily checks

|q(0)| = ∏_{i=1}^{r} λ_{i}.

Since q has integer coefficients, we conclude det(r_{A}^{(2)}) ≥ 1. ⊓ ⊔

13.2.1 The General Strategy

In this subsection we prove in Theorem 13.19 a general result about approximations. It will be applied to the various cases, we will have to deal with, when we will for instance show assertion (3) of Theorem 13.3 that the class of groups, for which the Determinant Conjecture 13.2 is true, is closed under the operations (1), (2), (3), (4) and (5) appearing in Definition 13.9.

Throughout this subsection we will consider the following data:

• R is a ring which satisfies ℤ ⊂ R ⊂ ℂ and is closed under complex conjugation;

• G is a group, A is a matrix in M_{d}(RG) and tr_{N (G)} : M_{d}(N (G)) → ℂ is the von Neumann trace (see Definition 1.2). Suppose that r_{A}^{(2)} : l^{2}(G)^{d} → l^{2}(G)^{d}, which is given by right multiplication with A, is positive;

• I is a directed set. For each i ∈ I we have a group G_{i}, A_{i} ∈ M_{d}(RG_{i}) and a finite normal trace tr_{i}: M_{d}(N (G_{i})) → ℂ (see Subsection 9.1.3) such that tr_{i}(I_{d}) = d holds for the unit matrix I_{d} ∈ M_{d}(N (G_{i})). Suppose that r_{A_{i}}^{(2)} : l^{2}(G_{i})^{d} → l^{2}(G_{i})^{d} is positive.

Let F: [0, ∞) → [0, ∞) be the spectral density function of the morphism r_{A}^{(2)} : l^{2}(G)^{d} → l^{2}(G)^{d} (see Definition 2.1). Let F_{i}: [0, ∞) → [0, ∞) be the spectral density function of the morphism r_{A_{i}}^{(2)} : l^{2}(G_{i})^{d} → l^{2}(G_{i})^{d} defined with respect to the trace tr_{i} (which may or may not be tr_{N (G_{i})}). Since r_{A}^{(2)} is positive, we get F (λ) = tr_{N (G)}(E_{λ}) for {E_{λ} | λ ∈ ℝ} the family of spectral projections of r_{A}^{(2)} (see Lemma 2.3 and Lemma 2.11 (11)). The analogous statement holds for F_{i}. Recall that for a directed set I and a net (x_{i})_{i∈I} of real numbers one defines

lim inf_{i∈I} x_{i} := sup{inf{x_{j} | j ∈ I, j ≥ i} | i ∈ I};
lim sup_{i∈I} x_{i} := inf{sup{x_{j} | j ∈ I, j ≥ i} | i ∈ I}.

Put

\underline{F} (λ) := lim inf_{i∈I} F_{i}(λ);  (13.13)
\overline{F} (λ) := lim sup_{i∈I} F_{i}(λ).  (13.14)

Notice that F and F_{i} are density functions in the sense of Definition 2.7, i.e. they are monotone non-decreasing and right-continuous. The functions \overline{F} and \underline{F} are monotone non-decreasing, but it is not clear whether they are rightcontinuous. To any monotone non-decreasing function f: [0, ∞) → [0, ∞) we can assign a density function

f^{+} : [0, ∞) → [0, ∞),  λ ↦ lim_{ϵ→0+} f(λ + ϵ).  (13.15)

We always have

\underline{F} (λ) ≤ \overline{F} (λ);  (13.16)
\underline{F} (λ) ≤ \underline{F}^{+}(λ);  (13.17)
\overline{F} (λ) ≤ \overline{F}^{+}(λ).  (13.18)

Define det(r_{A}^{(2)}) as in Definition 3.11 and analogously define det_{i}(r_{A_{i}}^{(2)}) by

exp (∫_{0+}^{∞} ln(λ) dF_{i}) if ∫_{0+}^{∞} ln(λ) dF_{i} > −∞ and by 0 if ∫_{0+}^{∞} ln(λ) dF_{i} = −∞. The main technical result of this subsection is

Theorem 13.19. Suppose that there is a constant K > 0 such that for the operator norms ||r_{A}^{(2)}|| ≤ K and ||r_{A_{i}}^{(2)}|| ≤ K hold for i ∈ I. Suppose that for any polynomial p with real coefficients we have

tr_{N (G)}(p(A)) = lim_{i∈I} tr_{i}(p(A_{i})).

Then

(1) For every λ ∈ ℝ

F (λ) = \underline{F}^{+}(λ) = \overline{F}^{+}(λ);

(2) If det_{i}(r_{A_{i}}^{(2)}) ≥ 1 for each i ∈ I, then

F (0) = lim_{i∈I} F_{i}(0);
det(r_{A}^{(2)}) ≥ 1.

Proof. (1) Fix λ ≥ 0. Choose a sequence of polynomials p_{n}(x) with real coefficients such that

χ_{[0,λ]}(x) ≤ p_{n}(x) ≤ χ_{[0,λ+1/n]}(x) + \frac{1}{n} · χ_{[0,K]}(x)  (13.20)

holds for 0 ≤ x ≤ K. Recall that χ_{S} is the characteristic function of a set S. Recall that the von Neumann trace is normal and satisfies tr_{N (G)}(f) ≤ tr_{N (G)}(g) for 0 ≤ f ≤ g, and each tr_{i} has the same properties by assumption. We conclude from (13.20) and the assumptions ||r_{A_{i}}^{(2)}|| ≤ K and tr_{i}(I_{d}) = d

F_{i}(λ) ≤ tr_{i}(r_{p_{n}(A_{i})}^{(2)}) ≤ F_{i}(λ + \frac{1}{n}) + \frac{d}{n}.

Taking the limit inferior or the limit superior we conclude using the assumption tr_{N (G)}(p_{n}(A)) = lim_{i∈I} tr_{i}(p(A_{i}))

\overline{F} (λ) ≤ tr_{N (G)}(r_{p_{n}(A)}^{(2)}) ≤ \underline{F} (λ + \frac{1}{n}) + \frac{d}{n}.  (13.21)

466 13. The Approximation Conjecture and the Determinant Conjecture

We conclude from (13.20) and the assumption ||r_{A}^{(2)}|| ≤ K

F (λ) ≤ tr_{N (G)}(r_{p_{n}(A)}^{(2)}) ≤ F (λ + \frac{1}{n}) + \frac{d}{n}.

This implies

lim_{n→∞} tr_{N (G)}(r_{p_{n}(A)}^{(2)}) = F (λ).

We get by applying lim_{n→∞} to (13.21)

\overline{F} (λ) ≤ F (λ) ≤ \underline{F}^{+}(λ).

Since \underline{F} is monotone non-decreasing, we get for ϵ > 0 using (13.16)

F (λ) ≤ \underline{F} (λ + ϵ) ≤ \overline{F} (λ + ϵ) ≤ F (λ + ϵ).

Since F is right continuous, we get by taking lim_{ϵ→0+}

F (λ) = \underline{F}^{+}(λ) = \overline{F}^{+}(λ).

This finishes the proof of assertion (1).

(2) We conclude from Lemma 3.15 (1) (the proof for det_{i} is analogous)

ln(det(r_{A}^{(2)})) = ln(K) · (F (K) − F (0)) − ∫_{0+}^{K} \frac{F (λ) − F (0)}{λ} dλ; (13.22)
ln(det_{i}(r_{A_{i}}^{(2)})) = ln(K) · (F_{i}(K) − F_{i}(0)) − ∫_{0+}^{K} \frac{F_{i}(λ) − F_{i}(0)}{λ} dλ. (13.23)

We get from the assumptions ln(det_{i}(r_{A_{i}}^{(2)})) ≥ 0 and tr_{i}(I_{d}) = d

∫_{0+}^{K} \frac{F_{i}(λ) − F_{i}(0)}{λ} dλ ≤ ln(K) · (F_{i}(K) − F_{i}(0)) ≤ ln(K) · d. (13.24)

We conclude

∫_{ϵ}^{K} \frac{\underline{F} (λ) − F (0)}{λ} dλ = ∫_{ϵ}^{K} \frac{\underline{F}^{+}(λ) − F (0)}{λ} dλ.  (13.25)

from the following calculation based on Lebesgue’s Theorem of majorized convergence and the fact that F (λ) = F (K) = d for λ ≥ K

|∫_{ϵ}^{K} \frac{\underline{F}^{+}(λ) − F (0)}{λ} dλ − ∫_{ϵ}^{K} \frac{\underline{F} (λ) − F (0)}{λ} dλ|

= |∫_{ϵ}^{K} \frac{lim_{n→∞} \underline{F} (λ + 1/n) − F (0)}{λ} dλ − ∫_{ϵ}^{K} \frac{\underline{F} (λ) − F (0)}{λ} dλ|
= |∫_{ϵ}^{K} lim_{n→∞} \frac{\underline{F} (λ + 1/n) − F (0)}{λ + 1/n} dλ − ∫_{ϵ}^{K} \frac{\underline{F} (λ) − F (0)}{λ} dλ|
= | lim_{n→∞} ∫_{ϵ}^{K} \frac{\underline{F} (λ + 1/n) − F (0)}{λ + 1/n} dλ − ∫_{ϵ}^{K} \frac{\underline{F} (λ) − F (0)}{λ} dλ|

= lim_{n→∞} |∫_{ϵ}^{K} \frac{\underline{F} (λ + 1/n) − F (0)}{λ + 1/n} dλ − ∫_{ϵ}^{K} \frac{\underline{F} (λ) − F (0)}{λ} dλ|

= lim_{n→∞} |∫_{ϵ+1/n}^{K+1/n} \frac{\underline{F} (λ) − F (0)}{λ} dλ − ∫_{ϵ}^{K} \frac{\underline{F} (λ) − F (0)}{λ} dλ|

≤ lim_{n→∞} (|∫_{ϵ}^{ϵ+1/n} \frac{\underline{F} (λ) − F (0)}{λ} dλ| + |∫_{K}^{K+1/n} \frac{\underline{F} (λ) − F (0)}{λ} dλ|)

≤ lim_{n→∞} (\frac{F (K) − F (0)}{ϵ · n} + \frac{F (K) − F (0)}{K · n})
= 0.

Since (13.25) holds for all ϵ > 0, we conclude from Levi’s Theorem of monotone convergence

∫_{0+}^{K} \frac{\underline{F} (λ) − F (0)}{λ} dλ = ∫_{0+}^{K} \frac{\underline{F}^{+}(λ) − F (0)}{λ} dλ.  (13.26)

Next we show

∫_{ϵ}^{K} \frac{lim inf_{i∈I}(F_{i}(λ) − F_{i}(0))}{λ} dλ ≤ lim inf_{i∈I} ∫_{ϵ}^{K} \frac{F_{i}(λ) − F_{i}(0)}{λ} dλ; (13.27)
∫_{0+}^{K} \frac{lim inf_{i∈I}(F_{i}(λ) − F_{i}(0))}{λ} dλ ≤ lim inf_{i∈I} ∫_{0+}^{K} \frac{F_{i}(λ) − F_{i}(0)}{λ} dλ. (13.28)

These follow from the following calculation for S = [ϵ, K] or S = (0, K]. Since sup {inf { F_{j}(λ) − F_{j}(0) | j ∈ I, i ≤ j} | i ∈ I} is a monotone nondecreasing function in λ ∈ S, we can find a sequence of step function t_{1} ≤ t_{2} ≤ t_{3} ≤ … which converges pointwise on S to it. Hence we can find a sequence i_{1} ≤ i_{2} ≤ … of elements in I such that t_{n} − \frac{1}{n} ≤ inf {F_{j}(λ) − F_{j}(0) | j ∈ I, i_{n} ≤ j} holds for n ≥ 0 and hence

sup {inf { F_{j}(λ) − F_{j}(0) | j ∈ I, i ≤ j} | i ∈ I}
= lim_{n→∞} (inf {F_{j}(λ) − F_{j}(0) | j ∈ I, i_{n} ≤ j}) .

We conclude from Levi’s Theorem of monotone convergence applied to the sequence (inf \{ \frac{F_{j}(λ)−F_{j}(0)}{λ} | j ∈ I, i_{n} ≤ j\})_{n}

∫_{A} \frac{lim inf_{i∈I}(F_{i}(λ) − F_{i}(0))}{λ} dλ

= ∫_{A} sup \{inf \{ \frac{F_{j}(λ) − F_{j}(0)}{λ} | j ∈ I, i ≤ j\} | i ∈ I\} dλ
= ∫_{A} lim_{n→∞} (inf \{ \frac{F_{j}(λ) − F_{j}(0)}{λ} | j ∈ I, i_{n} ≤ j\}) dλ
= lim_{n→∞} ∫_{A} inf \{ \frac{F_{j}(λ) − F_{j}(0)}{λ} | j ∈ I, i_{n} ≤ j\} dλ

≤ sup \{∫_{A} inf \{ \frac{F_{j}(λ) − F_{j}(0)}{λ} | j ∈ I, i ≤ j\} dλ | i ∈ I\}
≤ sup \{inf \{ ∫_{A} \frac{F_{j}(λ) − F_{j}(0)}{λ} dλ | j ∈ I, i ≤ j\} | i ∈ I\}
= lim inf_{i∈I} ∫_{A} \frac{F_{j}(λ) − F_{j}(0)}{λ} dλ.

We get for ϵ > 0 from assertion (1) and equations (13.18), (13.24), (13.25) and (13.27)

∫_{ϵ}^{K} \frac{F (λ) − F (0)}{λ} dλ = ∫_{ϵ}^{K} \frac{\underline{F}^{+}(λ) − F (0)}{λ} dλ
= ∫_{ϵ}^{K} \frac{\underline{F} (λ) − F (0)}{λ} dλ
≤ ∫_{ϵ}^{K} \frac{\underline{F} (λ) − \overline{F} (0)}{λ} dλ
= ∫_{ϵ}^{K} \frac{lim inf_{i∈I} F_{i}(λ) − lim sup_{i∈I} F_{i}(0)}{λ} dλ
≤ ∫_{ϵ}^{K} \frac{lim inf_{i∈I}(F_{i}(λ) − F_{i}(0))}{λ} dλ
≤ lim inf_{i∈I} ∫_{ϵ}^{K} \frac{F_{i}(λ) − F_{i}(0)}{λ} dλ
≤ lim inf_{i∈I} ∫_{0+}^{K} \frac{F_{i}(λ) − F_{i}(0)}{λ} dλ
≤ ln(K) · d.

Since this holds for all ϵ > 0, we conclude from Levi’s Theorem of monotone convergence

∫_{0+}^{K} \frac{F (λ)−F (0)}{λ} dλ ≤ ∫_{0+}^{K} \frac{\underline{F} (λ)−\overline{F} (0)}{λ} dλ ≤ ln(K) · d.

In particular we get ∫_{0+}^{K} \frac{\underline{F} (λ)−\overline{F} (0)}{λ} dλ < ∞ which is only possible if

\underline{F}^{+}(0) − \overline{F} (0) = lim_{λ→0+} \underline{F} (λ) − \overline{F} (0) = 0.

This and assertion (1) implies \overline{F} (0) = F (0). The same argument applies to any directed subset J ⊂ I, so that we get for any directed subset J ⊂ I

lim sup_{i∈J} F_{i}(0) = F (0).  (13.29)

lim_{i∈I} F_{i}(0) = F (0),  (13.30)

because otherwise we could construct ϵ > 0 and a sequence i_{1} ≤ i_{2} ≤ i_{3} ≤ … of elements in I with F_{i_{n}}(0) ≤ F (0) − ϵ for n ≥ 1, what would contradict (13.29).

We conclude using F (K) = d = F_{i}(K) from assertion (1) and equations (13.22), (13.23), (13.26), (13.28) and (13.30)

ln(det(r_{A}^{(2)}))

= ln(K) · (F (K) − F (0)) − ∫_{0+}^{K} \frac{F (λ) − F (0)}{λ} dλ
= ln(K) · (F (K) − F (0)) − ∫_{0+}^{K} \frac{\underline{F}^{+}(λ) − F (0)}{λ} dλ
= ln(K) · (F (K) − F (0)) − ∫_{0+}^{K} \frac{\underline{F} (λ) − F (0)}{λ} dλ
= ln(K) · (F (K) − lim_{i∈I} F_{i}(0)) − ∫_{0+}^{K} \frac{lim inf_{i∈I} F_{i}(λ) − lim_{i∈I} F_{i}(0))}{λ} dλ
= ln(K) · (F (K) − lim_{i∈I} F_{i}(0)) − ∫_{0+}^{K} \frac{lim inf_{i∈I}(F_{i}(λ) − F_{i}(0))}{λ} dλ
≥ ln(K) · (F (K) − lim_{i∈I} F_{i}(0)) − lim inf_{i∈I} ∫_{0+}^{K} \frac{F_{i}(λ) − F_{i}(0)}{λ} dλ

= lim sup_{i∈I} (ln(K) · (F (K) − F_{i}(0)) − ∫_{0+}^{K} \frac{F_{i}(λ) − F_{i}(0)}{λ} dλ)

= lim sup_{i∈I} (ln(K) · (F_{i}(K) − F_{i}(0)) − ∫_{0+}^{K} \frac{F_{i}(λ) − F_{i}(0)}{λ} dλ)
= lim sup_{i∈I} ln(det_{i}(r_{A_{i}}^{(2)}))
≥ 0.

This finishes the proof of Theorem 13.19. ⊓ ⊔

13.2.2 Limits of Inverse Systems

Throughout this subsection we consider an inverse system of groups {G_{i} | i ∈ I} over the directed set I. Denote its limit by G = lim_{i∈I} G_{i}. Let ψ_{i}: G → G_{i} for i ∈ I and ϕ_{i,j}: G_{j} → G_{i} for i, j ∈ I, i ≤ j be the structure maps. Let ℤ ⊂ R ⊂ ℂ be a ring closed under complex conjugation. Let A ∈ M(m, n, RG) be a matrix. Denote by A_{i} ∈ M(m, n, RG) the matrix obtained by applying elementwise the ring homomorphism RG → RG_{i} induced by ψ_{i} for i ∈ I. Let F be the spectral density function and det(r_{A}^{(2)}) be the Fuglede-Kadison determinant of the morphism of finitely generated Hilbert N (G) modules r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}. Let F_{i} be the spectral density function and det(r_{A_{i}}^{(2)}) be the Fuglede-Kadison determinant of the morphism of finitely generated Hilbert N (G_{i})-modules r_{A_{i}}^{(2)} : l^{2}(G_{i})^{m} → l^{2}(G_{i})^{n}. Define \underline{F}, \overline{F}, \underline{F}^{+} and \overline{F}^{+} as in (13.13), (13.14) and (13.15). We want to prove

Theorem 13.31. We get in the notation above:

(1) For every λ ∈ ℝ

F (λ) = \underline{F}^{+}(λ) = \overline{F}^{+}(λ);

(2) If det_{i}(r_{A_{i}}^{(2)}) ≥ 1 for each i ∈ I, then

dim_{N (G)}(ker(r_{A}^{(2)})) = lim_{i∈I} dim_{N (G_{i})}(ker(r_{A_{i}}^{(2)}));
det(r_{A}^{(2)}) ≥ 1.

In order to prove this we want to apply Theorem 13.19, but we first need the following preliminaries. Define for a matrix B ∈ M(m, n, ℂG) a (very secret Russian) real number

K^{G}(B) := mn · max{||b_{i,j}||_{1} | 1 ≤ i ≤ m, 1 ≤ j ≤ n},  (13.32)

where for a = ∑_{g∈G} λ_{g}g ∈ ℂG its L^{1}-norm ||a||_{1} is defined by ∑_{g∈G} |λ_{g}|.

Lemma 13.33. We get for any B ∈ M(m, n, RG)

||r_{B}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}|| ≤ K^{G}(B).

Proof. We get for u ∈ l^{2}(G) and b = ∑_{g∈G} λ_{g} · g ∈ ℂG

||ub||_{l^{2}} = ||∑_{g∈G} λ_{g} · ug||_{l^{2}} ≤ ∑_{g∈G} ||λ_{g} · ug||_{l^{2}} ≤ ∑_{g∈G} |λ_{g}| · ||u||_{l^{2}} = ||b||_{1} · ||u||_{l^{2}}.

This implies for x ∈ l^{2}(G)^{m}

||r_{B}^{(2)}(x)||_{l^{2}}^{2} = ∑_{j=1}^{n} ||∑_{i=1}^{m} x_{i}B_{i,j}||_{l^{2}}^{2}

≤ ∑_{j=1}^{n} (∑_{i=1}^{m} ||x_{i}B_{i,j}||_{l^{2}})^{2}

≤ ∑_{j=1}^{n} (∑_{i=1}^{m} ||x_{i}||_{l^{2}} · ||B_{i,j}||_{1})^{2}

≤ ∑_{j=1}^{n} (∑_{i=1}^{m} ||x_{i}||_{l^{2}} · max{||B_{i,j}||_{1} | 1 ≤ i ≤ m})^{2}

= ∑_{j=1}^{n} max{||B_{i,j}||_{1} | 1 ≤ i ≤ m}^{2} · (∑_{i=1}^{m} ||x_{i}||_{l^{2}})^{2}

≤ ∑_{j=1}^{n} max{||B_{i,j}||_{1} | 1 ≤ i ≤ m}^{2} · m^{2} · ∑_{i=1}^{m} ||x_{i}||_{l^{2}}^{2}
≤ n · max{||B_{i,j}||_{1} | 1 ≤ i ≤ m, 1 ≤ j ≤ n}^{2} · m^{2} · ||x||_{l^{2}}^{2}
≤ (nm · max{||B_{i,j}||_{1} | 1 ≤ i ≤ m, 1 ≤ j ≤ n} · ||x||_{l^{2}})^{2}
= (K^{G}(B) · ||x||_{l^{2}})^{2} .

This finishes the proof of Lemma 13.33. ⊓ ⊔

Lemma 13.34. Let p(x) be a polynomial with real coefficients and B ∈ M_{m}(ℂG). Then there is i_{0} ∈ I with

tr_{N (G)}(p(B)) = tr_{N (G_{i})}(p(B_{i})) for i ≥ i_{0}.

Proof. Let p(B) be the matrix (∑_{g∈G} λ_{g}(i, j) · g)_{i,j}. Then

tr_{N (G)}(p(B)) = ∑_{i=1}^{m} λ_{1}(i, i);

tr_{N (G_{i})}(p(B_{i})) = ∑_{i=1}^{m} ∑_{g∈G,ψ_{i}(g)=1} λ_{g}(i, i).

Only finitely many of the numbers λ_{g}(i, j) are different from zero. Since G is lim_{i∈I} G_{i}, there is an index i_{0} such that ψ_{i}(g) = 1 together with λ_{g}(i, i) ≠ 0 implies g = 1 for all i ≥ i_{0}. Now Lemma 13.34 follows. ⊓ ⊔

We want to apply Theorem 13.19 to AA^{∗}, where A^{∗} is obtained by transposing A and applying elementwise the involution of RG, which sends ∑_{g∈G} λ_{g} ·g to ∑_{g∈G} \overline{λ_{g}} ·g^{−1}. The conditions appearing in Theorem 13.19 are satisfied, where we take the constant K appearing in Theorem 13.19 to be K^{G}(A^{∗}A) (see (13.32)). This follows from Lemma 13.33 and Lemma 13.34, since K^{G_{i}}(ψ_{i}(B)) ≤ K^{G}(B) holds for any matrix B ∈ M(m, n, ℂG).

Recall that r_{A^{∗}}^{(2)} = (r_{A}^{(2)})^{∗}. Hence r_{AA^{∗}}^{(2)} is positive, the spectral density

functions of r_{AA^{∗}}^{(2)} and r_{A}^{(2)} differ by a factor 2 (see Lemma 2.11 (11)) and det(r_{AA^{∗}}^{(2)}) = det(r_{A}^{(2)})^{2} (see Theorem 3.14 (1) and (3)). The analogous statement holds for A_{i} for i ∈ I. Hence Theorem 13.31 follows. ⊓ ⊔

13.2.3 Colimits of Directed Systems

Throughout this subsection we consider a directed system of groups {G_{i} | i ∈ I} over the directed set I. Denote its colimit by G = colim_{i∈I} G_{i}. Let ψ_{i}: G_{i} → G for i ∈ I and ϕ_{i,j}: G_{i} → G_{j} for i, j ∈ I, i ≤ j be the structure maps. Let A ∈ M(m, n, ℤG) be a matrix. We want to show Proposition 13.35. We get in the notation above:

(1) Suppose that each G_{i} is of det ≥ 1-class. Then the same is true for G and there are matrices A_{i} ∈ M(m, n, ℤG_{i}) such that

dim_{N (G)} (ker (r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}))

= lim_{i∈I} dim_{N (G)} (ker (r_{A_{i}}: l^{2}(G_{i})^{m} → l^{2}(G_{i})^{n}));

(2) If ϕ^{G_{i}}: Wh(G_{i}) → ℝ, [A] ↦ ln(det(r_{A}^{(2)})) (see (3.92)) is trivial for each i ∈ I, then ϕ^{G}: Wh(G) → ℝ is trivial.

Proof. (1) Consider a matrix A ∈ M(m, n, ℤG). Write A_{i,j} = ∑_{g∈G} λ_{g}(i, j) · g. Let V be the set of elements g ∈ G for which λ_{g}(i, j) ≠ 0 holds for some (i, j). The set V is finite. Since G is colim_{i∈I} G_{i}, we can find i_{0} ∈ I with V ⊂ im(ψ_{i_{0}}). Choose for each g ∈ V an element \overline{g} ∈ G_{i_{0}} with ψ_{i_{0}}(\overline{g}). Define a matrix A_{i_{0}} ∈ M_{m}(ℤG_{i_{0}}) by requiring that its (i, j)-th entry is ∑_{g∈V} λ_{g}(i, j)· \overline{g}. Then A = ψ_{i_{0}}(A_{i_{0}}), where we denote by the same letter ψ_{i_{0}} the ring homomorphism M_{m}(ℤG_{i}) → M_{m}(ℤG) induced by ψ_{i_{0}} Notice that this lift A_{i_{0}} is not unique. Let A_{i} for i ≥ i_{0} be the matrix in M_{m}(ℤG_{i}) obtained from ϕ_{i_{0},i} and A_{i_{0}}. Put B = AA^{∗}, B_{i_{0}} = A_{i_{0}}A_{i_{0}}^{∗} and B_{i} = A_{i}A_{i}^{∗} for i ≥ i_{0}.

By construction we get K^{G_{i}}(B_{i}) ≤ K^{G_{i_{0}}} (B_{i_{0}}) for all i ≥ i_{0} and K^{G}(B) ≤ K^{G_{i_{0}}} (B_{i_{0}}). If we put K = K^{G_{i_{0}}} (B_{i_{0}}), we conclude from Lemma 13.33

||r_{B}^{(2)}|| ≤ K;  (13.36)
||r_{B_{i}}^{(2)}|| ≤ K  for i ≥ i_{0}.  (13.37)

Let p(x) be a polynomial with real coefficients. Then ψ_{i_{0}}(p(B_{i_{0}})) = p(B). Choose i_{1} ∈ I with i_{0} ≤ i_{1} such that for any (of the finitely many) g ∈ G_{i_{1}}, for which for some entry in p(B_{i_{1}}) the coefficient of g is non-trivial, the implication ψ_{i_{1}}(g) = 1 ⇒ g = 1 holds. Then we get

tr_{N (G_{i})}(p(B_{i})) = tr_{N (G)}(p(B))  for i ≥ i_{1}.  (13.38)

Because of (13.36), (13.37) and (13.38) the conditions of Theorem 13.19 are satisfied. Since by assumption det(r_{B_{i}}^{(2)}) ≥ 1 for all i ∈ I and r_{B} and r_{B_{i}} for i ∈ I are positive, we conclude det(r_{B}^{(2)}) ≥ 1 and

dim_{N (G)} (ker (r_{B}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}))
= lim_{i∈I} dim_{N (G)}(r_{B_{i}}^{(2)} : l^{2}(G_{i})^{m} → l^{2}(G_{i})^{n}).

Since det(r_{A}^{(2)}) = det(r_{B}^{(2)})^{2} holds by Theorem 3.14 (1) and (3) and the kernels of r_{A}^{(2)} and r_{B}^{(2)} and the kernels of r_{B_{i}}^{(2)} and r_{A_{i}}^{(2)} for i ∈ I agree, assertion (1) follows.

(2) Suppose that A ∈ M_{m}(ℤG) is invertible. We use the same notation and construction as before and apply it to B = AA^{∗} and to B^{−1} = (A^{−1})^{∗}A^{−1}. The lifts B_{i_{0}} and B_{i_{0}}^{−1} are not inverse to one another. But we can find an index i_{1} ≥ i_{0} such that B_{i} and B_{i}^{−1} are inverse to one another for all i ≥ i_{1}. By assumption det(r_{B_{i}}^{(2)}) = det(r_{B_{i}^{−1}}^{(2)}) = 1. We conclude from Theorem 13.19

as explained above that det(r_{B}^{(2)}) ≥ 1 and det(r_{B^{−1}}^{(2)}) ≥ 1 hold. Theorem 3.14 (1) and (3) implies det(r_{B}^{(2)}) = 1 and hence det(r_{A}^{(2)}) = 1. This finishes the proof of Theorem 13.35. ⊓ ⊔

13.2.4 Amenable Extensions

This subsection is devoted to the proof of

Proposition 13.39. Let H ⊂ G be a subgroup such that H is of det ≥ 1class and G/H is an amenable discrete homogeneous space. Then G is of det ≥ 1-class.

Let (X, d) be a metric space such that any bounded set consists of finitely many elements. Define for K ≥ 0 and A ⊂ X the set N_{K}(A) = B_{K}(A) ∩ B_{K}(X − A). A nested sequence of finite subsets E_{1} ⊂ E_{2} ⊂ E_{2} ⊂ … of X is called an amenable exhaustion if X = ⋃_{n≥1} E_{n} and for all K > 0 and ϵ > 0 there is an integer n(K, ϵ) such that |N_{K}(E_{n})| ≤ ϵ·|E_{n}| holds for n ≥ n(K, ϵ).

Lemma 13.40. Let H ⊂ G be a subgroup such that G/H is an amenable discrete homogeneous space. Then it possesses an amenable exhaustion.

Proof. By assumption we can find for positive integers n, K a subset A_{n,K} with |N_{K}(A_{n,K})| ≤ \frac{1}{n} · |A_{n,K}|. Since G acts transitively on G/H and the metric is G-invariant, we may assume after translation that the element 1H ∈ G/H belongs to A_{n,K}. We construct the desired exhaustion (E_{n})_{n≥1} inductively over n. Put E_{1} = A_{1,1}. Suppose that we have already constructed E_{1}, E_{2}, …, E_{n} with |N_{k}(E_{k})| ≤ \frac{1}{k} · |E_{k}| for k = 1, 2, …, n. Choose a positive integer d such that E_{n} ⊂ B_{d}(1H) and d ≥ n + 1 holds. Define E_{n+1} := E_{n} ∪ B_{n}(1H) ∪ A_{n+1,2d}. We conclude B_{d}(E_{n+1}) ⊂ B_{2d}(A_{n+1,2d}) from the triangle inequality and 1H ∈ A_{n+1,2d}. We have B_{d}(G/H − E_{n+1}) ⊂ B_{2d}(G/H−A_{n+1,2d}). This implies N_{n+1}(E_{n+1}) ⊂ N_{d}(E_{n+1}) ⊂ N_{2d}(A_{n+1,2d}). We conclude

|N_{n+1}(E_{n+1})| ≤ |N_{2d}(A_{n+1,2d})| ≤ \frac{1}{n + 1} · |A_{n+1,2d}| ≤ \frac{1}{n + 1} · |E_{n+1}|.

Since B_{n}(1G) ⊂ E_{n+1} by construction, we have G/H = ⋃_{n≥1} E_{n}. ⊓ ⊔

Fix an amenable exhaustion E_{1} ⊂ E_{2} ⊂ E_{2} ⊂ … of G/H. For B ∈ M_{d}(N (H)) put

tr_{n}(B) := \frac{tr_{N (H)}(B)}{|E_{n}|}.

Consider A ∈ M_{d}(N (G)) such that the associated morphism r_{A}: l^{2}(G)^{d} → l^{2}(G)^{d} of Hilbert N (G)-modules is positive. Let q_{n}: l^{2}(G)^{d} → l^{2}(G)^{d} be ⨁_{i=1}^{d} q′_{n}, where q′_{n}: l^{2}(G) → l^{2}(G) is the orthogonal projection onto the subspace of l^{2}(G) generated by the preimage of E_{n} under the canonical projection pr: G → G/H. Notice that q_{n} is not G-equivariant in general, but is H-equivariant. The image of q_{n} is a finitely generated Hilbert N (H)- module, namely, after a choice of a preimage \overline{x} of each element x in E_{n} under the projection pr: G → G/H we get an obvious isometric H-isomorphism u_{n}: im(q_{n}) ⟶^{≅} l^{2}(H)^{d|E_{n}|}. Let A_{n} ∈ M_{d|E_{n}|}(N (H)) be the matrix uniquely determined by the property that r_{A_{n}}: l^{2}(H)^{d|E_{n}|} → l^{2}(H)^{d|E_{n}|} is the morphism u ∘ q_{n} ∘ r_{A} ∘ u^{−1}. In the sequel we identify im(q_{n}) and l^{2}(H)^{d|E_{n}|} by u. In particular r_{A_{n}} = q_{n}r_{A}q_{n} and r_{A_{n}} is positive. Since ||q_{n}|| ≤ 1 holds for any projection, we conclude

Lemma 13.41. We have ||r_{A_{n}}|| ≤ ||r_{A}|| for all n ≥ 1.

Lemma 13.42. Consider a matrix A ∈ M_{d}(N (G)). Let p be a polynomial with real coefficients. Then

tr_{N (G)}(p(A)) = lim_{n→∞} tr_{n}(p(A_{n})).

Proof. By linearity it suffices to treat the case p(x) = x^{s} for some positive integer s. For k = 1, 2, …, d let e_{k} ∈ l^{2}(G)^{d} be the element, whose components are zero except for the k-th entry, which is the unit element of G considered as element in l^{2}(G). Recall that pr: G → G/H denotes the canonical projection.

Fix g ∈ G with pr(g) ∈ E_{n} and k ∈ {1, 2, …, d}. Since q_{n}(ge_{k}) = ge_{k} and q_{n} is selfadjoint, we get

⟨r_{A_{n}}^{s}(ge_{k}), ge_{k}⟩ = ⟨(q_{n}r_{A}q_{n})^{s}(ge_{k}), ge_{k}⟩ = ⟨r_{A}q_{n}r_{A}q_{n} … q_{n}r_{A}(ge_{k}), ge_{k}⟩.

We have the telescope sum

r_{A}q_{n}r_{A}q_{n} … q_{n}r_{A} = r_{A}^{s} − r_{A}(1 − q_{n})r_{A}^{s−1} − r_{A}q_{n}r_{A}(1 − q_{n})r_{A}^{s−2}
− … − r_{A}q_{n}r_{A}q_{n} … r_{A}(1 − q_{n})r_{A}.

This implies

|⟨r_{A}^{s}(ge_{k}), ge_{k}⟩ − ⟨r_{A_{n}}^{s}(ge_{k}), ge_{k}⟩|
≤ ∑_{i=1}^{s−1} |⟨(1 − q_{n})r_{A}^{i}(ge_{k}), (r_{A}^{∗}q_{n})^{s−i}(ge_{k})⟩|

≤ ∑_{i=1}^{s−1} ||(1 − q_{n})r_{A}^{i}(ge_{k})||_{l^{2}} · ||r_{A}^{∗}||^{s−i}.  (13.43)

13.2 Partial Proofs of the Approximation and Determinant Conjecture 475

Fix ϵ > 0 and i ∈ {1, 2, …, s}. For k = 1, 2, …, d we write

r_{A}^{i}(e_{k}) = (∑_{g∈G} λ_{g}(k, l) · g)_{l=1}^{d} ∈ l^{2}(G)^{d}.

Since this element belongs to l^{2}(G)^{d}, we can choose R = R(ϵ) ≥ 0 such that

||r_{A}^{i}(e_{k}) − (  ∑_{g∈G,pr(g)∈B_{R}(1H)} λ_{g}(k, l) · g)_{l=1}^{d}||_{l^{2}} ≤ ϵ.

Suppose that g_{0} ∈ G satisfies B_{R}(pr(g_{0})) ⊂ E_{n}. Then

{g ∈ G, pr(g_{0}g) ∉ E_{n}} ⊂ {g ∈ G, pr(g) ∉ B_{R}(1H)}.

We conclude ||(1 − q_{n})r_{A}^{i}(g_{0}e_{k})||_{l^{2}}

= ||(1 − q_{n}) (∑_{g∈G} λ_{g}(k, l) · g_{0}g)_{l=1}^{d}||_{l^{2}}
= ||(  ∑_{g∈G,pr(g_{0}g)∉E_{n}} λ_{g}(k, l) · g_{0}g)_{l=1}^{d}||_{l^{2}}
≤ ||(  ∑_{g∈G,pr(g)∉B_{R}(1H)} λ_{g}(k, l) · g_{0}g)_{l=1}^{d}||_{l^{2}}
= ||r_{A}^{i}(g_{0}e_{k}) − (  ∑_{g∈G,pr(g)∈B_{R}(1H)} λ_{g}(k, l) · g_{0}g)_{l=1}^{d}||_{l^{2}}
= ||r_{A}^{i}(e_{k}) − (  ∑_{g∈G,pr(g)∈B_{R}(1H)} λ_{g}(k, l) · g)_{l=1}^{d}||_{l^{2}}
≤ ϵ.  (13.44)

Choose for any x ∈ E_{n} an element \overline{x} ∈ G with pr(\overline{x}) = x. We conclude from (13.43) and (13.44)

|tr_{N (G)}(A^{s}) − tr_{n}(A_{n}^{s})|

= |\frac{1}{|E_{n}|} ∑_{k=1}^{d} ∑_{x∈E_{n}}⟨r_{A}^{s}(\overline{x}e_{k}), \overline{x}e_{k}⟩ − ⟨r_{A_{n}}^{s}(\overline{x}e_{k}), \overline{x}e_{k}⟩|

≤ \frac{1}{|E_{n}|} ∑_{k=1}^{d} ∑_{x∈E_{n}} |⟨r_{A}^{s}(\overline{x}e_{k}), \overline{x}e_{k}⟩ − ⟨r_{A_{n}}^{s}(\overline{x}e_{k}), \overline{x}e_{k}⟩|

≤ \frac{1}{|E_{n}|} ∑_{k=1}^{d} ∑_{x∈E_{n}} ∑_{j=1}^{s−1} ||(1 − q_{n})r_{A}^{j}(\overline{x}e_{k})||_{l^{2}} · ||r_{A}^{∗}||^{s−j}

≤ \frac{1}{|E_{n}|} ∑_{k=1}^{d} ∑_{x∈E_{n},x∉N_{R}(E_{n})} ∑_{j=1}^{s−1} ||(1 − q_{n})r_{A}^{j}(\overline{x}e_{k})||_{l^{2}} · ||r_{A}^{∗}||^{s−j}

+\frac{1}{|E_{n}|} ∑_{k=1}^{d} ∑_{x∈E_{n},x∈N_{R}(E_{n})} ∑_{j=1}^{s−1} ||(1 − q_{n})r_{A}^{j}(\overline{x}e_{k})||_{l^{2}} · ||r_{A}^{∗}||^{s−j}

≤ \frac{1}{|E_{n}|} ∑_{k=1}^{d} ∑_{x∈E_{n},x∉N_{R}(E_{n})} ∑_{j=1}^{s−1} ϵ · ||r_{A}^{∗}||^{s−j}

+\frac{|N_{R}(E_{n})|}{|E_{n}|} · ds · ||1 − q_{n}|| · max\{||r_{A}||^{j} · ||r_{A}^{∗}||^{s−j} | j = 1, 2 …, s − 1\}

≤ ϵ · ds · max{||r_{A}^{∗}||^{j} | j = 1, 2, …, s}

+\frac{|N_{R}(E_{n})|}{|E_{n}|} · ds · 2 · max\{||r_{A}||^{j} · ||r_{A}^{∗}||^{s−j} | j = 1, 2 …, s − 1\}.

Put

C_{1} := ds · max{||r_{A}^{∗}||^{j} | j = 1, 2, …, s};
C_{2} := ds · 2 · max{||r_{A}||^{j} · ||r_{A}^{∗}||^{s−j} | j = 1, 2 …, s − 1}.

Then C_{1} and C_{2} are independent of ϵ and n and we get

|tr_{N (G)}(A^{s}) − tr_{n}(A_{n}^{s})| ≤ C_{1} · ϵ + C_{2} · \frac{|N_{R}(E_{n})|}{|E_{n}|}.

Since (E_{n})_{n≥1} is an amenable exhaustion, we can find n(ϵ) such that \frac{|N_{R}(E_{n})|}{|E_{n}|} ≤ ϵ for n ≥ n(ϵ) holds. Hence we get for n ≥ n(ϵ)

|tr_{N (G)}(A^{s}) − tr_{n}(A_{n}^{s})| ≤ C_{1} · ϵ + C_{2} · ϵ.

Since ϵ > 0 was arbitrary, Lemma 13.42 follows. ⊓ ⊔

Now we can finish the proof of Proposition 13.39. Namely, because of Lemma 13.41 and Lemma 13.42 the assumptions of Theorem 13.19 are satisfied and Proposition 13.39 follows.

13.2.5 Quotients with Finite Kernels

This subsection is devoted to the proof of

Lemma 13.45. Let 1 → K → G ⟶^{p} Q → 1 be a group extension such that K is finite. Let f: M → N be a morphism of finitely generated Hilbert N (Q)-modules.

(1) The Hilbert space res_{p} M together with the G-action obtained from the given Q-action and p: G → Q is a finitely generated Hilbert N (G)- module. The morphism f yields a morphism of finitely generated Hilbert N (G)-modules res_{p} f: res_{p} M → res_{p} N;

(2) If M = N, then the von Neumann traces of f and res_{p} f satisfy tr_{N (G)}(res_{p} f) = \frac{1}{|K|} · tr_{N (Q)}(f);

(3) We get for the spectral density functions of f and res_{p} f

F^{G}(res_{p} f) = \frac{1}{|K|} · F^{Q}(f);

(4) dim_{N (G)}(ker(res_{p} f)) = \frac{1}{|K|} · dim_{N (Q)}(ker(f));

(5) The Novikov-Shubin invariants of f and res_{p} f satisfy

α(res_{p} f) = α(f);

(6) The Fuglede-Kadison determinants of f and res_{f} satisfy

det_{N (G)}(res_{p} f) = (det_{N (Q)}(f))^{\frac{1}{|K|}} ;

(7) The group Q is of det ≥ 1-class, if G is of det ≥ 1-class.

Proof. (1) Obviously it suffices to check this for M = l^{2}(Q). Let N_{K} ∈ ℂG be the element ∑_{k∈K} k. Then right multiplication with |K|^{−1} · N_{K} yields an orthogonal projection r_{|K|^{−1}·N_{K}}: l^{2}(G) → l^{2}(G). We get an isometric G-isomorphism v: res_{p} l^{2}(Q) → im(r_{|K|^{−1}·N_{K}}) by sending ∑_{q∈Q} λ_{q} · q to ∑_{q∈Q} λ_{q} · |K|^{−1/2} · ∑_{g∈p^{−1}(q)} g.

(2) Obviously it suffices to treat the case M = N = l^{2}(Q). Denote by e_{Q} ∈ Q and e_{G} ∈ G the unit element. Let f(e_{Q}) = ∑_{q∈Q} λ_{q} ·q. If i: im(r_{|K|^{−1}·N_{K}}) → l^{2}(G) is the inclusion, we get

i ∘ v ∘ res_{p} f ∘ v^{−1} ∘ r_{|K|^{−1}·N_{K}} (e_{G}) = ∑_{q∈Q} \frac{λ_{q}}{|K|} · ∑_{g∈p^{−1}(q)} g.

This implies

tr_{N (Q)}(f) = ⟨f(e_{Q}), e_{Q}⟩_{l^{2}(Q)}
= λ_{e_{Q}}
= |K| · ⟨i ∘ v ∘ f ∘ v^{−1} ∘ r_{|K|^{−1}·N_{K}} (e_{G}), e_{G}⟩_{l^{2}(G)}
= |K| · tr_{N (G)}(res_{p} f).

(3) This follows from assertions (1) and (2). (4), (5) and (6) follow from assertions (1) and (3).

(7) Let A ∈ M_{d}(ℤQ) be a matrix. It induces a morphism r_{A}^{(2)} : l^{2}(Q)^{d} → l^{2}(Q)^{d} of finitely generated Hilbert N (Q)-modules by right multiplication. We have to show det(r_{A}^{(2)}) ≥ 1 provided that G is of det ≥ 1-class.

Let n be any positive integer. We get a morphism res_{p} r_{A^{n}}^{(2)} : res_{p} l^{2}(Q)^{d} → res_{p} l^{2}(Q)^{d} of finitely generated Hilbert N (G)-modules. Notice that we have the orthogonal sum decomposition

l^{2}(G) = im(r_{|K|^{−1}·N_{K}}) ⊕ im(r_{1−|K|^{−1}·N_{K}}).

Consider the morphism

(v^{d} ∘ r_{A^{n}}^{(2)} ∘ (v^{−1})^{d}) ⊕ id_{im(r_{1−|K|^{−1}·N_{K}})}^{d}: l^{2}(G)^{d} → l^{2}(G)^{d},

where the isomorphism v and the orthogonal projection r_{|K|^{−1}·N_{K}} are taken from the proof of assertion (1). We conclude from Assertion (6), Theorem 3.14 (1) and Lemma 3.15 (7)

(det_{N (Q)}(r_{A}^{(2)}))^{n}
= det_{N (Q)}(r_{A^{n}}^{(2)})
= \frac{1}{|K|} · det_{N (G)}(res_{p} r_{A^{n}}^{(2)})
= \frac{1}{|K|} · det_{N (G)} (v^{d} ∘ r_{A^{n}}^{(2)} ∘ (v^{−1})^{d}) · det_{N (G)} ((id_{im(r_{1−|K|^{−1}·N_{K}})})^{d})
= \frac{1}{|K|} · det_{N (G)} (v^{d} ∘ r_{A^{n}}^{(2)} ∘ (v^{−1})^{d} ⊕ (id_{im(r_{1−|K|^{−1}·N_{K}})})^{d}) . (13.46)

For u = ∑_{q∈Q} λ_{q} · q in ℤQ let \overline{u} ∈ ℤG be the element ∑_{q∈Q} λ_{q} · ∑_{g∈p^{−1}(q)} g. Define B = (b_{i,j})_{i,j} ∈ M_{d}(ℤG) to be the matrix obtained from A^{n} = (a_{i,j})_{i,j} by putting b_{i,j} = \overline{a_{i,j} − 1}. One easily checks

\frac{1}{|K|} · r_{|K|·I_{d}+B}^{(2)} = v^{d} ∘ r_{A^{n}}^{(2)} ∘ (v^{−1})^{d} ⊕ (id_{im(r_{1−|K|^{−1}·N_{K}})})^{d},

where I_{d} is the identity matrix in M_{d}(ℤG). Notice that |K| · I_{d} + B lies in M_{d}(ℤG) so that by assumption det_{N (G)} (r_{|K|·I_{d}+B}^{(2)}) ≥ 1 holds. Theorem 3.14 (1) implies

det_{N (G)} (v^{d} ∘ r_{A^{n}}^{(2)} ∘ (v^{−1})^{d} ⊕ (id_{im(r_{1−|K|^{−1}·N_{K}})})^{d})
= det_{N (G)} (\frac{1}{|K|} · r_{|K|·I_{d}+B}^{(2)})
= \frac{1}{|K|^{d}} · det_{N (G)} (r_{|K|·I_{d}+B}^{(2)})
≥ \frac{1}{|K|^{d}} .  (13.47)

We conclude from (13.46) and (13.47) that for any positive integer n

det_{N (Q)}(r_{A}^{(2)}) ≥ |K|^{−(d+1)/n}

holds. Hence det_{N (Q)}(r_{A}^{(2)}) ≥ 1. This finishes the proof of Lemma 13.45. ⊓ ⊔ Finally we can give the proof of Theorem 13.3.

Proof. (1) Because of Lemma 13.4 it suffices to prove assertion (3) of Lemma 13.4. The natural homomorphism i: G → lim_{i∈I} G/G_{i} is injective because of ⋂_{i≥1} G_{i} = {1}. We conclude

dim_{N (G)} (ker (r_{A}^{(2)})) = dim_{N (lim_{i∈I} G/G_{i})} (ker (r_{i_{∗}A}^{(2)}))

from Lemma 1.24. Since the composition of i: G → lim_{i∈I} G/G_{i} with the structure map ψ_{i}: lim_{i∈I} G/G_{i} → G/G_{i} is the canonical projection G → G/G_{i}, the claim follows from Theorem 13.31 (2).

(3) This follows from Theorem 3.14 (6), Theorem 13.31 (2), Proposition 13.35 Proposition 13.39 and Lemma 13.45 (7).

(2) follows from assertions (1) and (3) as soon as we have shown that the trivial group is of det ≥ 1-class. This has already been done in Lemma 13.12.

(4) This follows from Theorem 3.14 (6), Theorem 13.31 (2) and Proposition 13.35. This finishes the proof of the main result of this chapter, namely, Theorem 13.3. ⊓ ⊔

13.3 Variations of the Approximation Results

In this sections we discuss some variations of the approximation results above (Theorem 13.48, Theorem 13.49 and Theorem 13.50) whose proofs are essentially modifications of the proof of Theorem 13.3.

We begin with a result of Dodziuk and Mathai [149, Theorem 0.1]. We will state it as presented by Eckmann [163], where a different proof is given.

Let G be an amenable group and let X be a finite free simplicial Gcomplex. We get a closed fundamental domain F as follows. For each simplex e of G\X choose a simplex ê in X, which lies in the preimage of e under the projection p: X → G\X and is mapped bijectively onto e under p. Let F be the union ⋃_{e} ê, where e runs through the simplices of G\X. This is a sub-CW -complex of X. Let E_{m} ⊂ X for m = 1, 2, … be a sub-CW -complex of X, which is the union of N_{m} translates (by elements of G) of F. Let ∂F be the topological boundary of F. Denote by ∂N_{m} the number of translates of F which meet ∂E_{m}. We call (E_{m})_{m≥1} a Følner exhaustion if E_{1} ⊂ E_{2} ⊂ …, X = ⋃_{m≥1} E_{m} and lim_{m→∞} \frac{∂N_{m}}{N_{m}} = 0. Since G is assumed to be amenable, such a Følner exhaustion exists. Let b_{p}(E_{m}) be the (ordinary) Betti number of the finite CW -complex E_{m}.

Theorem 13.48. We get for the amenable group G and the finite free simplicial G-complex X with Følner exhaustion (E_{m})_{m≥1}

b_{p}^{(2)}(X; N (G)) = lim_{m→∞} \frac{b_{p}(E_{m})}{N_{m}}.

The proof of Theorem 13.48 is analogous to the one of Proposition 13.39. The modified versions of Lemma 13.41 and Lemma 13.42 are proved in [149, Lemma 2.2 and Lemma 2.3].

Next we explain two results of Farber [183, Theorem 0.3 and Theorem 0.4].

Theorem 13.49. Let G be a group with a sequence of (not necessarily normal) subgroups of finite index [G : G_{i}]

G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ G_{3} ⊃ …

such that ⋂_{i≥0} G_{i} = {1}. Let n_{i} be the number of subgroups in G which are conjugate to G_{i}. For a given g ∈ G let n_{i}(g) be the number of subgroups of G which are conjugate to G_{i} and contain g. Suppose that for any g ∈ G with g ≠ 1

lim_{i→∞} \frac{n_{i}(g)}{n_{i}} = 0.

Let X be a free G-CW -complex of finite type. Then

b_{p}^{(2)}(X; N (G)) = lim_{i→∞} \frac{b_{p}(G_{i}\X)}{[G : G_{i}]}.

If each G_{i} is normal in G, then the condition lim_{i→∞} \frac{n_{i}(g)}{n_{i}} = 0 is obviously satisfied and Theorem 13.49 reduces to [328, Theorem 0.1] and can be viewed as a special case of Theorem 13.3.

Notice that one can construct out of the given nested sequence G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ … a new one G = G′_{0} ⊃ G′_{1} ⊃ G′_{2} ⊃ … such that each G′_{i} ⊂ G is a normal subgroup of finite index and ⋂_{i∈I} G′_{i} = {1}. Namely, take G′_{i} = ⋂_{g∈G} g^{−1}G_{i}g. So the group G appearing in Theorem 13.49 is residually finite.

The proof of Theorem 13.49 is an application of Theorem 13.19. One has to prove the relevant versions of Lemma 13.33 and Lemma 13.34. Lemma 13.33 carries over directly. For Lemma 13.34 one has to use the following easily verified equation

\frac{1}{[G : G_{i}]} · tr_{ℂ}(r_{g}) = \frac{n_{i}(g)}{n_{i}},

where g ∈ G and r_{g}: ℂ[G_{i}\G] → ℂ[G_{i}\G] is given by right multiplication with g. Notice that det(r_{A_{i}}) ≥ 1 by the argument appearing in the proof of Theorem 13.3 (2).

The proof of the next result (and of some generalizations of it) can be found in [183, page 360].

Theorem 13.50. Let G be a group with a sequence of normal subgroups of finite index [G : G_{i}]

G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ G_{3} ⊃ …

such that ⋂_{i≥0} G_{i} = {1}. Let K ⊂ ℂ be an algebraic number field which is closed under complex conjugation. Let R ⊂ K be the ring of algebraic integers in K. Consider a unitary G-representation ρ: G → M_{m}(R). Let X be a free G-CW -complex of finite type. Denote by V^{i} the flat vector bundle over G_{i}\X given by the representation ρ restricted to G_{i}. Then

b_{p}^{(2)}(X; N (G)) = lim_{i→∞} \frac{dim_{ℂ}(H_{p}(G_{i}\X; V^{i}))}{dim_{ℂ}(V^{i}) · [G : G_{i}]}.

If one takes V to be ℂ with the trivial G-action, Theorem 13.50 reduces to [328, Theorem 0.1].

One can also deal with the signature and the L^{2}-signature instead of the Betti numbers and the L^{2}-Betti numbers. The next result is taken from [347], where also the (rather obvious) notions of a finite Poincaré pair (X, Y), of the L^{2}-signature of a G-covering (\overline{X}, \overline{Y}) and of the signature of (X, Y) are explained.

Theorem 13.51. Let G be a group with a sequence of normal subgroups of finite index [G : G_{i}]

G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ G_{3} ⊃ …

such that ⋂_{i≥0} G_{i} = {1}. Let (X, Y) be a 4n-dimensional finite Poincaré pair. Let p: \overline{X} → X be a G-covering of X and \overline{Y} be the preimage of Y under p. Denote by (X_{i}, Y_{i}) the 4n-dimensional finite Poincaré pair given by (G_{i}\\overline{X}, G_{i}\\overline{Y}). Then

sign^{(2)}(\overline{X}, \overline{Y}) = lim_{i→∞} \frac{sign(X_{i}, Y_{i})}{[G : G_{i}]}.

Some generalizations and variations of Theorem 13.51 and a version for signatures of Theorem 13.48 can be found in [347]. From the Atiyah-Patodi-Singer index theorem [11, Theorem 4.14] and its L^{2}-version due to Cheeger and Gromov [105, (0.9)] it is clear that Theorem 13.51 implies a similar result for eta-invariants and L^{2}-eta-invariants of the spaces \overline{Y} and Y_{k}. Such a convergence result has been proved without the assumption that Y appears in a pair (X, Y) in [500, Theorem 3.12]. Namely, if M is a closed (4n − 1)- dimensional manifold and p: \overline{M} → M is a G-covering, then we get for M_{i} = G_{i}\M

η^{(2)}(\overline{M}) = lim_{i→∞} \frac{η(M_{i})}{[G : G_{i}]}.

Question 13.52. (Approximating Fuglede-Kadison determinants by determinants). Is there also an approximation results for the (generalized) Fuglede-Kadison determinant?

This seems to be a difficult question and to involve more input. At least we can show the following very special result.

Lemma 13.53. Let p ∈ ℚ[ℤ] be an element different from zero and let f: ℚ[ℤ] ⟶ ℚ[ℤ] be the ℚ[ℤ]-map given by multiplication with p. Write p as a product

p = c · t^{k} · ∏_{i=1}^{r}(t − a_{i})

for non-zero complex numbers c, a_{1}, …, a_{r} and an integer k. Then we get

ln(det(f^{(2)})) = lim_{n→∞} \frac{ln(det(f[n]))}{n} = ln(|c|) + ∑_{\begin{matrix}i=1,2,...,r \\ |a_{i}|>1 \end{matrix}} ln(|a_{i}|),

where det(f^{(2)}) is the Fuglede-Kadison-determinant of the morphism of Hilbert N (ℤ)-modules f^{(2)} : l^{2}(ℤ) → l^{2}(ℤ) obtained from f by tensoring with l^{2}(ℤ) over ℚ[ℤ], and det(f[n]) is the Fuglede-Kadison determinant of the morphism of Hilbert N (1)-modules f[n]: ℂ[ℤ/n] → ℂ[ℤ/n] obtained from f by taking the tensor product with ℂ[ℤ/n] over ℚ[ℤ].

Proof. We obtain from (3.23)

ln(det(f^{(2)})) = ln(|c|) + ∑_{\begin{matrix}i=1,2,...,r \\ |a_{i}|>1 \end{matrix}} ln(|a_{i}|).  (13.54)

Next we compute det(f[n]). The regular ℤ/n-representation ℂ[ℤ/n] is the orthogonal direct sum ⨁_{j=0}^{n−1} V_{j} of one-dimensional ℤ/n-representations V_{j}, where the generator of ℤ/n acts on V_{j} = ℂ by multiplication with ζ_{n}^{j} for ζ_{n} = exp(2πi/n). Then f[n] is the direct sum ⨁_{j=0}^{n−1} f[n]_{j}, where f[n]_{j}: V_{j} ⟶ V_{j} is multiplication with p(ζ_{n}^{j}). We get from Example 3.12 and Lemma 3.15 (7)

det(f[n]) = ∏_{j=0}^{n−1} det(f[n]_{j});

det(f[n]_{j}) = \{ \begin{matrix}|c| · ∏_{i=1}^{r} |ζ_{n}^{j} − a_{i}| & if ζ_{n}^{j} ∉ \{a_{1}, a_{2}, …, a_{r}\} \\ 1 & otherwise \end{matrix} ;

det(f[n]) = ∏_{\begin{matrix}j=0,...,(n−1) \\ ζ_{n}^{j}∉\{a_{1},a_{2},...,a_{r}\} \end{matrix}} |c| · ∏_{i=1}^{r} |ζ_{n}^{j} − a_{i}|.  (13.55)

Since n − r ≤ #{j ∈ {0, 1, …, (n − 1)} | ζ_{n}^{j} ∉ {a_{1}, a_{2}, …, a_{r}}} ≤ n and lim_{n→∞} \frac{n−r}{n} = lim_{n→∞} \frac{n}{n} = 1 holds, we get

lim_{n→∞} \frac{1}{n} · ln (  ∏_{\begin{matrix}j=0,...,(n−1) \\ ζ_{n}^{j}∉\{a_{1},a_{2},...,a_{r}\} \end{matrix}} |c|) = ln(|c|).  (13.56)

Notice that c is the highest coefficient of p and hence a rational number. We conclude from (13.54), (13.55) and (13.56) that we can assume without loss of generality c = 1.

Equation (13.55) implies

det(f[n]) = (  ∏_{\begin{matrix}j=0,1,...,(n−1) \\ ζ_{n}^{j}∈\{a_{1},a_{2},...a_{r}\} \end{matrix}} ∏_{\begin{matrix}i=1,...,r \\ ζ_{n}^{j}≠a_{i} \end{matrix}} |ζ_{n}^{j} − a_{i}|)^{−1} · ∏_{j=0}^{n−1} ∏_{\begin{matrix}i=1,...,r \\ a_{i}^{n}≠1 \end{matrix}} |ζ_{n}^{j} − a_{i})|

· ∏_{j=0}^{n−1} ∏_{\begin{matrix}i=1,...,r \\ a_{i}^{n}=1,a_{i}≠ζ_{n}^{j} \end{matrix}} |ζ_{n}^{j} − a_{i})| .

= ( ∏_{\begin{matrix}l=1,...,r \\ a_{l}^{n}=1 \end{matrix}} ∏_{\begin{matrix}i=1,...,r \\ a_{l}≠a_{i} \end{matrix}} |a_{l} − a_{i}|)^{−1} · ∏_{\begin{matrix}i=1,...,r \\ a_{i}^{n}≠1 \end{matrix}} ∏_{j=0}^{n−1} |ζ_{n}^{j} − a_{i})|

· ∏_{\begin{matrix}i=1,...,r \\ a_{i}^{n}=1 \end{matrix}} ∏_{\begin{matrix}j=0,1,...,(n−1) \\ a_{i}≠ζ_{n}^{j} \end{matrix}} |ζ_{n}^{j} − a_{i})| .  (13.57)

From the identities of polynomials with complex coefficients

∏_{j=0}^{n−1}(t − ζ_{n}^{j}) = t^{n} − 1;

∏_{j=1}^{n−1}(t − ζ_{n}^{j}) = ∑_{j=0}^{n−1} t^{j},

we conclude

∏_{\begin{matrix}i=1,...,r \\ a_{i}^{n}≠1 \end{matrix}} ∏_{j=0}^{n−1} |ζ_{n}^{j} − a_{i})| = ∏_{\begin{matrix}i=1,...,r \\ a_{i}^{n}≠1 \end{matrix}} |a_{i}^{n} − 1|;  (13.58)

∏_{\begin{matrix}i=1,...,r \\ a_{i}^{n}=1 \end{matrix}} ∏_{\begin{matrix}j=0,1,...,(n−1) \\ a_{i}≠ζ_{n}^{j} \end{matrix}} |ζ_{n}^{j} − a_{i})| = ∏_{\begin{matrix}i=1,...,r \\ a_{i}^{n}=1 \end{matrix}} n.  (13.59)

We conclude from (13.57), (13.58) and (13.59)

\frac{ln(det(f[n]))}{n} = ∑_{\begin{matrix}i=1,...,r \\ a_{i}^{n}≠1 \end{matrix}} \frac{ln(|a_{i}^{n} − 1|)}{n} + ∑_{\begin{matrix}i=1,...,r \\ a_{i}^{n}=1 \end{matrix}} \frac{ln(n)}{n}

− ∑_{\begin{matrix}l=1,...,r \\ a_{l}^{n}=1 \end{matrix}} ∑_{\begin{matrix}i=1,...,r \\ a_{l}≠a_{i} \end{matrix}} \frac{ln(|a_{l} − a_{i}|)}{n}.  (13.60)

We have

lim_{n→∞} ∑_{\begin{matrix}l=1,...,r \\ a_{l}^{n}=1 \end{matrix}} ∑_{\begin{matrix}i=1,...,r \\ a_{l}≠a_{i} \end{matrix}} \frac{ln(|a_{l} − a_{i}|)}{n} = 0;  (13.61)

lim_{n→∞} \frac{ln(|a_{i}^{n} − 1|)}{n} = ln(|a_{i}|) if |a_{i}| > 1;  (13.62)
lim_{n→∞} \frac{ln(|a_{i}^{n} − 1|)}{n} = 0  if |a_{i}| < 1;  (13.63)
lim_{n→∞} \frac{ln(n)}{n} = 0.  (13.64)

Notice that p is by assumption a polynomial with rational coefficients and hence each root a_{i} is an algebraic integer. The main input, which is not needed when dealing with Betti numbers or signatures, is the following result taken from [472, Corollary B1 on page 30]. Namely, there is a constant D > 0 (depending only on p but not on n) such that for n ≥ 2 and a_{i} with a_{i}^{n} ≠ 1 the equation

|a_{i}^{n} − 1| ≥ n^{−D}  (13.65)

holds. Hence we get for n ≥ 2 from (13.65) for a_{i} with a_{i}^{n} ≠ 1 and |a_{i}| = 1

\frac{−D · ln(n)}{n} ≤ \frac{ln(|a_{i}^{n} − 1|)}{n} ≤ \frac{ln(2)}{n}.  (13.66)

Equations (13.64) and (13.66) imply

lim_{n→∞} ∑_{\begin{matrix}i=1,...,r \\ a_{i}^{n}≠1,|a_{i}|=1 \end{matrix}} \frac{ln(|a_{i}^{n} − 1|)}{n} = 0.  (13.67)

We conclude from equations (13.60), (13.61), (13.62), (13.63), (13.64) and (13.67)

lim_{n→∞} \frac{det(f[n])}{n} = ∑_{i=1,2,...r,|a_{i}|>1} ln(|a_{i}|).  (13.68)

Now Lemma 13.53 follows from (13.54) and (13.68). ⊓ ⊔

Next we show that Lemma 13.53 is not true in general for polynomials p ∈ ℂ[ℤ] with complex coefficients. Hence the condition that the coefficients are rational (or at least algebraic) is essential.

Example 13.69. Fix a sequence of positive integers 2 = n_{1} < n_{2} < n_{3} < … such that for each k ≥ 1

\frac{1}{n_{k+1}} ≤ \frac{1}{2^{k}n_{k} exp(n_{k})}.  (13.70)

Choose a sequence of positive integers (m_{k})_{k} with m_{1} = 1 such that for all k ≥ 1

0 < \frac{m_{k+1}}{n_{k+1}} − \frac{m_{k}}{n_{k}} ≤ \frac{1}{n_{k+1}} .  (13.71)

Namely, define m_{k+1} inductively as the smallest integer for which 0 < \frac{m_{k+1}}{n_{k+1}} − \frac{m_{k}}{n_{k}} holds. We get from (13.70) and (13.71)

∑_{l=k}^{∞} (\frac{m_{l+1}}{n_{l+1}} − \frac{m_{l}}{n_{l}} ) ≤ ∑_{l=k+1}^{∞} \frac{1}{2^{l}n_{l} exp(n_{l})}

≤ \frac{exp(−n_{k})}{n_{k}} · ∑_{l=k+1}^{∞} 2^{−l}

≤ \frac{exp(−n_{k})}{n_{k}}.

This shows that we can define a real number

s = lim_{k→∞} \frac{m_{k}}{n_{k}}

and that the following holds

0 ≤ n_{k}s − m_{k} ≤ exp(−n_{k}).  (13.72)

Put a = exp(2πis). The number s is not rational (in fact it is not even algebraic) and hence a^{n} ≠ 1 for all n ≥ 1, since otherwise (13.72) would imply that \frac{m_{k}}{n_{k}} becomes stationary for large k, a contradiction to (13.71). We conclude from the estimate | exp(iu) − 1| ≤ 2 · \sqrt{|u|} and (13.60) and (13.72)

\frac{ln(| det((t − a): ℂ[ℤ/n_{k}] → ℂ[ℤ/n_{k}])|)}{n_{k}} = \frac{ln(|a^{n_{k}} − 1|)}{n_{k}}
= \frac{ln(| exp(2πin_{k}s) − 1|)}{n_{k}}
= \frac{ln(| exp(2πi(n_{k}s − m_{k})) − 1|)}{n_{k}}
≤ \frac{ln(2 · \sqrt{2π(n_{k}s − m_{k})})}{n_{k}}
≤ \frac{ln(2 · \sqrt{2π exp(−n_{k})})}{n_{k}}
= −\frac{1}{2} + \frac{ln(2) + ln(\sqrt{2π})}{n_{k}}.

Hence the sequence \frac{ln(| det((t−a) : ℂ[ℤ/n_{k}]→ℂ[ℤ/n_{k}])|)}{n_{k}} cannot converge to zero. On the other hand we get det ((t − a): l^{2}(ℤ) ⟶ l^{2}(ℤ)) = 0 from (3.23).

We mention without proof that Lemma 13.53 can be extended to matrices A ∈ M(m, n, ℚ[ℤ]).

13.4 Miscellaneous

Some special cases of the results of this chapter have been proved in [110],

[153] and [527]. Further variations of the approximation results can be found in Farber [183], where von Neumann categories and Dixmier traces are investigated. Generalizations of some of the results of this chapter from ℚ as coefficients to the field of algebraic numbers \overline{ℚ} ⊂ ℂ can be found in [148]. Let G be a group with a sequence of normal subgroups of finite index [G : G_{i}]

G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ G_{3} ⊃ …

such that ⋂_{i≥0} G_{i} = {1}. Estimates for the individual Betti numbers

b_{p}(G_{i}\X) are given in [111] in the case that b_{p}^{(2)}(X) = 0 or that b_{p}^{(2)}(X) = 0 and α(X) = ∞^{+}.

Lemma 13.53 motivates the following question, which is linked to Question 13.52.

Question 13.73 (Approximating L^{2}-torsion by torsion). Let M be a closed Riemannian manifold and let

π_{1}(M) = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ …

be a nested sequences of normal subgroups of finite index with ⋂_{i≥0} G_{i} = {1}.

Equip M̃ and G_{i}\M̃ with the induced Riemannian metric. Let ρ_{an}^{(2)}(M̃) be the analytic L^{2}-torsion of M̃ (see Definition 3.128) and let ρ_{an}(G_{i}\M̃) be the analytic Ray-Singer torsion of G_{i}\M̃ with respect to the trivial representation ℂ (see (3.9)).

Under which circumstances does

ρ_{an}^{(2)}(M̃) = lim_{i→∞} \frac{ρ_{an}(G_{i}\M̃)}{[π_{1}(M) : G_{i}]}

hold?

Maybe there is a link between the Volume Conjecture 4.8 and Question 13.73 above.

Exercises

13.1. Show that the maps Φ^{G}: Wh(G) → ℝ and Φ^{G×ℤ}: Wh(G × ℤ) → ℝ have the same image.

13.2. Let f: U → U be a positive endomorphism of a finitely generated Hilbert N (G)-module. Show for the spectral density function F of f

dim_{N (G)}({u ∈ U | f(u) = λu}) = F (λ) − lim_{ϵ→0+} F (λ − ϵ).

13.3. Consider the situation described in the beginning of Subsection 13.2.1 for R = ℂ. Suppose that there is a constant K > 0 such that for the operator norms ||r_{A}^{(2)}||_{∞} ≤ K and ||r_{A_{i}}^{(2)}||_{∞} ≤ K hold for i ∈ I. Suppose that for any polynomial p with real coefficients we have

tr_{N (G)}(p(A)) = lim_{i∈I} tr_{i}(p(A_{i})).

Let λ ≥ 0 be a number such that there is no x ≠ 0 with r_{A}^{(2)}(x) = λ · x. Prove

F (λ) = lim_{i∈I} F_{i}(λ).

13.4. Let {G_{i} | i ∈ I} be an inverse system of normal subgroups of the group G directed by inclusion over the directed set I such that ⋂_{i∈I} G_{i} = {1}.

Suppose that G is countable. Show that there is a sequence i_{1} ≤ i_{2} ≤ i_{3} ≤ … of elements in I such that ⋂_{n=1}^{∞} G_{i_{n}} = {1}.

13.5. Let G be a group which is of det ≥ 1-class. Show for any finite subgroup H ⊂ G that its Weyl group WH = NH/H is of det ≥ 1-class. Show that the homomorphism

K_{1}(WH) → ℝ,

which sends the class of a matrix A ∈ GL_{n}(ℤ[WH]) to the Fuglede-Kadison determinant of the morphism of finitely generated Hilbert N (G)-modules r_{A}^{(2)} : l^{2}(G/H)^{n} → l^{2}(G/H)^{n} given by right multiplication with A, is trivial.

13.6. If G contains a residually finite or residually amenable subgroup of finite index, then G is residually finite or residually amenable.

13.7. Let F_{g} be a closed orientable surface of genus g ≥ 2. Put X = S^{1} × F_{g}. Construct a sequence π_{1}(X) = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ … of normal subgroups of finite index with ⋂_{i=0}^{∞} G_{i} = {1} such that

lim_{i→∞} \frac{b_{1}(G_{i}\X̃)}{[π_{1}(X) : G_{i}]} = 0,

but for any ϵ > 0

lim_{i→∞} \frac{b_{1}(G_{i}\X̃)}{[π_{1}(X) : G_{i}]^{1−ϵ}} = ∞

holds.

13.8. Let N be a closed Riemannian manifold. Consider nested sequences of normal subgroups of finite index π_{1}(S^{1}) ⊃ G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ … and π_{1}(N) ⊃ H_{0} ⊃ H_{1} ⊃ H_{2} ⊃ … with ⋂_{i≥0} H_{i} = ⋂_{i≥0} G_{i} = {1}. Fix a Riemannian metric on S^{1} and equip S^{1} × N with the product Riemannian metric. Equip each covering of S^{1} × N with the induced Riemannian metric. Show

ρ_{an}^{(2)}({ \atop S^{1} × N}) = lim_{i→∞} \frac{ρ_{an}((G_{i} × H_{i})\({ \atop S^{1} × N}))}{[π_{1}(S^{1} × N) : (G_{i} × H_{i})]}.

14. L^{2}-Invariants and the Simplicial Volume

Introduction

In this chapter we give a survey on bounded cohomology and Gromov’s notion of the simplicial volume ||M|| ∈ ℝ of a closed connected orientable manifold and discuss the following conjecture.

Conjecture 14.1 (Simplicial volume and L^{2}-invariants). Let M be an aspherical closed orientable manifold of dimension ≥ 1. Suppose that its simplicial volume ||M|| vanishes. Then M̃ is of determinant class and

b_{p}^{(2)}(M̃) = 0 for p ≥ 0;
ρ^{(2)}(M̃) = 0.

The part about the L^{2}-Betti numbers is taken from [237, section 8A on page 232], whereas the part about the L^{2}-torsion appears in [330, Conjecture 3.2].

In Section 14.1 we give a survey on the basic definitions and properties of bounded cohomology and simplicial volume. In Section 14.2 we discuss Conjecture 14.1 and give some evidence for it. Conjecture 14.1 suggests an interesting connection between two rather different notions. It is illuminating to draw conclusions from it, which partially have already been proved or whose direct proof should be easier than a possible proof of Conjecture 14.1.

To understand this chapter, it is only required to be familiar with Sections 1.2 and 3.4.

14.1 Survey on Simplicial Volume

In this section we define the notions of simplicial volume and bounded cohomology and give a survey about their basic properties.

14.1.1 Basic Definitions

Let X be a topological space and let C_{∗}^{sing}(X; ℝ) be its singular chain complex with real coefficients. Recall that a singular p-simplex of X is a continuous map σ: Δ_{p} → X, where here Δ_{p} denotes the standard p-simplex (and not the Laplace operator). Let S_{p}(X) be the set of all singular p-simplices. Then C_{p}^{sing}(X; ℝ) is the real vector space with S_{p}(X) as basis. The p-th differential ∂_{p} sends the element σ given by a p-simplex σ: Δ_{p} → X to ∑_{i=0}^{p}(−1)^{i} ·σ ∘s_{i}, where s_{i}: Δ_{p−1} → Δ_{p} is the i-th face map. Define the L^{1}-norm of an element x ∈ C_{p}^{sing}(X; ℝ), which is given by the (finite) sum ∑_{σ∈S_{p}(X)} λ_{σ} · σ, by

||x||_{1} := ∑_{σ} |λ_{σ}|.  (14.2)

Define the L^{1}-seminorm of an element y in the p-th singular homology H_{p}^{sing}(X; ℝ) := H_{p}(C_{∗}^{sing}(X; ℝ)) by

||y||_{1} := inf{||x||_{1} | x ∈ C_{p}^{sing}(X; ℝ), ∂_{p}(x) = 0, y = [x]}.  (14.3)

Notice that ||y||_{1} defines only a seminorm on H_{p}^{sing}(X; ℝ), it is possible that ||y||_{1} = 0 but y ≠ 0. The next definition is taken from [232, page 8].

Definition 14.4 (Simplicial volume). Let M be a closed connected orientable manifold of dimension n. Define its simplicial volume to be the nonnegative real number

||M|| := ||j([M])||_{1} ∈ [0, ∞)

for any choice of fundamental class [M] ∈ H_{n}^{sing}(M; ℤ) and j: H_{n}^{sing}(M; ℤ) → H_{n}^{sing}(M; ℝ) the change of coefficients map associated to the inclusion ℤ → ℝ.

Let ϕ ∈ C_{sing}^{p}(X; ℝ) := hom_{ℝ}(C_{p}^{sing}(X; ℝ), ℝ) = map(S_{p}(X), ℝ) be a singular cochain. Define its L^{∞}-norm by

||ϕ||_{∞} := sup{|ϕ(σ)| | σ ∈ S_{p}(X)}.  (14.5)

Define the L^{∞}-seminorm of an element ψ in the p-th singular cohomology H_{sing}^{p}(X; ℝ) := H^{p}(C_{sing}^{∗}(X; ℝ)) by

||ψ||_{∞} := inf{||ϕ||_{∞} | ϕ ∈ C_{sing}^{p}(X; ℝ), δ^{p}(ϕ) = 0, ψ = [ϕ]}, (14.6)

where δ^{p}: C_{sing}^{p}(X; ℝ) → C_{sing}^{p+1}(X; ℝ) is the p-th differential. Notice that ||ϕ||_{∞} and ||ψ||_{∞} can take the value ∞.

Definition 14.7 (Bounded cohomology). Let Ĉ^{∗}(X) be the subcochain complex of C_{sing}^{∗}(X; ℝ) consisting of singular cochains ϕ with ||ϕ||_{∞} < ∞. The bounded cohomology Ĥ^{p}(X) is defined to be the cohomology H^{p}(Ĉ^{∗}(X)) of Ĉ^{∗}(X). For a group G we put Ĥ^{∗}(G) := Ĥ^{∗}(BG).

Notice that Ĥ^{∗}(BG) is independent of the choice of model of BG because of homotopy invariance of bounded cohomology. Sometimes the bounded cohomology of a group is defined in terms of the obvious bounded cochain complex of the real cochain complex given by the bar-resolution. These two notions agree (see [232, page 49], [275, (3.4) and Corollary 3.6.1]).

In some situations we want to extend the definition of simplicial volume to the case where M is not compact. The locally finite homology H_{p}^{lf}(X; ℤ) of a topological space X is the homology of the chain complex C_{∗}^{lf}(X) of (formal possibly infinite) sums ∑_{σ} λ_{σ} ·σ of singular simplices with coefficients λ_{σ} ∈ ℤ which are locally finite, i.e. for any compact subset C ⊂ X the number of singular simplices σ, which meet C and satisfy λ_{σ} ≠ 0, is finite. The notion of L^{1}-norm carries over to elements in the locally finite homology if one also allows the value ∞. Recall that for an orientable manifold M without boundary of dimension n the n-th locally finite homology H_{n}^{lf}(M; ℤ) is isomorphic to ℤ and that a choice of an orientation corresponds to a choice of a generator [M] called fundamental class.

Definition 14.8 (Simplicial volume). Let M be a connected (not necessarily compact) orientable manifold of dimension n without boundary. Define its simplicial volume by

||M|| := ||j([M])||_{1} ∈ [0, ∞],

where j: H_{n}^{lf}(M; ℤ) → H_{n}^{lf}(M; ℝ) is the change of coefficients map associated to the inclusion ℤ → ℝ and [M] ∈ H_{n}^{lf}(M; ℤ) is any choice of fundamental class.

14.1.2 Elementary Properties

Bounded cohomology is not a cohomology theory. It is natural, i.e. a continuous map f: X → Y induces a homomorphism Ĥ^{p}(f): Ĥ^{p}(Y) → Ĥ^{p}(X) such that Ĥ^{p}(g ∘ f) = Ĥ^{p}(f) ∘ Ĥ^{p}(g) and Ĥ^{p}(id) = id hold. It satisfies homotopy invariance, i.e. two homotopic maps induce the same homomorphism on bounded cohomology, and the dimension axiom, i.e. Ĥ^{p}({∗}) is zero for p ≥ 1 and ℝ for p = 0. But excision does not hold and there is no long exact sequence associated to a pair and no Mayer-Vietoris sequence. Therefore bounded cohomology is much harder to compute than ordinary cohomology. Moreover, we will see that the bounded cohomology Ĥ^{p}(X) of a finite CW - complex X can be non-trivial for some p > dim(X).

Notice that the inclusion i_{∗}: Ĉ^{∗}(X) → C_{sing}^{∗}(C; ℝ) induces homomorphisms

i^{p}: Ĥ^{p}(X) → H_{sing}^{p}(X; ℝ) .  (14.9)

The Kronecker product ⟨ψ, y⟩ of ψ ∈ H_{sing}^{p}(X; ℝ) and y ∈ H_{p}^{sing}(X; ℝ) is defined by ϕ(x) for any choice of elements ϕ ∈ ker(δ^{p}) and x ∈ ker(∂_{p}) with ψ = [ϕ] and y = [x]. For ψ ∈ Ĥ^{p}(X) and x ∈ H_{p}^{sing}(X; ℝ) we define ⟨ψ, y⟩ by ⟨i^{p}(ψ), y⟩.

Lemma 14.10. (1) Let X be a topological space. Consider x ∈ H_{p}^{sing}(X; ℝ). We have ||x||_{1} = 0 if and only if for any ψ ∈ Ĥ^{p}(X) the Kronecker product satisfies ⟨ψ, x⟩ = 0. Otherwise ||x||_{1} > 0 and

\frac{1}{||x||_{1}} = inf\{||ψ||_{∞} | ψ ∈ Ĥ^{p}(X), ⟨ψ, x⟩ = 1\};

(2) Let M be a closed connected orientable manifold of dimension n. Then the simplicial volume ||M|| vanishes if and only if i^{n}: Ĥ^{n}(M) → H_{sing}^{n}(M; ℝ) is trivial. Otherwise we get

\frac{1}{||M||} = ||α||_{∞},

where α ∈ H_{sing}^{n}(M; ℝ) is the image of the fundamental cohomology class of M under the change of rings homomorphism H_{sing}^{n}(M; ℤ) → H_{sing}^{n}(M; ℝ).

Proof. (1) This is proved for instance in [35, Proposition F.2.2 on page 278].

(2) This follows from (1). ⊓ ⊔

Lemma 14.11. Let f: M → N be a map of closed connected oriented manifolds of the same dimension n. Let deg(f) be the degree of f. Then

||M|| ≥ | deg(f)| · ||N||.

In particular the simplicial volume is a homotopy invariant.

Proof. For any x ∈ C_{n}^{sing}(X; ℝ) we get ||C_{n}^{sing}(f)(x)||_{1} = ||x||_{1}. ⊓ ⊔

Corollary 14.12. If f: M → M is a selfmap of a closed connected orientable manifold of degree different from −1, 0 and 1, then ||M|| = 0.

Lemma 14.13. Let p: M → N be a d-sheeted covering of closed connected orientable manifolds. Then

||M|| = d · ||N||.

Proof. Since p is a map of degree d, it suffices to prove because of Lemma 14.11 that ||M|| ≤ d · ||N||. For a singular simplex σ: Δ_{n} → N let L(σ) be the set of singular simplices σ̃: Δ_{n} → M with p ∘ σ̃ = σ. It consists of precisely d elements. If the singular chain u = ∑_{σ} λ_{σ} · σ represents [N], then ũ = ∑_{σ} λ_{σ} · ∑_{σ̃∈L(σ)} σ̃ represents [M] and ||ũ||_{1} = d · ||u||_{1}. ⊓ ⊔

Lemma 14.14. Let X be a topological space. We have ||y||_{1} = 0 for any y ∈ H_{1}(X; ℝ). Moreover, Ĥ^{1}(X) = 0.

Proof. Any element y ∈ H_{1}(X; ℝ) can be represented by an ℝ-linear combination of elements of the shape [σ: S^{1} → X] for continuous maps σ: S^{1} → X. If σ_{k}: S^{1} → X is the composition of σ with the map S^{1} → S^{1} sending z to z^{k} for some integer k > 0, then σ and \frac{1}{k} · σ_{k} represent the same element in H_{1}(X; ℝ) and ||\frac{1}{k} · σ_{k}||_{1} = \frac{1}{k} · ||σ||_{1}. This implies ||y||_{1} = 0.

Consider an element ψ ∈ Ĥ^{1}(X). Fix a representative ϕ ∈ ker(δ̂^{1}). We conclude from Lemma 14.10 (1) that i^{1} : Ĥ^{1}(X) → H_{sing}^{1}(X; ℝ) is trivial. Hence there is u ∈ C_{sing}^{0}(X) with δ^{0}(u) = ϕ. We have |u(σ_{0})−u(σ_{1})| ≤ ||ϕ||_{∞} for any two singular 0-simplices σ_{0} and σ_{1} whose images lie in the same path component of X. Since ϕ ∈ Ĉ^{1}(X), we can arrange u ∈ Ĉ^{0}(X). Hence ψ = 0.

⊓ ⊔

14.1.3 Bounded Cohomology and Amenable Groups

The next result is due to Gromov [232, page 40], [275, Theorem 4.3].

Theorem 14.15. (Mapping theorem for bounded cohomology). Let f: X → Y be a map of path-connected topological spaces which induces an epimorphism π_{1}(X) → π_{1}(Y) with an amenable group as kernel. Then the induced map Ĥ^{p}(f): Ĥ^{p}(Y) → Ĥ^{p}(X) is an isometric isomorphism.

Theorem 14.15 and Lemma 14.10 (1) imply

Corollary 14.16. (1) Let X be a path-connected topological space with fundamental group π and classifying map f: X → Bπ. Then f induces for p ≥ 0 an isometric isomorphism Ĥ^{p}(f): Ĥ^{p}(Bπ) ⟶^{≅} Ĥ^{p}(X);

(2) Let X be a path-connected topological space with amenable fundamental group. Then Ĥ^{p}(X) is zero for p ≥ 1 and ℝ for p = 0;

(3) Let M be a connected closed orientable manifold of dimension n with fundamental group π and classifying map f: M → Bπ. Then

||M|| = ||H_{n}^{sing}(f)([M])||_{1};

(4) Let M be a closed connected orientable manifold of dimension ≥ 1 with amenable fundamental group. Then

||M|| = 0.

14.1.4 The Simplicial Volume of Hyperbolic and Low-Dimensional Manifolds

Define the positive real number v_{n} to be the supremum of the volumes of all n-dimensional geodesic simplices, i.e. the convex hull of (n + 1) points in general position, in the n-dimensional hyperbolic space ℍ^{n}. This is the same as the maximum over the volumes of all simplices in \overline{ℍ}^{n}, which is obtained from ℍ^{n} by adding its natural boundary. Any regular ideal simplex in \overline{ℍ}^{n} has volume v_{n} and any simplex in \overline{ℍ}^{n} of volume v_{n} is a regular ideal simplex [243]. We have (see [243])

v_{2} = π;
v_{3} = \frac{3}{2} · ∑_{j≥1} \frac{sin(2πj/3)}{j^{2}} ≃ 1.10149;

v_{4} = \frac{10π}{3} · arcsin(1/3) − \frac{π^{2}}{3} ≃ 0.26889;
v_{n} ≤ \frac{π}{(n − 1)!};
lim_{n→∞} \frac{n! · v_{n}}{\sqrt{n} · e} = 1.

Theorem 14.17 (Simplicial volume of hyperbolic manifolds). Let M be a closed hyperbolic orientable manifold of dimension n. Then

||M|| = \frac{vol(M)}{v_{n}}.

If n = 2r + 1 is odd and C_{n} > 0 is the constant introduced in (3.151), then M̃ is L^{2}-det-acyclic and

||M|| = \frac{(−1)^{r}}{C_{n} · v_{n}} · ρ^{(2)}(M̃).

Proof. The equation ||M|| = \frac{vol(M)}{v_{n}} follows from work of Gromov and Thurston (see [232, page 11]). The second equation follows from Theorem 3.152. ⊓ ⊔

Theorem 14.18 (Simplicial volume of low-dimensional manifolds).

(1) We have ||S^{1}|| = 0;

(2) We have ||S^{2}|| = ||T^{2}|| = 0. Let F_{g} be the closed connected orientable surface of genus g ≥ 1. Then

||F_{g}|| = 2 · |χ(F_{g})| = 4g − 4;

(3) Let M be a compact connected orientable irreducible 3-manifold with infinite fundamental group such that the boundary of M is empty or a disjoint union of incompressible tori. Suppose that M satisfies Thurston’s Geometrization Conjecture, i.e. there is a decomposition along a minimal family of pairwise non-isotopic not boundary-parallel incompressible 2-sided tori in M whose pieces are Seifert manifolds or hyperbolic manifolds. Let M_{1}, M_{2}, …, M_{r} be the hyperbolic pieces. They all have finite volume [385, Theorem B on page 52]. Then M is det-L^{2}-acyclic and

||M|| = \frac{1}{v_{3}} · ∑_{i=1}^{r} vol(M_{i}) = \frac{−6π}{v_{3}} · ρ^{(2)}(M̃).

In particular, ρ^{(2)}(M̃) = 0 if and and only if ||M|| = 0.

Proof. (1) This follows from Corollary 14.12 or Lemma 14.14.

(2) This follows for S^{2} and T^{2} from Corollary 14.12. The other cases are proved in [232, page 9].

(3) The claim for ||M|| is proved in [479], [491]. The claim for ρ^{(2)}(M̃) has already been proved in Theorem 4.3. ⊓ ⊔

14.1.5 Volume and Simplicial Volume

In this subsection we discuss the relation of the simplicial volume and the volume of a closed Riemannian manifold.

The next result is due to Thurston. Its proof can be found for instance in [232, Section 1.2]. (See also [274], [491].)

Theorem 14.19 (Thurston’s estimate on the simplicial volume). Let M be a complete connected orientable Riemannian manifold of dimension n, which has finite volume and whose sectional curvature satisfies −k ≤ sec(M) ≤ −1 for some real number k ≥ 1. Then there is a constant C_{n} > 0, which depends only on n = dim(M) but not on M such that

vol(M) ≤ C_{n} · ||M||.

In particular ||M|| > 0.

One knows that C_{n} ≤ \frac{π}{(n−1)!}. Certainly C_{n} ≥ v_{n} by Theorem 14.17. It is unknown whether one can take C_{n} = v_{n}.

Next we want to get reverse estimates, i.e. we want to estimate the volume by the simplicial volume from below. This is motivated by the following result due to Cheeger [102].

Theorem 14.20 (Cheeger’s finiteness theorem). For any given numbers D > 0 and v > 0 there are only finitely many diffeomorphism classes of closed Riemannian manifolds M of fixed dimension n such that for the sectional curvature | sec(M)| ≤ 1, for the diameter diam(M) ≤ D and for the volume vol(M) ≥ v hold.

Definition 14.21 (Minimal volume). Let M be a smooth manifold. Define its minimal volume minvol(M) to be the infimum over all volumes vol(M, g), where g runs though all complete Riemannian metrics on M, for which the sectional curvature satisfies | sec(M, g)| ≤ 1.

Let M be a closed connected orientable Riemannian manifold of dimension n. Then for any x̃ ∈ M̃ the following limit exists

h_{vol}(M) := lim_{R→∞} \frac{ln(vol(B_{R}(x̃))}{R},  (14.22)

where B_{R}(x̃) is the geodesic ball in the universal covering M̃ with center x̃ of radius R. This limit is independent of the choice of x̃ and called the (volume) entropy of M. One always has the inequality h_{vol(M)}(M) ≤ (dim(M) − 1) by Bishop’s inequality if sec(M) ≥ −1.

The entropy h(G, S) of a finitely generated group G with respect to a finite set of generators S is defined by

h(G, S) := lim inf_{R→∞} \frac{ln(N(R))}{R},

where N(R) is the number of elements of G which can be written as a word of length ≤ R in the generators of S or their inverses. The entropy h(G) of a finitely generated group G is the infimum over all numbers h(G, S), where S runs through all finite sets of generators. If M is a closed Riemannian manifold of diameter diam(M), then h(π_{1}(M)) ≤ 2 · diam(M) · h_{vol}(M) (see [238, Theorem 5.16 on page 282] or [377] for a proof).

The next result is taken from [232, page 12 and page 37], [238, 5.39 on page 307]. For a real number k and a complete Riemannian manifold M of dimension n the bound on the sectional curvature sec(M) ≥ −k^{2} implies the bound on its Ricci curvature

Ricci_{x}(v, v) ≥ −(n − 1) · k^{2} · ⟨v, v⟩_{x}  (14.23)

for all x ∈ M and v ∈ T_{x}M.

Theorem 14.24 (Gromov’s estimate on the simplicial volume).

(1) Let M be a closed connected orientable Riemannian manifold of dimension n whose Ricci curvature satisfies for all x ∈ M and v ∈ T_{x}M

Ricci_{x}(v, v) ≥ −\frac{1}{n − 1} · ⟨v, v⟩_{x}.

Then there exists a constant C_{n} with 0 < C_{n} < n!, which depends on n but not on M, such that

||M|| ≤ C_{n} · vol(M);

(2) Let M be a closed connected orientable Riemannian manifold of dimension n. Then there exists a constant C_{n} with 0 < C_{n} < n!, which depends on n but not on M, such that

||M|| ≤ C_{n} · h_{vol}(M)^{n} · vol(M);

(3) Let M be a closed connected orientable Riemannian manifold of dimen-

sion n. Then

||M|| ≤ (n − 1)^{n} · n! · minvol(M).

Example 14.25. Obviously any closed flat Riemannian manifold has vanishing minimal volume. Hence we get

minvol(T^{n}) = ||T^{n}|| = 0.

Let F_{g} be the closed orientable surface of genus g, then

minvol(F_{g}) = 2π · |χ(F_{g})| = 2π · |2 − 2g| = π · ||F_{g}||

by the following argument. The Gauss-Bonnet formula implies for any Riemannian metric on F_{g} whose sectional curvature satisfies | sec | ≤ 1

vol(F_{g}) ≥ ∫_{F_{g}} | sec | dvol ≥ |∫_{F_{g}} sec dvol| = |2π · χ(F_{g})|.

If g ≠ 1 and we take the Riemannian metric whose sectional curvature is constant 1 or −1, then the Gauss-Bonnet Theorem shows

|2π · χ(F_{g})| = |∫_{F_{g}} sec dvol| = vol(F_{g}).

Now the claim follows.

Notice that ||S^{2}|| = 0 and minvol(S^{2}) ≠ 0. We have (see [32], [232, page 93])

minvol(ℝ^{2}) = 2π(1 + \sqrt{2});
minvol(ℝ^{n}) = 0  for n ≥ 3.

The next result is taken from [232, page 6].

Theorem 14.26 (Minimal volume and characteristic classes). Let M be a closed connected orientable manifold of dimension n. Denote by p_{I}(M) the Pontryagin number of M for a fixed partition I of n (see [379, page 185]). Then there is a constant C(I, n), which depends on I and n but not on M, such that

|p_{I}(M)| ≤ C(I, n) · minvol(M).

There is also a dimension constant E_{n} satisfying

|χ(M)| ≤ E_{n} · minvol(M).

Proof. The Chern-Weil-Theorem (see [379, Corollary 1 on page 308]) implies that for an appropriate polynomial P_{I}(x)

|p_{I}(M)| = |∫_{M} P_{I}(Ω)| ≤ sup{||P_{I}(Ω)||_{x} | x ∈ M} · vol(M),

where Ω is the curvature tensor. If the sectional curvature satisfies | sec(M)| ≤ 1, then one can find a constant C(I, n), which depends only on I and n but not on M, such that sup{||P_{I}(Ω)||_{x} | x ∈ M} ≤ C(I, n). The proof for the Euler characteristic is similar and based on the Gauss-Bonnet Theorem. ⊓ ⊔

A discussion of examples of Riemannian manifolds with vanishing minimal volume is given in [232, Appendix 2].

14.1.6 Simplicial Volume and Betti Numbers

The next theorem is taken from [232, page 12].

Theorem 14.27 (Betti numbers and simplicial volume). Let M be a complete connected orientable Riemannian manifold of dimension n with finite volume. Let k_{1} ≥ k_{2} > 0 be positive constants such that the sectional curvature satisfies −k_{1} ≤ sec(M) ≤ −k_{2}. Then there is a constant C(n, k_{1}/k_{2}), which depends only on n and the ratio k_{1}/k_{2} but not on M, such that

∑_{i≥0} b_{p}(M) ≤ C(n, k_{1}/k_{2}) · ||M||.

Notice that both Theorem 14.19 and Theorem 14.27 imply that ||M|| > 0 holds for a closed connected orientable Riemannian manifold with negative sectional curvature.

The next example is illuminating since it shows how information about the fundamental class gives bounds on the sum of the Betti numbers.

Example 14.28. Let X be a simplicial complex with N simplices. Then

∑_{p≥0} b_{p}(X) ≤ ∑_{p≥0} dim_{ℤ} C_{p}(X) ≤ N.

Let M be a closed manifold of dimension n. Suppose that M admits a triangulation with t simplices of dimension n. Then the total number of simplices in this triangulation is bounded by 2^{t} since any p-simplex σ is the intersection of the n-simplices which contain σ. This implies

∑_{p≥0} b_{p}(M) ≤ 2^{t}.

One can also get an estimate in terms of singular simplices. Let M be a closed oriented manifold M of dimension n. Suppose that [M] ∈ H_{n}(M; ℤ) can be represented by a ℤ-linear combination ∑_{i=1}^{t} λ_{i} · σ_{i} of t singular simplices σ_{i}. Then we get

∑_{p≥0} b_{p}(M) ≤ t · 2^{n}.

This follows from Poincaré duality by the following argument. Let S_{p}(M) be the set of singular p-simplices. Let S_{p}(M)′ be the subset of S_{p}(M) consisting of those singular p-simplices which are obtained from one of the singular nsimplices σ_{i}: Δ_{n} → M above by composition with some face map Δ_{p} → Δ_{n}. Let ψ ∈ H^{p}(M; ℝ) be any element which can be represented by a cocyle ϕ ∈ C_{sing}^{p}(M; ℝ) = map(S_{p}(M), ℝ) which vanishes on any singular simplex in S_{p}(M)′. Then

ψ ∩ [M] = [ϕ ∩ (∑_{i=1}^{t} λ_{i} · σ_{i})] = [∑_{i=1}^{t} λ_{i} · ϕ ∩ σ_{i}] = [∑_{i=1}^{t} λ_{i} · 0] = 0

follows from the definition of the cap-product [226, 24.19 on page 152]. Since ? ∩ [M]: H^{p}(M; ℝ) → H_{n−p}(M, ℝ) is bijective by Poincaré duality, this implies ψ = 0. We conclude that the dimension of H^{p}(M; ℝ) is at most the cardinality of S_{p}(M)′. Now the claim follows.

Remark 14.29. Notice, however, that in general the minimum over all the numbers t for which [M] ∈ H_{n}(M; ℤ) can be represented by a ℤ-linear combination ∑_{i=1}^{t} λ_{i} · σ_{i} of t singular simplices (see Example 14.28 above ) is not related to ||M||. It is bounded from above by the infimum of the L^{1}-seminorms of all integral singular cycles representing [M] but this infimum has not the nice properties as ||M|| and is very hard to handle. A better version is defined by Gromov [238, page 306] using ideas of Thurston. Gromov introduces an extension of the simplicial volume to foliations with transverse measures and defines a new invariant ||[M]_{ℤ}||_{Δ}^{F} for a closed orientable manifold M. It satisfies

||M|| ≤ ||[M]_{ℤ}||_{Δ}^{F}.  (14.30)

There seems to be no known counterexample to the equality ||M|| = ||[M]_{ℤ}||_{Δ}^{F}.

The next result appears as an exercise in [238, page 307].

Theorem 14.31. Let M be an aspherical closed orientable manifold of dimension n. Then

∑_{p≥0} b_{p}^{(2)}(M̃) ≤ 2^{n} · ||[M]_{ℤ}||_{Δ}^{F}.

14.1.7 Simplicial Volume and S^{1}-Actions

Theorem 14.32. Let M be a connected closed orientable manifold. If M carries a non-trivial S^{1}-action, then

||M|| = 0.

If M carries an S^{1}-action with M^{S^{1}} = ∅, then

minvol(M) = 0.

Proof. The claim for the simplicial volume is proved in [232, Section 3.1, page 41], [526], the one for the minimal volume is taken from [232, page 7]. ⊓ ⊔

14.1.8 Some Information about the Second Bounded Cohomology of Groups

We mention the following two results due to Epstein-Fujiwara [178], [209] and Bestvina-Fujiwara [41].

Theorem 14.33. The second bounded cohomology Ĥ^{2}(BG) is a real vector space of infinite dimension if G is a non-elementary word-hyperbolic group or if G is a subgroup, which is not virtually abelian, of the mapping class group of a compact orientable surface of genus g with p punctures.

This shows in particular that the second bounded cohomology of B(ℤ ∗ ℤ) = S^{1}∨S^{1} is non-trivial despite the fact that S^{1}∨S^{1} is a 1-dimensional finite CW -complex. We see that there are no Mayer-Vietoris sequences for bounded cohomology in general. The behaviour of the second bounded cohomology of a group under free amalgamated products and HNN-extensions is analysed in [210]. Burger and Monod [76], [77, Theorem 20 and Theorem 21] show that the natural map from the second bounded cohomology group to the second cohomology group of G is injective if G is an irreducible lattice in a connected semisimple Lie group with finite center, no compact factors and of rank ≥ 2.

14.1.9 Further Properties of the Simplicial Volume

We begin by stating a proportionality principle for the simplicial volume [232, page 11] which is analogous to the one for L^{2}-invariants (see Theorem 3.183).

Theorem 14.34 (Proportionality principle for the simplicial volume). Let M and N be closed connected orientable Riemannian manifolds, whose universal coverings are isometrically diffeomorphic. Then

\frac{||M||}{vol(M)} = \frac{||N||}{vol(N)}.

14.2 Simplicial Volume and L^{2}-Invariants 501

The proof of the next result can be found for instance in [35, Theorem F.2.5. on page 279] and [232, page 10].

Lemma 14.35. Let M and N be closed connected orientable manifolds of dimensions m and n. There exists for each integer k a constant C(k) which depends on k but not on M and N, such that

||M × N|| ≤ C(m + n) · ||M|| · ||N||;
||M|| · ||N|| ≤ ||M × N||.

The next result is taken from [232, page 10].

Theorem 14.36. Let M and N be closed connected orientable manifolds of the same dimension n ≥ 3. Then we get for their connected sum M#N

||M#N|| = ||M|| + ||N||.

14.2 Simplicial Volume and L^{2}-Invariants of Universal Coverings of Closed Manifolds

In this section we give some evidence for Conjecture 14.1 and discuss some consequences.

Remark 14.37. Conjecture 14.1 is definitely false if one drops the condition “aspherical” since the simplicial volume of a simply connected closed orientable manifold M is zero (see Corollary 14.16 (4)), but b_{0}^{(2)}(M̃) = b_{0}(M) = 1. One may ask whether for a (not necessarily aspherical) det-L^{2}-acyclic closed orientable manifold M with vanishing simplicial volume the L^{2}-torsion ρ^{(2)}(M̃) must be trivial. The answer is again negative. Namely, let M be a 3-dimensional hyperbolic manifold. Put N = S^{2} × M. Then ||N|| = 0 by Corollary 14.12, the universal covering is det-L^{2}-acyclic and ρ^{(2)}(Ñ) ≠ 0 by Theorem 3.96 (4) and Theorem 3.152.

14.2.1 Hyperbolic Manifolds and 3-Manifolds

Let M be a hyperbolic orientable manifold of dimension n or a closed connected orientable manifold of dimension 3 satisfying the assumptions of Theorem 14.18 (3). Then M is aspherical and of determinant class and we get for a dimension constant D_{n} which is different from zero for n odd and zero for n even (see Theorem 4.1, Theorem 3.96 (3), Theorem 14.17 and Theorem 14.18 (3))

b_{p}^{(2)}(M̃) = 0  for p ≥ 0 if ||M|| = 0;
ρ^{(2)}(M̃) = D_{n} · ||M||.

Hence for such manifolds Conjecture 14.1 is true.

The considerations above lead to the question whether the relation ρ^{(2)}(M̃) = D_{n} · ||M|| may hold for all aspherical closed orientable manifolds of dimension n for a dimension constant D_{n}. This is not the case as the following example shows.

Example 14.38. Let M be a 3-dimensional orientable closed hyperbolic 3manifold. Let N be M × M × M and F be an orientable closed hyperbolic surface. Let F^{d} be the d-fold cartesian product of F with itself. We get from Theorem 3.96 (4), Theorem 14.17 and Theorem 14.35

ρ^{(2)}({ \atop N × F^{d}}) = 0;
ρ^{(2)}({ \atop M × F^{d+3}}) ≠ 0;
||N × F^{d}|| ≠ 0.

All the manifolds appearing in the list above are orientable closed aspherical manifolds of dimension 9 + 2d. If ρ^{(2)}(M̃) = D_{9+2d} · ||M|| holds for all of them, we get D_{9+2d} = 0 from the first and third equation and D_{9+2d} ≠ 0 from the second, a contradiction.

14.2.2 S^{1}-Actions

Let M be an aspherical closed orientable manifold with non-trivial S^{1}-action. Then ||M|| = 0 by Theorem 14.32. Hence Conjecture 14.1 predicts that M̃ is det-L^{2}-acyclic and ρ^{(2)}(M̃) = 0. This has already been proved in Theorem 3.111.

14.2.3 Amenable Fundamental Groups

Let M be an aspherical closed orientable manifold with amenable fundamental group. Then ||M|| = 0 by Corollary 14.16 (4). Hence Conjecture 14.1 predicts that M̃ is det-L^{2}-acyclic and ρ^{(2)}(M̃) = 0. This has already been proved in Theorem 3.113 in the special case where the fundamental group is elementary amenable.

Actually, it suffices in Theorem 3.113 to require that π_{1}(M) contains a normal infinite elementary amenable subgroup. This raises

Question 14.39. (Simplicial volume and normal infinite amenable subgroups). For which closed connected orientable manifolds, whose fundamental group contains an amenable infinite normal subgroup, does the simplicial volume vanish?

14.2.4 Selfmaps of Degree Different from −1, 0 and 1

Let f: M → M be a selfmap of an aspherical closed orientable manifold M whose degree is different from −1, 0 and 1. Then ||M|| = 0 by Corollary 14.12. Hence Conjecture 14.1 predicts that b_{p}^{(2)}(M̃) = 0 for all p ≥ 0, M̃ is of determinant class and ρ^{(2)}(M̃) = 0. There is the conjecture that f is homotopic to a covering with deg(f) sheets and then this conclusion would follow from Theorem 1.35 (9) and Theorem 3.96 (5) provided that M̃ is of determinant class. Without this conjecture we can at least prove the claim for the L^{2}-Betti numbers under certain assumptions about the fundamental group.

A group G is Hopfian if any surjective group homomorphism f: G → G is an isomorphism. Examples of Hopfian groups are residually finite groups [356], [399, Corollary 41.44]. It is not true that a subgroup of finite index of a Hopfian group is again Hopfian [31, Theorem 2]. At least any subgroup of finite index in a residually finite group is again residually finite and hence Hopfian.

Theorem 14.40. Let M be an aspherical closed orientable manifold. Suppose that any normal subgroup of finite index of its fundamental group is Hopfian and there is a selfmap f: M → M of degree deg(f) different from −1, 0, and 1. Then

b_{p}^{(2)}(M̃) = 0 for p ≥ 0.

Proof. Fix an integer n ≥ 1. Let p: \overline{M} → M be the covering of M associated to the image of π_{1}(f^{n}): π_{1}(M) → π_{1}(M). By elementary covering theory there is a map \overline{f^{n}}: M → \overline{M} satisfying p∘\overline{f^{n}} = f^{n}. Since deg(f^{n}) = deg(\overline{f^{n}})· deg(p) and deg(p) = [π_{1}(M): im(π_{1}(f)], we conclude

[π_{1}(M) : im(π_{1}(f^{n}))] ≤ | deg(f^{n})| = | deg(f)|^{n}.  (14.41)

Consider a map g: N_{1} → N_{2} of two closed connected oriented n-dimensional manifolds of degree deg(g) ≠ 0. Abbreviate G_{1} = π_{1}(N_{1}) and G_{2} = π_{1}(N_{2}). Then there is a diagram of ℤG_{2}-chain complexes which commutes up to ℤG_{2}-homotopy

hom_{ℤG_{1}}(C_{n−∗}(Ñ_{1}), ℤG_{1}) ⊗_{ℤ[π_{1}(g)]} ℤG_{2} ⟵^{g^{∗}} hom_{ℤG_{2}}(C_{n−∗}(Ñ_{2}), ℤG_{2})
(?∩[N_{1}])⊗_{ℤ[π_{1}(g)]}id↓  ^{?∩g_{∗}([N_{1}])}↓
C_{∗}(Ñ_{1}) ⊗_{ℤ[π_{1}(g)]} ℤG_{2}  ⟶^{g_{∗}}  C_{∗}(Ñ_{2})

Tensoring with l^{2}(G_{2}) and then applying (reduced) L^{2}-homology yields a commutative diagram of finitely generated Hilbert N (G_{2})-modules

H_{(2)}^{n−p} (hom_{ℤG_{1}}(C_{∗}(Ñ_{1}), ℤG_{1}) ⊗_{ℤ[π_{1}(g)]} l^{2}(G_{2})) ⟵^{g^{∗}} H_{(2)}^{n−p}(Ñ_{2}; N (G_{2}))
H_{∗}^{(2)}((?∩[N_{1}])⊗_{ℤ[π_{1}(g)]}id)↓  ^{H_{∗}^{(2)}(?∩g_{∗}([N_{1}]))}↓
H_{p}^{(2)} (C_{∗}(Ñ_{1}) ⊗_{ℤ[π_{1}(g)]} l^{2}(G_{2}))  ⟶^{g_{∗}} H_{p}^{(2)}(Ñ_{2}; N (G_{2}))

The right vertical arrow is bijective since deg(g) is invertible in ℝ, g_{∗}([N_{1}]) = deg(g) · [N_{2}] and ? ∩ [N_{2}]: hom_{ℤG_{2}}(C_{n−∗}(Ñ_{2}), ℤG_{2}) → C_{∗}(Ñ_{2}) is a ℤ[G_{2}]- chain homotopy equivalence by Poincaré duality. Hence the morphism of finitely generated Hilbert N (G_{2})-modules

g_{∗}: H_{p}^{(2)} (C_{∗}(Ñ_{1}) ⊗_{ℤ[π_{1}(g)]} l^{2}(G_{2})) → H_{p}^{(2)}(Ñ_{2}; N (G_{2}))

is surjective. Theorem 1.12 (2) implies

dim_{N (G_{2})} (H_{p}^{(2)} (C_{∗}(Ñ_{1}) ⊗_{ℤ[π_{1}(g)]} l^{2}(G_{2}))) ≥ b_{p}^{(2)}(Ñ_{2}).

Let e_{p}(N_{1}) be the number of p-cells in a fixed triangulation of N_{1}. We conclude from Theorem 1.12 (2)

dim_{N (G_{2})} (H_{p}^{(2)} (C_{∗}(Ñ_{1}) ⊗_{ℤ[π_{1}(g)]} l^{2}(G_{2})))
≤ dim_{N (G_{2})} (C_{p}(Ñ_{1}) ⊗_{ℤ[π_{1}(g)]} l^{2}(G_{2}))
≤ e_{p}(N_{1}).

This shows

b_{p}^{(2)}(Ñ_{2}) ≤ e_{p}(N_{1})  for p ≥ 0.  (14.42)

If we apply (14.42) to g = \overline{f^{n}}, we get

b_{p}^{(2)}(˜\overline{M}) ≤ e_{p}(M).

Since b_{p}^{(2)}(˜\overline{M}) = [π_{1}(M) : im(π_{1}(f^{n}))] · b_{p}^{(2)}(M̃) holds by Theorem 1.35 (9), we get

b_{p}^{(2)}(M̃) ≤ \frac{e_{p}(M)}{[π_{1}(M) : im(π_{1}(f^{n}))]}.  (14.43)

Hence it suffices to show that there is no integer n such that im(π_{1}(f^{n})) = im(π_{1}(f^{k})) for all k ≥ n since then the limit for n → ∞ of the right-hand side of (14.43) is zero. Suppose that such n exists. Then the composition \overline{f^{n}} ∘ p: \overline{M} → \overline{M} induces an epimorphism on π_{1}(\overline{M}). Since π_{1}(\overline{M}) is Hopfian by assumption, \overline{f^{n}} ∘ p: \overline{M} → \overline{M} is an isomorphism. Since \overline{M} is aspherical, \overline{f^{n}} ∘ p is a homotopy equivalence. This implies

1 = deg(\overline{f^{n}} ∘ p) = deg(\overline{f^{n}}) · deg(p).

This shows that | deg(p)| = 1 and hence π_{1}(f^{n}) is surjective. Since π_{1}(M) is Hopfian and M aspherical, we conclude | deg(f^{n})| = | deg(f)^{n}| = 1. This contradicts the assumption deg(f) ∉ {−1, 0, 1}. This finishes the proof of Theorem 14.40. ⊓ ⊔

14.2.5 Negative Sectional Curvature and Locally Symmetric Spaces

There is the following conjecture which is attributed to Gromov in [457, page 239].

Conjecture 14.44. (Gromov’s Conjecture about positivity of the simplicial volume for non-negative sectional and negative Ricci curvature). Let M be a closed connected orientable Riemannian manifold whose sectional curvature is non-negative and whose Ricci curvature is everywhere negative. Then

||M|| > 0.

Suppose that M is a closed connected orientable manifold with negative sectional curvature. Then the Singer Conjecture 11.1 and Conjecture 11.3 predict that either one of the L^{2}-Betti numbers b_{p}^{(2)}(M̃) or ρ^{(2)}(M̃) is different from zero. Hence together with Conjecture 14.1 they predict ||M|| > 0. Indeed ||M|| > 0 follows from Theorem 14.19 or from Theorem 14.27 for such M.

Conjecture 14.44 implies the following

Conjecture 14.45. (Positivity of the simplicial volume of locally symmetric spaces of non-compact type). Let M be a closed connected Riemannian manifold whose universal covering is a symmetric space of noncompact type. Then

||M|| > 0.

Conjecture 14.45 has been proved by Savage [457, page 239] in the case where the universal covering M̃ is SL(n, ℝ)/SO(n).

Thus Conjecture 14.1 is true for aspherical locally symmetric spaces if Conjecture 14.45 is true (see Lemma 5.10 and equations (5.13) and (5.15)).

14.2.6 Simplicial Volume and L^{2}-Invariants

The following result follows from Gromov [238, page 300].

Theorem 14.46 (L^{2}-Betti numbers and volume). Let M be an aspherical closed Riemannian manifold of dimension n. Suppose that for some δ > 0 the Ricci curvature satisfies for all x ∈ M and v ∈ T_{x}M

Ricci_{x}(v, v) ≥ −δ · ⟨v, v⟩_{x}.

Then there is a constant C(n, δ), which depends on n and on δ but not on M, such that

∑_{p≥0} b_{p}^{(2)}(M̃) ≤ C(n, δ) · vol(M).

Theorem 14.46 implies together with (14.23)

Theorem 14.47 (L^{2}-Betti numbers and minimal volume). Let M be an aspherical closed Riemannian manifold with minvol(M) = 0. Then for p ≥ 0

b_{p}^{(2)}(M̃) = 0.

The essential property of Riemannian manifolds with Ricci_{x}(v, v) ≥ −δ · ⟨v, v⟩_{x} comes from packing inequalities as explained in [238, 5.31 on page 294]. A metric space X has packing type P for a function P (R, r) with values in the positive integers if every ball of radius R contains at most P (R, r) pairwise disjoint balls of radius r. Bishop’s inequality implies for a Riemannian manifold M of dimension n with Ricci_{x}(v, v) ≥ −(n − 1) · ⟨v, v⟩_{x} that P (R, r) ≤ C · (R/r)^{n} for 0 ≤ r ≤ R ≤ 1 and P (R, r) ≤ exp(α_{n} · R) for 1/2 ≤ r ≤ R and 1 ≤ R, where C is a universal constant and α_{n} a dimension constant. These packing inequalities are used to describe the manifold M in question as a homotopy retract of a simplicial complex Q which comes from the nerve of a covering of M and whose number of simplices is controlled.

Some evidence for Conjecture 14.1 comes from Theorem 14.24 (3) and Theorem 14.47. It raises the following question

Question 14.48. (Minimal volume and L^{2}-torsion for aspherical manifolds). Let M be an aspherical closed Riemannian manifold with minvol(M) = 0. Is then M̃ of determinant class and

ρ^{(2)}(M̃) = 0 ?

Some further evidence for Conjecture 14.1 comes from Remark 14.29 and Theorem 14.31.

14.3 Miscellaneous

More information about the simplicial volume can be found for instance in [76], [77] [178], [227], [232], [238], [275], [412] and [480].

One may ask

Question 14.49 (Simplicial volume and S^{1}-foliations). Does the simplicial volume ||M|| of a connected closed orientable manifold M vanish, if M carries an S^{1}-foliation?

This question has a positive answer if one considers special S^{1}-foliations of certain 3-manifolds [52]. In view of of Conjecture 14.1 this is related to Question 3.186.

We mention the following result due to Besson-Courtois-Gallot [40, page 733].

Theorem 14.50. Let M be a closed connected manifold M of dimension n ≥ 3 with Riemannian metric g, whose Ricci curvature satisfies Ricci_{x}(v, v) ≥ −(n − 1)⟨v, v⟩_{x} for x ∈ M and v ∈ T_{x}M and whose volume vol(M, g) coincides with its minimal volume minvol(M). Then (M, g) is hyperbolic, i.e. sec(M, g) = −1.

We mention the following improvement by Smillie and Gromov (see [232, page 23], [35, Proposition F.4.12 on page 293], [478]) of a result of Milnor [372] and Sullivan [487].

Theorem 14.51 (Flat bundles and simplicial volume). Let E → B be a flat orientable n-dimensional vector bundle over a topological space B. Then its Euler class e(E) ∈ H^{n}(B; ℝ) (see [379, page 98]) satisfies

||e(E)||_{∞} ≤ 2^{−n}.

In particular one gets for a flat n-dimensional bundle E → M over a closed connected orientable manifold M of dimension n

χ(E) ≤ 2^{−n} · ||M||,

where χ(E) = ⟨e(E), [M]⟩ is the Euler number of E.

We mention the following conjecture due to Sullivan [487, page 187]. A manifold M of dimension n has an affine structure if there is an atlas whose change of coordinates maps are restrictions of affine isomorphisms ℝ^{n} → ℝ^{n}.

Conjecture 14.52. (Sullivan’s Conjecture on affine manifolds and Euler characteristic). Let M be an n-dimensional closed connected orientable manifold which admits an affine structure. Then χ(M) = 0.

Notice that the tangent bundle T M of a manifold M with affine structure is flat and that the Euler number χ(T M) of the tangent bundle of a closed manifold M is equal to its Euler characteristic χ(M) [379, Corollary 11.12 on page 130]. Hence for an n-dimensional closed connected orientable manifold M, which admits an affine structure, the vanishing of ||M|| implies χ(M) = 0 by Theorem 14.51. This raises

Question 14.53. (Vanishing of the simplicial volume for affine manifolds). For which closed connected orientable manifolds M, which admit an affine structure, does ||M|| = 0 hold?

A proof of the following result using bounded cohomology is presented in [35, Theorem F.6.5 on page 307]. The original proof appeared in [261].

Theorem 14.54. Let M be an n-dimensional closed connected orientable manifold which admits an affine structure. Suppose that the range of the holonomy π_{1}(M) → GL(n, ℝ)^{+} of the flat tangent bundle T M is amenable. Then χ(M) = 0.

The key idea is to show that the projection p: ST M → M induces an injection Ĥ^{n}(M) → Ĥ^{n}(ST M) and then use the observation that the pullback p^{∗}T M has a non-vanishing section and hence p^{∗}e(T M) = e(p^{∗}(T M)) = 0.

Because of the Euler-Poincaré formula (see Theorem 1.35 (2)) Sullivan’s Conjecture 14.52 follows from the following conjecture which is posed as a question in [237, page 232].

Conjecture 14.55. (Gromovs’s Conjecture on affine structures and L^{2}-Betti numbers). Let M be an n-dimensional closed connected orientable manifold which admits an affine structure. Then b_{p}^{(2)}(M̃) = 0 for p ≥ 0.

We have discussed the Volume Conjecture 4.8 already in Section 4.3 which links the simplicial volume ||S^{3} − K|| (or, equivalently, ρ^{(2)}(K) of Definition 4.5 by Theorem 14.18 (3)) for a knot K ⊂ S^{3} to the colored Jones polynomial.

Exercises

14.1. Let M be a closed connected orientable manifold of dimension n. Let m be the minimal number of n-dimensional simplices appearing in a smooth triangulation of M. Show ||M|| ≤ m.

14.2. Prove ||ℝ|| = ∞.

14.3. Let M and N be oriented manifolds without boundary. Suppose that there is a proper map f: M → N of degree d. Show that then ||M|| ≥ d·||N||. Conclude ||ℝ^{n}|| = 0 for n ≥ 2.

14.4. Show for a path-connected topological space X that Ĥ^{0}(X) ≅ ℝ.

14.5. Let X be a simplicial complex. Let C^{∗}(X; ℝ) be the cellular cochain complex with coefficients in ℝ. Let I_{p} be a cellular basis for C^{p}(X). Then C^{p}(X) = map(I_{p}, ℝ). Define for ϕ ∈ C^{p}(X) its L^{∞}-norm by

||ϕ||_{∞} := sup{ϕ(b) | b ∈ I_{p}}.

Show that this is independent of the choice of cellular basis. Let l^{∞}C^{p}(X) ⊂ C^{p}(X) be the submodule given by elements ϕ with ||ϕ||_{∞} < ∞. Show that this defines a subchain complex l^{∞}C^{∗}(X) of C^{∗}(X; ℝ) so that H^{∗}(l^{∞}C^{∗}(X)) is defined. Show that in general H^{∗}(l^{∞}C^{∗}(X)) is not isomorphic to the bounded cohomology Ĥ^{p}(X).

14.6. Let M be a closed connected orientable Riemannian manifold of dimension n with residually finite fundamental group. Let k_{1} ≥ k_{2} > 0 be positive constants such that the sectional curvature satisfies −k_{1} ≤ sec(M) ≤ −k_{2}. Then there is a constant C(n, k_{1}/k_{2}), which depends only on n and the ratio k_{1}/k_{2} but not on M, such that

∑_{i≥0} b_{p}^{(2)}(M̃) ≤ C(n, k_{1}/k_{2}) · ||M||.

14.7. Let M be a closed connected oriented Riemannian manifold of dimension n. Let dvol_{M} be its volume form. Show that the cochain ϕ ∈ C_{sing,C^{∞}}^{n}(M; ℝ) sending a singular smooth simplex σ: Δ_{n} → M to \frac{1}{vol(M)} · ∫_{M} σ^{∗} dvol_{M} is a cocycle and represents an element in H_{sing}^{n}(M; ℝ) which is the image of the cohomological fundamental class of M under the change of rings map H^{n}(M; ℤ) → H^{n}(M; ℝ). Prove that ||ϕ||_{∞} = ∞.

14.8. Let f: X → Y be a map of topological spaces. Show for any y ∈ H_{p}^{sing}(X; ℝ) and ψ ∈ H_{sing}^{p}(Y; ℝ)

||H_{p}^{sing}(f)(y)||_{1} ≤ ||y||_{1};
||H_{sing}^{p}(f)(ψ)||_{∞} ≤ ||ψ||_{∞}.

14.9. Let M be an aspherical closed orientable Riemannian manifold of dimension n. Suppose that for some δ > 0 the Ricci curvature satisfies for all x ∈ M and v ∈ T_{x}M

Ricci_{x}(v, v) ≥ −δ · ⟨v, v⟩_{x}.

Show that there is a constant C(n, δ), which depends on n and on δ but not on M, such that

χ(M) ≤ C(n, δ) · vol(M);
sign(M) ≤ C(n, δ) · vol(M).

14.10. Show for the free group F_{g} on g generators

h(F_{g}) = ln(2g − 1).

14.11. Let M and N be closed connected oriented manifolds. Let f: M → N be a map of degree 1. Show that π_{1}(f) is surjective. Prove that ||M|| = ||N|| if the kernel of π_{1}(f) is amenable.

14.12. Let M be an aspherical closed connected orientable manifold which carries a non-trivial S^{1}-action. Show minvol(M) = 0.

14.13. Let M be a closed connected orientable manifold. Suppose that it carries an S^{1}-action such that M^{S^{1}} is empty. Show that then all Pontryagin numbers vanish.

14.14. Give an example of a closed connected orientable manifold M with infinite fundamental group whose minimal volume minvol(M) is zero and at least one L^{2}-Betti number of its universal covering is positive.

14.15. Let F → E ⟶^{p} B be a fibration of closed connected orientable manifolds with dim(F) ≥ 1. Suppose that π_{1}(F) is amenable. Show ||E|| = 0.

14.16. Show that the torus T^{2} is the only closed connected orientable 2dimensional manifold which admits an affine structure.

14.17. Let M be a closed connected manifold of dimension n. Suppose that for any ϵ > 0 there are positive integers d_{ϵ} and t_{ϵ} satisfying \frac{t_{ϵ}}{d_{ϵ}} ≤ ϵ together with a d_{ϵ}-sheeted covering p_{ϵ}: M_{ϵ} → M such that M_{ϵ} has a smooth triangulation with t_{ϵ} simplices of dimension n. Prove

||M|| = 0;
b_{p}^{(2)}(M̃) = 0.

Assume furthermore that the fundamental group π_{1}(M) is of det ≥ 1-class. Then show that M̃ is det-L^{2}-acyclic and

ρ^{(2)}(M̃) = 0.

15. Survey on Other Topics Related to L^{2}-Invariants

Introduction

In this chapter we discuss very briefly some topics which are related to L^{2}- invariants but cannot be covered in this books either because the author does not feel competent enough to write more about them or either because it would require to write another book to treat them in detail. The interested reader should look at the references cited below. Neither the list of references nor the list of topics is complete.

15.1 L^{2}-Index Theorems

The starting point for investigating L^{2}-invariants was the L^{2}-version of the index theorem by Atiyah [9]. Let P be an elliptic differential operator on a closed Riemannian manifold M and let \overline{M} → M be a regular covering of M with G as group of deck transformations. Then we can lift the Riemannian metric and the operator P to \overline{M}. The operator P is Fredholm and its index is defined by

ind(P) = dim_{ℂ}(ker(P)) − dim_{ℂ}(ker(P^{∗})).  (15.1)

Using the von Neumann trace one can define the L^{2}-index of the lifted operator \overline{P} analogously

ind_{N (G)}(\overline{P}) = dim_{N (G)}(ker(\overline{P})) − dim_{N (G)}(ker(\overline{P}^{∗})).  (15.2)

Then the L^{2}-index theorem says

ind_{N (G)}(\overline{P}) = ind(P).  (15.3)

If one puts elliptic boundary conditions on the operator, this result was generalized to the case where M is compact and has a boundary by Schick [459]. This generalization is the L^{2}-version of the index theorem in [10].

These boundary conditions are local. Atiyah, Patodi and Singer [11], [12],

512 15. Survey on Other Topics Related to L^{2}-Invariants

[13] prove versions of the index theorem for manifolds with boundary using global boundary conditions, which, in contrast to the local conditions, apply to important geometrically defined operators such as the signature operator. This index theorem involves as a correction term the eta-invariant. The L^{2}-version of the eta-invariant has been defined and studied by Cheeger and Gromov [105], [106]. The L^{2}-version of this index theorem for manifolds with boundary using global boundary conditions has been proved by Ramachandran [427] for Dirac type operators.

The difference ρ^{(2)}(M) := η^{(2)}(M̃) − η(M) is the relative von Neumann eta-invariant. This invariant depends only on the smooth structure of M, not on the Riemannian metric [105]. If π_{1}(M) is torsionfree and the Baum-Connes Conjecture holds for the maximal group C^{∗}-algebra of π_{1}(M), then Keswani [291] shows that ρ^{(2)}(M) is a homotopy invariant. See also [359]. Chang and Weinberger [98] use ρ^{(2)}(M) to prove their Theorem 0.10.

The L^{2}-index Theorem of Atiyah is a consequence of the index theorem of Mishchenko-Fomenko [380], where an index is defined which takes values in K_{0}(C_{r}^{∗}(G)). If one applies the composition of the change of rings homomorphism K_{0}(C_{r}^{∗}(G)) → K_{0}(N (G)) and the dimension homomorphism dim_{N (G)} : K_{0}(N (G)) → ℝ to the index of Mishchenko and Fomenko, one obtains the L^{2}-index of Atiyah.

Some further references on L^{2}-index theory and related topics are [7], [21], [71], [72], [120], [121], [122], [154], [339], [386], [391], [392], [394], [437], [438], [439], [440], [460], [474], [485], [486].

Connections between L^{2}-cohomology and discrete series of representations of Lie groups are investigated, for instance, in [14], [123], [131], [135], [464].

15.2 L^{p}-Cohomology

One can also define and investigate L^{p}-cohomology for p ∈ [1, ∞], as done for instance by Gromov [237, section 8] and Pansu [409], [410], [413]. One systematically replaces the L^{2}-condition by a L^{p}-condition and l^{2}(G) by l^{p}(G). Thus one gets L^{p}-cohomology for each p ∈ [1, ∞]. However, only for p = 2 the reduced L^{p}-cohomology is a Hilbert space and one can associate to the L^{p}-cohomology a real number, its dimension. There exists a L^{p}-version of the de Rham Theorem. The question whether for an element α ∈ ℂG with α ≠ 0 there is β ∈ l^{p}(G) with β ≠ 0 and α ∗ β = 0 is studied in [312], [422]. Notice that the Atiyah-Conjecture predicts that for p = 2 and G torsionfree such an element β does not exists, but for p > 2 such elements are constructed in the case of finitely generated free and finitely generated free abelian groups in [312], [422].

Further references on L^{p}-cohomology are [2], [101], [170], [222], [223], [535].

15.4 Knot Concordance and L^{2}-signature 513

15.3 Intersection Cohomology

When we have investigated L^{2}-invariants of manifolds M, we mostly have considered the case where there is a proper and cocompact action of a group G on M. Of course L^{2}-cohomology can also be interesting and has been investigated for complete non-necessarily compact Riemannian manifolds without such an action. For instance algebraic and arithmetic varieties have been studied. In particular, the Cheeger-Goresky-MacPherson Conjecture [104] and the Zucker Conjecture [531] have created a lot of activity. They link the L^{2}-cohomology of the regular part with the intersection homology of an algebraic variety. Some references concerning these topics are [53], [55], [95], [174], [266], [315], [319], [367], [397], [398], [402], [403], [414], [415], [451], [452], [453], [454], [455], [530], [531], [532], [533], [534].

15.4 Knot Concordance and L^{2}-signature

A knot in the 3-sphere is slice if there exists a locally flat topological embedding of the 2-disk into D^{4} whose restriction to the boundary is the given knot. For a long time Casson-Gordon invariants [97] have been the only known obstructions for a knot to be slice. Cochran, Orr and Teichner give in [113] new obstructions for a knot to be slice using L^{2}-signatures. They construct a hierarchy of obstructions depending on one another and on certain choices. The obstructions take values in relative L-groups associated to Ore localizations of certain group rings. These rings map to the von Neumann algebra N (G) of certain groups G and the Ore localizations map to the algebra U(G) of affiliated operators. Recall that U(G) is the Ore localization of N (G) with respect to the multiplicative set of non-zero-divisors in N (G). Therefore they can use L^{2}-signatures to detect these obstructions in certain cases. Thus they can construct an explicit knot, which is not slice, but whose Casson-Gordon invariants are all trivial.

16. Solutions of the Exercises

Chapter 1

1.1. If H is finite, then use the projection l^{2}(G) → l^{2}(G) given by right multiplication with \frac{1}{|H|} · ∑_{h∈H} h. Suppose that l^{2}(G/H) is a Hilbert N (G)- module. Since l^{2}(G/H)^{H} is non-trivial, there must be a non-trivial Hilbert space V such that (V ⊗ l^{2}(G))^{H} ≠ 0. Hence l^{2}(G)^{H} ≠ 0. This implies |H| < ∞.

1.2. Because of Theorem 1.9 (5) it suffices to prove for a weak isomorphism of finite dimensional Hilbert N (G)-modules u: V → W and positive endomorphisms f: V → V and g: W → W with u ∘ f = g ∘ u that tr_{N (G)}(f) = tr_{N (G)}(g). By polar decomposition of u one can achieve U = V and that u is positive. Let {E_{λ} | λ ∈ [0, ∞)} be the spectral family of u. Consider the decomposition V = im(E_{λ}) ⊕ im(1 − E_{λ}).

One writes f as a matrix with respect to this decomposition ( \begin{matrix}f_{1,1} & f_{1,2} \\ f_{2,1} & f_{2,2} \end{matrix} )

and similar for g. Since u respects this decomposition, the morphisms f_{2,2} and g_{2,2} are conjugated by the automorphism of im(1 − E_{λ}) induced by u. Hence tr_{N (G)}(f_{2,2}) = tr_{N (G)}(g_{2,2}). We have tr_{N (G)}(f) − tr_{N (G)}(f_{2,2}) = tr_{N (G)}(f_{1,1}) and tr_{N (G)}(g) − tr_{N (G)}(g_{2,2}) = tr_{N (G)}(g_{1,1}). Since tr_{N (G)}(f_{1,1}) and tr_{N (G)}(g_{1,1}) are bounded by max{||f||, ||g||}·dim_{N (G)}(im(E_{λ})) and Theorem 1.12 (4) implies

lim_{λ→0} dim_{N (G)}(im(E_{λ})) = dim_{N (G)}(ker(u)) = 0,

the claim follows.

1.3. Put I = {1, 2, …}. Let V_{n} be the Hilbert N (G)-module given by the Hilbert sum ⨁_{m=n}^{∞} l^{2}(G). Then dim_{N (G)}(V_{n}) = ∞ for all n but ⋂_{n≥0} V_{n} = {0}.

1.4. Replace U_{1} and V_{1} by U_{1}/ ker(u_{1}) and V_{1}/ ker(v_{1}) and replace U_{5} and V_{5} by clos(im(u_{4})) and clos(im(v_{4})). Then split the problem into three diagrams whose rows are short weakly exact sequences. It suffices to prove the claim for these diagrams. One reduces the claim to the case, where the rows are short exact sequences, by replacing a weakly exact sequence 0 → V_{0} → V_{1} ⟶^{q} V_{2} → 0 by the exact sequence 0 → ker(q) → V_{1} ⟶^{pr} ker(q)^{⊥} → 0. Then the claim follows from the long weakly exact homology sequence (see Theorem 1.21).

1.5. Use (f ∘ g)^{∗} = g^{∗} ∘ f^{∗} for composable bounded operators of Hilbert spaces.

1.6. Existence of finite self-covering (see Example 1.37), computation in terms of the quotient field of the group ring (see Lemma 1.34), S^{1}-action (see Theorem 1.40), mapping torus (see Theorem 1.39).

1.7. One can construct a fibration S^{1} → X → B. The main difficulty is to show that B is up to homotopy of finite type (see [331, Lemma 7.2]). Then one can apply Theorem 1.41. A more general statement will be proved in Theorem 7.2 (1) and (2).

1.8. See [434, Theorem VIII.3 on page 256].

1.9. One easily checks that this is a spectral family and then use the characterization of ∫_{−∞}^{∞} g(λ)dE_{λ} given in (1.65).

1.10. The graph norm is the L^{2}-norm on the kernel of an operator.

1.11. Take M = ℝ − {0} with the Riemannian metric coming from ℝ. Take ω = χ_{(0,∞)} and η = exp(−x^{2}) · dx.

1.12. Write ℂ ⊗_{ℤ} C_{∗}(X) as a direct sum of suspensions of 1-dimensional chain complexes of the shape ℂ[ℤ]^{n} → ℂ[ℤ]^{n} with non-trivial differentials and of zero-dimensional chain complexes with ℂ[ℤ]^{n} in dimension zero (see the proof of Lemma 2.58). The 1-dimensional chain complexes do not contribute to the L^{2}-Betti number (see Lemma 1.34 (1)). The contribution of the zero-dimensional chain complexes to the L^{2}-Betti numbers and to the Betti numbers are the same.

1.13. The zero-th and first L^{2}-Betti numbers of E(ℤ/2 ∗ ℤ/2) vanish since ℤ/2 ∗ ℤ/2 contains ℤ as subgroup of finite index (see Lemma 1.34 (2) and Theorem 1.35 (9)). Hence the zero-th and first L^{2}-Betti numbers of M̃ vanish by Theorem 1.35 (1). From the Euler-Poincaré formula and Poincaré duality (see Theorem 1.35 (2) and (3)) we get χ(M) = b_{2}^{(2)}(M̃). Since b_{1}(B(ℤ/2 ∗ ℤ/2)) = 0, one shows analogously χ(M) = 2 + b_{2}(M).

1.14. Use Example 1.38 and truncate the relevant CW -complexes of finite type BG and X occuring there above dimension n + 1. Notice that b_{p}^{(2)}({ \atop X ∨ Y}) = b_{p}^{(2)}(X̃) + b_{p}^{(2)}(Ỹ) holds for p ≥ 1 if π_{1}(X) and π_{1}(Y) are infinite (see Theorem 1.35 (5) and (8)) and that b_{p}(X ∨ Y) = b_{p}(X) + b_{p}(Y) holds for p ≥ 1.

Chapter 2

2.1. Assertion i.) is obvious. Assertion ii.) is proved as follows. If f and g are Fredholm and weak isomorphisms, g ∘ f is a weak isomorphism and Fredholm by Lemma 2.13 (3). Suppose that g∘f and f are Fredholm and weak isomorphisms. Injectivity and Fredholmness of g follow from Lemma 2.13 (2), density of the image of g is obvious. Suppose that g and g ∘ f are Fredholm and weak isomorphisms. Then f is a weak isomorphism and Fredholm since f^{∗} is a weak isomorphism and Fredholm by the argument above applied to f^{∗} ∘ g^{∗} and g^{∗}. Notice that f is Fredholm and a weak isomorphism if and only if f^{∗} is Fredholm and a weak isomorphism (see Lemma 2.4). The desired counterexample is constructed in Example 1.19

2.2. For G = ℤ we find u by Lemma 2.58. By induction (see (2.57)) we get the desired u for each group G with ℤ ⊂ G. Now construct X as follows. Start with the finite 2-skeleton of EG. Then attach for p = 3, 4, … trivially a p-cell and then a (p + 1)-cell, where the attaching map is given by an element u_{p} corresponding to α_{p} and involves only the trivially attached p-cell.

2.3. The matrix A lives already over a finitely generated subgroup H of G. Now use induction (see (2.57)) and the fact that over a finite group all Novikov-Shubin invariants are ∞^{+} (see Example 2.5).

2.4. The condition α_{1}(X̃) = 1 implies that π is virtually nilpotent with growth rate precisely 1 (see Theorem 2.55 (5)). Hence π_{1}(X) contains ℤ as subgroup of finite index by the formula for the growth rate appearing in Subsection 2.1.4. We conclude from Theorem 2.55 (1) and (6) that α_{2}(X̃) = α_{2}(Eπ) = α_{2}(Eℤ) = ∞^{+}.

2.5. The fundamental group π of M is finitely presented. Hence there is a model for Bπ with finite 2-skeleton. Let f: M → Bπ be the classifying map. If we apply Theorem 2.55 (1) to its restriction to the 2-skeletons M_{2} → Bπ_{2} we conclude that π determines α_{p}(M̃) = α for p = 1, 2. Now apply Poincaré duality Theorem 2.55 (2) and the fact that a compact 3-manifold M with non-empty boundary is homotopy equivalent to a finite 2-dimensional CW - complex.

2.6. If π_{2}(B) → π_{1}(S^{1}) is not trivial, then π_{1}(E) is finite and all Novikov-Shubin invariants must be ∞^{+} (see Example 2.5). Suppose that π_{2}(B) → π_{1}(S^{1}) is trivial. Then π_{1}(S^{1}) → π_{1}(E) is an isomorphism. Now apply Lemma 2.58 and Theorem 2.61.

2.7. Suppose that the ℂ[ℤ^{n}]-module l^{2}(ℤ^{n}) is flat. Then im(c_{2}^{(2)}) = ker(c_{1}^{(2)}) holds for the differentials of C_{∗}^{(2)}(T̃^{n}) and im(c_{2}^{(2)}) must be closed. We conclude from Lemma 2.11 (8) and (9) that α_{2}(T̃^{n}) = ∞^{+}. Example 2.59 implies n = 1.

It remains to show for each ℂ[ℤ]-module M that Tor_{p}^{ℂ[ℤ]}(M, l^{2}(ℤ)) = 0 for p ≥ 1. The functor Tor commutes in both variables with colimits over directed systems [94, Proposition VI.1.3. on page 107]. Hence we can assume without loss of generality that M is a finitely generated ℂ[ℤ]-module since each module is the directed union of its finitely generated submodules. Recall that ℂ[ℤ] is a principal ideal domain (see [15, Proposition V.5.8 on page 151 and Corollary V.8.7 on page 162]). Hence M is a finite sum of principal ideals. Therefore it suffices to show that for any non-trivial p ∈ ℂ[ℤ] multiplication with p induces an injection l^{2}(ℤ) → l^{2}(ℤ). Since the von Neumann dimension of this kernel is zero by Lemma 2.58), the kernel is trivial by Theorem 1.12 (1).

2.8. Use the product formula (see Theorem 1.35 (4) and Theorem 2.55 (3)) and Examples 2.59 and 2.69.

2.9. Use the product formula (see Theorem 1.35 (4) and Theorem 2.55 (3)) and Examples 2.59 and 2.70.

2.10. Use the product formula (see Theorem 2.55 (3)).

2.11. See Theorem 3.100 and Remark 3.184.

2.12. There exist up to isomorphism only countably many finitely presented groups. Moreover, for a given finitely presented group G and m, n ≥ 0 there are only countably many elements in M(m, n, ℤG). Hence A is countable. We get {r | r ∈ ℚ, r ≥ 0} ⊂ A from the product formula (see Theorem 2.55 (3)), Lemma 2.58 and Example 2.59.

Chapter 3

3.1. First show for a chain homotopy equivalence f: C_{∗} → D_{∗} of finite based free ℤ-chain complexes that

ρ^{ℤ}(D_{∗}) − ρ^{ℤ}(C_{∗}) = τ(f) − ∑_{p≥0}(−1)^{p} · ln (det_{ℤ}(H_{p}(f))) = 0.

The proof is an easy version of the one of Theorem 3.35 (5). Then prove that both sides of the equation are homotopy invariants and additive under direct sums of ℤ-chain complexes. Construct a ℤ-chain complex D_{∗} which is ℤhomotopy equivalent to C_{∗} and which is a direct sum of iterated suspensions of 0-dimensional ℤ-chain complexes with ℤ as zero-th chain module and of 1-dimensional ℤ-chain complexes of the shape ℤ ⟶^{n} ℤ for some n ∈ ℤ, n ≠ 0.

This reduces the claim to these special chain complexes, where the proof is trivial.

3.2. Follow the proof of Theorem 3.35 (2) or see [116, (23.1)].

3.3. Follow the proof of Theorem 3.35 (4).

3.4. Follow the proof of Theorem 3.35 (6c) or see [116, (23.2)].

3.5. See [327, Corollary 5.5].

3.6. See [327, Example 1.15].

3.7. Use Lemma 3.15 (1) and ∫_{ϵ}^{a} \frac{1}{λ·(− ln(− ln(λ))·ln(λ))} dλ = − ln(ln(− ln(a)))+ ln(ln(− ln(ϵ))) for 0 < ϵ < a < e^{−e}.

3.8. Since ℂ[ℤ^{n}]_{(0)} is a field, we can transform A over ℂ[ℤ^{n}]_{(0)} by elementary row and column operations into a diagonal matrix. Hence we can find over ℂ[ℤ^{n}] matrices E_{1}, E_{2}, …, E_{m+n} and a diagonal matrix D such that each E_{i} is a triangular matrix with non-zero diagonal entries or a permutation matrix and

E_{1} · E_{2} · . . . · E_{m} · A · E_{m+1} · E_{2} · . . . · E_{m+n} = D.

The claim is easily verified for each matrix E_{i} and the matrix D and then proved for A using Lemma 1.34 (1), Theorem 3.14 (1) and (2) and Lemma 3.37 (1) and (2).

3.9. All claims except for ρ^{(2)}(X) have already been proved in Lemma 2.58.

Notice that the determinant of R_{n}: l^{2}(ℤ) → l^{2}(ℤ) for n ∈ ℤ, n > 0 is n and the homology of the 1-dimensional ℂ[ℤ]-chain complex ℂ[ℤ] ⟶^{R_{n}} ℂ[ℤ] is zero for n ∈ ℤ, n > 0. For each n ∈ ℤ, n > 0 one constructs a free ℤ-CW - complex which is obtained from S̃^{1} by attaching trivially an ℤ-equivariant 3-cell and then a ℤ-equivariant 4-cell such that the 4-th differential of C_{∗}(X_{n}) looks like R_{n}. Then X_{m} and X_{n} for m ≠ n leads to the example of two finite ℤ-CW -complexes whose homology with ℂ-coefficients vanishes in all dimensions exept in dimension zero and hence is isomorphic in all dimensions but whose L^{2}-torsion are different.

Let C_{∗} be a finite free ℤ[ℤ]-chain complex such that ℂ ⊗_{ℤ} H_{p}(C_{∗}) is a torsion module over ℂ[ℤ] for all p ∈ ℤ. Then l^{2}(ℤ) ⊗_{ℤ[ℤ]} C_{∗} is det-L^{2}-acyclic. After a choice of a ℤ[ℤ]-basis for C_{∗} we can define ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} C_{∗}) ∈ ℝ.

Let f_{∗}: C_{∗} → D_{∗} be a ℤ[ℤ]-chain homotopy equivalence of finite based free ℤ[ℤ]-chain complexes for which ℂ⊗_{ℤ}H_{p}(C_{∗}) and ℂ⊗_{ℤ}H_{p}(D_{∗}) are torsion modules over ℂ[ℤ] for p ∈ ℤ. We get

ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} C_{∗}) − ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} D_{∗}) = Φ^{ℤ}(τ(f_{∗}))

for the homomorphism Φ^{ℤ}: Wh(ℤ) → ℝ defined in (3.92). This follows from the chain complex version of Theorem 3.93 (1). Since Wh(ℤ) = {1}, we conclude

ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} C_{∗}) = ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} D_{∗}).

Hence ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} C_{∗}) depends only on the ℤ[ℤ]-chain homotopy type of C_{∗} and is in particular independent of the choice of the ℤ[ℤ]-basis. Let 0 → C_{∗} → D_{∗} → E_{∗} → 0 be an exact sequence of finite free ℤ[ℤ]-chain complexes for which ℂ⊗_{ℤ} H_{p}(C_{∗}), ℂ⊗_{ℤ} H_{p}(D_{∗}) and ℂ⊗_{ℤ} H_{p}(E_{∗}) are torsion modules over ℂ[ℤ] for p ≥ 0. From Additivity of L^{2}-torsion (see Theorem 3.35 (1)) we conclude

ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} C_{∗}) − ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} D_{∗}) + ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} E_{∗}) = 0.

Let M be a finitely generated ℤ[ℤ]-module such that ℂ ⊗_{ℤ} M is a torsion ℂ[ℤ]-module. The ring ℤ[ℤ] is Noetherian. Any finitely generated ℤ[ℤ]- module M has a finite free ℤ[ℤ]-resolution F_{∗}. Define

ρ^{(2)}(M) = ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} F_{∗})

for any finite free ℤ[ℤ]-resolution F_{∗} of M. The choice of F_{∗} does not matter since two such resolutions are ℤ[ℤ]-chain homotopy equivalent.

Let C_{∗} be any finite based free ℤ[ℤ]-chain complex such that ℂ⊗_{ℤ} H_{p}(C_{∗}) is a torsion ℂ[ℤ]-module for p ∈ ℤ. Without loss of generality we can assume C_{p} = 0 for p < 0. Then one can show by induction over the number n for which H_{p}(C_{∗}) = 0 for p ≥ n

ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} C_{∗}) = ∑_{p≥0}(−1)^{p} · ρ^{(2)}(H_{p}(C_{∗})).

The induction beginning n = 0 follows from the fact that the vanishing of H_{p}(C_{∗}) for all p ≥ 0 implies that C_{∗} is ℤ[ℤ]-chain homotopy equivalent to the trivial ℤ[ℤ]-chain complex. In the induction step from (n − 1) to n choose a finite free resolution F_{∗} of H_{n}(C_{∗}) and construct a ℤ[ℤ]-chain map Σ^{n}F_{∗} → C_{∗} which induces an isomorphism on the n-th homology. We get an exact sequence of finite free ℤ[ℤ]-chain complexes 0 → C_{∗} → cone(f_{∗}) → Σ^{n+1}F_{∗} → 0. We conclude that H_{p}(cone(f_{∗})) = H_{p}(C_{∗}) for p ≤ n − 1 and H_{p}(cone(f_{∗})) = 0 for p ≥ n − 1. Moreover, we get

ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} C_{∗}) = ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} cone(f_{∗})) − ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} Σ^{n+1}P_{∗}).

From the definitions we get

−ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} Σ^{n+1}P_{∗}) = (−1)^{n} · ρ^{(2)}(H_{n}(C_{∗})).

This finishes the induction step.

We can write

ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} C_{∗}) = − \frac{1}{2} · ∑_{p≥0}(−1)^{p} · p·

ln (det (id_{l^{2}(ℤ)} ⊗r_{A_{p}}: l^{2}(ℤ) ⊗_{ℤ[ℤ]} ℤ[ℤ]^{s_{p}} → l^{2}(ℤ) ⊗_{ℤ[ℤ]} ℤ[ℤ]^{s_{p}}))

for appropriate matrices A_{p} ∈ M_{s_{p}}(ℤ[ℤ]). If d_{p} = det_{[ℤ[ℤ]}(A_{p}), we get from the previous Exercise 3.8

det (id_{l^{2}(ℤ)} ⊗r_{A_{p}}: l^{2}(ℤ) ⊗_{ℤ[ℤ]} ℤ[ℤ]^{s_{p}} → l^{2}(ℤ) ⊗_{ℤ[ℤ]} ℤ[ℤ]^{s_{p}})

= det (id_{l^{2}(ℤ)} ⊗r_{d_{p}}: l^{2}(ℤ) ⊗_{ℤ[ℤ]} ℤ[ℤ] → l^{2}(ℤ) ⊗_{ℤ[ℤ]} ℤ[ℤ]) .

We can write ln (det (id_{l^{2}(ℤ)} ⊗r_{d_{p}}: l^{2}(ℤ) ⊗_{ℤ[ℤ]} ℤ[ℤ] → l^{2}(ℤ) ⊗_{ℤ[ℤ]} ℤ[ℤ])) as ln(|a|) − ln(|b|) for algebraic integers a, b with |a|, |b| ≥ 1 because of Example 3.22. Hence the same is true for ρ^{(2)}(l^{2}(ℤ) ⊗_{ℤ[ℤ]} C_{∗}).

3.10. Consider an endomorphism a: V → V of a finitely generated Hilbert N (G)-module which is a weak isomorphism and not of determinant class. Let C_{∗} be the 1-dimensional N (G)-chain complex V ⊕ V ⟶^{pr_{1}} V and D_{∗} be the N (G)-chain complex with D_{1} = V and D_{p} = 0 for p ≠ 1. Let f_{∗}: C_{∗} → D_{∗} be the chain map given by f_{1} = id ⊕a and f_{0} = 0. Then H_{1}^{(2)}(f_{∗}) can be identified with a and f_{p} is of determinant class for all p ∈ ℤ. Hence H_{1}^{(2)}(f_{∗}) is not of determinant class and f_{∗} cannot be of determinant class by Theorem 3.35 (5).

3.11. Without loss of generality the double chain complex is in the first quadrant. It suffices to prove the claim for the alternating sum of the rows, otherwise skip the double complex. Use induction over the width n, i.e. the smallest number n ≥ −1 such that C_{p,q} = 0 for p > n. The induction beginning n ≤ 1 follows from Theorem 3.35 (1) applied to the obvious exact sequence 0 → C_{0,∗} → T_{∗} → ΣC_{1,∗} → 0. In the induction step from n − 1 ≥ 1 to n we can consider the double subcomplex C′_{∗,∗} ⊂ C_{∗,∗} with C′_{n,∗} = C_{n,∗}, C′_{n−1,∗} = clos(im(C_{n,∗} → C_{n−1,∗})) and C′_{p,∗} = 0 for p ≠ n − 1, n and define C′′_{∗,∗} to be C_{∗,∗}/C′_{∗,∗}. We have an exact sequence of double complexes 0 → C′_{∗,∗} → C_{∗,∗} → C′′_{∗,∗} → 0 which induces also an exact sequence of the associated total complexes. Then for p ∈ ℤ the columns and rows of C′_{∗,∗} and C′′_{∗,∗} are det-L^{2}-acyclic (see Lemma 3.14 (2), Lemma 3.15 (3) and Theorem 3.35 (1)). Now apply Theorem 3.35 (1) and the induction hypothesis to C′_{∗,∗} and C′′_{∗,∗}.

3.12. Statement (1) is equivalent to the corresponding statement about finitely generated groups since any matrix over ℤG comes by induction of a matrix over a finitely generated subgroup and Theorem 3.14 (6) holds. Because of Theorem 3.93 (1) it suffices to prove for a finitely generated group G that for any element in x ∈ Wh(G) we can find a G-homotopy equivalence f: M → N of cocompact free proper G-manifolds without boundary such that τ^{G}(f) = x+∗(x) since ϕ^{G}(x+∗(x)) = 2·ϕ^{G}(x). One can find a cocompact free proper G-manifold W of even dimension such that ∂W = ∂_{0}W ∐ ∂_{1}W, the inclusion i_{k}: ∂_{k}W → W is a G-homotopy equivalence for k = 0, 1 and τ^{G}(i_{0}) = x. Poincaré duality implies τ^{G}(i_{1}^{−1} ∘ i_{0}) = x + ∗(x).

3.13. This follows from Example 3.12 and Theorem 3.14 (6) and the obvious fact that the (classical) determinant of an invertible matrix over ℤ takes values in {±1}.

3.14. The set of matrices over ℤG of finite size is a countable set. Hence the three sets are countable.

The first set is closed under addition because of b_{p}^{(2)} (X ∪_{G} Y; N (G)) = b_{p}^{(2)}(X; N (G)) + b_{p}^{(2)}(Y; N (G)) for p ≥ 1 and |G| = ∞ (see Theorem 1.35 (5) and (8)) and the first set is {n · |G|^{−1} | n ∈ ℤ, n ≥ 0} for |G| < ∞ by Example 1.32.

The third set can be identified with the subgroup of ℝ

{ln(det(r_{A})) − ln(det(r_{B}))}

where A and B run through all matrices A ∈ M_{k}(ℤG) and B ∈ M_{l}(ℤG) for k, l ≥ 1 such that r_{A}: l^{2}(G)^{k} → l^{2}(G)^{k} and r_{B}: l^{2}(G)^{l} → l^{2}(G)^{l} are weak isomorphisms of determinant class. Namely, fix a det-L^{2}-acyclic finite free proper G-CW -complex X. Then X × S^{3} is a finite, free, proper and det-L^{2}-acyclic with ρ^{(2)}(X × S^{3}) = 0 by Theorem 3.93 (4). Given such a matrix A one can attach to X × S^{3} free G-cells in two consecutive dimensions such that the resulting G-CW -complex Z_{A} or Z_{B} respectively is finite, free, proper and det-L^{2}-acyclic and ρ^{(2)}(Z_{A}; N (G)) = ln(det(r_{A})) or ρ^{(2)}(Z_{B}; N (G)) = − ln(det(r_{B})) respectively (use Theorem 3.35 (1)). Now consider Z = Z_{A} ∪_{X×S^{3}} Z_{B}. It is a det-L^{2}-acyclic finite free proper G-CW - complex with ρ^{(2)}(Z; N (G)) = ln(det(r_{A})) − ln(det(r_{B})) by Theorem 3.93 (2).

3.15. Use the exercise before and the facts that there are only countably many finitely presented groups and the fundamental group of a connected finite CW -complex is finitely presented.

3.16. Consider a connected finite CW -complex X with fundamental group π and A ∈ M_{k}(ℤπ) such that [A] ∈ Wh(π) is different from zero. Attach trivially k 4-cells to X and call the result Y. Construct an extension of id: X → X to a self map f: Y → Y such that C_{∗}(f̃, ĩd) is given by A. Then τ(f) = [A] holds in Wh(G) and f induces the identity on π. We obtain a fibration Y → T_{f} → S^{1} such that θ(p) ∈ H^{1}(S^{1}; Wh(π_{1}(T_{f}))) = Wh(π_{1}(T_{f})) defined in (3.98) is the image of τ(f) under the homomorphism i_{∗}: Wh(π) → Wh(π_{1}(T_{f})) induced by the inclusion i: Y → T_{f}. Since i induces the obvious split injection on the fundamental groups π → π × ℤ = π_{1}(T_{f}), θ(p) is nontrivial.

3.17. The fundamental group π_{1}(T_{f}) has a presentation as an HNNextension

⟨π, s | sws^{−1} = f_{∗}(w) for w ∈ π⟩.

This follows from the Theorem of Seifert and van Kampen applied to an appropriate decomposition of the mapping torus into two connected pieces with connected intersection.

3.18. This follows from Example 3.107.

3.19. Use the exercise before.

3.20. (1) We get from Lemma 3.139 (2)

θ_{F} (t) = t · ∫_{0}^{∞} e^{−tλ} · F (λ) dλ
= t · ∫_{0}^{K} e^{−tλ} · F (λ) dλ + F (K) · t · ∫_{K}^{∞} e^{−tλ} dλ
= t · ∫_{0}^{K} e^{−tλ} · F (λ) dλ + F (K) · e^{−tK}
= t · ∫_{0}^{K} ∑_{n≥0} \frac{(−λ)^{n}}{n!} · t^{n} · F (λ) dλ + F (K) · ∑_{n≥0} \frac{(−K)^{n}}{n!} · t^{n}

= ∑_{n≥0} \frac{(−1)^{n}}{n!} · (− ∫_{0}^{K} n · λ^{n−1} · F (λ) dλ + F (K) · K^{n}) · t^{n}.

We get from the definitions

a_{0} = θ_{F} (0) = ∫_{0}^{∞} e^{0·λ} dF (λ) = F (K) − F (0) = F (K).

(2) We can write θ_{F} (t) = ∑_{n=0}^{N} a_{n} · t^{n} + R(t) for a function R(t) satisfying lim_{t→0} \frac{R(t)}{t^{N}} = 0. We get

\frac{1}{Γ (s)} · ∫_{0}^{ϵ} t^{s−1} · θ_{F} (t) dt = ∑_{n=0}^{N} a_{n} · \frac{1}{Γ (s)} · ∫_{0}^{ϵ} t^{s+n−1} dt

+\frac{1}{Γ (s)} · ∫_{0}^{ϵ} t^{s−1} · R(t) dt

= ∑_{n=0}^{N} a_{n} · \frac{1}{Γ (s + 1)} · \frac{s}{s + n} · ϵ^{s+n}

+\frac{1}{Γ (s)} · ∫_{0}^{ϵ} t^{s−1} · R(t) dt.

The function \frac{1}{Γ (s)} · ∫_{0}^{ϵ} t^{s−1} · R(t) dt is holomorphic for ℜ(s) > −N.

(3) For any α with 0 < α < α(F) we can find t_{0} such that θ_{F} (t) ≤ t^{−α} holds for t ≥ t_{0}. Now go through the calculation (3.131) in the case δ = ∞.

(4) We get from partial integration (3.16)

∫_{ϵ+}^{K} λ^{−s} dF (λ) = −s · ∫_{ϵ}^{K} λ^{−s−1} · F (λ) dλ + K^{−s} · F (K) − ϵ^{−s−1} · F (ϵ).

For any α with 0 < α < α(F) there is λ(α) such that F (λ) ≤ λ^{α} holds for 0 ≤ λ ≤ λ(α). We conclude from Lebesgue’s Theorem of majorized convergence

∫_{0}^{∞} λ^{−s} dF (λ) = −s · ∫_{0}^{K} λ^{−s−1} · F (λ) dλ + K^{−s} · F (K).

This implies using Lemma 3.15 (1)

\frac{d}{ds} ∫_{0}^{∞} λ^{−s} dF (λ)|_{0} = (−s · \frac{d}{ds} ∫_{0}^{K} λ^{−s−1} · F (λ) dλ

− ∫_{0}^{K} λ^{−s−1} · F (λ) dλ + ln(K) · K^{−s} · F (K))|_{s=0}

= − ∫_{0}^{K} \frac{F (λ)}{λ} dλ + ln(K) · F (K)
= ∫_{0}^{∞} ln(λ) dF (λ).

(5) This is analogous to (3.133) using (3.132).

(6) This follows from the other assertions.

3.21. The heat kernel of Δ_{0} on ℝ is given by

e^{−tΔ_{0}}(x, y) = \frac{1}{4πt} · e^{−(x−y)^{2}/4t}.

Since [0, 1] is a fundamental domain for the ℤ-action on ℝ = S̃^{1}, we get for t > 0

θ_{0}(S̃^{1})(t) = ∫_{0}^{1} \frac{1}{4πt} · e^{−(x−x)^{2}/4t} dx = \frac{1}{4πt}.

From ∗ ∘ Δ_{1} = Δ_{0} ∘ ∗ we get θ_{1}(S̃^{1}) = θ_{0}(S̃^{1}). We have for ℜ(s) > 1

\frac{1}{Γ (s)} · ∫_{0}^{1} t^{s−1} · θ_{1}(S̃^{1}) dt = \frac{1}{4π} · s · \frac{1}{(s − 1) · Γ (s + 1)}.

\frac{d}{ds} \frac{1}{Γ (s)} · ∫_{0}^{1} t^{s−1} · θ_{1}(S̃^{1}) dt|_{s=0} = −\frac{1}{4π} .

A direct computation shows

∫_{1}^{∞} t^{−1} · θ_{1}(S̃^{1}) dt = \frac{1}{4π} .

3.22. See [327, Example 1.18].

3.23. From Poincaré duality (see Theorem 1.35 (3)) and from the long exact weak homology sequence (see Theorem 1.21) we conclude

b_{p}^{(2)}(M) = b_{p}^{(2)}(M, ∂M) = b_{p}^{(2)}(∂M).

From the version of Poincaré duality for manifolds with boundary (cf. Theorem 3.93 (3) and its proof) we conclude that (M, ∂M) is det-L^{2}-acyclic and satisfies

ρ^{(2)}(M) = −ρ^{(2)}(M, ∂M).

We conclude from Theorem 3.35 (1) that ∂M is det-L^{2}-acyclic and we have

ρ^{(2)}(M) = ρ^{(2)}(M, ∂M) + ρ^{(2)}(∂M).

3.24. Let A be the (1, 1)-matrix over ℂ[ℤ] with entry (z − λ) for λ ∈ ℝ^{≥0}. The operator norm of R_{A} is 1 + λ. We compute:

tr ((1 − \frac{AA^{∗}}{(1 + λ)^{2}} )^{p})
= tr ((1 − \frac{(z − λ)(z^{−1} − λ)}{(1 + λ)^{2}})^{p})
= tr ((1 − \frac{1 + λ^{2} − λ · (z + z^{−1})}{(1 + λ)^{2}})^{p})
= tr ((\frac{2 · λ}{(1 + λ)^{2}} + \frac{λ}{(1 + λ)^{2}} · (z + z^{−1}))^{p})
= \frac{λ^{p}}{(1 + λ)^{2p}} · tr ((2 + (z + z^{−1}))^{p})

= \frac{λ^{p}}{(1 + λ)^{2p}} · tr (∑_{k=0}^{p} \binom{p}{k} · 2^{p−k} · (z + z^{−1})^{k})

= \frac{(2 · λ)^{p}}{(1 + λ)^{2p}} · ∑_{k=0}^{p} \binom{p}{k} · 2^{−k} · tr ((z + z^{−1})^{k})

= \frac{(2 · λ)^{p}}{(1 + λ)^{2p}} · ∑_{k=0}^{p} \binom{p}{k} · 2^{−k} · tr (∑_{j=0}^{k} \binom{k}{j} · z^{j} · z^{−(k−j)})

= \frac{(2 · λ)^{p}}{(1 + λ)^{2p}} · ∑_{k=0}^{p} \binom{p}{k} · 2^{−k} · ∑_{j=0}^{k} \binom{k}{j} · tr (z^{2j−k})

= \frac{(2 · λ)^{p}}{(1 + λ)^{2p}} · ∑_{k=0,k even}^{p} \binom{p}{k} · 2^{−k} · \binom{k}{k/2}

= \frac{(2 · λ)^{p}}{(1 + λ)^{2p}} · ∑_{k=0}^{[p/2]} \binom{p}{2k} · 4^{−k} · \binom{2k}{k}

= \frac{(2 · λ)^{p}}{(1 + λ)^{2p}} · ∑_{k=0}^{[p/2]} 4^{−k} · \frac{p!}{(p − 2k)! · k! · k!}

Now apply (3.24) and Theorem 3.172.

Chapter 4

4.1. Suppose that M has a prime decomposition whose factors are homotopy spheres with precisely one exception which is D^{3} or an irreducible manifold with infinite fundamental group. By the Sphere Theorem [252, Theorem 4.3] an irreducible 3-manifold is aspherical if and only if it is a 3-disk or has infinite fundamental group. Hence M is aspherical.

Suppose that M is aspherical. Then its fundamental group is torsionfree. Hence each summand in its prime decomposition M = M_{1}#M_{2}# … #M_{r} has trivial or torsionfree infinite fundamental group. We can assume without loss of generality that no M_{i} is homotopy equivalent to S^{3} since the connected sum with such manifolds does not affect the homotopy type. Since S^{3} is not aspherical, we have r ≥ 1. There is an obvious 2-connected map #_{i=1}^{r}M_{i} →

⋁_{i=1}^{r} M_{i}. Hence π_{2} (⋁_{i=1}^{r} M_{i}) ≅ π_{2} \binom{}{⋁_{i=1}^{r} M_{i}} ≅ H_{2} \binom{}{⋁_{i=1}^{r} M_{i}} is trivial. A

Mayer Vietoris argument implies H_{2} ( ; ⋁_{i=1}^{r} M_{i}|_{M_{i}}) = 0 for i = 1, 2, …, r.

Since the inclusion of M_{i} into ⋁_{i=1}^{r} M_{i} is injective, we conclude H_{2}(M̃_{i}) = 0 for i = 1, 2, …, r. This shows that each M_{i} is different from S^{1} × S^{2}. Hence each M_{i} is either an irreducible manifold with infinite fundamental group or D^{3}. In particular each M_{i} is aspherical. The obvious map #_{i=1}^{r}M_{i} → ⋁_{i=1}^{r} M_{i} is a map of aspherical spaces inducing an isomorphism on the fundamental groups. Hence it is a homotopy equivalence. It induces a homeomorphism ∂M → ∐_{i=1}^{r} ∂M_{i}. Hence it induces an isomorphism

H_{3}(M, ∂M)) ≅ ℤ → H_{3} (⋁_{i=1}^{r} M_{i}, ∐_{i=1}^{r} ∂M_{i}) ≅ ⨁_{i=1}^{r} H_{3}(M_{i}, ∂M_{i}) ≅ ℤ^{r}.

This implies r = 1.

4.2. Theorem 3.183 implies for two closed connected 3-manifolds M and N with the same geometry that α_{p}(M̃) = α_{p}(Ñ) for p ≥ 1 and that b_{p}^{(2)}(M̃) = 0 ⇔ b_{p}^{(2)}(Ñ) = 0 for p ≥ 0. If M has a geometry, it is either aspherical or the geometry is S^{3} or S^{2} ×ℝ. Since we assume π_{1}(M) to be infinite, the geometry S^{3} cannot occur. If M is aspherical, it is an irreducible manifold with infinite fundamental group by Exercise 4.1 and hence b_{p}^{(2)}(M̃) = 0 for all p ≥ 0 by Theorem 4.1. If M has the geometry S^{2} × ℝ, we conclude b_{p}^{(2)}(M̃) = 0 for all p ≥ 0, since S^{1} ×S^{2} has geometry S^{2} ×ℝ and b_{p}^{(2)}({ \atop S^{1} × S^{2}}) = 0 for all p ≥ 0. We conclude from Theorem 4.2 that one can distinguish the geometries from one another by the Novikov-Shubin invariants of the universal coverings of closed 3-manifolds except ℍ^{2} × ℝ and { \atop SL_{2}(ℝ)}.

4.3.

(1) ℍ^{3}: The fundamental group of a closed hyperbolic manifold does not contain ℤ ⊕ ℤ as subgroup [466, Corollary 4.6 on page 449]. It cannot be virtually cyclic because this would imply that a finite covering of M is homotopy equivalent to S^{1} or because this would imply α_{1}(M̃) = 1 by Theorem 2.55 (5a) contradicting Theorem 4.2 (2) which says α_{1}(M̃) = ∞^{+};

(2) S^{3}: π is finite if and only if M̃ is compact;

(3) S^{2} × ℝ: M is finitely covered by S^{1} × S^{2} and hence has infinite virtually cyclic fundamental group. Since α_{1}(M) = 1 is only true for the geometry S^{2} × ℝ by Theorem 4.2 (2), the fundamental groups for all the other geometries cannot be infinite virtually cyclic by Theorem 2.55 (5a);

(4) ℝ^{3}: M is finitely covered by T^{3} and hence contains ℤ^{3} as subgroup of finite index. Since α_{1}(M) = 3 is only true for the geometry ℝ^{3} by Theorem 4.2 (2), the fundamental groups of all the other geometries cannot contain ℤ^{3} as subgroup of finite index by Theorem 2.55 (5a);

(5) Nil: M is finitely covered by a closed manifold \overline{M} which is a S^{1}-fibration S^{1} → \overline{M} → T^{2}. Hence π contains a subgroup of finite index G which can be written as an extension 1 → ℤ → G → ℤ^{2} → 1;

(6) ℍ^{2} × ℝ: M is finitely covered by S^{1} × F_{g} for a closed surface of genus g ≥ 2. Hence π contains a subgroup of finite index which is isomorphic to ℤ × G for some group G which is not solvable. Hence π is not solvable. The fundamental groups of all geometries except ℍ^{3}, ℍ^{2} × ℝ and { \atop Sl_{2}(ℝ)} are solvable;

(7) { \atop Sl_{2}(ℝ)}: M is finitely covered by a closed manifold \overline{M} which is a S^{1}- prinicipal bundle S^{1} → \overline{M} → F_{g} for a closed surface of genus g ≥ 2. This shows that π is not solvable and contains ℤ⊕ℤ as subgroup. Next we show that π does not contain a subgroup of finite index of the shape ℤ×G for a group G. Suppose the contrary. Then we can arrange by possibly passing to a finite covering over \overline{M} that π_{1}(\overline{M}) = ℤ × G. Since S^{1} → \overline{M} → F_{g} is a S^{1}-principal bundle, the center of π_{1}(\overline{M}) contains the image of π_{1}(S^{1}) → π_{1}(\overline{M}). Since Z(π_{1}(F_{g})) does not contain ℤ ⊕ ℤ and hence has trivial center, the center of π_{1}(\overline{M}) is the image of π_{1}(S^{1}) → π_{1}(\overline{M}). The center of ℤ × G is ℤ × Z(G). This implies that the map π_{1}(S^{1}) → π_{1}(\overline{M}) has a retraction which comes from the projection ℤ × G → ℤ. Hence the S^{1}-bundle S^{1} → \overline{M} → F_{g} is trivial. This implies that \overline{M} is diffeomorphic to S^{1} × F_{g} and hence carries the geometry ℍ^{2} × ℝ, a contradiction;

(8) Sol: Any Sol-manifold M has a finite covering \overline{M} which can be written as a bundle T^{2} → \overline{M} → S^{1} [466, Theorem 5.3 on page 447]. Hence we get a subgroup G ⊂ π of finite index which is an extension 0 → ℤ^{2} → G → ℤ → 0. The group π cannot be virtually abelian, since otherwise π contains ℤ^{3} as subgroup of finite index and carries the geometry ℝ^{3}.

4.4. All Novikov-Shubin invariants are ∞^{+} (see Example 2.5) and we have b_{p}^{(2)}(M̃) = |π|^{−1}·b_{p}(M̃) (see Theorem 1.35 (9)). If M is closed, M̃ is homotopy equivalent to S^{3}. If M has boundary, then H_{p}(M̃) = 0 for p = 1, 3 and we get dim_{ℂ}(H_{2}(M̃; ℂ)) = χ(M̃) − 1. Hence

b_{0}^{(2)}(M̃) = |π|^{−1};
b_{1}^{(2)}(M̃) = 0;
b_{2}^{(2)}(M̃) = χ(M) − |π|^{−1} if ∂M ≠ ∅;
b_{2}^{(2)}(M̃) = 0  if ∂M = ∅;
b_{3}^{(2)}(M̃) = 0  if ∂M ≠ ∅;
b_{3}^{(2)}(M̃) = |π|^{−1}  if ∂M = ∅.

4.5. We have to check that it is not true that b_{p}^{(2)}(M̃) = 0 for all p ≥ 0 and α(M̃) = ∞^{+} for all p ≥ 1 holds. Suppose that it is true. Then we conclude from Theorem 4.1 and Theorem 4.2 (2) and (4) that M is an irreducible 3-manifold with non-empty compressible torus boundary. Now use the construction in the proof of [322, Lemma 6.4 on page 52] based on the Loop Theorem [252, Theorem 4.2 on page 39] and the formula for the second Novikov-Shubin invariants under wedges (cf. Theorem 2.55 (4)) to get a contradiction also in this remaining case.

4.6. See [322, Proposition 6.5 on page 54].

4.7. This follows from Theorem 3.106. Notice that π_{1}(T_{f}) belongs to the class G (cf. the paragraph after the proof of Theorem 3.106).

Chapter 5

5.1. See [251, Lemma 3.1 in IV.3 on page 205].

5.2. See see [293, page 4 and 5].

5.3. See [293, page 7].

5.4. See [293, page 8].

5.5. Since GL(n, ℂ)×GL(m, ℂ) is a subgroup of GL(m+n, ℂ) closed under taking conjugate transpose and similar for ℝ, we get that the product G×H of two connected linear reductive Lie groups G and H is again a connected linear reductive Lie group. Since we get for the center Z(G × H) = Z(G) × Z(H), the claim follows by the previous Exercise 5.4.

5.6. Since the spaces in question are homogeneous, it suffices to construct the global symmetry t_{x} in some model at one point x.

5.7. One can identify G_{p}(ℝ^{p+q}) with SO(p + q)/SO(p) × SO(q). Now apply Example 5.11.

5.8. By Theorem 5.12 (2) one can read off dim(M) − f-rk(M̃) from α_{p}(M̃) for p ≥ 1 which depends only on the homotopy type (see Theorem 2.55 (1)). Notice that M and N are aspherical by Lemma 5.10 and hence homotopic if they have isomorphic fundamental groups.

5.9. Amenability of π_{1}(M) is equivalent to α_{1}(M̃) ≠ ∞^{+} by Theorem 2.55 (5b) provided that π_{1}(M) is infinite. We conclude from (5.14) and the fact

that \frac{dim({ \atop M_{ncp}})−f-rk({ \atop M_{ncp}})}{2} ≥ 1 if { \atop M_{ncp}} ≠ \{∗\} (see Theorem 5.12 (2)) that { \atop M_{ncp}} = \{∗\} if π_{1}(M) is amenable.

Suppose that { \atop M_{ncp}} = \{∗\}. Then M carries a Riemannian metric with nonnegative sectional curvature (see Theorem 5.9) and hence with non-negative Ricci curvature. Then π_{1}(M) has polynomial growth (see [218, Theorem 3.106 on page 144]). Hence π_{1}(M) is virtually nilpotent and in particular amenable.

5.10. Let X^{n} for n = 2, 3, 5 be a closed hyperbolic manifold of dimension n. Put M = X^{5} × X^{5} and N = X^{3} × X^{3} × X^{2} × X^{2}. We have ℤ^{4} ⊂ π_{1}(N), but ℤ^{4} cannot be a subgroup of π_{1}(M) since any abelian subgroup of π_{1}(X^{5}) is trivial or ℤ. Hence M and N have non-isomorphic fundamental groups. All L^{2}-Betti numbers, all Novikov-Shubin invariants and the L^{2}-torsion of M̃ and Ñ with respect to the π_{1}(M)- and π_{1}(N)-action agree. This follows from the various product formulas and the known values for hyperbolic manifolds or directly from Theorem 5.12.

5.11. Since M is aspherical, M̃ = M̃_{Eucl} × M̃_{ncp} by Lemma 5.10. Now apply Theorem 1.35 (2) and (5.15).

5.12. Suppose that b_{p}^{(2)}(M̃) = 0 for all p ≥ 0. Since M carries a Riemannian metric with non-positive sectional curvature (see Theorem 5.9 and Lemma 5.10), the Whitehead group of its fundamental group vanishes [192, page 61]. Hence also Ñ is det-L^{2}-acyclic and ρ^{(2)}(M̃) = ρ^{(2)}(Ñ) by Theorem 3.96 (1).

Suppose that b_{p}^{(2)}(M̃) ≠ 0 for some p ≥ 0. Then M has even dimension by Theorem 5.12 (1). But ρ^{(2)}(Ñ) = 0 for any closed even-dimensional Riemannian manifold (see Subsection 3.4.3).

Chapter 6

6.1. Since M is the directed union of its finitely generated submodules, we can assume without loss of generality because of Theorem 6.7 (1) and (4c) that M is finitely generated projective. Now apply Lemma 6.28 (3).

6.2. Choose an exact sequence 0 → K → N (G)^{n} → M → 0. We conclude from Theorem 6.7 (4b) that dim_{N (G)}(K) + dim_{N (G)}(M) = n. Since K is the directed union of its finitely generated submodules, we get from Theorem 6.7 (1) and (4c) the existence of a finitely generated projective N (G)- module P ⊂ K with dim_{N (G)}(P) ≥ dim_{N (G)}(K) − ϵ/2. Hence N (G)^{n}/P is a finitely presented N (G)-module which satisfies dim_{N (G)}(N (G)^{n}/P) ≤ dim_{N (G)}(M) + ϵ/2 because of Theorem 6.7 (4b) and maps surjectively onto M. Therefore we can assume without loss of generality that M is finitely presented.

Because of Theorem 6.7 (3) we may assume M = TM.

From Lemma 6.28 (4), we get an exact sequence 0 → N (G)^{n} ⟶^{f} N (G)^{n} → M → 0 for some positive N (G)-map f. Let {E_{λ} | λ ≥ 0} be the spectral family of ν(f) (see (6.22)). Then ν(f) splits as the orthogonal sum of a weak isomorphism g_{λ}: im(E_{λ}) → im(E_{λ}) and an automorphism g_{λ}^{⊥}: im(E_{λ})^{⊥} →

im(E_{λ})^{⊥}. Thus we obtain an exact sequence 0 → ν^{−1}(im(E_{λ})) ⟶^{ν(g_{λ})} ν^{−1}(im(E_{λ})) → M → 0 and dim_{N (G)}(ν^{−1}(im(E_{λ}))) = dim_{N (G)}(im(E_{λ})) from Theorem 6.24. Since ker(ν(f)) = 0 by Theorem 6.24 (3) we conclude from Theorem 1.12 (4) that lim_{λ→0} dim_{N (G)}(im(E_{λ})) = 0. Hence we can find λ > 0 with dim_{N (G)}(ν^{−1}(im(E_{λ}))) ≤ ϵ. Since M is a quotient of ν^{−1}(im(E_{λ})), the claim follows.

6.3. From the previous Exercise 6.2 we get a finitely generated projective N (G)-module Q with an epimorphism f: Q → P/M such that dim_{N (G)}(Q) ≤ dim_{N (G)}(P/M) + ϵ holds. Choose an N (G)-map \overline{p}: P → Q such that f ∘ \overline{p} = p holds for the canonical projection p: P → P/M. Because of Theorem 6.7 (1) P/ ker(\overline{p}) is a finitely generated projective N (G)-module and hence ker(\overline{p}) is a direct summand in P. Theorem 6.7 (4b) implies dim_{N (G)}(M) ≤ dim_{N (G)}(ker(\overline{p})) + ϵ. Take P′ = ker(\overline{p}).

6.4. If M is finitely generated and projective, M and M^{∗} are isomorphic, an isomorphism is given by any inner product (see Lemma 6.23). If M is finitely generated, the claim follows from Theorem 6.7 (3) and (4e). The countably generated case is reduced to the finitely generated case as follows.

Since M is countably generated, we can find a family {M_{n} | n = 0, 1, 2, …} of finitely generated submodules of M such that M_{m} ⊂ M_{n} holds for m ≤ n and M is the union ⋃_{n≥0} M_{n}. Then

M = colim_{n→∞} M_{n};
M^{∗} = lim_{n→∞} M_{n}^{∗}.

Theorem 6.13 (1) and Theorem 6.18 imply

dim(M) = sup {dim(M_{n}) | n ≥ 0};
dim(M^{∗}) = sup {inf {dim(im(M_{n}^{∗} → M_{m}^{∗})) | n = m, m + 1, …} | m ≥ 0} .

Hence it suffices to show that dim(im(M_{n}^{∗} → M_{m}^{∗})) = dim(M_{m}^{∗}) for m ≤ n. As the canonical projection induces an isomorphism (PM_{n})^{∗} → M_{n}^{∗} it suffices to show for an inclusion P ⊂ Q of finitely generated projective N (G)-modules that dim(im(Q^{∗} → P^{∗})) = dim(P^{∗}).

Since \overline{P} is a direct summand in Q, it suffices to treat the case Q = \overline{P}. Theorem 6.7 (4d) implies dim(P) = dim(\overline{P}) and hence dim(P^{∗}) = dim(\overline{P}^{∗}). We get an exact sequence 0 → (\overline{P} /P)^{∗} → \overline{P}^{∗} ⟶^{i^{∗}} P^{∗} → coker(i^{∗}) → 0. Since (\overline{P} /P)^{∗} is zero, we conclude dim(coker(i^{∗})) = 0 and hence the claim from Theorem 6.7 (4b).

6.5. Each map ϕ_{J,K} is injective and hence its image has dimension 1 by Theorem 6.7 (4b). Theorem 6.13 (2) implies dim_{N (ℤ)}(M) = 1. Notice that ϕ_{J,K}^{∗} is given by multiplication with ∏_{u∈K−J}(z − u). Since M^{∗} is the limit over the inverse system given by the maps ϕ_{J,K}^{∗}, we get an isomorphism

M^{∗} = ⋂_{u∈S^{1}}(z − u),

where (z − u) is the ideal generated by (z − u). We have already shown in Example 6.19 that ⋂_{u∈S^{1}}(z − u) = {0}.

6.6. We first show that N (ℤ) is not Noetherian. Identify N (ℤ) = L^{∞}(S^{1}) (see Example 1.4). Consider the sequence of L^{∞}(S^{1})-submodules P_{1} ⊂ P_{2} ⊂ P_{3} ⊂ … ⊂ L^{∞}(S^{1}) such that P_{n} is the ideal generated by the characteristic function of {exp(2πit) | t ∈ [1/n, 1]}. One easily checks that P = ⋃_{n≥1} P_{n} is a submodule of L^{∞}(S^{1}) which can be written as the infinite sum of nontrivial L^{∞}(S^{1})-modules Q_{n} for n ≥ 1, where Q_{n} is the ideal generated by the characteristic function of {exp(2πit) | t ∈ [1/(n + 1), 1/n]}. Hence P cannot be finitely generated.

If i: ℤ → G is an inclusion, i_{∗}P is a submodule of i_{∗}N (ℤ) = N (G) and is the infinite sum of non-trivial N (G)-modules i_{∗}Q_{n} by Theorem 6.29 (1).

We will later show in an exercise of Chapter 9 that N (G) is Noetherian if and only if G is finite.

6.7. Choose a free AG-module F and AG-maps i: P → F and r: F → P with r ∘i = id. Since M ⊂ P is finitely generated, there is a finitely generated free direct summand F_{0} ⊂ F with i(M) ⊂ F_{0} and F_{1} = F/F_{0} free. Hence i induces a map f: P/M → F_{1}. It suffices to show that f is trivial because then i(P) ⊂ F_{0} and the restriction of r to F_{0} yields an epimorphism F_{0} → P.

Let g: AG → P/M be any AG-map. Since the von Neumann algebra N (G) is semihereditary (see Theorem 6.7 (1)), the image of N (G)⊗_{AG} (f ∘g) is a finitely generated projective N (G)-module. Its von Neumann dimension is zero by Theorem 6.7 (4b) since N (G) ⊗_{AG} P/M has trivial dimension by assumption. We conclude from Lemma 6.28 (3) that the image of N (G) ⊗_{AG} (f ∘ g) is trivial. Hence N (G) ⊗_{AG} (f ∘ g) is the zero map. Since AG → N (G) is injective, f ∘ g is trivial. This implies that f is trivial since g was arbitrary.

6.8. We conclude from Theorem 6.7 (4b) that T_{dim}M is the largest N (G)- submodule of M with vanishing von Neumann dimension.

This definition of T_{dim}M coincides with the one of Definition 6.1 of TM by Theorem 6.7 (3), (4b) and (4e) and Lemma 6.28 (3), provided that M is finitely generated.

A counterexample for not finitely generated M comes from Exercise 6.5, where a module M was constructed with dim(M) = 1 and M^{∗} = 0.

6.9. Let M be a ℂH-module. If P_{∗} is a projective ℂH-resolution for M, then ℂG ⊗_{ℂH} P_{∗} is a projective ℂG-resolution for ℂG ⊗_{ℂH} M. We have N (G) ⊗_{ℂG} ℂG ⊗_{ℂH} P_{∗} ≅_{N (G)} N (G) ⊗_{N (H)} N (H) ⊗_{ℂH} P_{∗}. Theorem 6.29 implies

dim_{N (H)} (Tor_{p}^{ℂH}(N (H), M)) = dim_{N (G)} (N (G) ⊗_{N (H)} Tor_{p}^{ℂH}(N (H), M))
= dim_{N (G)} (Tor_{p}^{ℂG}(N (G), ℂG ⊗_{ℂH} M)) .

6.10. Let ℤ ⊂ G be an infinite cyclic subgroup of finite index. Let M be a ℂG-module. Consider the canonical ℂG-map p: ℂG ⊗_{ℂ[ℤ]} res_{G}^{ℤ} M → M sending g ⊗ m to gm, where res_{G}^{ℤ} M is the restriction of the ℂG-module M to a ℂ[ℤ]-module. Define a ℂG-map i: M → ℂG ⊗_{ℂ[ℤ]} res_{G}^{ℤ} M by sending m to \frac{1}{[G:ℤ]} · ∑_{gℤ∈G/ℤ} g ⊗ g^{−1}m. One easily checks that it is independent of the choice of representative g for gℤ and satisfies p ∘ i = id. Since [G : ℤ] is finite, N (G) is isomorphic as N (ℤ)-module to ⨁_{[G:ℤ]} N (ℤ). This implies that Tor_{p}^{ℂG}(N (G); M) is a direct summand in

Tor_{p}^{ℂG}(N (G), ℂG ⊗_{ℂ[ℤ]} res_{G}^{ℤ} M) ≅ ⨁_{[G:ℤ]} Tor_{p}^{ℂ[ℤ]}(N (ℤ), res_{G}^{ℤ} M).

Hence it suffices to prove the claim for G = ℤ.

The functor Tor commutes in both variables with colimits over directed systems [94, Proposition VI.1.3. on page 107]. Hence it suffices to prove for a finitely generated ℂℤ-module M that Tor_{p}^{ℂ[ℤ]}(N (ℤ), M) = 0 for p ≥ 1. Since ℂ[ℤ] is a principal ideal domain [15, Proposition V.5.8 on page 151 and Corollary V.8.7 on page 162], we can write

M = ℂ[ℤ]^{n_{p}} ⊕ (⨁_{i_{p}=1}^{s_{p}} ℂ[ℤ]/((z − a_{p,i_{p}})^{r_{p,ip}}))

for a_{p,i_{p}} ∈ ℂ and n_{p}, s_{p}, r_{p,i_{p}} ∈ ℤ with n_{p}, s_{p} ≥ 0 and r_{p,i_{p}} ≥ 1, where z is a fixed generator of ℤ. Hence we can assume without loss of generality that M = ℂ[ℤ]/(z −a)^{n} for appropriate a ∈ ℂ and n ≥ 1. Using the obvious ℂ[ℤ]- resolution one reduces the claim to the assertion that N (ℤ) ⟶^{r_{(z−a)^{n}}} N (ℤ) is injective which is obvious in view of the identification N (ℤ) = L^{∞}(S^{1}) (see Example 1.4).

6.11. (1) This is a modification of the corresponding argument for dimensionflatness appearing in the solution to Exercise 6.9.

(2) Show that Tor_{p}^{ℂK}(N (K), M) is a direct summand in

Tor_{p}^{ℂK}(N (K), ℂK ⊗_{ℂG} res_{K}^{G} M) ≅ ⨁_{[K:G]} Tor_{p}^{ℂG}(N (G), res_{K}^{G} M)

for any ℂK-module M. The argument is analogous to the one appearing in the solution of the previous Exercise 6.10.

(3) Since N (G) is flat over ℂG, we get for p ≥ 1

H_{p}^{G}(EG; N (G)) = N (G) ⊗_{ℂG} H_{p}(EG) = 0.

(4) This follows from (3).

(5) This follows from (3) and Lemma 6.98.

(6), (7) (8) These follow from (1), (4) and (5) and the following statements. We get from Theorem 1.35 (2) and (8) and Example 2.59

α_{1}(Eℤ^{n}; N (ℤ^{n})) = n;
b_{1}^{(2)}(E(ℤ ∗ ℤ); N (ℤ ∗ ℤ)) = χ(S^{1} ∨ S^{1}) − b_{0}^{(2)}(E(ℤ ∗ ℤ); N (ℤ ∗ ℤ)) = 1.

If M is an aspherical manifold whose universal covering is a symmetric space, then we get for G = π_{1}(M) from Corollary 5.16 (2) that one of the L^{2}-Betti numbers b_{p}^{(2)}(G) is different from zero or one of the Novikov Shubin invariants α_{p}(EG; N (G)) is different from ∞^{+}.

(9) Because of (1) and (3) it suffices to prove for the fundamental group π of a connected sum M = M_{1}# … #M_{r} of (compact connected orientable) non-exceptional prime 3-manifolds M_{j} that H_{p}^{G}(Eπ; N (π)) is different from zero for at least one p ∈ {0, 1, 2}. We can assume without loss of generality that ∂M contains no S^{2}, otherwise glue D^{3}-s to M. It suffices to treat the case |π| = ∞ because H_{0}(EG; N (G)) is non-zero for finite G (see Theorem 6.54 (8b)). We conclude from Theorem 4.1 that either b_{1}^{(2)}(M̃) ≠ 0 or M is ℝℙ^{3}#ℝℙ^{3} or M is a prime manifold with infinite fundamental group whose boundary is empty or a disjoint union of tori. Since the classifying map M → Bπ is 2-connected we conclude from Lemma 6.98 that H_{1}(Eπ; N (π)) ≠ 0 if b_{1}^{(2)}(M̃) ≠ 0 or α_{1}(M̃) ≠ 0. Since α_{1}({ \atop ℝℙ^{3} ∗ ℝℙ^{3}}) = α_{1}({ \atop S^{1} × S^{2}}) = 1 by Theorem 4.2 (2), only the case of an non-exceptional irreducible 3-manifold with infinite fundamental group whose boundary is empty or a disjoint union of tori is left. Notice that M is aspherical by Exercise 4.1. Hence it suffices to show that α_{2}(M̃) ≠ ∞^{+} because of Lemma 6.98. This follows in the case, where the boundary is empty or a union of incompressible tori from Theorem 4.2. Now reduce the case, where the boundary contains a compressible torus, to the case, where the boundary is a union of incompressible tori, using the construction in the proof of [322, Lemma 6.4 on page 52] based on the Loop Theorem [252, Theorem 4.2 on page 39] and the formula for the second Novikov-Shubin invariants under wedges (cf. Theorem 2.55 (4)).

6.12. See [334, Theorem 5.3 on page 233].

6.13. See [107, Section 4].

6.14. Fix a natural number n. Then the number of cells with isotropy group whose order is less or equal to n must be finite since otherwise

m(X; N (G)) ≥ ∑_{c∈I(X),|G_{c}|≤n} |G_{c}|^{−1} ≥ ∑_{c∈I(X),|G_{c}|≤n} n^{−1} ≥ ∞.

6.15. Use induction over the cells of B. In the induction step apply the sum formula (see Theorem 6.80 (2)) to the pushout on the level of total spaces which is obtained from a pushout on the base space level, which is given by attaching a cell, by the pullback construction.

6.16. Let K be the kernel of the projection p: ℤ ∗ ℤ → ℤ which sends the two generators of ℤ ∗ ℤ to the generator of ℤ. We obtain a fibration BK → B(ℤ ∗ ℤ) → Bℤ. Hence B(ℤ ∗ ℤ) is homotopy equivalent to the mapping torus of an appropriate selfhomotopy equivalence BK → BK. But b_{1}^{(2)}(ℤ ∗ ℤ) = 1.

6.17. We can assume without loss of generality that the boundary of M contains no S^{2}-s, otherwise attach D^{3}-s to M.

If M is a prime manifold with infinite fundamental group, then either M is irreducible and hence aspherical by Exercise 4.1 and we get χ^{(2)}(π_{1}(M)) = χ(M) from Theorem 1.35 (2) or M = S^{1} × S^{2} and hence χ^{(2)}(π_{1}(S^{1} × S^{2})) = χ^{(2)}(S̃^{1}) = χ(S^{1}) = χ(S^{1} × S^{2}) = 0 by Theorem 1.35 (2). Notice that π_{1}(M) = ∗_{i=1}^{r}π_{1}(M_{i}) and χ(M) = ∑_{i=1}^{r} χ(M_{i}). We conclude χ^{(2)}(M̃) = 1 − r+∑_{i=1}^{r} χ^{(2)}(M̃_{i}) from Theorem 6.80 (2) and (8). We have χ^{(2)}(EG; N (G)) = |G|^{−1} for a finite group G by Theorem 6.80 (7). Now the claim follows from Theorem 4.1.

6.18. b_{p}^{(2)}(G) = b_{p}^{(2)}(E(G; FIN); N (G)) follows from Theorem 6.54 (2) since EG × E(G; FIN) is a model for EG.

6.19. ℂ⊗_{ℤ}C_{∗}(E(G; FIN)) is a projective ℂG-resolution of ℂ equipped with the trivial G-action.

6.20. This follows from Theorem 6.54 (2) since EG × X is a model for EG.

6.21. There is an obvious exact sequence of ℤ[ℤ/5] modules 0 → ℤ → ℤ[ℤ/5] → ℤ[ℤ/5]/(N) → 0, where ℤ carries the trivial ℤ/5-action. The associated long cohomology sequence implies

H^{1}(ℤ/5; ℤ[ℤ/5]/(N)) ≅ H^{2}(ℤ/5; ℤ) ≅ ℤ/5.

Notice that the obvious ℤ/5-action on ℤ[ℤ/5]/(N) has no non-trivial fixed point. Now apply Example 6.94.

6.22. Since im(c_{p}) and ker(c_{p}) are submodules of free ℂ[F_{g}]-modules, they are free. Since 0 → ker(c_{p}) → C_{p} → im(c_{p}) → 0 is exact, we conclude that im(c_{p}) and ker(c_{p}) are finitely generated free ℂ[F_{g}]-modules. In particular we see that H_{p}(C_{∗}(X)) has a finite free 1-dimensional resolution, for instance 0 → im(c_{p+1}) → ker(c_{p}) → H_{p}(C_{∗}(X)) → 0. Now fix any finite based free ℂ[F_{g}]-chain complex F [p]_{∗} which is concentrated in dimension p and p+1 and satisfies H_{n}(F [p]_{∗}) = 0 for n ≠ p and H_{p}(F [p]_{∗}) ≅_{ℂ[F_{g}]} H_{p}(C_{∗}(X)). Notice that the ℂ[F_{g}]-chain homotopy type of F [p]_{∗} depends only on the ℂ[F_{g}]- isomorphism class of H_{p}(C_{∗}(X)). One easily constructs a ℂ[F_{g}]-chain map f_{∗}: ⨁_{p≥0} F [p]_{∗} → C_{∗}(X) which induces an isomorphism on homology and hence is a ℂ[G]-chain equivalence. We conclude from the homotopy invariance of the Novikov-Shubin invariants and the L^{2}-Betti numbers (see Theorem 2.19)

b_{p}^{(2)}(X; N (G)) = b_{p}(l^{2}(G) ⊗_{ℂ[F_{g}]} F [p]_{∗}) + b_{p}(l^{2}(G) ⊗_{ℂ[F_{g}]} F [p − 1]_{∗});
α_{p}(X; N (G)) = α_{p}(l^{2}(G) ⊗_{ℂ[F_{g}]} F [p − 1]_{∗})

and that b_{p}(l^{2}(G) ⊗_{ℂF_{g}]} F [p]_{∗}) and α_{p+1}(l^{2}(G) ⊗_{ℂF_{g}]} F [p]_{∗}) depend only on the ℂ[F_{g}]-isomorphism class of H_{p}(C_{∗}(X)).

6.23. We know b_{0}^{(2)}(ℤ ∗ ℤ) = 0 and α_{1}(ℤ ∗ ℤ) = ∞^{+} from Theorem 1.35 (8) and Theorem 2.55 (5). Let G_{r} ⊂ G be the subgroup ∏_{i=1}^{r} ℤ ∗ ℤ. We conclude from the product formulas Theorem 1.35 (4) and Theorem 2.55 (3) that b_{p}^{(2)}(G_{r}) = 0 for p ≤ r − 1 and α_{p}(G_{r}) = ∞^{+} for p ≤ r holds. Lemma 6.98 implies H_{p}^{G_{r}}(EG_{r}; N (G_{r})) = 0 for p ≤ r − 1.

Let G′_{r} be the subgroup ∏_{i=r+1}^{∞} ℤ ∗ ℤ of G. Obviously G = G_{r} × G′_{r}. Hence EG_{r} × EG′_{r} with the obvious G = G_{r} × G′_{r}-operation is a model for EG. Hence we get an N (G)-chain isomorphism

N (G)⊗_{N (G_{r})⊗N (G′_{r})}((N (G_{r}) ⊗_{ℂG_{r}} C_{∗}(EG_{r})) ⊗_{ℂ} (N (G′_{r}) ⊗_{ℂG′_{r}} C_{∗}(EG′_{r})))

≅ N (G) ⊗_{ℂG} C_{∗}(EG).

Since H_{p}(N (G_{r})⊗_{ℂG_{r}} C_{∗}(EG_{r})) = 0 for p ≤ r−1, and N (G_{r})⊗_{ℂG_{r}} C_{∗}(EG_{r}) is a free N (G_{r})-chain complex, N (G_{r}) ⊗_{ℂG_{r}} C_{∗}(EG_{r}) is N (G_{r})-chain homotopy equivalent to a projective N (G_{r})-chain complex P_{∗} with P_{p} = 0 for p ≤ r − 1. Hence N (G) ⊗_{ℂG} C_{∗}(EG) is N (G)-chain homotopy equivalent to an N (G)-chain complex Q_{∗} with Q_{p} = 0 for p ≤ r − 1. This implies H_{n}(N (G) ⊗_{ℂG} C_{∗}(EG)) = 0 for p ≤ r − 1. Since r can be chosen arbitrarily large, the assertion follows.

A more general statement will be proved in Lemma 12.11 (5).

6.24. We get H_{p}^{ℤ}(S̃^{1}; N (ℤ)) = {0} for p ≥ 1 and H_{1}^{ℤ^{2}}(T̃^{2}; N (ℤ^{2})) ≠ {0} from Lemma 6.98 since b_{p}^{(2)}(S̃^{1}) = 0 for p ≥ 1, α_{p}(S̃^{1}) = ∞^{+} for p ≥ 2 and α_{2}(T̃^{2}) = 2 holds by Example 2.59.

6.25. If N = N (H), Theorem 6.104 follows from Theorem 6.29 applied to the inclusion G → G × H. Now Theorem 6.104 follows for any projective N (H)- module N by Theorem 6.7 (4b) and (4c). Since the statement in Theorem 6.104 (2) is symmetric in M and N, it is now proved in the case, where M or N is projective.

If N is a finitely generated N (H)-module, it splits as PN⊕TN, where PN is finitely generated projective and dim_{N (H)}(TN) = 0 (see Theorem 6.7 (3) and (4e)). Since dim_{N (H)}(TN) = 0 and each N (G)-module M is a quotient of a free N (G)-module, we conclude dim_{N (G×H)}(N (G × H) ⊗_{N (G)⊗_{ℂ}N (H)} M ⊗_{ℂ} TN) = 0 for any N (G)-module M from Theorem 6.7 (4b). Hence Theorem 6.104 follows for finitely generated N.

Since N is the directed colimit of its finitely generated submodules and N (G×H)⊗_{N (G)⊗_{ℂ}N (H)}M ⊗_{ℂ}− commutes with directed colimits and directed colimits are exact, Theorem 6.104 follows from Additivity (see Theorem 6.7 (4b)) and Theorem 6.13.

6.26. Let C_{∗} and D_{∗} be positive N (G) and N (H)-chain complexes. There are canonical isomorphisms of N (G) ⊗_{ℂ} N (H)-modules

α: ⨁_{p+q=n} H_{p}(C_{∗}) ⊗_{ℂ} H_{q}(D_{∗}) ⟶^{≅} H_{n}(C_{∗} ⊗ D_{∗}).

Next we want to show that the composition of the canonical map

N (G×H)⊗_{N (G)⊗N (H)}H_{p}(C_{∗}⊗_{ℂ}D_{∗}) → H_{p}(N (G×H)⊗_{N (G)⊗N (H)}(C_{∗}⊗_{ℂ}D_{∗}))

with N (G × H) ⊗_{N (G)⊗N (H)} α is a dim-isomorphism, i.e. its kernel and cokernel have zero dimension. Obviously we may assume that D_{∗} is finite dimensional. We use induction over the dimension d of D_{∗} If D_{∗} is zero-dimensional, this follows from Theorem 6.104 (1). In the induction step use the long dimexact sequences associated to 0 → D_{∗}|_{d−1} → D_{∗} → d[D_{d}] → 0 and a Five-Lemma for dim-isomorphisms for dim-exact sequences whose proof is an easy consequence of Theorem 6.7 (4b). Here dim-exact has the obvious meaning and follows from Theorem 6.104 (1).

As explained in the proof of 6.54 (5), we can assume that X is a G-CW -complex and Y is an H-CW -complex. Now apply the results above to C_{∗} = N (G) ⊗_{ℤG} C_{∗}(X) and D_{∗} = N (H) ⊗_{ℤH} C_{∗}(Y). The claim follows from Theorem 6.104 (2) since there is a ℤ[G × H]-chain isomorphism C_{∗}(X) ⊗_{ℂ} C_{∗}(Y) → C_{∗}(X × Y).

Chapter 7

7.1. This follows from an iterated application of Theorem 7.2 (6).

7.2. This follows from the previous Exercise 7.1.

7.3. We can write BG as the colimit of the directed system

BG_{1} ⊂ BG_{1} ∨ BG_{2} ⊂ BG_{1} ∨ BG_{2} ∨ BG_{3} ⊂ ….

Notice that each of the structure maps has a retraction. If p: EG → BG is the universal covering, we can write EG as the colimit of the directed system of free G-CW -complexes

EG|_{BG_{1}} ⊂ EG|_{BG_{1}∨BG_{2}} ⊂ EG|_{BG_{1}∨BG_{2}∨BG_{3}} ⊂ ….

We conclude from Theorem 6.7 (4c) and Theorem 6.29

b_{p}^{(2)}(G) = lim_{r→∞} b_{p}^{(2)} ( ⋁_{n=1}^{r} G_{i}) .

We have for p ≥ 2

b_{p}^{(2)} ( ⋁_{n=1}^{r} G_{i}) = ∑_{n=1}^{r} b_{p}^{(2)}(G_{i})

and

b_{1}^{(2)} ( ⋁_{n=1}^{r} G_{i}) − b_{0}^{(2)} ( ⋁_{n=1}^{r} G_{i}) = r − 1 + ∑_{n=1}^{r}(b_{1}^{(2)}(G_{i}) − b_{0}^{(2)}(G_{i})).

It suffices to prove this in the case r = 2, where it follows from the long exact homology sequence … → H_{p}^{G}(G; N (G)) → H_{p}^{G}(G ×_{G_{1}} EG_{1}; N (G)) ⊕ H_{p}^{G}(G ×_{G_{2}} EG_{2}; N (G)) → H_{p}^{G}(EG; N (G)) → H_{p−1}^{G}(G; N (G)) → …, Theorem 6.7 (4b) and Theorem 6.29. Notice that b_{0}^{(2)}(G_{i}; N (G_{i})) ≤ 1/2 holds for all i ∈ I by Theorem 6.54 (8b).

7.4. This follows from an iterated application of Theorem 6.54 (5).

7.5. It suffices to show that G ∈ B_{d} for each d ≥ 0. Since B_{0} is the class of infinite groups by Theorem 6.54 (8b), this is clear for d = 0. The induction step from d to d + 1 is done as follows. We can write G = G_{1} × G_{2}, where G_{1} and G_{2} are of the shape ∏_{j∈J} H_{j} for an infinite index set J and non-trivial groups H_{j}. Hence G_{1} ∈ B_{d} and G_{2} ∈ B_{0} by induction hypothesis. Then G = G_{1} × G_{2} ∈ B_{d+1} by Theorem 6.54 (5).

As G = ∏_{i∈I} ℤ is abelian and hence amenable, we get H_{0}^{G}(EG; N (G)) ≠ 0 from Theorem 6.54 (8c).

7.6. An infinite locally finite group is an infinite amenable group. Hence H_{0}^{G}(EG; N (G)) ≠ 0 by Theorem 6.54 (8c) and b_{p}^{(2)}(G) = 0 for all p ≥ 0 by Theorem 7.2 (1). We have H_{p}^{H}(EH; N (H)) = H_{p}^{G}(G ×_{H} EH; N (G)) = 0 for p ≥ 1 for any finite subgroup H ⊂ G by Theorem 6.29 (1). Since G is the colimit of the system of its finite subgroups, H_{p}^{G}(EG; N (G)) = 0 for p ≥ 1 by a colimit argument.

7.7. Suppose that K is finite. Then the claim follows from Theorem 6.54 (6b).

Suppose that H is finite. It suffices to show for any fibration BH → E_{0} ⟶^{f} E_{1} of connected CW -complexes that

b_{p}^{(2)}(Ẽ_{1}) = |H| · b_{p}^{(2)}(Ẽ_{0}).

The map h induces a ℂ[π_{1}(E_{0})]-chain homotopy equivalence f_{∗}: C_{∗}(Ẽ_{0}) → res C_{∗}(Ẽ_{1}), where res denotes the restriction coming from the epimorphism π_{1}(f). It yields an N (π_{1}(E_{0}))-chain equivalence

N (π_{1}(E_{0})) ⊗_{ℂ[π_{1}(E_{0})]} C_{∗}(Ẽ_{0}) ⟶^{≃} res (N (π_{1}(E_{1})) ⊗_{ℂ[π_{1}(E_{1})]} C_{∗}(Ẽ_{1})) .

Since for any N (π_{1}(E_{1}))-module M we have (see Lemma 13.45)

|H| · dim_{N (π_{1}(E_{0}))}(res M) = dim_{N (π_{1}(E_{1}))}(M),

the claim follows.

7.8. Let f: B → B(π_{1}(B)) be the classifying map of B. We can find a fibration F_{0} ⟶^{j} E_{0} ⟶^{q} B(π_{1}(B)) together with a homotopy equivalence u: E_{0} → E such that f ∘ p ∘ u = q (see [521, Theoren 7.30 in I.7 on page 42]). Since the L^{2}-Betti numbers are homotopy invariants (see Theorem 6.54 (1)) and π_{1}(B) ∈ B_{d} by assumption, it suffices to prove that b_{p}^{(2)}(F̃_{0}) < ∞ for p ≤ d. There is also a fibration F ⟶^{k} F_{1} ⟶^{r} B̃ and a homotopy equivalence v: F_{1} → F_{0} such that u ∘ j ∘ v ∘ k ≃ i. It remains to prove that b_{p}^{(2)}(F̃_{1}) < ∞ for p ≤ d.

The Serre spectral sequence converges to H_{p+q}^{π_{1}(F_{1})}(F̃_{1}; N (π_{1}(F_{1}))) and has as E^{2}-term

E_{p,q}^{2} = H_{p}(B̃) ⊗_{ℂ} H_{q} (N (π_{1}(F_{1})) ⊗_{ℂ[π_{1}(F_{1})]} ℂ[π_{1}(F_{1})] ⊗_{ℂ[π_{1}(k)]} C_{∗}(F̃))

since B̃ is simply connected. Because of Additivity (see Theorem 6.7 (4b)) and the assumption that H_{p}(B̃; ℂ) is finite dimensional for p ≤ d, it suffices to show that for q ≤ d

dim_{N (π_{1}(F_{1}))} (H_{q} (N (π_{1}(F_{1})) ⊗_{ℂ[π_{1}(F_{1})]} ℂ[π_{1}(F_{1})] ⊗_{ℂ[π_{1}(k)]} C_{∗}(F̃))) < ∞.

But this is b_{q}^{(2)}(π_{1}(E) ×_{π_{1}(i)} F̃; N (π_{1}(E))) by Theorem 6.29.

7.9. We have already explained in the solution of Exercise 7.3 that

b_{1}^{(2)}(G ∗ H) − b_{0}^{(2)}(G ∗ H) = 1 + b_{1}^{(2)}(G) − b_{0}^{(2)}(G) + b_{1}^{(2)}(H) − b_{0}^{(2)}(H).

Since b_{0}^{(2)}(G) = |G|^{−1} by Theorem 6.54 (8), the claim follows.

7.10. Theorem 6.80 (4) and Lemma 6.87 show

ch_{{1}}^{g}(χ^{G}(E(G; FIN))) = χ^{(2)}(E(G; FIN); N (G)) = χ^{(2)}(G).

By definition ch_{{1}}^{g} ⊂ {r ∈ ℝ | d · r ∈ ℤ}.

7.11. SL(2, ℤ) is isomorphic to ℤ/6 ∗_{ℤ/2} ℤ/4. We have χ^{(2)}(G) = |G|^{−1} for finite G. Now apply (7.13).

7.12. The proof is analogous to the one of Lemma 7.16.

7.13. (1) The Seifert-van Kampen theorem applied to an appropriate decomposition of T_{Bf} into two connected pieces with connected intersection implies that π_{1}(T_{Bf}) is isomorphic to the quotient of the group G ∗ ℤ, which is obtained by introducing for a fixed generator t ∈ ℤ and each g ∈ G the relation tgt^{−1} = f(g). This group is isomorphic to K ⋊ ℤ. Analogous to the proof in Theorem 7.10 one shows that the universal covering of T_{Bf} has trivial homology and hence T_{Bf} is a model for B(K ⋊ ℤ).

(2) We conclude from Theorem 1.39 that b_{p}({ \atop B(K ⋊ ℤ)}) = b_{p}(T̃_{Bf}) = 0 for p ≥ 0. Since G is of det ≥ 1-class by assumption, K ⋊ ℤ is of det ≥ 1-class by Theorem 13.3 (3). The invariant is well-defined because of Lemma 13.6.

(3) obvious.

(4) The proofs of assertion (1), (2), (3) and (4) of Theorem 7.27 carry directly over to the case of an endomorphism. The proofs of assertions (5) of Theorem 7.27 carries over since there is a canonical bijection K/K_{0} ⟶^{≅} G/G_{0} if we assume that f induces a bijection G/G_{0} ⟶^{≅} G/G_{0}. The proofs of assertions (6) and (7) of Theorem 7.27 carry directly over to the case of an endomorphism if one reformulates them as stated.

Chapter 8

8.1. We have to show for a Hilbert space H and a densely defined closed operator f: dom(f) ⊂ H → H that f^{∗}f = 0 implies f = 0. This follows from Lemma 2.2 (1) and the equation ⟨f^{∗}f(x), x⟩ = ⟨f(x), f(x)⟩ for x ∈ dom(f^{∗}f).

8.2. We get a commutative diagram of ℂ-categories

{N (G)^{n}} ⟶^{ν} {l^{2}(G)^{n}}
↓  ↓
{U(G)^{n}} ⟶^{ν_{U}} {l^{2}(G)^{n}}_{U}

whose vertical arrows are the obvious inclusions and whose horizontal arrows are isomorphisms of ℂ-categories. The map ν has already been introduced in (6.22). The categories {U(G)^{n}} and {l^{2}(G)^{n}}_{U} and the functor ν_{U} are defined analogously. Now one can apply the functor idempotent completion Idem to it. This yields a commutative diagram with isomorphisms of ℂ-categories as horizontal maps. Now identify this diagram with the desired diagram by equivalences of ℂ-categories. This has been done for the upper horizontal arrow in the proof of Theorem 6.24 and is done for the lower horizontal arrow analogously using Lemma 8.3 (4).

8.3. The inclusions G ⊂ ℤG ⊂ ℂG ⊂ l^{1}(G) are obvious. Given u = ∑_{g∈G} λ_{g}· g ∈ l^{1}(G), we obtain a bounded G-operator r_{u}: l^{2}(G) → l^{2}(G) which is uniquely determined by g_{0} ↦ ∑_{g∈G} λ_{g} ·g_{0}g. The operator norm of r_{u} satisfies ||r_{u}||_{∞} ≤ ||u||_{1}. Since C_{r}^{∗}(G) is the closure in the norm topology of the image of the map ℂG → B(l^{2}(G)), u ↦ r_{u}, we get an inclusion l^{1}(G) ⊂ C_{r}^{∗}(G). Since N (G) is defined as B(l^{2}(G))^{G} and B(l^{2}(G))^{G} is closed in the norm topology in B(l^{2}(G)), we get an inclusion C_{r}^{∗}(G) ⊂ N (G). We get an inclusion N (G) ⊂ l^{2}(G) by f ↦ f(e) for e ∈ G the unit element. For v ∈ l^{2}(G) we get a densely defined operator r_{v}: dom(r_{v}) = ℂG ⊂ l^{2}(G) → l^{2}(G) which maps u = ∑_{i=1}^{n} λ_{i} · g_{i} to ∑_{i=1}^{n} λ_{i} · g_{i} · v. Since its adjoint is densely defined, this operator is closable. Its minimal closure is an element in U(G). This gives the inclusion l^{2}(G) ⊂ U(G).

8.4. Since U(G) is regular (see Theorem 8.22 (3)), im(f) ⊂ Q and ker(f) ⊂ P are direct summands by Lemma 8.18. We get from Theorem 8.29

ker(f) = 0 ⇔ dim_{U(G)}(ker(f)) = 0 ⇔ dim_{U(G)}(im(f)) = dim_{U(G)}(Q) ⇔ im(f) = Q.

8.5. Let 0 → M_{0} ⟶^{i} M_{1} ⟶^{p} M_{2} → 0 be an exact sequence of N (G)- modules. Then 0 → T_{dim}M_{0} ⟶^{T_{dim}i} T_{dim}M_{1} ⟶^{T_{dim}p} T_{dim}M_{2} is exact since im(i)∩T_{dim}M_{1} = i(T_{dim}M_{0}) follows from Additivity (see Theorem 6.7 (4b)). Hence T_{dim} is left exact.

In Exercise 6.5 we have constructed an N (ℤ)-module M = colim_{J∈I} N_{j} satisfying

dim_{N (ℤ)}(M) = 1;
M^{∗} = 0.

The structure map of N_{j} for j = ∅ ∈ I induces an injection i: N (ℤ) → M. Thus we obtain an exact sequence 0 → N (ℤ) ⟶^{i} M ⟶^{p} coker(i) → 0. Obviously TN (ℤ) = 0. We conclude TM = M from M^{∗} = 0 and hence dim(TM) = 1. We conclude from Additivity (see Theorem 6.7 (4b)) that dim_{N (G)}(coker(i)) = 0 and hence dim_{N (G)}(T coker(i)) = 0 and that the sequence TN (ℤ) ⟶^{Ti} TM ⟶^{Tp} T coker(i) cannot be exact at TM. Hence T is not left exact.

8.6. The following is taken from Example 8.34. Let I_{1} ⊂ I_{2} ⊂ … ⊂ N (G) be a nested sequence of ideals which are direct summands in N (ℤ) such that dim_{N (ℤ)}(I_{n}) ≠ 1 and lim_{n→∞} dim_{N (ℤ)}(I_{n}) = 1. Let I be the ideal ⋃_{n=1}^{∞} I_{n}. Put M = N (ℤ)/I = colim_{n→∞} N (G)/I_{n}. Then Additivity (see Theorem 6.7 (4b)) and Theorem 6.13 (1) imply dim(M) = 0 and hence M = T_{dim}M. Since N (ℤ)/I_{n} is projective and N (ℤ) is semihereditary (see Theorem 6.5 and Theorem 6.7 (1)), we conclude from Theorem 6.13 (1) and Lemma 6.28 (3) that T_{dim} (N (G)/I_{n}) = 0. Hence T_{dim} (colim_{n→∞} N (G)/I_{n}) = M ≠ {0} is different from colim_{n→∞} T_{dim} (N (G)/I_{n}) = {0}. This shows that T_{dim} does not commute with colimits over directed systems in general.

However, it does commute with colimits over directed systems with injective maps as structure maps since for any submodule N ⊂ M we get T_{dim}N = T_{dim}M ∩ N from Additivity (see Theorem 6.7 (4b)).

8.7. In Exercise 6.5 we have constructed an N (ℤ)-module M = colim_{i∈I} N_{i} satisfying

dim_{N (ℤ)}(M) = 1;
M^{∗} = 0;

Take M_{i} to be the image of the injective structure map ψ_{i}: N_{i} → M. Then M = TM follows from M^{∗} = {0} and TM_{i} = 0 from M_{i} ≅ N (ℤ) since N (ℤ) is semihereditary (see Theorem 6.5 and Theorem 6.7 (1)) and Theorem 6.13 (1) and Lemma 6.28 (3) hold.

8.8. Consider an exact sequence 0 → M_{0} → M_{1} → M_{2} → 0 of N (G)- modules. It stays exact after applying U(G)⊗_{N (G)}− by Theorem 8.22 (2) and we obtain by the various maps j_{M_{i}} a map from the first to the second exact sequence. Now apply the snake lemma to get the desired six-term sequence.

8.9. Let G be a locally finite group G which is infinite. Then G is an infinite amenable group and we get b_{p}^{(2)}(G; N (G)) = b_{p}^{(2)}(G; U(G)) = 0 for all p ≥ 0 from Theorem 7.2 (1) and Theorem 8.31.

Let I be the directed set of finite subgroups of G. Consider the directed system over I which assigns to H, K ∈ I with H ⊂ K the canonical projection ℂ[G/H] → ℂ[G/K]. We obtain an obvious epimorphism of ℂ[G]- modules α: colim_{H∈I} ℂ[G/H] → ℂ. Since G is locally finite, I is the same as the directed set of finitely generated subgroups and hence α is bijective. As U(G) ⊗_{ℂ[G]} − commutes with colimits, we get an isomorphism

β: colim_{H∈I} U(G) ⊗_{ℂ[G]} ℂ[G/H] ⟶^{≅} U(G) ⊗_{ℂG} ℂ.

Since dim_{U(G)}(U(G) ⊗_{ℂ[G]} ℂ[G/K]) = |K|^{−1} is different from zero for finite K ⊂ G, the element 1_{U(G)} ⊗_{ℂG} 1K in U(G) ⊗_{ℂ[G]} ℂ[G/K] is different from zero for each finite subgroup K ⊂ G. Hence the source of β is non-trivial and therefore U(G) ⊗_{ℂG} ℂ ≠ {0}. This shows

H_{0}(EG; U(G)) = U(G) ⊗_{ℂG} ℂ ≠ {0}.

Chapter 9

9.1. Suppose that tr is a faithful finite normal trace on the von Neumann algebra A. We want to show that A is finite, i.e. for any projection p ∈ A with p ∼ 1 we have p = 1. Choose u ∈ A with p = uu^{∗} and 1 = u^{∗}u. Then

tr(1 − p) = tr(1) − tr(p) = tr(u^{∗}u) − tr(uu^{∗}) = 0.

Since tr is faithful, we conclude p = 1.

Suppose that A is finite. Then there is the universal trace tr^{u}: A → Z(A) (see Theorem 9.5). Since Z(A) is an abelian von Neumann algebra, we get from Example 9.6 an identification Z(A) = L^{∞}(X, ν). We obtain a faithful finite normal trace

tr: Z(A) = L^{∞}(X, ν) → ℂ, f ↦ ∫_{X} fdν.

Then tr ∘ tr^{u} is a faithful finite normal trace on A.

9.2. Since M is hyperbolic, ℤ ⊕ ℤ is not a subgroup of π_{1}(M) (see [65, Corollary 3.10 on page 462]). Since M is aspherical of dimension ≥ 2, its fundamental group cannot be virtually cyclic. Hence π_{1}(M) is not virtually abelian. Since M is closed, π_{1}(M) is finitely generated. Now apply Lemma 9.4 (3).

9.3. Since M is closed, π_{1}(M) is finitely generated. We conclude from Lemma 9.4 (3) that either N (π_{1}(M)) is of type II_{1} or π_{1}(M) is virtually abelian and N (π_{1}(M)) is of type I_{f}. Suppose that π_{1}(M) is virtually abelian. Then there is a finite covering \overline{M} → M such that π_{1}(\overline{M}) ≅ ℤ^{d} for some integer d ≥ 0. Up to homotopy we can assume that the prime decomposition of \overline{M} contains no D^{3} and no homotopy sphere. Hence any factor in the prime decomposition of \overline{M} has non-trivial fundamental group. Since π_{1}(\overline{M}) is abelian, this implies that \overline{M} is prime. Hence either \overline{M} is S^{1} × S^{2} or an irreducible manifold with fundamental group ℤ^{d} for some integer d ≥ 1. This implies that \overline{M} is aspherical. Hence it is homotopy equivalent to T^{d}. Since H_{n}(M) = 0 for n ≥ 4, we must have d = 1, 2, 3.

9.4. The two isomorphisms, inverse to one another, are given by tensoring with the R-M_{n}(R)-bimodule _{R}R^{n}_{M_{n}(R)} and the M_{n}(R)-R-bimodule M_{n}(R)R^{n}_{R}. Notice that there is an isomorphism of M_{n}(R)-M_{n}(R)-bimodules (_{M_{n}(R)}R^{n}_{R}) ⊗_{R} (_{R}R^{n}_{M_{n}(R)}) ≅ M_{n}(R) and an isomorphism of R-R-bimodules (_{R}R^{n}_{M_{n}(R)}) ⊗_{M_{n}(R)} (_{M_{n}(R)}R^{n}_{R}) ≅ R.

9.5. From Example 9.14 we get an isomorphism

dim^{u}: K_{0}(L^{∞}(X, ν)) → L^{∞}(X, ν, ℤ).

Obviously dim^{u}(A) is the constant function with value 1 which cannot be written as 2-times an element in L^{∞}(X, ν, ℤ). If we compose the canonical map K_{1}(ℤ) → K_{1}(A) with the map given by the ordinary determinant of commutative rings det: K_{1}(A) → A^{inv}, we see that the element in K_{1}(ℤ) given by (−1) is mapped to the non-trivial element −1 in A. Hence K_{1}(A) → K̃_{1}(A) is not injective.

Consider the matrices

P_{1} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) , P_{2} = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) , U = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

Then P_{i}^{2} = P_{i} for i = 1, 2, P_{1} + P_{2} = I and UP_{1}U^{−1} = P_{2}. Hence we get for the M_{2}(A)-module A^{2}

A^{2} ⊕ A^{2} ≅ (P_{1}) ⊕ (P_{1}) ≅ (P_{1}) ⊕ (P_{2}) ≅ (I_{2}) = M_{2}(A).

Hence A^{2} is finitely generated projective and we get [M_{2}(A)] = 2 · [A^{2}] in K_{0}(A). The generator of K_{1}(ℤ) is sent under the canonical map K_{1}(ℤ) → K_{1}(M_{2}(A)) to zero because of

( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) · U · ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} )^{−1} · U^{−1}.

9.6. See [51, Example 8.1.2 on page 67], [514, Example 7.1.11 on page 134].

9.7. This follows from Theorem 9.13 since N (G) = ℂG for a finite group G. See also [470, Corollary 2 in Chapter 2 on page 16].

9.8. Let f_{∗}: C_{∗} → D_{∗} be an R-chain homotopy equivalence of finite based free R-chain complexes such that H_{n}(C_{∗}) and H_{n}(D_{∗}) are S-torsion for each n. Since RS^{−1} is flat as R-module, the RS^{−1}-chain complexes RS^{−1} ⊗_{R} C_{∗} and RS^{−1} ⊗_{R} D_{∗} are finite based free acyclic RS^{−1}-chain complexes. Hence ρ(RS^{−1} ⊗_{R} C_{∗}) and ρ(RS^{−1} ⊗_{R} D_{∗}) are defined and we get in K_{1}(RS^{−1})

ρ(RS^{−1} ⊗_{R} D_{∗}) − ρ(RS^{−1} ⊗_{R} C_{∗}) = τ(RS^{−1} ⊗_{R} f_{∗}).

Since K_{1}(R) → K_{1}(RS^{−1}) sends τ(f_{∗}) to τ(RS^{−1}⊗_{R}f_{∗}), we see that the class of ρ(RS^{−1} ⊗_{R} C_{∗}) in coker(i_{1}) depends only on the R-chain homotopy type of C_{∗} and is in particular independent of the choice of R-basis. Notice that there is a canonical isomorphism from coker(i_{1}) to the image of K_{1}(RS^{−1}) → K_{0}(R → S).

Fix based free 1-dimensional chain complexes F_{∗}[n] such that H_{i}(F_{∗}[n]) = 0 for i ≠ 0 and H_{0}(F_{∗}[n]) is R-isomorphic to H_{n}(C_{∗}). Let d be the dimension of C_{∗}. One easily constructs an R-chain homotopy equivalence from ⨁_{n=0}^{d} Σ^{n}F_{∗}[n] to C_{∗}. This implies that ρ(F_{∗}[n]) and ∑_{n=0}^{d}(−1)^{n} · ρ(F_{∗}[n]) have the same image under the map K_{1}(RS^{−1}) → K_{0}(R → S). One easily checks that ρ(F_{∗}[n]) is mapped to [H_{n}(C_{∗})] under the map K_{1}(RS^{−1}) → K_{0}(R → S).

9.9. Obviously RS^{−1} = ℚ. The determinant of commutative rings induces isomorphisms K_{1}(ℤ) ⟶^{≅} {±1} and K_{1}(ℚ) ⟶^{≅} ℚ^{inv} and the rank of an abelian group and the dimension of a rational vector space induce isomorphisms K_{0}(ℤ) ⟶^{≅} ℤ and K_{0}(ℚ) ⟶^{≅} ℤ. This follows from the fact that ℤ has a Euclidean algorithm and that ℚ is a field. A ℤ-module M is ℚ-torsion and has a 1-dimensional finite free resolution if and only if it is a finite abelian group. This follows from the fact that a finitely generated abelian group is a direct sum of finitely many copies of cyclic groups. The isomorphism K_{0}(ℤ → ℚ) ⟶^{≅} ℚ^{+,inv} sends a finite abelian group M to its order.

9.10. Since ℂG = l^{2}(G) any ℂG-submodule of ℂG is a direct summand. Hence ℂG is semisimple. By Wedderburn’s Theorem we have ℂG = ∏_{i=1}^{r} M_{n_{i}}(ℂ) since any finite-dimensional skew field over ℂ is ℂ itself. We have

Z(ℂG) ≅ ∏_{i=1}^{r} Z(M_{n_{i}}(ℂ)) ≅ ∏_{i=1}^{r} ℂ.

This shows that r is the number of conjugacy classes of elements in G. We have

K_{0}(ℂG) ≅ K_{0} (∏_{i=1}^{r} M_{n_{i}}(ℂ)) ≅ ∏_{i=1}^{r} K_{0}(ℂ) ≅ ℤ^{r}.

Analogously we get

K_{1}(ℂG) ≅ K_{1} (∏_{i=1}^{r} M_{n_{i}}(ℂ)) ≅ ∏_{i=1}^{r} K_{1}(ℂ) ≅ ∏_{i=1}^{r} ℂ^{inv}.

We conclude from Theorem 9.31 that

L_{0}^{p}(ℂG) = K_{0}(ℂG) = ℤ^{r};
L_{1}^{p}(ℂG) = 0;
L_{1}^{h}(ℂG) = 0;
L_{1}^{s}(ℂG) = ℤ/2;

and that L_{0}^{s}(ℂG) is a subgroup of finite index in L_{0}^{h}(ℂG), and L_{0}^{h}(ℂG) is a subgroup of finite index in K_{0}(ℂG). This implies

L_{0}^{h}(ℂG) = ℤ^{r};
L_{0}^{s}(ℂG) = ℤ^{r}.

9.11. An Ore localization of a Noetherian ring is again Noetherian [448, Poposition 3.1.13 on page 278]. Hence U(G) is Noetherian if N (G) is (see Theorem 8.22 (1)). If U(G) is Noetherian, it must be semisimple since it is von Neumann regular (see Lemma 8.20 (2) and Theorem 8.22 (3)). For any semisimple ring its K_{0}-group is finitely generated. We conclude K_{0}(N (G)) = K_{0}(U(G)) from Theorem 9.20 (1).

Suppose that K_{0}(N (G)) is finitely generated. We can split N (G) = N (G)_{I} ×N (G)_{II}. Since K_{0}(N (G)_{II}) → Z(N (G)_{II})^{ℤ/2} is surjective by Theorem 9.13 (2), N (G)_{II} must be trivial. Hence N (G) is of type I. Lemma 9.4 (1) implies that G is virtually abelian. Since the image of dim_{N (G)} : K_{0}(N (G)) → ℝ is finitely generated and contains |H|^{−1} for any finite subgroup H ⊂ G, there must be a bound on the orders of finite subgroups of G. Suppose that ℤ ⊂ G. Since dim_{N (ℤ)} : K_{0}(N (Z)) → ℝ is surjective and dim is compatible with induction, dim_{N (G)} : K_{0}(N (G)) → ℝ is surjective and therefore K_{0}(N (G)) is not finitely generated. Hence G is a virtually abelian group which does not contain ℤ as subgroup and has a bound on the orders of its finite subgroups. This implies that G is finite.

9.12. Suppose that R is semihereditary. Let f: P → P be a nilpotent Rendomorphism of a finitely generated projective R-module. Then im(f) ⊂ P is a finitely generated submodule of a finitely generated projective one and hence projective. We have exact sequences 0 → (im(f), f) → (P, f) ⟶^{pr} (P/ im(f), 0) → 0 and 0 → (im(f), 0) → (P, 0) ⟶^{pr} (P/ im(f), 0) → 0. Let (M, g) be the pullback of (P, f) ⟶^{pr} (P/ im(f), 0)) ⟵^{pr} (P, 0). We obtain exact sequences 0 → (im(f), f) → (M, g) → (P, 0) → 0 and 0 → (im(f), 0) → (M, g) → (P, f) → 0. Notice that M is finitely generated projective (what is not necessarily true for P/ im(f)) and g is nilpotent as f is nilpotent. We get in Nil(R)

[im(f), f] + [P, 0] = [M, g] = [im(f), 0] + [P, f].

This implies [P, f] = [im(f), f] ∈ Ñil(R). We can iterate this argument and get [P, im(f)] = [im(f^{n}), f] ∈ Ñil(R) for all natural numbers n. Choose n large enough that f^{n} = 0 and we conclude [P, f] = 0 ∈ Ñil(R).

Both N (G) and U(G) are semihereditary (see Theorem 6.5, Theorem 6.7 (1), Lemma 8.18 and Theorem 8.22 (3).

9.13. Suppose Wh(G) is trivial. Consider a finite subgroup H of the center of G. We derive from Theorem 9.38 that ℚ ⊗_{ℤ} Wh(H) is trivial. The rank of the finitely generated abelian group Wh(H) is the difference r − q, where r is the number of ℝ-conjugacy classes and q the number of ℚ-conjugacy classes in G (see Subsection 3.1.1). We conclude r = q. A finite abelian group H satisfies r = q if and only if any cyclic subgroup has at most two generators. A non-trivial cyclic group ℤ/n has at most two generators if and only if n = 2, 3, 4, 6. Hence H is a finite product of copies of ℤ/2, ℤ/3 or of ℤ/2 and ℤ/4.

9.14. We conclude from Theorem 9.62 that the image of K_{0}(N (ℤG)) → K_{0}(N (G)) is generated by [N (G)]. We conclude from Lemma 9.56 for any finitely generated projective ℤG-module P that the image of [P] under K_{0}(ℤG) → K_{0}(N (G)) is HS(ℂ ⊗_{ℤ} P)(1) · [N (G)].

9.15. This follows from Theorem 9.54 (3).

9.16. This follows from Theorem 9.13 (2), Lemma 9.56 and Theorem 9.63.

9.17. The map d_{A} sends [M] to [PM]. The proof that it is well-defined is similar to the one of Theorem 9.64. The argument for U is analogous using Lemma 8.27.

9.18. The map G_{0}(ℂF) → G_{0}(ℂF) given by V ⊗_{ℂ} − and the diagonal F - action is well-defined and sends the class of ℂ[F/H] to the class of V ⊗_{ℂ} ℂ[F/H]. Hence it suffices to show that [ℂ[F/H]] = 0 in G_{0}(ℂF). If H is trivial, this has already been proved in Theorem 9.66 (2). Since any subgroup of a free group is free, it remains to treat the case, where H is a non-trivial free group.

The map given by induction G_{0}(ℂH) → G_{0}(ℂF) sends [ℂ] to [ℂ[G/H]], where ℂ is viewed as ℂH-module by the trivial action. Let f: H → ℤ be an epimorphism. Restriction with f induces a homomorphism G_{0}(ℂℤ) → G_{0}(ℂH) which sends [ℂ] to [ℂ]. We have already shown in the proof of Theorem 9.66 (2) that [ℂ] = 0 in G_{0}(ℂℤ).

9.19. Since G is finite, ℚG is semisimple and hence any finitely generated ℚG-module is finitely generated projective. The functor sending a ℤG-module M to the ℚG-module ℚG ⊗_{ℤG} M ≅_{ℚG} ℚ ⊗_{ℤ} M is exact. Hence j is welldefined.

Consider a finitely generated projective ℚG-module P. Choose an idempotent matrix A ∈ M_{n}(ℚG) such that the image of the map r_{A}: ℚG^{n} → ℚG^{n} given by right multiplication with A is ℚG-isomorphic to P. Choose an integer l ≥ 1 such that B := l · A is a matrix over ℤG. Let M be the cokernel of the map r_{B}: ℤG^{n} → ℤG^{n}. Then G_{0}(ℤG) → K_{0}(ℚG) maps [M] to [P]. Hence G_{0}(ℤG) → K_{0}(ℚG) is surjective.

A more general result is proved in Lemma 10.70 (1).

Chapter 10

10.1. Analogous to the proof of Lemma 10.5, where here Y should be a finite 2-dimensional CW -complex with π_{1}(Y) ≅ G and p: \overline{Y} → Y be its universal covering.

10.2. We get a G-covering X → X/G with finite G\X. We get an exact sequence π_{1}(G\X) → G → π_{0}(G\X). Since π_{1}(G\X) is finitely generated and π_{0}(X) is a finite set, G contains a finitely generated subgroup of finite index. Hence G is finitely generated.

Suppose that there is an epimorphism f′: F_{g} → G from the free group of rank g to G. Choose a map f: #_{i=1}^{g}S^{1} × S^{2} → BG which induces f′ on the fundamental groups. Take M to be the pullback of EG → BG with f. Then M is a cocompact free proper G-manifold. The long exact homotopy sequence implies that M is connected.

10.3. Suppose that d(Λ) = 0. Consider r ∈ ℝ and ϵ > 0. We can find g ∈ Λ with 0 < g < ϵ. Choose n ∈ ℤ with n · g ≤ r ≤ (n + 1) · g. Hence 0 ≤ r − n · g ≤ ϵ. Hence Λ is dense in ℝ. This shows that Λ = ℝ if and only if Λ is closed and satisfies d(Λ) = 0.

Suppose that d(Λ) > 0 and Λ is closed. Then d(Λ) ∈ Λ. Consider g ∈ Λ. Choose n ∈ ℤ with n · d(Λ) ≤ g < (n + 1) · d(Λ). Since 0 ≤ g − n · d(Λ) < d(Λ) and g − n · d(Λ) ∈ Λ, we conclude g = n · d(Λ).

10.4. Without loss of generality we can assume that C_{∗} is finite dimensional. Since any module is the colimit of its finitely generated submodules, we can find a directed system of finite F G-subchain complexes C_{∗}[i] ⊂ C_{∗} such that C_{∗} = colim_{i∈I} C_{∗}[i]. Since the functor colimit over directed systems is exact and N (G) ⊗_{F G} − commutes with colimits, we conclude

H_{p}(N (G) ⊗_{F G} H_{p}(C_{∗})) = colim_{i∈I} H_{p}(N (G) ⊗ C_{∗}[i]).

Because of Theorem 6.13 (2) it suffices to show for any F G-chain map f_{∗}: D_{∗} → E_{∗} of finite F G-chain complexes

dim_{N (G)} (im (H_{p} (id ⊗_{F G}f_{∗}: N (G) ⊗_{F G} D_{∗} → N (G) ⊗_{F G} E_{∗}))) ∈ Λ.

Since we have the long exact homology sequence associated to the exact sequence 0 → D_{∗} → cone(f_{∗}) → ΣC_{∗} → 0, we conclude from Additivity (see Theorem 6.7 (4b)) that it suffices to show for any finite F G-chain complex F_{∗} and p ∈ ℤ that

dim_{N (G)} (H_{p} (N (G) ⊗_{F G} F_{∗})) ∈ Λ.

Again from Additivity (see Theorem 6.7 (4b)) we conclude that it suffices to show for any F G-map g: M_{0} → M_{1} of finitely generated F G-modules that

dim_{N (G)} (im (id ⊗_{F G}g: N (G) ⊗_{F G} M_{0} → N (G) ⊗_{F G} M_{1})) ∈ Λ.

Since N (G) ⊗_{F G} M_{0} ⟶^{id ⊗_{F G}g} N (G) ⊗_{F G} M_{1} ⟶^{pr} N (G) ⊗_{F G} coker(g) → 0 is

exact, Additivity (see Theorem 6.7 (4b)) implies

dim_{N (G)} (im (id ⊗_{F G}g: N (G) ⊗_{F G} M_{0} → N (G) ⊗_{F G} M_{1}))

= dim_{N (G)} (N (G) ⊗_{F G} M_{1}) − dim_{N (G)} (N (G) ⊗_{F G} coker(g)) .

But dim_{N (G)} (N (G) ⊗_{F G} M_{1}) and dim_{N (G)} (N (G) ⊗_{F G} coker(g)) belong to Λ because of Lemma 10.7 and Lemma 10.10 (3) since G satisfies by assumption the Atiyah Conjecture 10.3 of order Λ with coefficients in F.

10.5. This follows from the previous Exercise 10.4 applied to N (G) ⊗_{ℤG} C_{∗}(EG) or from Theorem 6.99.

10.6. Pass to the sequence of p-Sylow subgroups. Now use the fact that a maximal proper subgroup of a p-group has index p and any proper subgroup is contained in a maximal proper subgroup to construct the desired sequence K_{1} ⊂ K_{2} ⊂ K_{3} ⊂ …. Now the argument is a variation of the argument appearing in Example 10.13. Notice that |K_{n}|^{−1} − |K_{n−1}|^{−1}| is (p − 1)/p. Use the p-adic expansion of r/(p − 1) and replace ℚG by (ℚG)^{p−1}. If the n-th coefficient of the p-adic expansion is k_{n}, use (I_{n})^{k_{n}} ⊂ (ℚG)^{k_{n}} ⊂ (ℚG)^{n}.

10.7. This is done by transfinite induction (see Lemma 10.40). Details can be found in [435, Theorem 7.7].

10.8. By the previous Exercise 10.7 C if closed under free products. For any positive integer k the group ∗_{n=1}^{k}G_{n} belongs to Linnell’s class C. Since any finite subgroup of ∗_{n=1}^{k}G_{n} is conjugate to a subgroup of one of the summands G_{i} (see [471, Theorem 8 in Section I.4 on page 36]), there is an upper bound on the orders of finite subgroups of ∗_{n=1}^{k}G_{n}. Hence ∗_{n=1}^{k}G_{n} satisfies the strong Atiyah Conjecture 10.2 by Theorem 10.19. Since G is the directed union of the subgroups ∗_{n=1}^{k}G_{n}, we conclude from Lemma 10.4 that G satisfies the strong Atiyah Conjecture 10.2. Now take G = ∗_{n=1}^{∞}ℤ/2^{n}. We could also take G = ⨁_{n∈ℤ} ℤ/2 which satisfies the strong Atiyah Conjecture 10.2 by Lemma 10.4 since it is locally finite.

10.9. Any manifold M which has a boundary or which is not compact is homotopy equivalent to a CW -complex of dimension dim(M) − 1. Any 1-dimensional CW -complex has a free group as fundamental group. Hence π_{1}(M) is free and hence belongs to C if M is a 2-dimensional manifold which has boundary or which is not compact.

Let M be a closed 2-dimensional manifold. If H_{1}(M)/ tors(M) is trivial, then π_{1}(M) is finite and hence belongs to C. Suppose H_{1}(M)/ tors(M) is not trivial. Then H_{1}(M)/ tors(M) is ℤ^{n} for some integer n ≥ 1. Let \overline{M} → M be the covering associated to the epimorphism π_{1}(M) → H_{1}(M)/ tors(H_{1}(M)). Then \overline{M} is a 2-dimensional non-compact manifold and hence has a free fundamental group. Because there is an exact sequence 1 → π_{1}(\overline{M}) → π_{1}(M) → ℤ^{n} → 1, the fundamental group π_{1}(M) belongs to C. Now apply Theorem 10.19.

10.10. The fundamental group of F belongs to C by the previous Exercise 10.9. Since C is closed under extensions with amenable quotient and ℤ and each finite group are amenable, π_{1}(M) belongs to C. Now apply Theorem 10.19.

10.11. Suppose that G contains a non-trivial element g such that the centralizer C_{G}(g) has finite index in G. Then b_{p}^{(2)}(G; N (G)) = 0 for all p ≥ 0 by Theorem 6.54 (6b) and Theorem 7.2 (1) and (2). We conclude H_{p}^{(2)}(EG; N (G)) = 0 from Theorem 6.7 (4e), Lemma 6.28 (3) and Lemma 6.52.

Suppose that G contains no non-trivial element g such that the centralizer C_{G}(g) has finite index in G. Then N (G) is a factor by Lemma 9.4 (4) and dim_{N (G)} and the universal center valued dimension dim_{N (G)}^{u} agree. By

assumption b_{p}^{(2)}(EG; N (G)) = dim_{N (G)}(H_{p}^{G}(EG; N (G)) is a non-negative integer n. We conclude from Theorem 6.7 (4e) and Theorem 9.13 (1) that PH_{p}^{G}(EG; N (G)) is N (G)-isomorphic to N (G)^{n}. Lemma 6.52 implies that the Hilbert N (G)-modules H_{p}^{(2)}(EG) and l^{2}(G)^{n} are isomorphic.

10.12. Suppose that π_{1}(M) belongs to C. We conclude b_{p}^{(2)}(M̃) = 0 for p ≥ 2 from Theorem 8.31 and (10.86). Since b_{n}^{(2)}(M̃) ≠ 0 by Theorem 1.62, we conclude n = 1. If n = 1, we get π_{1}(M) ∈ C from Exercise 10.9.

10.13. Let 0 → ⨁_{n∈ℤ} ℤ/2 → L → ℤ → 0 be the canonical exact sequence for the lamplighter group L. The lamplighter group does not satisfy the strong Atiyah Conjecture 10.2 by Theorem 10.23. Since ⨁_{ℤ} ℤ/2 is locally finite, it satisfies the strong Atiyah Conjecture 10.2 because of Lemma 10.4. The group ℤ satisfies the strong Atiyah Conjecture 10.2 by Lemma 1.34 (1).

10.14. Since each S(G_{i}) is von Neumann regular by assumption, S(G) is von Neumann regular. Since K_{0} commutes with directed unions of rings and each S(G_{i}) satisfies K′, the union S(G) satisfies K′ by Theorem 3.14 (6). Let G be the locally finite group ⨁_{n∈ℤ} ℤ/2 and {G_{i} | i ∈ I} be the directed system of its finitely generated subgroups. Each G_{i} is finite and hence D(G_{i}) = ℂG_{i} is semisimple for each i ∈ I. We get D(G) = ⋃_{i∈I} D(G_{i}) from Lemma 10.83. But D(G) cannot be semisimple by Lemma 10.28 (1).

10.15. The proof that the map is well-defined is analogous to (6.4). The integrality of its image follows from Lemma 10.46.

10.16. Fix a set-theoretic section s: G/H → G of the projection G → G/H. Then a ℂH-basis for ℂG considered as ℂH-module is given by {s(gH) | gH ∈ G/H}. The image of this basis under the obvious inclusion ℂG ⊂ D(G) is a D(H)-basis for D(G) considered as D(H)-module (see Lemma 10.59). The image of this D(H)-basis under the inclusion D(G) → U(G) is a U(H)-basis for U(G) considered as U(H)-module. This shows that the left and middle vertical arrow are well-defined and that the diagram commutes.

Let G be a virtually finitely generated free group G. If d is the least common multiple of the orders of finite subgroups of G, then one can find a finitely generated free subgroup F ⊂ G of index d (see for instance [24, Theorem 8.3 and Theorem 8.4], [461, Theorem 5 in Section 7 on page 747]). Then D(H) is semisimple by Lemma 10.39 and Theorem 10.43. Hence the image of the composition

K_{0}(D(G)) ⟶^{i} K_{0}(U(G)) ⟶^{dim_{U(G)}} ℝ

is contained in \frac{1}{|FIN (G)|}ℤ, since this is true for F by Lemma 10.51 and the diagram appearing in the exercise commutes. Since any virtually free group G is the directed union of its finitely generated subgroups {G_{i} | i ∈ I}, each G_{i} is again virtually finitely generated free, D(G) = ⋃_{i∈I} D(G_{i}) by Lemma 10.83 and K_{0} commutes with directed unions, the claim follows.

10.17. One easily checks that f is injective. We have

ψ(y^{−1}z − x^{−1}y) = (xu)^{−1}xu^{2} − x^{−1}xu = u − u = 0.

Since L is a skewfield, the division closure D of im(f) is itself a skewfield. Since the skewfield generated by im(f) is division closed, it agrees with D, i.e. f yields a field of fractions f: ℂ⟨x, y, z⟩ → D. It cannot be a universal field of fractions by Lemma 10.77. Moreover, f: ℂ⟨x, y, z⟩ → D and ℂ⟨x, y, z⟩ → K cannot be isomorphic fields of fractions.

10.18. Let G be a countable group which is not finitely generated, for instance the free group on a countable infinite set of generators. Let {G_{i} | i ∈ I} be the directed system of its finitely generated subgroups. Then G = ⋃_{i∈I} G_{i} but G ≠ G_{i} for all i ∈ I. Choose a numeration G = {g_{1}, g_{2}, g_{3}, …}. We get an element u = ∑_{n=1}^{∞} \frac{1}{n^{2}} · g_{n} in l^{1}(G). We have already seen in Exercise 8.3 that l^{1}(G) ⊂ N (G), namely, right multiplication with u defines a bounded G-equivariant operator r_{u}: l^{2}(G) → l^{2}(G). If this element lies in N (G_{i}) for a finitely generated subgroup G_{i}, then r_{u}(1) ∈ l^{2}(G_{i}). This yields a contradiction because G_{i} ≠ G. This shows N (G) ≠ ⋃_{i∈I} N (G_{i}). This implies also U(G) ≠ ⋃_{i∈I} U(G_{i}) since N (G) ∩ U(G_{i}) = N (G_{i}) by Lemma 8.3 (4).

10.19. This follows from Theorem 9.20 (1), Theorem 10.38 and Theorem 10.87.

Chapter 11

11.1. Take M = T^{2} × F_{g} for a closed connnected orientable surface F_{g} of genus g ≥ 2 and equip M with the product of the flat Riemannian metric on T^{2} and the hyperbolic Riemannian metric on F_{g}. Theorem 3.105 implies b_{p}^{(2)}(M̃) = 0 for p ≥ 0.

11.2. The universal covering of a flat closed manifold M is isometrically diffeomorphic to ℝ^{n}. Recall that T̃^{n} is det-L^{2}-acyclic and ρ^{(2)}(T̃^{n}) = 0 (see Theorem 3.105). Now apply Theorem 3.183.

11.3. Suppose that π_{1}(F) and π_{1}(B) are finite. Then π_{1}(E) is finite. The values of the zero-th and first L^{2}-Betti number of Ẽ can be derived from Theorem 1.35 (1) and (8). Now apply the Euler-Poincaré formula and Poincaré duality (see Theorem 1.35 (2) and (3)).

Suppose that exactly one of the groups π_{1}(F) and π_{1}(B) is finite. Then π_{1}(E) is infinite and hence b_{0}^{(2)}(Ẽ) = 0 by Theorem 1.35 (8). A spectral sequence argument shows b_{2}^{(2)}(M̃) = 0, the corresponding entries on the E_{2}- term have vanishing von Neumann dimension. We get b_{1}^{(2)}(M̃) = b_{3}^{(2)}(M̃) and b_{0}^{(2)}(M̃) = b_{4}^{(2)}(M̃) = 0 from Poincaré duality (see Theorem 1.35 (3)). Now apply the Euler-Poincaré formula (see Theorem 1.35 (2)) and the product formula for the Euler characteristic χ(M) = χ(F) · χ(B).

Suppose that both π_{1}(B) and π_{1}(F) are infinite. We conclude b_{p}^{(2)}(M̃) = 0 for p = 0, 1 from Theorem 6.67. Now apply Euler-Poincaré formula and Poincaré duality (see Theorem 1.35 (2) and (3)).

If M is aspherical, either F and B are closed surfaces of genus g ≥ 1 or one of them is S^{1}.

11.4. Suppose that dim(M) = 2. Then M is a closed surface of genus ≥ 2 by the Gauss-Bonnet formula and the claim follows from Example 1.36 and Example 2.70.

Suppose dim(M) ≥ 4. We conclude from Theorem 11.6 that b_{p}^{(2)}(M̃) = 0 and α_{p}^{Δ}(M̃) = ∞^{+} holds for p < dim(M)/2 since for some b with 1− \frac{dim(M)}{2} < b < 1 and a = 1 we have −a^{2} ≤ sec(M) ≤ −b^{2}. Now apply Poincaré duality (see Theorem 1.35 (3) and Theorem 2.55 (2)) and Theorem 2.66 (2).

11.5. See [520, Proposition 4.6 on page 190] for M a Kähler manifold. Kähler hyperbolicity is proved analogously.

11.6. See [520, (1.12) on page 158].

11.7. Suppose such N ⊂ M̃ exists. Then N is closed Kähler with Kähler form ω|_{N} if ω is the Kähler form of M. Then (ω|_{N})^{dim(N)/2} is a multiple of the volume form of N [520, (1.12) on page 158] and cannot be exact although ω is exact.

11.8. The Kähler structure is given in [520, Example 4.4 on page 189]. Since b_{n}^{(2)}(T̃^{2n}) = 0, Theorem 11.14 implies that there is no Kähler hyperbolic structure on T^{2n}.

11.9. The desired isomorphism is induced by complex conjugation (see [520, Theorem 5.1 in V.5 on page 197]).

11.10. Because of Theorem 6.54 (6b) it suffices to treat the case G = G_{0}. Obviously G acts properly on the contractible space Eπ for π = π_{1}(F). We conclude from Example 1.36, Theorem 6.54 (2) and Theorem 13.45 (4)

b_{1}^{(2)}(G) = b_{1}^{(2)}(Eπ; N (G)) = k · b_{1}^{(2)}(Eπ; N (π)) = k · b_{1}^{(2)}(π) ≠ 0

if k is the order of the kernel of G_{0} → π.

11.11. See [3, page 3]).

Chapter 12

12.1. See [496, Satz 18.2 in §18 on page 223].

12.2. Because of Lemma 12.3 and homotopy invariance (see Theorem 1.35 (1) and Theorem 2.55 (1)) it suffices to show that it is not possible that both b_{p}^{(2)}(M̃) = b_{p}^{(2)}(π) vanishes for p = 0, 1 and α_{p}(M̃) = α_{p}(π) is ∞^{+} for p = 1, 2. Suppose the contrary.

We can assume without loss of generality that ∂M has no boundary component which is homotopy equivalent to S^{2}. Otherwise we can attach finitely many copies of D^{3} to M, what does not affect the fundamental group of π of M. Furthermore we can assume that the prime decomposition contains no homotopy sphere because removing such factor from the prime decomposition does not alter the homotopy type.

Since χ(∂M) = 2 · χ(M) implies χ(M) ≤ 0, we conclude from Theorem 4.1 that b_{p}^{(2)}(M̃) = 0 for p = 0, 1, 2, 3 and hence M is homotopy equivalent to RP^{3}#RP^{3} or a prime 3-manifold with infinite fundamental group whose boundary is empty or a union of tori. Theorem 4.2 (2) implies that M is not homotopy equivalent to ℝℙ^{3}#ℝℙ^{3} or S^{1} ×S^{2}. Hence we are left with the case that M is irreducible and non-exceptional and has infinite fundamental group. The case that M is closed is excluded by Theorem 4.2 (2) and (4). The case that ∂M contains an incompressible torus is ruled out by Theorem 4.2 (4). The case, where ∂M contains an compressible torus is reduced to the case, where ∂M contains no compressible torus by the construction appearing in the proof of [322, Lemma 6.4 on page 52] based on the Loop Theorem [252, Theorem 4.2 on page 39] and the formula for first L^{2}-Betti number under wedges (see Theorem 1.35 (5)).

12.3. By Lemma 5.10 we have M̃ = M̃_{Eucl} × M̃_{ncp}. Now the claim follows from (5.14) and (5.15) and Lemma 12.3.

12.4. Consider r > 0. Choose r_{0} > 0 and x̃_{0} ∈ M̃ such that M̃ = ⋃_{w∈π_{1}(M)} B_{r_{0}}(w · x̃_{0}) and r_{0} ≥ r. This is possible since M is compact. Since

M̃ is contractible, we can find R > 2r_{0} such that B_{2r_{0}}(x̃_{0}) is contractible in B_{R}(x̃_{0}). Then B_{2r_{0}}(w · x̃_{0}) is contractible in B_{R}(w · x̃_{0}). For each x̃ ∈ M̃ we can find w ∈ π_{1}(M) such that x̃ ∈ B_{r_{0}}(w · x̃_{0}). Hence we get for appropriate w ∈ π_{1}(M) that B_{r}(x̃) ⊂ B_{2r_{0}}(w · x̃_{0}). This shows that B_{r}(x̃) is contractible in B_{R}(w · x̃_{0}). Since B_{R}(w · x̃_{0}) ⊂ B_{2R}(x̃), we conclude that B_{r}(x̃) is contractible in B_{2R}(x̃) for a number 2R which does not depend on x̃.

12.5. See [341, Lemma 11.6 (5) on page 806].

12.6. This follows from a spectral sequence argument applied to homology with coefficients in N (π_{1}(E)) and Lemma 12.3.

12.7. Since π is not amenable, H_{0}^{π}(Eπ; N (π)) = 0 by Lemma 12.11 (4). The condition χ(X) = 0 and H_{p}^{π}(X̃; N (π)) = H_{p}^{π}(Eπ; N (π)) = 0 for p = 0, 1 imply b_{2}^{(2)}(X̃) = 0 by the Euler-Poincaré formula (see Theorem 1.35 (2)). Since N (π) is semihereditary (see Theorem 6.5 and Theorem 6.7 (1)) and H_{2}(X̃; N (π)) is the kernel of the second differential c_{2}^{N (π)} of N (π) ⊗_{ℤ[π]} C_{∗}(X̃), H_{2}(X̃; N (π)) is direct mmand in N (π) ⊗_{ℤ[π]} C_{2}(X̃) ≅ N (π)^{s}. Since H_{2}(X̃; N (π)) has trivial von Neumann dimension, it must be zero by Lemma 6.28 (3). Since c_{2}^{N (π)} is injective, c_{2} is injective. Hence H_{p}(X̃) = 0 for p ≥ 2 and X̃ must be contractible.

12.8. We conclude def(G) ≤ 1 from Lemma 7.22.

Suppose that def(G) = 1. Then we can find a connected finite 2dimensional CW -complex X with G ≅ π_{1}(X) and χ(X) = 0. Lemma 12.11 (4) implies that G is non-amenable. Now apply the previous Exercise 12.7 to X.

12.9. We get q(G) ≥ 0 from the Euler-Poincaré formula (see Theorem 1.35 (2)) and Poincaré duality (Theorem 1.35 (3)). The claim in the case q(G) = 0 follows from Lemma 12.5.

12.10. We get H_{n}^{G×H}(E(G × H); N (G × H)) = 0 for n ≤ d + e + 1 from Lemma 12.11 (3). We conclude H_{l+e+2}^{G×H}(E(G × H); N (G × H)) ≠ 0 from Lemma 12.3 and the product formulas Lemma 2.35 (1) and Theorem 6.54 (5) using Lemma 2.17 (1). (Notice that in the case of the Novikov-Shubin invariants we do not need the assumption that the spectral density functions have the limit property. This enters only when we want to get numerical values but not when we want to decide whether the Novikov-Shubin invariant is ∞^{+}.)

12.11. G = ℤ/2, ℤ^{n}, ∏_{i=1}^{∞}(ℤ/2 ∗ ℤ/2): Since G is in all cases an extension of two abelian groups and hence amenable, we conclude s = −1 from Lemma 12.11 (4).

G = ∗_{i=1}^{r}ℤ for r ≥ 2: This group is non-amenable and b_{1}^{(2)}(G) = r − 1 ≠ 0. We conclude s = 0 from Lemma 12.3 and Lemma 12.11 (4).

G = ∏_{i=1}^{r} ℤ ∗ ℤ: Since ℤ ∗ ℤ is non-amenable and b_{r}^{(2)}(G) ≠ 0 by the product formula (see Theorem 6.54 (5)), we conclude s = r − 1 from Lemma 12.3 and Lemma 12.11 (3). G = ℤ × (ℤ ∗ ℤ): We have already computed s for ℤ and ℤ ∗ ℤ. Now s = 0 follows from the previous Exercise 12.10.

∏_{i=1}^{∞}(ℤ/2 ∗ ℤ/3): The group ℤ/2 ∗ ℤ/3 is not amenable since it contains a non-abelian free group of finite index. Hence s = ∞ by Lemma 12.11 (5).

12.12. (1) is clear when one works with differential forms and uses a collar of ∂N in N.

(2) Since the signature of N is non-zero, the image of H^{2k}(N; ∂N; ℂ) → H^{2k}(N; ℂ) and hence the image of H_{c}^{2k}(M; ℂ) → H^{2k}(M; ℂ) are non-trivial. Lemma 1.92 implies that H_{(2)}^{p}(M) is different from zero. From Theorem 1.72 we conclude that the kernel of (Δ_{2k})_{min} : dom ((Δ_{2k})_{min}) ⊂ L^{2}Ω^{2k}(M) → L^{2}Ω^{2k}(M) is non-trivial. Hence this operator contains zero in its spectrum.

Chapter 13

13.1. The Bass-Heller Swan decomposition for (ℤ[G])[ℤ] = ℤ[G × ℤ] yields an isomorphisms

K_{0}(ℤG) ⊕ K_{1}(ℤG) ⊕ Nil(ℤG) ⊕ Nil(ℤG) ⟶^{≅} K_{1}(ℤ[G × ℤ]).

Its reduced version is the isomorphism

K̃_{0}(ℤG) ⊕ Wh(G) ⊕ Ñil(ℤG) ⊕ Ñil(ℤG) ⟶^{≅} Wh(G × ℤ).

We have analogous Bass-Heller-Swan isomorphisms for N (G)[ℤ]. These are natural with respect to the change of rings map ℤG → N (G).

The map K̃_{0}(ℤG) → K̃_{0}(N (G)) is trivial (see Theorem 9.62). We get Ñil(N (G)) = 0 from Exercise 9.12. The map K_{1}(ℤ[G × ℤ]) → K_{1}(N (G × ℤ)) factorizes through K_{1}(N (G)[ℤ]). The composition Wh(G) → Wh(G ×

ℤ) ⟶^{Φ^{G×ℤ}} ℝ agrees with ϕ^{G} by Theorem 3.14 (6).

13.2. One checks for x ∈ im(E_{λ}) ∩ ⋂_{μ<λ} im(E_{μ})^{⊥} using (1.65) and Lemma 2.2 (2)

⟨f(x) − λx, f(x) − λx⟩ = ||f(x)||^{2} − 2λ · ⟨f(x), x⟩ + λ^{2} · ||x||^{2}

= λ^{2}||x||^{2} − 2λ^{2} · ||x||^{2} + λ^{2} · ||x||^{2} = 0.

This implies

im(E_{λ}) ∩ ⋂_{μ<λ} im(E_{μ})^{⊥} = {u ∈ U | f(u) = λ}.

Now apply Additivity and Continuity of dim_{N (G)} (see Theorem 1.12 (2) and (4)).

13.3. We conclude from the previous Exercise 13.2 that F is continuous at λ. Theorem 13.19 (1) implies

F (λ) = \underline{F}^{+}(λ) = \overline{F}^{+}(λ).

If \underline{F} (λ) = F (λ) − ϵ, we get F (μ) = \underline{F}^{+}(μ) ≤ \underline{F} (λ) ≤ F (λ) − ϵ for μ < λ. Continuity of F at λ implies ϵ = 0. Hence \underline{F} (λ) = \overline{F} (λ) = F (λ).

13.4. Choose a numeration {g_{n} | n = 0, 1, 2, …} of G with g_{0} = 1. We construct inductively a sequence i_{0} ≤ i_{1} ≤ i_{2} ≤ … such that {g_{0}, g_{2}, …, g_{n}}∩ ⋂_{k=1}^{n} G_{i_{k}} = {1}. The induction beginning n = 0 is trivial. In the induction step we can find an index i′_{n} ∈ I such that g_{n} ∉ G_{i′_{n}}. Choose i_{n} ∈ I such that i′_{n} ≤ i_{n} and i_{n−1} ≤ i_{n}.

13.5. Since G is of det ≥ 1-class, NH and WH are of det ≥ 1-class by Theorem 13.3 (3). Hence Φ^{WH}: Wh(WH) → ℝ is trivial by Lemma 13.6. The following diagram commutes by Theorem 3.14 (6) and Theorem 13.45

K_{1}(ℤWH) ⟶^{res} K_{1}(ℚNH) ⟶^{ind} K_{1}(ℚG)
↓  ↓  ↓
K_{1}(N (WH)) ⟶^{res} K_{1}(N (NH)) ⟶^{ind} K_{1}(N (G))
↓  ↓  ↓
ℝ  ⟶^{· \frac{1}{|H|}}  ℝ  ⟶^{id}  ℝ

The map in question is obtained by starting at the left upper corner, going to the right upper corner and then going down to the right lower corner. Hence it factorizes through Φ^{WH} which agrees with the composition of the map, which is obtained by going from the upper left corner down to the left lower corner, and the projection K_{1}(ℤWH) → Wh(WH).

13.6. Let H ⊂ G be a subgroup of finite index which is residually finite or residually amenable respectively. Then H′ = ⋂_{g∈G} gHg^{−1} is a normal subgroup of G of finite index in both H and G. Consider g ∈ G. If g does not lie in H′, then the projection G → G/H′ maps g to an element different from 1 and G/H′ is finite and amenable. Suppose g ∈ H′. Then we can find an epimorphism f: H → L for some group L, which is finite or amenable respectively, such that g is not mapped to 1 ∈ L. Let K be the kernel of f which is a normal subgroup in H. Then K′∩⋂_{g∈G} gKg^{−1} ⊂ G has finite index and G/K′ is finite or amenable respectively and the projection G → G/K′ sends g to an element different from 1 in G/K′.

13.7. Choose a sequence of normal subgroups of finite index

π_{1}(F_{g}) = H_{0} ⊃ H_{1} ⊃ H_{2} ⊃ …

with ⋂_{i=0}^{∞} H_{i} = {1} and a sequence of integers n_{1} ≤ n_{2} ≤ … such that [π_{1}(F_{g}) : H_{i}] ≥ exp(i · n_{i}) holds for i ≥ 1. Put G_{i} = 2^{n_{i}} · ℤ × H_{i}. Then

\frac{b_{1}(G_{i}\X̃)}{[π_{1}(X) : G_{i}]^{1−ϵ}} = \frac{(2g − 2) · [π_{1}(F_{g}) : H_{i}] + 3}{[π_{1}(F_{g}) : H_{i}]^{1−ϵ} · 2^{n_{i}·(1−ϵ)}}
= \frac{(2g − 2) · [π_{1}(F_{g}) : H_{i}]^{ϵ}}{2^{n_{i}·(1−ϵ)}} + \frac{3}{[π_{1}(F_{g}) : H_{i}]^{1−ϵ} · 2^{n_{i}·(1−ϵ)}} .

If ϵ = 0, this converges for i → ∞ to zero. Suppose ϵ > 0, we get

\frac{b_{1}(G_{i}\X̃)}{[π_{1}(X) : G_{i}]^{1−ϵ}} ≥ \frac{(2g − 2) · exp(i · n_{i} · ϵ)}{2^{n_{i}·(1−ϵ)}}

and hence \frac{b_{1}(G_{i}\X̃)}{[π_{1}(X):G_{i}]^{1−ϵ}} converges to ∞ for i → ∞.

13.8. We get from Exercise 3.5 and the product formula for analytic torsion (which can be viewed as a special case of (3.126) in combination with Theorem 3.149) and Theorem 3.105

ρ_{an}(G_{i}\S̃^{1}) = ln(vol(G_{i}\S^{1}));
ρ_{an}((G_{i} × H_{i})\({ \atop S^{1} × N})) = ρ_{an}(G_{i}\S̃^{1}) · χ(H_{i}\Ñ);
ρ_{an}^{(2)}({ \atop S^{1} × N}) = 0.

Since lim_{i→∞} \frac{ln(vol(G_{i}\S^{1}))}{[π_{1}(S^{1}):G_{i}]} = 0 and \frac{χ(H_{i}\N)}{[π_{1}(N):H_{i}]} = χ(N) hold the claim follows.

Chapter 14

14.1. Let σ_{1}, σ_{2}, …, σ_{m} be the n-simplices in an oriented triangulation. Denote by σ′_{1}, σ′_{2}, …, σ′_{m} the associated singular n-simplices. Then ||M|| is represented by ∑_{i=1}^{m} σ′_{i} and ||M|| ≤ || ∑_{i=1}^{m} σ′_{i}||_{1} = m.

14.2. Let u = ∑_{σ∈I} λ_{σ} · σ for some subset I ⊂ S_{1}(ℝ) be a locally finite singular 1-chain which is in the kernel of the first differential and represents [ℝ] ∈ H_{1}^{lf}(ℝ; ℤ). We can arrange without changing the properties above and its L^{1}-norm that each singular simplex σ: [0, 1] → ℝ appearing in I is an affine orientation preserving embedding and λ_{σ} ≠ 0 holds for σ ∈ I. Notice that I must be countable. Since u represents [ℝ], we get for any x ∈ ℝ such that x does not meet σ({0, 1}) for σ ∈ I

∑_{\begin{matrix}σ, \\ x∈σ(0,1) \end{matrix}} λ_{σ} = 1.

Since u is locally finite, we can choose a sequence (x_{n})_{n≥0} of real numbers such that for any σ ∈ I and n ≥ 0 we have x_{n} ∉ σ({0, 1}) and for any σ ∈ I the implication x_{m}, x_{n} ∈ im(σ) ⇒ n = m holds . Then we get for an integer k > 0

|| ∑_{σ∈I} λ_{σ} · σ||_{1} = ∑_{σ∈I} |λ_{σ}| ≥ ∑_{i=1}^{k} ∑_{\begin{matrix}σ∈I, \\ x_{i}∈σ(0,1) \end{matrix}} |λ_{σ}| ≥ ∑_{i=1}^{k} 1 = k.

14.3. Since f is proper it induces a chain map C_{∗}^{lf}(f): C_{∗}^{lf}(M; ℝ) → C_{∗}^{lf}(N; ℝ). This chain map preserves the L^{1}-norm. Since f has degree d, we get H_{n}^{lf}(f)([M]) = d · [N]. This implies ||M|| ≥ d · ||N||.

There exists a proper selfmap ℝ^{n} → ℝ^{n} of degree d ≥ 2 provided that n ≥ 2.

14.4. Consider ϕ ∈ Ĉ^{0}(X) with δ^{0}(ϕ) = 0. Then ϕ: S_{0}(X) → ℝ is the constant function since X is path-connected by assumption.

14.5. The condition that X is a simplicial complex (and not only a CW - complex) ensures that l^{∞}C^{∗}(X) is indeed a subcomplex.

If X is finite, l^{∞}C^{∗}(X) = C^{∗}(X), but in general H^{∗}(X; ℝ) and Ĥ^{∗}(X) are different for an finite simplicial complex X.

14.6. This follows from 13.3 (1) and (2), Lemma 14.13 and Theorem 14.27.

14.7. The cochain ϕ is the image under the de Rham cochain map of the element which is given by the volume form in the de Rham complex multiplied with vol(M)^{−1}. Hence it is a cocycle which represents the image of the cohomological fundamental class of M under the change of rings map H^{n}(M; ℤ) → H^{n}(M; ℝ) (see Theorem 1.47).

Choose an orientation preserving embedding i: S^{1} × [0, 3]^{n−1} → M. Let p_{n} : S^{1} × [0, 3]^{n−1} → S^{1} × [0, 3]^{n−1} be the map which sends (z, t_{1}, …, t_{n−1}) to (z^{n}, t_{1}, …, t_{n−1}). Let σ: Δ_{n} → S^{1} × [0, 3]^{n−1} be an orientation preserving embedding whose image contains S^{1} × [1, 2]^{n−1}. Then

|ϕ(i ∘ p_{n} ∘ σ)| = ∫_{Δ_{p}}(i ∘ p_{n} ∘ σ)^{∗} dvol_{M}
≥ ∫_{S^{1}×[1,2]^{n−1}}(i ∘ p_{n})^{∗} dvol_{M}
= n · ∫_{S^{1}×[1,2]^{n−1}} i^{∗} dvol_{M} .

Since ∫_{S^{1}×[1,2]^{n−1}} i^{∗} dvol_{M} > 0 and n can be chosen arbitrary large, we conclude ||ϕ||_{∞} = ∞.

14.8. The chain map C_{p}^{sing}(f): C_{p}^{sing}(X; ℝ) → C_{p}^{sing}(Y; ℝ) is isometric with respect to the L^{1}-norm. The cochain map C_{sing}^{p}(f): C_{sing}^{p}(Y; ℝ) → C_{sing}^{p}(X; ℝ) is norm decreasing with respect to the L^{∞}-norm.

14.9. This follows from Theorem 14.46 using the Euler Poincaré formula (see Theorem 1.35 (2)), the obvious inequality

sign^{(2)}(M̃) ≤ b_{2m}^{(2)}(M̃)

for dim(M) = 4m and the equality

sign^{(2)}(M̃) = sign(M)

which is a conclusion of Atiyah’s L^{2}-index theorem (see [9]) applied to the signature operator.

14.10. See [238, Example 5.13 on page 281].

14.11. Denote by p: \overline{N} → N the covering associated to the subgroup im(π_{1}(f)) of π_{1}(N). Then we can find a map \overline{f}: M → \overline{N} satisfying f = p ∘ \overline{f}. Since f has degree one, H_{n}(p): H_{n}(\overline{N}) → H_{n}(N) must be surjective for n = dim(M). Since H_{n}(\overline{N}) = 0 for non-compact \overline{N}, the space \overline{N} must be compact and hence the index of im(π_{1}(f)) in π_{1}(N) is finite. Since this index is the degree of p, it must be 1 and hence π_{1}(f) must be surjective. Now apply Lemma 14.10 (1) and Theorem 14.15.

14.12. This follows from Corollary 1.43 and Theorem 14.32.

14.13. This follows from Theorem 14.26 and Theorem 14.32.

14.14. Let F_{g} be a closed orientable surface with g ≥ 2. Then b_{p}^{(2)}({ \atop S^{3} × F_{g}}) ≠ 0 for p = 1, 4 by Example 1.32, Theorem 1.35 (4) and Example 1.36. Since S^{3} and hence S^{3} ×F_{g} carries a free S^{1}-action, the minimal volume of S^{3} ×F_{g} vanishes by Theorem 14.32.

14.15. Since the quotient of an amenable group is amenable again, the map π_{1}(E) → π_{1}(B) is surjective with an amenable group as kernel. Hence p induces an isometric isomorphism Ĥ^{dim(E)}(B) ⟶^{≅} Ĥ^{dim(E)}(E) by Theorem 14.15.

Recall that for any closed oriented manifold M of dimension n the n-th homology H_{n}(M; ℤ) is ℤ. Suppose that the π_{1}(B)-action on H_{dim(F)}(F; ℤ) given by the fiber transport is trivial. A spectral sequence argument shows that H_{n}(E; ℤ) = 0 for n > dim(F) + dim(B) and H_{n}(E; ℤ) = ℤ for n = dim(F) + dim(B). This implies dim(E) = dim(F) + dim(B). If the π_{1}(B)- action is non-trivial, pass to the 2-sheeted covering \overline{B} → B given by the subgroup of elements in π_{1}(B) which act trivially on H_{dim(F)}(F; ℤ) and apply the same argument to the pullback fibration F → \overline{E} → \overline{B}. Hence we get in all cases dim(B) < dim(E) since we assume dim(F) ≥ 1. We conclude that the map H_{dim(E)}(p) : H_{dim(E)}(E; ℤ) → H_{dim(E)}(B; ℤ) is trivial and maps in particular [E] to 0.

Now ||E|| = 0 follows from Lemma 14.10 (1).

14.16. Suppose that F_{g} carries an affine structure. Then its tangent bundle is flat. We conclude from Theorem 14.51 that 4 · |χ(F_{g})| ≤ ||F_{g}||. We get ||F_{g}|| = 2 · |χ(F_{g})| for g ≥ 1 and ||S^{2}|| = 0 from Theorem 14.18 (2). Hence χ(F_{g}) = 2 − 2g must be zero. This shows F_{g} = T^{2}. Obviously T^{2} carries an affine structure.

14.17. Fix ϵ > 0. Let p_{ϵ}: M_{ϵ} → M be a d_{ϵ}-sheeted covering and let K_{ϵ} be a triangulation for M_{ϵ} with t_{ϵ} simplices of dimension n such that t_{ϵ} ≤ d_{ϵ} · ϵ holds. The number of p-simplices in K_{ϵ} is bounded by ( \begin{matrix}n + 1 \\ p \end{matrix} )·t_{ϵ}. Since each

simplex has at most (n + 1) faces, the norm of the p-th differential c_{p}^{(2)}[ϵ] in C_{∗}^{(2)}(K̃_{ϵ}) is bounded by (n+1) for all p ≥ 0. Let Δ_{p}[ϵ]: C_{p}^{(2)}(K̃_{ϵ}) → C_{p}^{(2)}(K̃_{ϵ}) be the Laplacian. We conclude for all p ≥ 0

||Δ_{p}[ϵ]|| ≤ 2 · (n + 1)^{2};
dim_{N (π_{1}(K_{ϵ}))}(C_{p}^{(2)}(K̃_{ϵ})) ≤ ( \begin{matrix}n + 1 \\ p \end{matrix} ) · t_{ϵ}.

This implies for all p ≥ 0 (see Lemma 3.15 (3) and (6))

dim_{N (π_{1}(K_{ϵ}))}(ker(Δ_{p}[ϵ])) ≤ ( \begin{matrix}n + 1 \\ p \end{matrix} ) · t_{ϵ};
det_{N (π_{1}(K_{ϵ}))}(Δ_{p}[ϵ]) ≤ ( \begin{matrix}n + 1 \\ p \end{matrix} ) · t_{ϵ} · 2 · (n + 1)^{2}.

Hence we get for all p ≥ 0

b_{p}^{(2)}(K̃_{ϵ}) ≤ ( \begin{matrix}n + 1 \\ p \end{matrix} ) · t_{ϵ};

ρ^{(2)}(K̃_{ϵ}) ≤ (∑_{p≥1} p) · ( \begin{matrix}n + 1 \\ p \end{matrix} ) · t_{ϵ} · 2 · (n + 1)^{2}.

Recall from Exercise 14.1 that ||M_{ϵ}|| ≤ t_{ϵ}. Since b_{p}^{(2)}(M̃_{ϵ}) = d_{ϵ} · b_{p}^{(2)}(M̃) holds by Theorem 1.35 (9) and ||M_{ϵ}|| = d_{ϵ} · ||M|| holds by Lemma 14.13, we conclude

b_{p}^{(2)}(M̃) ≤ ( \begin{matrix}n + 1 \\ p \end{matrix} ) · ϵ;
||M|| ≤ ϵ.

Since ϵ > 0 was arbitrary, we conclude

b_{p}^{(2)}(M̃) = 0;
||M|| = 0.

If we assume that π_{1}(M) is of det ≥ 1-class, then M̃ is det-L^{2}-acyclic and ρ^{(2)}(M̃) is given by ρ^{(2)}(K̃) for any choice of triangulation (see Lemma 13.6). Because of Theorem 3.96 (5) the same is true for M̃_{ϵ} and we get

ρ^{(2)}(M̃) = \frac{ρ^{(2)}(K̃_{ϵ})}{d_{ϵ}} ≤ (∑_{p≥1} p) · ( \begin{matrix}n + 1 \\ p \end{matrix} ) · ϵ · 2 · (n + 1)^{2}.

Since this holds for all ϵ > 0 we get ρ^{(2)}(M̃) = 0.

References

1. H. Abels. A universal proper G-space. Math. Z., 159(2):143–158, 1978.

2. D. Alexandru-Rugină. Harmonic forms and L_{p}-cohomology. Tensor (N.S.), 57(2):176–191, 1996.

3. J. Amorós, M. Burger, K. Corlette, D. Kotschick, and D. Toledo. Fundamental groups of compact Kähler manifolds. American Mathematical Society, Providence, RI, 1996.

4. D. R. Anderson and H. J. Munkholm. Boundedly controlled topology. Springer-Verlag, Berlin, 1988. Foundations of algebraic topology and simple homotopy theory.

5. M. T. Anderson. L^{2} harmonic forms and a conjecture of Dodziuk-Singer. Bull. Amer. Math. Soc. (N.S.), 13(2):163–165, 1985.

6. M. T. Anderson. L^{2} harmonic forms on complete Riemannian manifolds. In Geometry and analysis on manifolds (Katata/Kyoto, 1987), pages 1–19. Springer-Verlag, Berlin, 1988.

7. N. Anghel. L^{2}-index formulae for perturbed Dirac operators. Comm. Math. Phys., 128(1):77–97, 1990.

8. P. Ara and D. Goldstein. A solution of the matrix problem for Rickart C^{∗}- algebras. Math. Nachr., 164:259–270, 1993.

9. M. F. Atiyah. Elliptic operators, discrete groups and von Neumann algebras. Astérisque, 32-33:43–72, 1976.

10. M. F. Atiyah and R. Bott. The index problem for manifolds with boundary. In Differential Analysis, Bombay Colloq., 1964, pages 175–186. Oxford Univ. Press, London, 1964.

11. M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambridge Philos. Soc., 77:43–69, 1975.

12. M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geometry. II. Math. Proc. Cambridge Philos. Soc., 78(3):405–432, 1975.

13. M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geometry. III. Math. Proc. Cambridge Philos. Soc., 79(1):71–99, 1976.

14. M. F. Atiyah and W. Schmid. A geometric construction of the discrete series for semisimple Lie groups. Invent. Math., 42:1–62, 1977.

15. M. Auslander and D. A. Buchsbaum. Groups, rings, modules. Harper & Row Publishers, New York, 1974. Harper’s Series in Modern Mathematics.

16. A. Avez. Variétés riemanniennes sans points focaux. C. R. Acad. Sci. Paris Sér. A-B, 270:A188–A191, 1970.

17. W. Ballmann. Lectures on spaces of nonpositive curvature. Birkhäuser Verlag, Basel, 1995. With an appendix by M. Brin.

18. W. Ballmann and J. Brüning. On the spectral theory of manifolds with cusps. J. Math. Pures Appl. (9), 80(6):593–625, 2001.

19. W. Ballmann and P. B. Eberlein. Fundamental groups of manifolds of nonpositive curvature. J. Differential Geom., 25(1):1–22, 1987.

20. W. Ballmann, M. Gromov, and V. Schroeder. Manifolds of nonpositive curvature. Birkhäuser Boston Inc., Boston, MA, 1985.

21. D. Barbasch and H. Moscovici. L^{2}-index and the Selberg trace formula. J. Funct. Anal., 53(2):151–201, 1983.

22. H. Bass. The degree of polynomial growth of finitely generated nilpotent groups. Proc. London Math. Soc. (3), 25:603–614, 1972.

23. H. Bass. Euler characteristics and characters of discrete groups. Invent. Math., 35:155–196, 1976.

24. H. Bass. Covering theory for graphs of groups. J. Pure Appl. Algebra, 89(1-2):3–47, 1993.

25. H. Bass, A. Heller, and R. G. Swan. The Whitehead group of a polynomial extension. Inst. Hautes Études Sci. Publ. Math., (22):61–79, 1964.

26. P. Baum and A. Connes. Geometric K-theory for Lie groups and foliations. Enseign. Math. (2), 46(1-2):3–42, 2000.

27. P. Baum, A. Connes, and N. Higson. Classifying space for proper actions and K-theory of group C^{∗}-algebras. In C^{∗}-algebras: 1943–1993 (San Antonio, TX, 1993), pages 240–291. Amer. Math. Soc., Providence, RI, 1994.

28. B. Baumslag and S. J. Pride. Groups with one more generator than relators. Math. Z., 167(3):279–281, 1979.

29. G. Baumslag. Automorphism groups of residually finite groups. J. London Math. Soc., 38:117–118, 1963.

30. G. Baumslag and P. B. Shalen. Amalgamated products and finitely presented groups. Comment. Math. Helv., 65(2):243–254, 1990.

31. G. Baumslag and D. Solitar. Some two-generator one-relator non-Hopfian groups. Bull. Amer. Math. Soc., 68:199–201, 1962.

32. C. Bavard and P. Pansu. Sur le volume minimal de ℝ^{2}. Ann. Sci. École Norm. Sup. (4), 19(4):479–490, 1986.

33. M. E. B. Bekka, P.-A. Cherix, and A. Valette. Proper affine isometric actions of amenable groups. In Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), pages 1–4. Cambridge Univ. Press, Cambridge, 1995.

34. M. E. B. Bekka and A. Valette. Group cohomology, harmonic functions and the first L^{2}-Betti number. Potential Anal., 6(4):313–326, 1997.

35. R. Benedetti and C. Petronio. Lectures on hyperbolic geometry. Springer-Verlag, Berlin, 1992.

36. S. K. Berberian. Baer *-rings. Springer-Verlag, New York, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 195.

37. S. K. Berberian. The maximal ring of quotients of a finite von Neumann algebra. Rocky Mountain J. Math., 12(1):149–164, 1982.

38. A. J. Berrick, I. Chatterji, and G. Mislin. From acyclic groups to the Bass conjecture for amenable groups. Preprint, 2001.

39. A. J. Berrick and M. E. Keating. The localization sequence in K-theory. K-Theory, 9(6):577–589, 1995.

40. G. Besson, G. Courtois, and S. Gallot. Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal., 5(5):731–799, 1995.

41. M. Bestvina and K. Fujiwara. Bounded cohomology of subgroups of mapping class groups. Preprint, 2000.

42. J.-M. Bismut. From Quillen metrics to Reidemeister metrics: some aspects of the Ray-Singer analytic torsion. In Topological methods in modern mathematics (Stony Brook, NY, 1991), pages 273–324. Publish or Perish, Houston, TX, 43. J.-M. Bismut and D. S. Freed. The analysis of elliptic families. I. Metrics and

connections on determinant bundles. Comm. Math. Phys., 106(1):159–176, 1986.

44. J.-M. Bismut and D. S. Freed. The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem. Comm. Math. Phys., 107(1):103–163, 1986.

45. J.-M. Bismut, H. Gillet, and C. Soulé. Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion. Comm. Math. Phys., 115(1):49–78, 1988.

46. J.-M. Bismut, H. Gillet, and C. Soulé. Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott-Chern forms. Comm. Math. Phys., 115(1):79–126, 1988.

47. J.-M. Bismut, H. Gillet, and C. Soulé. Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants. Comm. Math. Phys., 115(2):301–351, 1988.

48. J.-M. Bismut and J. Lott. Flat vector bundles, direct images and higher real analytic torsion. J. Amer. Math. Soc., 8(2):291–363, 1995.

49. J.-M. Bismut and W. Zhang. An extension of a theorem by Cheeger and Müller. Astérisque, (205):235, 1992. With an appendix by F. Laudenbach.

50. J.-M. Bismut and W. Zhang. Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle. Geom. Funct. Anal., 4(2):136–212, 1994.

51. B. Blackadar. K-theory for operator algebras. Springer-Verlag, New York, 1986.

52. M. Boileau, S. Druck, and E. Vogt. A vanishing theorem for the Gromov volume of 3-manifolds with an application to circle foliations. Math. Ann., 322(3):493–524, 2002.

53. A. Borel. L^{2}-cohomology and intersection cohomology of certain arithmetic varieties. In Emmy Noether in Bryn Mawr (Bryn Mawr, Pa., 1982), pages 119–131. Springer-Verlag, New York, 1983.

54. A. Borel. The L^{2}-cohomology of negatively curved Riemannian symmetric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math., 10:95–105, 1985.

55. A. Borel and W. Casselman. Cohomologie d’intersection et L^{2}-cohomologie de variétés arithmétiques de rang rationnel 2. C. R. Acad. Sci. Paris Sér. I Math., 301(7):369–373, 1985.

56. A. Borel and N. R. Wallach. Continuous cohomology, discrete subgroups, and representations of reductive groups. Princeton University Press, Princeton, N.J., 1980.

57. R. Bott and L. W. Tu. Differential forms in algebraic topology. Springer-Verlag, New York, 1982.

58. J.-P. Bourguignon and H. Karcher. Curvature operators: pinching estimates and geometric examples. Ann. Sci. École Norm. Sup. (4), 11(1):71–92, 1978.

59. J.-P. Bourguignon and A. Polombo. Intégrands des nombres caractéristiques et courbure: rien ne va plus dès la dimension 6. J. Differential Geom., 16(4):537– 550 (1982), 1981.

60. T. P. Branson and P. B. Gilkey. The asymptotics of the Laplacian on a manifold with boundary. Comm. Partial Differential Equations, 15(2):245– 272, 1990.

61. C. Bratzler. L^{2}-Betti Zahlen und Faserungen. Diplomarbeit, Johannes Gutenberg-Universität Mainz, 1997. http://www.math.unimuenster.de/u/lueck/publ/diplome/bratzler.dvi.

62. M. Braverman. Witten deformation of analytic torsion and the spectral sequence of a filtration. Geom. Funct. Anal., 6(1):28–50, 1996.

63. G. Bredon. Topology and geometry, volume 139 of Graduate texts in Mathematics. Springer, 1993.

64. G. E. Bredon. Introduction to compact transformation groups. Academic Press, New York, 1972. Pure and Applied Mathematics, Vol. 46.

65. M. R. Bridson and A. Haefliger. Metric spaces of non-positive curvature. Springer-Verlag, Berlin, 1999. Die Grundlehren der mathematischen Wissenschaften, Band 319.

66. M. G. Brin and C. C. Squier. Groups of piecewise linear homeomorphisms of the real line. Invent. Math., 79(3):485–498, 1985.

67. T. Bröcker and K. Jänich. Einführung in die Differentialtopologie. Springer-Verlag, Berlin, 1973. Heidelberger Taschenbücher, Band 143.

68. R. Brooks. The fundamental group and the spectrum of the Laplacian. Comment. Math. Helv., 56(4):581–598, 1981.

69. K. Brown. Cohomology of groups, volume 87 of Graduate texts in Mathematics. Springer, 1982.

70. K. S. Brown and R. Geoghegan. An infinite-dimensional torsion-free fp_{∞} group. Invent. Math., 77(2):367–381, 1984.

71. J. Brüning. L^{2}-index theorems on certain complete manifolds. J. Differential Geom., 32(2):491–532, 1990.

72. J. Brüning. On L^{2}-index theorems for complete manifolds of rank-one-type. Duke Math. J., 66(2):257–309, 1992.

73. J. Brüning and M. Lesch. Hilbert complexes. J. Funct. Anal., 108(1):88–132, 1992.

74. U. Bunke. Equivariant torsion and G-CW-complexes. Geom. Funct. Anal., 9(1):67–89, 1999.

75. G. Burde and H. Zieschang. Knots. Walter de Gruyter & Co., Berlin, 1985.

76. M. Burger and N. Monod. Bounded cohomology of lattices in higher rank Lie groups. J. Eur. Math. Soc. (JEMS), 1(2):199–235, 1999.

77. M. Burger and N. Monod. Continuous bounded cohomology and applications to rigidity theory. preprint, to appear in GAFA, 2001.

78. M. Burger and A. Valette. Idempotents in complex group rings: theorems of Zalesskii and Bass revisited. J. Lie Theory, 8(2):219–228, 1998.

79. D. Burghelea, L. Friedlander, and T. Kappeler. Meyer-Vietoris type formula for determinants of elliptic differential operators. J. Funct. Anal., 107(1):34– 65, 1992.

80. D. Burghelea, L. Friedlander, and T. Kappeler. Asymptotic expansion of the Witten deformation of the analytic torsion. J. Funct. Anal., 137(2):320–363, 1996.

81. D. Burghelea, L. Friedlander, and T. Kappeler. Witten deformation of the analytic torsion and the Reidemeister torsion. In Voronezh Winter Mathematical Schools, pages 23–39. Amer. Math. Soc., Providence, RI, 1998. Trans. Ser. 2 184 (1998).

82. D. Burghelea, L. Friedlander, and T. Kappeler. Torsions for manifolds with boundary and glueing formulas. Math. Nachr., 208:31–91, 1999.

83. D. Burghelea, L. Friedlander, and T. Kappeler. Relative torsion. Commun. Contemp. Math., 3(1):15–85, 2001.

84. D. Burghelea, L. Friedlander, T. Kappeler, and P. McDonald. Analytic and Reidemeister torsion for representations in finite type Hilbert modules. Geom. Funct. Anal., 6(5):751–859, 1996.

85. J. W. Cannon, W. J. Floyd, and W. R. Parry. Introductory notes on Richard Thompson’s groups. Enseign. Math. (2), 42(3-4):215–256, 1996.

86. J. Cao and F. Xavier. Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature. Math. Ann., 319(3):483–491, 2001.

87. S. E. Cappell. On connected sums of manifolds. Topology, 13:395–400, 1974.

88. S. E. Cappell. A splitting theorem for manifolds. Invent. Math., 33(2):69–170, 1976.

89. S. E. Cappell and J. L. Shaneson. Nonlinear similarity. Ann. of Math. (2), 113(2):315–355, 1981.

90. S. E. Cappell and J. L. Shaneson. On 4-dimensional s-cobordisms. J. Differential Geom., 22(1):97–115, 1985.

91. S. E. Cappell, J. L. Shaneson, M. Steinberger, S. Weinberger, and J. E. West. The classification of nonlinear similarities over Z_{2^{r}}. Bull. Amer. Math. Soc. (N.S.), 22(1):51–57, 1990.

92. A. L. Carey, M. Farber, and V. Mathai. Determinant lines, von Neumann algebras and L^{2} torsion. J. Reine Angew. Math., 484:153–181, 1997.

93. A. L. Carey and V. Mathai. L^{2}-torsion invariants. J. Funct. Anal., 110(2):377– 409, 1992.

94. H. Cartan and S. Eilenberg. Homological algebra. Princeton University Press, Princeton, N. J., 1956.

95. W. Casselman. Introduction to the L^{2}-cohomology of arithmetic quotients of bounded symmetric domains. In Complex analytic singularities, pages 69–93. North-Holland, Amsterdam, 1987.

96. A. J. Casson and S. A. Bleiler. Automorphisms of surfaces after Nielsen and Thurston. Cambridge University Press, Cambridge, 1988.

97. A. J. Casson and C. M. Gordon. On slice knots in dimension three. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pages 39–53. Amer. Math. Soc., Providence, R.I., 1978.

98. S. Chang and S. Weinberger. On invariants of Hirzebruch and Cheeger-Gromov. preprint, 2001.

99. T. A. Chapman. Compact Hilbert cube manifolds and the invariance of Whitehead torsion. Bull. Amer. Math. Soc., 79:52–56, 1973.

100. T. A. Chapman. Topological invariance of Whitehead torsion. Amer. J. Math., 96:488–497, 1974.

101. M. Chayet and N. Lohoue. Sur la cohomologie L^{p} des variétés. C. R. Acad. Sci. Paris Sér. I Math., 324(2):211–213, 1997.

102. J. Cheeger. Finiteness theorems for Riemannian manifolds. Amer. J. Math., 92:61–74, 1970.

103. J. Cheeger. Analytic torsion and the heat equation. Ann. of Math. (2), 109(2):259–322, 1979.

104. J. Cheeger, M. Goresky, and R. MacPherson. L^{2}-cohomology and intersection homology of singular algebraic varieties. In Seminar on Differential Geometry, pages 303–340. Princeton Univ. Press, Princeton, N.J., 1982.

105. J. Cheeger and M. Gromov. Bounds on the von Neumann dimension of L^{2}- cohomology and the Gauss-Bonnet theorem for open manifolds. J. Differential Geom., 21(1):1–34, 1985.

106. J. Cheeger and M. Gromov. On the characteristic numbers of complete manifolds of bounded curvature and finite volume. In Differential geometry and complex analysis, pages 115–154. Springer-Verlag, Berlin, 1985.

107. J. Cheeger and M. Gromov. L_{2}-cohomology and group cohomology. Topology, 25(2):189–215, 1986.

108. S.-s. Chern. On curvature and characteristic classes of a Riemann manifold. Abh. Math. Sem. Univ. Hamburg, 20:117–126, 1955.

109. P. R. Chernoff. Essential self-adjointness of powers of generators of hyperbolic equations. J. Functional Analysis, 12:401–414, 1973.

110. B. Clair. Residual amenability and the approximation of L^{2}-invariants. Michigan Math. J., 46(2):331–346, 1999.

111. B. Clair and K. Whyte. Growth of Betti numbers. Preprint, 2001.

112. G. Cliff and A. Weiss. Moody’s induction theorem. Illinois J. Math., 32(3):489–500, 1988.

113. T. Cochran, K. Orr, and P. Teichner. Knot concordance, Whitney towers and L^{2}-signatures. Preprint, to appear in Ann. of Math., 1999.

114. D. E. Cohen. Combinatorial group theory: a topological approach. Cambridge University Press, Cambridge, 1989.

115. J. M. Cohen. von Neumann dimension and the homology of covering spaces. Quart. J. Math. Oxford Ser. (2), 30(118):133–142, 1979.

116. M. M. Cohen. A course in simple-homotopy theory. Springer-Verlag, New York, 1973. Graduate Texts in Mathematics, Vol. 10.

117. P. M. Cohn. Free ideal rings. J. Algebra, 1:47–69, 1964.

118. P. M. Cohn. Free rings and their relations. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, second edition, 1985.

119. P. M. Cohn. Algebra. Vol. 3. John Wiley & Sons Ltd., Chichester, second edition, 1991.

120. A. Connes. Noncommutative geometry. Academic Press Inc., San Diego, CA, 1994.

121. A. Connes and H. Moscovici. The L^{2}-index theorem for homogeneous spaces. Bull. Amer. Math. Soc. (N.S.), 1(4):688–690, 1979.

122. A. Connes and H. Moscovici. The L^{2}-index theorem for homogeneous spaces of Lie groups. Ann. of Math. (2), 115(2):291–330, 1982.

123. A. Connes and H. Moscovici. L^{2}-index theory on homogeneous spaces and discrete series representations. In Operator algebras and applications, Part I (Kingston, Ont., 1980), pages 419–433. Amer. Math. Soc., Providence, R.I., 1982.

124. P. Connor and F. Raymond. Actions of compact Lie groups on aspherical manifolds. In Topology of manifolds, pages 227–264. Markham, Chicago, 1970. Cantrell and Edwards.

125. C. W. Curtis and I. Reiner. Methods of representation theory. Vol. II. John Wiley & Sons Inc., New York, 1987. With applications to finite groups and orders, A Wiley-Interscience Publication.

126. X. Dai and H. Fang. Analytic torsion and R-torsion for manifolds with boundary. Asian J. Math., 4(3):695–714, 2000.

127. X. Dai and R. Melrose. Adiabatic limit of the analytic torsion. Preprint.

128. J. F. Davis and W. Lück. Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory. K-Theory, 15(3):201–252, 1998.

129. M. Davis and I. Leary. The L^{2}-cohomology of Artin groups. Preprint, 2001.

130. M. W. Davis and B. Okun. Vanishing theorems and conjectures for the ℓ^{2}- homology of right-angled Coxeter groups. Geom. Topol., 5:7–74 (electronic), 2001.

131. D. L. de George and N. R. Wallach. Limit formulas for multiplicities in L^{2}(γ\G). Ann. Math. (2), 107(1):133–150, 1978.

132. P. de la Harpe. Topics in geometric group theory. University of Chicago Press, Chicago, IL, 2000.

133. G. de Rham. Reidemeister’s torsion invariant and rotations of S^{n}. In Differential Analysis, Bombay Colloq., pages 27–36. Oxford Univ. Press, London, 134. G. de Rham. Differentiable manifolds. Springer-Verlag, Berlin, 1984. Forms,

currents, harmonic forms, Translated from the French by F. R. Smith, With an introduction by S. S. Chern.

135. D. L. DeGeorge and N. R. Wallach. Limit formulas for multiplicities in L^{2}(γ\G). II. The tempered spectrum. Ann. of Math. (2), 109(3):477–495, 1979.

136. A. Deitmar. Higher torsion zeta functions. Adv. Math., 110(1):109–128, 1995.

137. A. Deitmar. A determinant formula for the generalized Selberg zeta function. Quart. J. Math. Oxford Ser. (2), 47(188):435–453, 1996.

138. A. Deitmar. Equivariant torsion of locally symmetric spaces. Pacific J. Math., 182(2):205–227, 1998.

139. A. Deitmar. Regularized and L^{2}-determinants. Proc. London Math. Soc. (3), 76(1):150–174, 1998.

140. T. Delzant. Sur l’anneau d’un groupe hyperbolique. C. R. Acad. Sci. Paris Sér. I Math., 324(4):381–384, 1997.

141. W. Dicks and P. H. Kropholler. Free groups and almost equivariant maps. Bull. London Math. Soc., 27(4):319–326, 1995.

142. W. Dicks and P. Menal. The group rings that are semifirs. J. London Math. Soc. (2), 19(2):288–290, 1979.

143. W. Dicks and T. Schick. The spectral measure of certain elements of the complex group ring of a wreath product. Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 159, Münster, to appear in Geometriae Dedicata, 2001.

144. J. Dixmier. von Neumann algebras. North-Holland Publishing Co., Amsterdam, 1981. With a preface by E. C. Lance, Translated from the second French edition by F. Jellett.

145. J. Dodziuk. de Rham-Hodge theory for L^{2}-cohomology of infinite coverings. Topology, 16(2):157–165, 1977.

146. J. Dodziuk. L^{2} harmonic forms on rotationally symmetric Riemannian manifolds. Proc. Amer. Math. Soc., 77(3):395–400, 1979.

147. J. Dodziuk. Sobolev spaces of differential forms and de Rham-Hodge isomorphism. J. Differential Geom., 16(1):63–73, 1981.

148. J. Dodziuk, P. A. Linnell, V. Mathai, T. Schick, and S. Yates. Approximating L^{2}-invariants, and the Atiyah conjecture. Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 170, Münster, 2001.

149. J. Dodziuk and V. Mathai. Approximating L^{2} invariants of amenable covering spaces: a combinatorial approach. J. Funct. Anal., 154(2):359–378, 1998.

150. J. Dodziuk and V. K. Patodi. Riemannian structures and triangulations of manifolds. J. Indian Math. Soc. (N.S.), 40(1-4):1–52 (1977), 1976.

151. S. K. Donaldson. Irrationality and the h-cobordism conjecture. J. Differential Geom., 26(1):141–168, 1987.

152. H. Donnelly. On L^{2}-Betti numbers for abelian groups. Canad. Math. Bull., 24(1):91–95, 1981.

153. H. Donnelly. On the spectrum of towers. Proc. Amer. Math. Soc., 87(2):322– 329, 1983.

154. H. Donnelly and C. Fefferman. L^{2}-cohomology and index theorem for the Bergman metric. Ann. of Math. (2), 118(3):593–618, 1983.

155. H. Donnelly and F. Xavier. On the differential form spectrum of negatively curved Riemannian manifolds. Amer. J. Math., 106(1):169–185, 1984.

156. J. L. Dupont. Curvature and characteristic classes. Springer-Verlag, Berlin, 1978. Lecture Notes in Mathematics, Vol. 640.

157. P. B. Eberlein. Geometry of nonpositively curved manifolds. University of Chicago Press, Chicago, IL, 1996.

158. P. B. Eberlein and J. Heber. A differential geometric characterization of symmetric spaces of higher rank. Inst. Hautes Études Sci. Publ. Math., (71):33–44, 1990.

159. B. Eckmann. Cyclic homology of groups and the Bass conjecture. Comment. Math. Helv., 61(2):193–202, 1986.

160. B. Eckmann. Amenable groups and Euler characteristic. Comment. Math. Helv., 67(3):383–393, 1992.

161. B. Eckmann. Projective and Hilbert modules over group algebras, and finitely dominated spaces. Comment. Math. Helv., 71(3):453–462, 1996.

162. B. Eckmann. 4-manifolds, group invariants, and l_{2}-Betti numbers. Enseign. Math. (2), 43(3-4):271–279, 1997.

163. B. Eckmann. Approximating l_{2}-Betti numbers of an amenable covering by ordinary Betti numbers. Comment. Math. Helv., 74(1):150–155, 1999.

164. B. Eckmann. Introduction to l_{2}-methods in topology: reduced l_{2}-homology, harmonic chains, l_{2}-Betti numbers. Israel J. Math., 117:183–219, 2000. Notes prepared by G. Mislin.

165. B. Eckmann. Idempotents in a complex group algebra, projective modules, and the von Neumann algebra. Arch. Math. (Basel), 76(4):241–249, 2001.

166. E. G. Effros. Why the circle is connected: an introduction to quantized topology. Math. Intelligencer, 11(1):27–34, 1989.

167. A. Efremov. Combinatorial and analytic Novikov Shubin invariants. Preprint, 1991.

168. J. Eichhorn. L_{2}-cohomology. In Geometric and algebraic topology, pages 215– 237. PWN, Warsaw, 1986.

169. G. Elek. Abelian coverings. J. Funct. Anal., 141(2):365–373, 1996.

170. G. Elek. Coarse cohomology and l_{p}-cohomology. K-Theory, 13(1):1–22, 1998.

171. G. Elek. Amenable groups, topological entropy and Betti numbers. Preprint, Alfred Renyi Mathematical Institute of the Hungarian Academy of Sciences, to appear in Israel J. Math., 1999.

172. G. Elek. On the analytic zero-divisor conjecture of Linnel. Preprint, 2001.

173. G. Elek. The rank of finitely generated modules over group algebras. Preprint, 2001.

174. K. D. Elworthy, X.-M. Li, and S. Rosenberg. Bounded and L^{2} harmonic forms on universal covers. Geom. Funct. Anal., 8(2):283–303, 1998.

175. I. Emmanouil. On a class of groups satisfying Bass’ conjecture. Invent. Math., 132(2):307–330, 1998.

176. I. Emmanouil. Solvable groups and Bass’ conjecture. C. R. Acad. Sci. Paris Sér. I Math., 326(3):283–287, 1998.

177. D. B. A. Epstein. Finite presentations of groups and 3-manifolds. Quart. J. Math. Oxford Ser. (2), 12:205–212, 1961.

178. D. B. A. Epstein and K. Fujiwara. The second bounded cohomology of wordhyperbolic groups. Topology, 36(6):1275–1289, 1997.

179. T. Fack and P. de la Harpe. Sommes de commutateurs dans les algèbres de von Neumann finies continues. Ann. Inst. Fourier (Grenoble), 30(3):49–73, 1980.

180. M. Falk and R. Randell. Pure braid groups and products of free groups. In Braids (Santa Cruz, CA, 1986), pages 217–228. Amer. Math. Soc., Providence, RI, 1988.

181. M. Farber. Singularities of the analytic torsion. J. Differential Geom., 41(3):528–572, 1995.

182. M. Farber. Homological algebra of Novikov-Shubin invariants and Morse inequalities. Geom. Funct. Anal., 6(4):628–665, 1996.

183. M. Farber. Geometry of growth: approximation theorems for L^{2} invariants. Math. Ann., 311(2):335–375, 1998.

184. M. Farber. Novikov-Shubin signatures. I. Ann. Global Anal. Geom., 18(5):477– 515, 2000.

185. M. Farber. von Neumann Betti numbers and Novikov type inequalities. Proc. Amer. Math. Soc., 128(9):2819–2827, 2000.

186. M. Farber. Novikov-Shubin signatures. II. Ann. Global Anal. Geom., 19(3):259–291, 2001.

187. M. Farber and S. Weinberger. On the zero-in-the-spectrum conjecture. Ann. of Math. (2), 154(1):139–154, 2001.

188. D. R. Farkas and P. A. Linnell. Zero divisors in group rings: something old, something new. In Representation theory, group rings, and coding theory, pages 155–166. Amer. Math. Soc., Providence, RI, 1989.

189. F. T. Farrell. The obstruction to fibering a manifold over a circle. Indiana Univ. Math. J., 21:315–346, 1971/1972.

190. F. T. Farrell and L. E. Jones. K-theory and dynamics. I. Ann. of Math. (2), 124(3):531–569, 1986.

191. F. T. Farrell and L. E. Jones. K-theory and dynamics. II. Ann. of Math. (2), 126(3):451–493, 1987.

192. F. T. Farrell and L. E. Jones. Rigidity and other topological aspects of compact nonpositively curved manifolds. Bull. Amer. Math. Soc. (N.S.), 22(1):59–64, 1990.

193. F. T. Farrell and L. E. Jones. Stable pseudoisotopy spaces of compact nonpositively curved manifolds. J. Differential Geom., 34(3):769–834, 1991.

194. F. T. Farrell and L. E. Jones. Isomorphism conjectures in algebraic K-theory. J. Amer. Math. Soc., 6(2):249–297, 1993.

195. F. T. Farrell and L. E. Jones. Topological rigidity for compact non-positively curved manifolds. In Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), pages 229–274. Amer. Math. Soc., Providence, RI, 1993.

196. F. T. Farrell and L. E. Jones. Rigidity for aspherical manifolds with π_{1} ⊂ GL_{m}(ℝ). Asian J. Math., 2(2):215–262, 1998.

197. F. T. Farrell and P. A. Linnell. Whitehead groups and the Bass conjecture. Preprint, 2000.

198. A. Fel′shtyn. Dynamical zeta functions, Nielsen theory and Reidemeister torsion. Mem. Amer. Math. Soc., 147(699):xii+146, 2000.

199. S. C. Ferry, A. A. Ranicki, and J. Rosenberg. A history and survey of the Novikov conjecture. In Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), pages 7–66. Cambridge Univ. Press, Cambridge, 1995.

200. R. H. Fox. Free differential calculus. I. Derivation in the free group ring. Ann. of Math. (2), 57:547–560, 1953.

201. W. Franz. Über die Torsion einer Überdeckung. J. Reine Angew. Math., 173:245–254, 1935.

202. M. H. Freedman. The topology of four-dimensional manifolds. J. Differential Geom., 17(3):357–453, 1982.

203. M. H. Freedman. The disk theorem for four-dimensional manifolds. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 647–663, Warsaw, 1984. PWN.

204. M. H. Freedman and P. Teichner. 4-manifold topology. I. Subexponential groups. Invent. Math., 122(3):509–529, 1995.

205. P. Freyd. Abelian categories. An introduction to the theory of functors. Harper & Row Publishers, New York, 1964.

206. D. Fried. Analytic torsion and closed geodesics on hyperbolic manifolds. Invent. Math., 84(3):523–540, 1986.

207. D. Fried. Torsion and closed geodesics on complex hyperbolic manifolds. Invent. Math., 91(1):31–51, 1988.

208. B. Fuglede and R. V. Kadison. Determinant theory in finite factors. Ann. of Math. (2), 55:520–530, 1952.

209. K. Fujiwara. The second bounded cohomology of a group acting on a Gromovhyperbolic space. Proc. London Math. Soc. (3), 76(1):70–94, 1998.

210. K. Fujiwara. The second bounded cohomology of an amalgamated free product of groups. Trans. Amer. Math. Soc., 352(3):1113–1129, 2000.

211. A. Furman. Gromov’s measure equivalence and rigidity of higher rank lattices. Ann. of Math. (2), 150(3):1059–1081, 1999.

212. A. Furman. Orbit equivalence rigidity. Ann. of Math. (2), 150(3):1083–1108, 1999.

213. D. Gabai. Convergence groups are Fuchsian groups. Bull. Amer. Math. Soc. (N.S.), 25(2):395–402, 1991.

214. D. Gaboriau. Invariants l^{2} de relation d’équivalence et de groupes. Preprint, Lyon, 2001.

215. D. Gaboriau. On orbit equivalence of measure preserving actions. Preprint, Lyon, 2001.

216. M. P. Gaffney. The harmonic operator for exterior differential forms. Proc. Nat. Acad. Sci. U. S. A., 37:48–50, 1951.

217. M. P. Gaffney. A special Stokes’s theorem for complete Riemannian manifolds. Ann. of Math. (2), 60:140–145, 1954.

218. S. Gallot, D. Hulin, and J. Lafontaine. Riemannian geometry. Springer-Verlag, Berlin, 1987.

219. R. Geroch. Positive sectional curvatures does not imply positive Gauss-Bonnet integrand. Proc. Amer. Math. Soc., 54:267–270, 1976.

220. P. B. Gilkey. Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Publish or Perish Inc., Wilmington, DE, 1984.

221. H. Gillet and C. Soulé. Analytic torsion and the arithmetic Todd genus. Topology, 30(1):21–54, 1991. With an appendix by D. Zagier.

222. V. M. Gol′dshteĭn, V. I. Kuz′minov, and I. A. Shvedov. The de Rham isomorphism of the L_{p}-cohomology of noncompact Riemannian manifolds. Sibirsk. Mat. Zh., 29(2):34–44, 216, 1988.

223. V. M. Gol′dshteĭn, V. I. Kuz′minov, and I. A. Shvedov. On Cheeger’s theorem: extensions to L_{p}-cohomology of warped cylinders. Siberian Adv. Math., 2(1):114–122, 1992.

224. K. R. Goodearl. von Neumann regular rings. Pitman (Advanced Publishing Program), Boston, Mass., 1979.

225. C. M. Gordon. Dehn surgery and satellite knots. Trans. Amer. Math. Soc., 275(2):687–708, 1983.

226. M. J. Greenberg. Lectures on algebraic topology. W. A. Benjamin, Inc., New York-Amsterdam, 1967.

227. R. I. Grigorchuk. Some results on bounded cohomology. In Combinatorial and geometric group theory (Edinburgh, 1993), pages 111–163. Cambridge Univ. Press, Cambridge, 1995.

228. R. I. Grigorchuk. An example of a finitely presented amenable group that does not belong to the class EG. Mat. Sb., 189(1):79–100, 1998.

229. R. I. Grigorchuk, P. A. Linnell, T. Schick, and A. Żuk. On a question of Atiyah. C. R. Acad. Sci. Paris Sér. I Math., 331(9):663–668, 2000.

230. R. I. Grigorchuk and A. Żuk. The Lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geom. Dedicata, 87(1-3):209–244, 231. M. Gromov. Groups of polynomial growth and expanding maps. Inst. Hautes

Études Sci. Publ. Math., (53):53–73, 1981.

232. M. Gromov. Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math., (56):5–99 (1983), 1982.

233. M. Gromov. Large Riemannian manifolds. In Curvature and topology of Riemannian manifolds (Katata, 1985), pages 108–121. Springer-Verlag, Berlin, 1986.

234. M. Gromov. Hyperbolic groups. In Essays in group theory, pages 75–263. Springer-Verlag, New York, 1987.

235. M. Gromov. Sur le groupe fondamental d’une variété kählérienne. C. R. Acad. Sci. Paris Sér. I Math., 308(3):67–70, 1989.

236. M. Gromov. Kähler hyperbolicity and L_{2}-Hodge theory. J. Differential Geom., 33(1):263–292, 1991.

237. M. Gromov. Asymptotic invariants of infinite groups. In Geometric group theory, Vol. 2 (Sussex, 1991), pages 1–295. Cambridge Univ. Press, Cambridge, 1993.

238. M. Gromov. Metric structures for Riemannian and non-Riemannian spaces. Birkhäuser Boston Inc., Boston, MA, 1999. Based on the 1981 French original [MR 85e:53051], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by S. M. Bates.

239. M. Gromov and H. B. Lawson, Jr. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Études Sci. Publ. Math., (58):83–196 (1984), 1983.

240. M. Gromov and M. A. Shubin. von Neumann spectra near zero. Geom. Funct. Anal., 1(4):375–404, 1991.

241. K. W. Gruenberg. Residual properties of infinite soluble groups. Proc. London Math. Soc. (3), 7:29–62, 1957.

242. A. Guichardet. Cohomologie des groupes topologiques et des algèbres de Lie. CEDIC, Paris, 1980.

243. U. Haagerup and H. J. Munkholm. Simplices of maximal volume in hyperbolic n-space. Acta Math., 147(1-2):1–11, 1981.

244. I. Hambleton and E. K. Pedersen. Non-linear similarity revisited. In Prospects in topology (Princeton, NJ, 1994), pages 157–174. Princeton Univ. Press, Princeton, NJ, 1995.

245. I. Hambleton and E. K. Pedersen. Topological equivalence of linear representations for cyclic groups. Preprint, MPI Bonn, 1997.

246. I. Hambleton, L. R. Taylor, and E. B. Williams. On G_{n}(RG) for G a finite nilpotent group. J. Algebra, 116(2):466–470, 1988.

247. D. Handelman. Rickart C^{∗}-algebras. II. Adv. in Math., 48(1):1–15, 1983.

248. Harish-Chandra. Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula. Ann. of Math. (2), 104(1):117– 201, 1976.

249. R. Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.

250. J.-C. Hausmann and S. Weinberger. Caractéristiques d’Euler et groupes fondamentaux des variétés de dimension 4. Comment. Math. Helv., 60(1):139–144, 1985.

251. S. Helgason. Differential geometry, Lie groups and symmetric spaces. Academic Press, 1978.

252. J. Hempel. 3-Manifolds. Princeton University Press, Princeton, N. J., 1976. Ann. of Math. Studies, No. 86.

253. J. Hempel. Residual finiteness for 3-manifolds. In Combinatorial group theory and topology (Alta, Utah, 1984), pages 379–396. Princeton Univ. Press, Princeton, NJ, 1987.

254. E. Hess and T. Schick. L^{2}-torsion of hyperbolic manifolds. Manuscripta Math., 97(3):329–334, 1998.

255. G. Higman. A finitely generated infinite simple group. J. London Math. Soc., 26:61–64, 1951.

256. N. Higson. The Baum-Connes conjecture. In Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), pages 637–646 (electronic), 1998.

257. N. Higson and G. Kasparov. E-theory and KK-theory for groups which act properly and isometrically on Hilbert space. Invent. Math., 144(1):23–74, 2001.

258. N. Higson, J. Roe, and T. Schick. Spaces with vanishing l^{2}-homology and their fundamental groups (after Farber and Weinberger). Geom. Dedicata, 87(1-3):335–343, 2001.

259. J. A. Hillman. Deficiencies of lattices in connected Lie groups. Preprint, 1999.

260. J. A. Hillman and P. A. Linnell. Elementary amenable groups of finite Hirsch length are locally-finite by virtually-solvable. J. Austral. Math. Soc. Ser. A, 52(2):237–241, 1992.

261. M. W. Hirsch and W. P. Thurston. Foliated bundles, invariant measures and flat manifolds. Ann. Math. (2), 101:369–390, 1975.

262. F. Hirzebruch. Automorphe Formen und der Satz von Riemann-Roch. In Symposium internacional de topología algebraica – International symposium on algebraic topology, pages 129–144. Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958.

263. N. Hitchin. Compact four-dimensional Einstein manifolds. J. Differential Geometry, 9:435–441, 1974.

264. C. Hog, M. Lustig, and W. Metzler. Presentation classes, 3-manifolds and free products. In Geometry and topology (College Park, Md., 1983/84), pages 154–167. Springer-Verlag, Berlin, 1985.

265. W. C. Hsiang and W. L. Pardon. When are topologically equivalent orthogonal transformations linearly equivalent? Invent. Math., 68(2):275–316, 1982.

266. W. C. Hsiang and V. Pati. L^{2}-cohomology of normal algebraic surfaces. I. Invent. Math., 81(3):395–412, 1985.

267. I. Hughes. Division rings of fractions for group rings. Comm. Pure Appl. Math., 23:181–188, 1970.

268. J. E. Humphreys. Introduction to Lie algebras and representation theory. Springer-Verlag, New York, 1972. Graduate Texts in Mathematics, Vol. 9.

269. D. Husemoller. Elliptic curves. Springer-Verlag, New York, 1987. With an appendix by R. Lawrence.

270. S. Illman. Whitehead torsion and group actions. Ann. Acad. Sci. Fenn. Ser. A I Math., (588):45, 1974.

271. S. Illman. Smooth equivariant triangulations of G-manifolds for G a finite group. Math. Ann., 233(3):199–220, 1978.

272. S. Illman. The equivariant triangulation theorem for actions of compact Lie groups. Math. Ann., 262(4):487–501, 1983.

273. S. Illman. Existence and uniqueness of equivariant triangulations of smooth proper G-manifolds with some applications to equivariant Whitehead torsion. J. Reine Angew. Math., 524:129–183, 2000.

274. H. Inoue and K. Yano. The Gromov invariant of negatively curved manifolds. Topology, 21(1):83–89, 1982.

275. N. Ivanov. Foundations of the theory of bounded cohomology. J. Soviet Math., 37:1090–1114, 1987.

276. W. H. Jaco and P. B. Shalen. Seifert fibered spaces in 3-manifolds. Mem. Amer. Math. Soc., 21(220):viii+192, 1979.

277. N. Jacobson. Lectures in abstract algebra. Vol. II. Linear algebra. D. Van Nostrand Co., Inc., Toronto-New York-London, 1953.

278. K. Johannson. Homotopy equivalences of 3-manifolds with boundaries. Springer-Verlag, Berlin, 1979.

279. F. E. A. Johnson and D. Kotschick. On the signature and Euler characteristic of certain four-manifolds. Math. Proc. Cambridge Philos. Soc., 114(3):431– 437, 1993.

280. J. Jost and Y. L. Xin. Vanishing theorems for L^{2}-cohomology groups. J. Reine Angew. Math., 525:95–112, 2000.

281. J. Jost and K. Zuo. Vanishing theorems for L^{2}-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry. Comm. Anal. Geom., 8(1):1–30, 2000.

282. R. V. Kadison and J. R. Ringrose. Fundamentals of the theory of operator algebras. Vol. I. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983. Elementary theory.

283. R. V. Kadison and J. R. Ringrose. Fundamentals of the theory of operator algebras. Vol. II. Academic Press Inc., Orlando, FL, 1986. Advanced theory.

284. D. A. Kajdan. On arithmetic varieties. In Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), pages 151– 217. Halsted, New York, 1975.

285. E. Kaniuth. Der Typ der regulären Darstellung diskreter Gruppen. Math. Ann., 182:334–339, 1969.

286. R. M. Kashaev. The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys., 39(3):269–275, 1997.

287. T. Kato. Pertubation theory for linear operators, volume 132 of Grundlehren der math. Wissenschaften. Springer, 1984.

288. M. A. Kervaire. Le théorème de Barden-Mazur-Stallings. Comment. Math. Helv., 40:31–42, 1965.

289. M. A. Kervaire. Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc., 144:67–72, 1969.

290. H. Kesten. Full Banach mean values on countable groups. Math. Scand., 7:146–156, 1959.

291. N. Keswani. Von Neumann eta-invariants and C^{∗}-algebra K-theory. J. London Math. Soc. (2), 62(3):771–783, 2000.

292. R. C. Kirby and L. C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations. Princeton University Press, Princeton, N.J., 1977. With notes by J. Milnor and M. F. Atiyah, Annals of Mathematics Studies, No. 88.

293. A. W. Knapp. Representation theory of semisimple groups. Princeton University Press, Princeton, NJ, 1986. An overview based on examples.

294. K. Kodaira. On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties). Ann. of Math. (2), 60:28–48, 1954.

295. K. Köhler. Holomorphic torsion on Hermitian symmetric spaces. J. Reine Angew. Math., 460:93–116, 1995.

296. K. Köhler. Equivariant Reidemeister torsion on symmetric spaces. Math. Ann., 307(1):57–69, 1997.

297. D. Kotschick. Four-manifold invariants of finitely presentable groups. In Topology, geometry and field theory, pages 89–99. World Sci. Publishing, River Edge, NJ, 1994.

298. M. Kreck, W. Lück, and P. Teichner. Counterexamples to the Kneser conjecture in dimension four. Comment. Math. Helv., 70(3):423–433, 1995.

299. M. Kreck, W. Lück, and P. Teichner. Stable prime decompositions of fourmanifolds. In Prospects in topology (Princeton, NJ, 1994), pages 251–269. Princeton Univ. Press, Princeton, NJ, 1995.

300. P. H. Kropholler, P. A. Linnell, and J. A. Moody. Applications of a new Ktheoretic theorem to soluble group rings. Proc. Amer. Math. Soc., 104(3):675– 684, 1988.

301. E. Laitinen and W. Lück. Equivariant Lefschetz classes. Osaka J. Math., 26(3):491–525, 1989.

302. T. Y. Lam. A first course in noncommutative rings. Springer-Verlag, New York, 1991.

303. C. Lance. On nuclear C^{∗}-algebras. J. Functional Analysis, 12:157–176, 1973.

304. B. Leeb. 3-manifolds with(out) metrics of nonpositive curvature. Invent. Math., 122(2):277–289, 1995.

305. H. W. Lenstra, Jr. Grothendieck groups of abelian group rings. J. Pure Appl. Algebra, 20(2):173–193, 1981.

306. J. Lewin. Fields of fractions for group algebras of free groups. Trans. Amer. Math. Soc., 192:339–346, 1974.

307. P. A. Linnell. Decomposition of augmentation ideals and relation modules. Proc. London Math. Soc. (3), 47(1):83–127, 1983.

308. P. A. Linnell. Zero divisors and L^{2}(G). C. R. Acad. Sci. Paris Sér. I Math., 315(1):49–53, 1992.

309. P. A. Linnell. Division rings and group von Neumann algebras. Forum Math., 5(6):561–576, 1993.

310. P. A. Linnell. Analytic versions of the zero divisor conjecture. In Geometry and cohomology in group theory (Durham, 1994), pages 209–248. Cambridge Univ. Press, Cambridge, 1998.

311. P. A. Linnell, W. Lück, and T. Schick. The Ore condition, affiliated operators, and the lamplighter group. Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 205 Münster, to appear in the Proceedings of the school/conference “High-dimensional manifold theory” in Trieste, May/June 2001, 2002.

312. P. A. Linnell and M. J. Puls. Zero divisors and L^{p}(G). II. New York J. Math., 7:49–58 (electronic), 2001.

313. P. A. Linnell and T. Schick. Finite group extensions and the Atiyah conjecture. Preliminary version.

314. N. Lohoue and S. Mehdi. The Novikov-Shubin invariants for locally symmetric spaces. J. Math. Pures Appl. (9), 79(2):111–140, 2000.

315. E. Looijenga. L^{2}-cohomology of locally symmetric varieties. Compositio Math., 67(1):3–20, 1988.

316. J. Lott. Heat kernels on covering spaces and topological invariants. J. Differential Geom., 35(2):471–510, 1992.

317. J. Lott. Equivariant analytic torsion for compact Lie group actions. J. Funct. Anal., 125(2):438–451, 1994.

318. J. Lott. The zero-in-the-spectrum question. Enseign. Math. (2), 42(3-4):341– 376, 1996.

319. J. Lott. L^{2}-cohomology of geometrically infinite hyperbolic 3-manifolds. Geom. Funct. Anal., 7(1):81–119, 1997.

320. J. Lott. Deficiencies of lattice subgroups of Lie groups. Bull. London Math. Soc., 31(2):191–195, 1999.

321. J. Lott. Delocalized L^{2}-invariants. J. Funct. Anal., 169(1):1–31, 1999.

322. J. Lott and W. Lück. L^{2}-topological invariants of 3-manifolds. Invent. Math., 120(1):15–60, 1995.

323. J. Lott and M. Rothenberg. Analytic torsion for group actions. J. Differential Geom., 34(2):431–481, 1991.

324. A. Lubotzky. Group presentation, p-adic analytic groups and lattices in SL_{2}(ℂ). Ann. of Math. (2), 118(1):115–130, 1983.

325. W. Lück. Eine allgemeine algebraische Beschreibung des Transfers für Faserungen auf projektiven Klassengruppen und Whiteheadgruppen. PhD thesis, Universität Göttingen, 1984.

326. W. Lück. Transformation groups and algebraic K-theory. Springer-Verlag, Berlin, 1989. Lecture Notes in Mathematics, Vol. 1408.

327. W. Lück. Analytic and topological torsion for manifolds with boundary and symmetry. J. Differential Geom., 37(2):263–322, 1993.

328. W. Lück. Approximating L^{2}-invariants by their finite-dimensional analogues. Geom. Funct. Anal., 4(4):455–481, 1994.

329. W. Lück. L^{2}-Betti numbers of mapping tori and groups. Topology, 33(2):203– 214, 1994.

330. W. Lück. L^{2}-torsion and 3-manifolds. In Low-dimensional topology (Knoxville, TN, 1992), pages 75–107. Internat. Press, Cambridge, MA, 1994.

331. W. Lück. Hilbert modules and modules over finite von Neumann algebras and applications to L^{2}-invariants. Math. Ann., 309(2):247–285, 1997.

332. W. Lück. L^{2}-Invarianten von Mannigfaltigkeiten und Gruppen. Jahresber. Deutsch. Math.-Verein., 99(3):101–109, 1997.

333. W. Lück. Dimension theory of arbitrary modules over finite von Neumann algebras and L^{2}-Betti numbers. I. Foundations. J. Reine Angew. Math., 495:135–162, 1998.

334. W. Lück. Dimension theory of arbitrary modules over finite von Neumann algebras and L^{2}-Betti numbers. II. Applications to Grothendieck groups, L^{2}- Euler characteristics and Burnside groups. J. Reine Angew. Math., 496:213– 236, 1998.

335. W. Lück. The universal functorial Lefschetz invariant. Fund. Math., 161(1-2):167–215, 1999. Algebraic topology (Kazimierz Dolny, 1997).

336. W. Lück. The type of the classifying space for a family of subgroups. J. Pure Appl. Algebra, 149(2):177–203, 2000.

337. W. Lück. A basic introduction to surgery theory. Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 197, Münster, to appear in the Proceedings of the summer school “High dimensional manifold theory” in Trieste May/June 2001, 2001.

338. W. Lück. L^{2}-invariants and their applications to geometry, group theory and spectral theory. In Mathematics Unlimited—2001 and Beyond, pages 859–871. Springer-Verlag, Berlin, 2001.

339. W. Lück. The relation between the Baum-Connes conjecture and the trace conjecture. Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 190, Münster, to appear in Invent. Math., 2001.

340. W. Lück. Chern characters for proper equivariant homology theories and applications to K- and L-theory. J. Reine Angew. Math., 543:193–234, 2002.

341. W. Lück. L^{2}-invariants of regular coverings of compact manifolds and CW - complexes. In Handbook of geometric topology, pages 735–817. Elsevier, Amsterdam, 2002.

342. W. Lück and D. Meintrup. On the universal space for group actions with compact isotropy. In Geometry and topology: Aarhus (1998), pages 293–305. Amer. Math. Soc., Providence, RI, 2000.

343. W. Lück, H. Reich, and T. Schick. Novikov-Shubin invariants for arbitrary group actions and their positivity. In Tel Aviv Topology Conference: Rothenberg Festschrift (1998), pages 159–176. Amer. Math. Soc., Providence, RI, 1999.

344. W. Lück and M. Rørdam. Algebraic K-theory of von Neumann algebras. K-Theory, 7(6):517–536, 1993.

345. W. Lück and M. Rothenberg. Reidemeister torsion and the K-theory of von Neumann algebras. K-Theory, 5(3):213–264, 1991.

346. W. Lück and T. Schick. L^{2}-torsion of hyperbolic manifolds of finite volume. Geometric and Functional Analysis, 9:518–567, 1999.

347. W. Lück and T. Schick. Approximating L^{2}-signatures by their finitedimensional analogues. Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 190, Münster, 2001.

348. W. Lück, T. Schick, and T. Thielmann. Torsion and fibrations. J. Reine Angew. Math., 498:1–33, 1998.

349. M. Lustig. On the rank, the deficiency and the homological dimension of groups: the computation of a lower bound via Fox ideals. In Topology and combinatorial group theory (Hanover, NH, 1986/1987; Enfield, NH, 1988), pages 164–174. Springer-Verlag, Berlin, 1990.

350. R. C. Lyndon and P. E. Schupp. Combinatorial group theory. Springer-Verlag, Berlin, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89.

351. S. MacLane. Categories for the working mathematician. Springer-Verlag, New York, 1971. Graduate Texts in Mathematics, Vol. 5.

352. I. Madsen. Reidemeister torsion, surgery invariants and spherical space forms. Proc. London Math. Soc. (3), 46(2):193–240, 1983.

353. I. Madsen and M. Rothenberg. On the classification of G-spheres. I. Equivariant transversality. Acta Math., 160(1-2):65–104, 1988.

354. W. Magnus. Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring. Math. Annalen, 111:259–280, 1935.

355. W. Magnus. Residually finite groups. Bull. Amer. Math. Soc., 75:305–316, 1969.

356. A. Malcev. On isomorphic matrix representations of infinite groups. Rec. Math. [Mat. Sbornik] N.S., 8 (50):405–422, 1940.

357. W. S. Massey. A basic course in algebraic topology. Springer-Verlag, New York, 1991.

358. V. Mathai. L^{2}-analytic torsion. J. Funct. Anal., 107(2):369–386, 1992.

359. V. Mathai. Spectral flow, eta invariants, and von Neumann algebras. J. Funct. Anal., 109(2):442–456, 1992.

360. V. Mathai. L^{2} invariants of covering spaces. In Geometric analysis and Lie theory in mathematics and physics, pages 209–242. Cambridge Univ. Press, Cambridge, 1998.

361. V. Mathai. K-theory of twisted group C^{∗}-algebras and positive scalar curvature. In Tel Aviv Topology Conference: Rothenberg Festschrift (1998), pages 203–225. Amer. Math. Soc., Providence, RI, 1999.

362. V. Mathai and M. Rothenberg. On the homotopy invariance of L^{2} torsion for covering spaces. Proc. Amer. Math. Soc., 126(3):887–897, 1998.

363. V. Mathai and M. A. Shubin. Twisted L^{2} invariants of non-simply connected manifolds and asymptotic L^{2} Morse inequalities. Russian J. Math. Phys., 4(4):499–526, 1996.

364. V. Mathai and S. G. Yates. Discrete Morse theory and extended L^{2} homology. J. Funct. Anal., 168(1):84–110, 1999.

365. T. Matumoto. On G-CW complexes and a theorem of J. H. C. Whitehead. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 18:363–374, 1971.

366. F. I. Mautner. The structure of the regular representation of certain discrete groups. Duke Math. J., 17:437–441, 1950.

367. R. Mazzeo and R. S. Phillips. Hodge theory on hyperbolic manifolds. Duke Math. J., 60(2):509–559, 1990.

368. C. T. McMullen. Iteration on Teichmüller space. Invent. Math., 99(2):425– 454, 1990.

369. C. T. McMullen. Renormalization and 3-manifolds which fiber over the circle. Princeton University Press, Princeton, NJ, 1996.

370. D. Meintrup. On the Type of the Universal Space for a Family of Subgroups. PhD thesis, Westfälische Wilhelms-Universität Münster, 2000.

371. R. J. Miatello. On the Plancherel measure for linear Lie groups of rank one. Manuscripta Math., 29(2-4):249–276, 1979.

372. J. Milnor. On the existence of a connection with curvature zero. Comment. Math. Helv., 32:215–223, 1958.

373. J. Milnor. Two complexes which are homeomorphic but combinatorially distinct. Ann. of Math. (2), 74:575–590, 1961.

374. J. Milnor. A duality theorem for Reidemeister torsion. Ann. of Math. (2), 76:137–147, 1962.

375. J. Milnor. Whitehead torsion. Bull. Amer. Math. Soc., 72:358–426, 1966.

376. J. Milnor. Infinite cyclic coverings. In Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), pages 115–133. Prindle, Weber & Schmidt, Boston, Mass., 1968.

377. J. Milnor. A note on curvature and fundamental group. J. Differential Geometry, 2:1–7, 1968.

378. J. Milnor. Introduction to algebraic K-theory. Princeton University Press, Princeton, N.J., 1971. Annals of Mathematics Studies, No. 72.

379. J. Milnor and J. D. Stasheff. Characteristic classes. Princeton University Press, Princeton, N. J., 1974. Annals of Mathematics Studies, No. 76.

380. A. S. Miščenko and A. T. Fomenko. The index of elliptic operators over C^{∗}- algebras. Izv. Akad. Nauk SSSR Ser. Mat., 43(4):831–859, 967, 1979. English translation in Math. USSR-Izv. 15 (1980), no. 1, 87–112.

381. E. E. Moise. Geometric topology in dimensions 2 and 3. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, Vol. 47.

382. B. Moishezon. On n-dimensional compact varieties with n algebraically independent meromorphic functions. AMS Translations, 63:51–177, 1967.

383. J. A. Moody. Induction theorems for infinite groups. Bull. Amer. Math. Soc. (N.S.), 17(1):113–116, 1987.

384. J. A. Moody. Brauer induction for G_{0} of certain infinite groups. J. Algebra, 122(1):1–14, 1989.

385. J. W. Morgan. On Thurston’s uniformization theorem for three-dimensional manifolds. In The Smith conjecture (New York, 1979), pages 37–125. Academic Press, Orlando, FL, 1984.

386. H. Moscovici. L^{2}-index of elliptic operators on locally symmetric spaces of finite volume. In Operator algebras and K-theory (San Francisco, Calif., 1981), pages 129–137. Amer. Math. Soc., Providence, R.I., 1982.

387. H. Moscovici and R. J. Stanton. R-torsion and zeta functions for locally symmetric manifolds. Invent. Math., 105(1):185–216, 1991.

388. G. D. Mostow. Strong rigidity of locally symmetric spaces. Princeton University Press, Princeton, N.J., 1973. Annals of Mathematics Studies, No. 78.

389. A. W. Mostowski. On the decidability of some problems in special classes of groups. Fund. Math., 59:123–135, 1966.

390. W. Müller. Analytic torsion and R-torsion of Riemannian manifolds. Adv. in Math., 28(3):233–305, 1978.

391. W. Müller. Manifolds with cusps of rank one. Springer-Verlag, Berlin, 1987. Spectral theory and L^{2}-index theorem.

392. W. Müller. L^{2}-index theory, eta invariants and values of L-functions. In Geometric and topological invariants of elliptic operators (Brunswick, ME, 1988), pages 145–189. Amer. Math. Soc., Providence, RI, 1990.

393. W. Müller. Analytic torsion and R-torsion for unimodular representations. J. Amer. Math. Soc., 6(3):721–753, 1993.

394. W. Müller. On the L^{2}-index of Dirac operators on manifolds with corners of codimension two. I. J. Differential Geom., 44(1):97–177, 1996.

395. H. Murakami and J. Murakami. The colored Jones polynomials and the simplicial volume of a knot. Acta Math., 186(1):85–104, 2001.

396. F. Murray and J. von Neumann. On rings of operators. Annals of Math., 37:116–229, 1936.

397. M. Nagase. L^{2}-cohomology and intersection homology of stratified spaces. Duke Math. J., 50(1):329–368, 1983.

398. M. Nagase. Remarks on the L^{2}-cohomology of singular algebraic surfaces. J. Math. Soc. Japan, 41(1):97–116, 1989.

399. H. Neumann. Varieties of groups. Springer-Verlag New York, Inc., New York, 1967.

400. S. P. Novikov and M. A. Shubin. Morse inequalities and von Neumann II_{1}- factors. Dokl. Akad. Nauk SSSR, 289(2):289–292, 1986.

401. S. P. Novikov and M. A. Shubin. Morse inequalities and von Neumann invariants of non-simply connected manifolds. Uspekhi. Matem. Nauk, 41(5):222– 223, 1986. in Russian.

402. T. Ohsawa. On the L^{2} cohomology of complex spaces. Math. Z., 209(4):519– 530, 1992.

403. T. Ohsawa. On the L^{2} cohomology groups of isolated singularities. In Progress in differential geometry, pages 247–263. Math. Soc. Japan, Tokyo, 1993.

404. M. Olbrich. L^{2}-invariants of locally symmetric spaces. Preprint, Göttingen, 2000.

405. A. Ol′ shanskii. Groups of bounded period with subgroups of prime order. Algebra und Logik, 21:369–418, 1982.

406. R. Oliver. Whitehead groups of finite groups. Cambridge University Press, Cambridge, 1988.

407. A. Y. Ol′shanskii and M. V. Sapir. Non-amenable finitely presented torsionby-cyclic groups. Electron. Res. Announc. Amer. Math. Soc., 7:63–71 (electronic), 2001.

408. D. S. Ornstein and B. Weiss. Ergodic theory of amenable group actions. I. The Rohlin lemma. Bull. Amer. Math. Soc. (N.S.), 2(1):161–164, 1980.

409. P. Pansu. Cohomologie L^{p} des variétés à courbure négative, cas du degré 1. Number Special Issue, pages 95–120 (1990), 1989. Conference on Partial Differential Equations and Geometry (Torino, 1988).

410. P. Pansu. Cohomologie L^{p}: Invariance sous quasiisometries. Preprint, Orsay, 1995.

411. P. Pansu. Introduction to L^{2} Betti numbers. In Riemannian geometry (Waterloo, ON, 1993), pages 53–86. Amer. Math. Soc., Providence, RI, 1996.

412. P. Pansu. Volume, courbure et entropie (d’après G. Besson, G. Courtois et S. Gallot). Astérisque, (245):Exp. No. 823, 3, 83–103, 1997. Séminaire Bourbaki, Vol. 1996/97.

413. P. Pansu. Cohomologie L^{p}, espaces homogènes et pinchement. Preprint, Orsay, 1999.

414. W. L. Pardon. The L_{2}-\overline{∂}-cohomology of an algebraic surface. Topology, 28(2):171–195, 1989.

415. W. L. Pardon and M. Stern. L^{2}–\overline{∂}-cohomology of complex projective varieties. J. Amer. Math. Soc., 4(3):603–621, 1991.

416. W. L. Paschke. An invariant for finitely presented cG-modules. Math. Ann., 301(2):325–337, 1995.

417. D. S. Passman. Semiprime and prime crossed products. J. Algebra, 83(1):158– 178, 1983.

418. D. S. Passman. Infinite crossed products. Academic Press Inc., Boston, MA, 1989.

419. A. L. T. Paterson. Amenability. American Mathematical Society, Providence, RI, 1988.

420. G. K. Pedersen. C^{∗}-algebras and their automorphism groups. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1979.

421. G. K. Pedersen. Analysis now. Springer-Verlag, New York, 1989.

422. M. J. Puls. Zero divisors and L^{p}(G). Proc. Amer. Math. Soc., 126(3):721–728, 1998.

423. D. Quillen. Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl., 19:31–35, 1986.

424. F. Quinn. Ends of maps. I. Ann. of Math. (2), 110(2):275–331, 1979.

425. F. Quinn. Ends of maps. II. Invent. Math., 68(3):353–424, 1982.

426. F. Quinn. Ends of maps. III. Dimensions 4 and 5. J. Differential Geom., 17(3):503–521, 1982.

427. M. Ramachandran. von Neumann index theorems for manifolds with boundary. J. Differential Geom., 38(2):315–349, 1993.

428. A. A. Ranicki. On the algebraic L-theory of semisimple rings. J. Algebra, 50(1):242–243, 1978.

429. A. A. Ranicki. The algebraic theory of surgery. I. Foundations. Proc. London Math. Soc. (3), 40(1):87–192, 1980.

430. A. A. Ranicki. Exact sequences in the algebraic theory of surgery. Princeton University Press, Princeton, N.J., 1981.

431. A. A. Ranicki. On the Hauptvermutung. In The Hauptvermutung book, pages 3–31. Kluwer Acad. Publ., Dordrecht, 1996.

432. D. B. Ray and I. M. Singer. R-torsion and the Laplacian on Riemannian manifolds. Advances in Math., 7:145–210, 1971.

433. D. B. Ray and I. M. Singer. Analytic torsion for complex manifolds. Ann. of Math. (2), 98:154–177, 1973.

434. M. Reed and B. Simon. Methods of modern mathematical physics. I. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, second edition, 1980. Functional analysis.

435. H. Reich. Group von Neumann algebras and related algebras. PhD thesis, Universität Göttingen, 1999. http://www.math.unimuenster.de/u/lueck/publ/diplome/reich.dvi.

436. K. Reidemeister. Homotopieringe und Linsenräume. Hamburger Abhandlungen 11, 1938.

437. J. Roe. An index theorem on open manifolds. I. J. Differential Geom., 27(1):87–113, 1988.

438. J. Roe. An index theorem on open manifolds. II. J. Differential Geom., 27(1):115–136, 1988.

439. J. Roe. Coarse cohomology and index theory on complete Riemannian manifolds. Mem. Amer. Math. Soc., 104(497):x+90, 1993.

440. J. Roe. Index theory, coarse geometry, and topology of manifolds. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 441. J. Rosenberg. Algebraic K-theory and its applications. Springer-Verlag, New

York, 1994.

442. J. Rosenberg. Analytic Novikov for topologists. In Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), pages 338–372. Cambridge Univ. Press, Cambridge, 1995.

443. M. Rothenberg. Torsion invariants and finite transformation groups. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pages 267–311. Amer. Math. Soc., Providence, R.I., 1978.

444. J. J. Rotman. An introduction to homological algebra. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1979.

445. C. P. Rourke and B. J. Sanderson. Introduction to piecewise-linear topology. Springer-Verlag, Berlin, 1982. Reprint.

446. L. H. Rowen. Ring theory. Vol. I. Academic Press Inc., Boston, MA, 1988.

447. L. H. Rowen. Ring theory. Vol. II. Academic Press Inc., Boston, MA, 1988.

448. L. H. Rowen. Ring theory. Academic Press Inc., Boston, MA, student edition, 1991.

449. R. Roy. The trace conjecture—a counterexample. K-Theory, 17(3):209–213, 1999.

450. M. Rumin. Sub-Riemannian limit of the differential form spectrum of contact manifolds. Geom. Funct. Anal., 10(2):407–452, 2000.

451. L. Saper. L_{2}-cohomology and intersection homology of certain algebraic varieties with isolated singularities. Invent. Math., 82(2):207–255, 1985.

452. L. Saper. L_{2}-cohomology of algebraic varieties. In Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pages 735–746, Tokyo, 1991. Math. Soc. Japan.

453. L. Saper. L_{2}-cohomology of Kähler varieties with isolated singularities. J. Differential Geom., 36(1):89–161, 1992.

454. L. Saper and M. Stern. L_{2}-cohomology of arithmetic varieties. Ann. of Math. (2), 132(1):1–69, 1990.

455. L. Saper and S. Zucker. An introduction to L^{2}-cohomology. In Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), pages 519– 534. Amer. Math. Soc., Providence, RI, 1991.

456. R. Sauer. Invariance properties of l^{2}-betti numbers and novikov-shubin invariants under orbit equivalence and quasi-isometry. PhD thesis, in preparation, 2002.

457. R. P. Savage, Jr. The space of positive definite matrices and Gromov’s invariant. Trans. Amer. Math. Soc., 274(1):239–263, 1982.

458. J. A. Schafer. The Bass conjecture and group von Neumann algebras. K-Theory, 19(3):211–217, 2000.

459. T. Schick. Analysis on ∂-manifolds of bounded geometry, Hodge-de Rham isomorphism and L^{2}-index theorem. Shaker Verlag, Aachen, 1996. PhD thesis, Johannes Gutenberg-Universität Mainz.

460. T. Schick. L^{2}-index theorem for boundary manifolds. Preprint 1998, to appear in Pacific J. Math., 1998.

461. T. Schick. Integrality of L^{2}-Betti numbers. Math. Ann., 317(4):727–750, 2000.

462. T. Schick. L^{2}-determinant class and approximation of L^{2}-Betti numbers. Trans. Amer. Math. Soc., 353(8):3247–3265 (electronic), 2001.

463. T. Schick. Erratum: “Integrality of L^{2}-Betti numbers” [Math. Ann. 317 (2000), no. 4, 727–750; 1777117]. Math. Ann., 322(2):421–422, 2002.

464. W. Schmid. L^{2}-cohomology and the discrete series. Ann. of Math. (2), 103(2):375–394, 1976.

465. A. H. Schofield. Representation of rings over skew fields. Cambridge University Press, Cambridge, 1985.

466. P. Scott. The geometries of 3-manifolds. Bull. London Math. Soc., 15(5):401– 487, 1983.

467. R. T. Seeley. Extension of C^{∞} functions defined in a half space. Proc. Amer. Math. Soc., 15:625–626, 1964.

468. R. T. Seeley. Complex powers of an elliptic operator. In Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pages 288–307. Amer. Math. Soc., Providence, R.I., 1967.

469. V. Sergiescu. A quick introduction to Burnside’s problem. In Group theory from a geometrical viewpoint (Trieste, 1990), pages 622–629. World Sci. Publishing, River Edge, NJ, 1991.

470. J.-P. Serre. Linear representations of finite groups. Springer-Verlag, New York, 1977. Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42.

471. J.-P. Serre. Trees. Springer-Verlag, Berlin, 1980. Translated from the French by J. Stillwell.

472. T. N. Shorey and R. Tijdeman. Exponential Diophantine equations. Cambridge University Press, Cambridge, 1986.

473. M. Shubin. Pseusodifferential operators and spectral theory. Springer, 1987.

474. M. A. Shubin. A Lefschetz fixed point formula in reduced L^{2}-cohomologies on manifolds with cylinders. In Symposium “Analysis on Manifolds with Singularities” (Breitenbrunn, 1990), pages 290–297. Teubner, Stuttgart, 1992.

475. M. A. Shubin. Semiclassical asymptotics on covering manifolds and Morse inequalities. Geom. Funct. Anal., 6(2):370–409, 1996.

476. I. M. Singer. Infinitesimally homogeneous spaces. Comm. Pure Appl. Math., 13:685–697, 1960.

477. I. M. Singer. Some remarks on operator theory and index theory. In K-theory and operator algebras (Proc. Conf., Univ. Georgia, Athens, Ga., 1975), pages 128–138. Lecture Notes in Math., Vol. 575. Springer-Verlag, Berlin, 1977.

478. J. Smillie. An obstruction to the existence of affine structures. Invent. Math., 64(3):411–415, 1981.

479. T. Soma. The Gromov invariant of links. Invent. Math., 64(3):445–454, 1981.

480. T. Soma. Bounded cohomology of closed surfaces. Topology, 36(6):1221–1246, 1997.

481. J. Stallings. Whitehead torsion of free products. Ann. of Math. (2), 82:354– 363, 1965.

482. N. E. Steenrod. A convenient category of topological spaces. Michigan Math. J., 14:133–152, 1967.

483. M. Steinberger and J. E. West. Equivariant h-cobordisms and finiteness obstructions. Bull. Amer. Math. Soc. (N.S.), 12(2):217–220, 1985.

484. B. Stenström. Rings of quotients. Springer-Verlag, New York, 1975. Die Grundlehren der Mathematischen Wissenschaften, Band 217, An introduction to methods of ring theory.

485. M. Stern. L^{2}-index theorems on locally symmetric spaces. Invent. Math., 96(2):231–282, 1989.

486. M. Stern. Index theory for certain complete Kähler manifolds. J. Differential Geom., 37(3):467–503, 1993.

487. D. Sullivan. A generalization of Milnor’s inequality concerning affine foliations and affine manifolds. Comment. Math. Helv., 51(2):183–189, 1976.

488. R. M. Switzer. Algebraic topology—homotopy and homology. Springer-Verlag, New York, 1975. Die Grundlehren der mathematischen Wissenschaften, Band 489. D. Tamari. A refined classification of semi-groups leading to generalized

polynomial rings with a generalized degree concept. In J. Gerretsen and J. de Groot, editors, Proceedings of the ICM, Amsterdam, 1954, volume 3, pages 439–440, Groningen, 1957.

490. E. Thoma. Über unitäre Darstellungen abzählbarer, diskreter Gruppen. Math. Ann., 153:111–138, 1964.

491. W. P. Thurston. Geometry and topology of 3-manifolds. Lecture notes, Princeton. http://www.msri.org/publications/books/gt3m, 1978.

492. W. P. Thurston. Three-dimensional geometry and topology. Vol. 1. Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy.

493. T. tom Dieck. Orbittypen und äquivariante Homologie. I. Arch. Math. (Basel), 23:307–317, 1972.

494. T. tom Dieck. Transformation groups and representation theory. Springer-Verlag, Berlin, 1979.

495. T. tom Dieck. Transformation groups. Walter de Gruyter & Co., Berlin, 1987.

496. H. Triebel. Höhere Analysis. VEB Deutscher Verlag der Wissenschaften, Berlin, 1972. Hochschulbücher für Mathematik, Band 76.

497. V. G. Turaev. Reidemeister torsion in knot theory. Uspekhi Mat. Nauk, 41(1(247)):97–147, 240, 1986. English translation in Russian Math. Surveys 41 (1986), no. 1, 119–182.

498. V. G. Turaev. Torsion invariants of spin^{c}-structures on 3-manifolds. Math. Res. Lett., 4(5):679–695, 1997.

499. C. Vafa and E. Witten. Eigenvalue inequalities for fermions in gauge theories. Comm. Math. Phys., 95(3):257–276, 1984.

500. B. Vaillant. Indextheorie für Überlagerungen. Diplomarbeit, Universität Bonn, http://styx.math.uni-bonn.de/boris/diplom.html, 1997.

501. A. Valette. Introduction to the Baum-Connes conjecture. To appear as ETHZ lecture notes, 2001.

502. N. T. Varopoulos. Random walks and Brownian motion on manifolds. In Symposia Mathematica, Vol. XXIX (Cortona, 1984), pages 97–109. Academic Press, New York, 1987.

503. S. M. Vishik. Analytic torsion of boundary value problems. Dokl. Akad. Nauk SSSR, 295(6):1293–1298, 1987.

504. S. M. Vishik. Analytic torsion of finite-dimensional local systems. Dokl. Akad. Nauk SSSR, 300(6):1295–1299, 1988.

505. S. M. Vishik. Generalized Ray-Singer conjecture. I. A manifold with a smooth boundary. Comm. Math. Phys., 167(1):1–102, 1995.

506. J. B. Wagoner. Diffeomorphisms, K_{2}, and analytic torsion. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pages 23–33. Amer. Math. Soc., Providence, R.I., 1978.

507. F. Waldhausen. On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2), 87:56–88, 1968.

508. F. Waldhausen. Algebraic K-theory of generalized free products. I, II. Ann. of Math. (2), 108(1):135–204, 1978.

509. C. T. C. Wall. Rational Euler characteristics. Proc. Cambridge Philos. Soc., 57:182–184, 1961.

510. C. T. C. Wall. Surgery on compact manifolds. Academic Press, London, 1970. London Mathematical Society Monographs, No. 1.

511. C. T. C. Wall. Norms of units in group rings. Proc. London Math. Soc. (3), 29:593–632, 1974.

512. C. T. C. Wall. Geometric structures on compact complex analytic surfaces. Topology, 25(2):119–153, 1986.

513. D. L. Webb. Higher G-theory of nilpotent group rings. J. Algebra, 116(2):457– 465, 1988.

514. N. E. Wegge-Olsen. K-theory and C^{∗}-algebras. The Clarendon Press Oxford University Press, New York, 1993. A friendly approach.

515. C. Wegner. L^{2}-invariants of finite aspherical CW -complexes with fundamental group containing a non-trivial elementary amenable normal subgroup. PhD thesis, Westfälische Wilhelms-Universität Münster, 2000.

516. C. Wegner. L^{2}-invariants of finite aspherical CW -complexes. Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 152, Münster, 2001.

517. B. A. F. Wehrfritz. Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices. Springer-Verlag, New York, 1973. Ergebnisse der Matematik und ihrer Grenzgebiete, Band 76.

518. C. A. Weibel. An introduction to homological algebra. Cambridge University Press, Cambridge, 1994.

519. S. Weinberger. The topological classification of stratified spaces. University of Chicago Press, Chicago, IL, 1994.

520. R. O. Wells, Jr. Differential analysis on complex manifolds. Springer-Verlag, New York, second edition, 1980.

521. G. W. Whitehead. Elements of homotopy theory. Springer-Verlag, New York, 1978.

522. H. Whitney. Geometric integration theory. Princeton University Press, Princeton, N. J., 1957.

523. K. Whyte. Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture. Duke Math. J., 99(1):93–112, 1999.

524. W. Woess. Random walks on infinite graphs and groups—a survey on selected topics. Bull. London Math. Soc., 26(1):1–60, 1994.

525. J. A. Wolf. Spaces of constant curvature. Publish or Perish Inc., Houston, TX, fifth edition, 1984.

526. K. Yano. Gromov invariant and S^{1}-actions. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 29(3):493–501, 1982.

527. S.-K. Yeung. Betti numbers on a tower of coverings. Duke Math. J., 73(1):201– 226, 1994.

528. G. Yu. The Novikov conjecture for groups with finite asymptotic dimension. Ann. of Math. (2), 147(2):325–355, 1998.

529. A. E. Zalesskii. On a problem of Kaplansky. Soviet. Math., 13:449–452, 1972.

530. S. Zucker. L_{2} cohomology of warped products and arithmetic groups. Invent. Math., 70(2):169–218, 1982/83.

531. S. Zucker. L_{2}-cohomology and intersection homology of locally symmetric varieties. In Singularities, Part 2 (Arcata, Calif., 1981), pages 675–680. Amer. Math. Soc., Providence, RI, 1983.

532. S. Zucker. L_{2}-cohomology and intersection homology of locally symmetric varieties. II. Compositio Math., 59(3):339–398, 1986.

533. S. Zucker. L^{2}-cohomology and intersection homology of locally symmetric varieties. III. Astérisque, (179-180):11, 245–278, 1989. Actes du Colloque de Théorie de Hodge (Luminy, 1987).

534. S. Zucker. L^{2}-cohomology of Shimura varieties. In Automorphic forms, Shimura varieties, and L-functions, Vol. II (Ann Arbor, MI, 1988), pages 377–391. Academic Press, Boston, MA, 1990.

535. S. Zucker. L^{p}-cohomology: Banach spaces and homological methods on Riemannian manifolds. In Differential geometry: geometry in mathematical physics and related topics (Los Angeles, CA, 1990), pages 637–655. Amer. Math. Soc., Providence, RI, 1993.

Notation

B_{K}(A), 460 b_{p}(X), 13 b^{(2)}(f), 73 b_{p}^{(2)}(C_{∗}), 24 b_{p}^{(2)}(G), 41, 263 b_{p}^{(2)}(X; N (G)), 34, 263 C_{r}^{∗}(G), 339 C^{∞}(M), 49 C^{∞}(ℂ ⊗_{ℝ} T M), 49 C_{∗}^{sing}(M), 50 C_{sing}^{∗}(M, ℂ), 50 C_{∗}^{sing,C^{∞}}(M), 50 C_{sing,C^{∞}}^{∗}(M; ℂ), 50 C_{∗}(X), 32 C_{∗}^{(2)}(X), 33 C_{(2)}^{∗}(X), 33

Ĉ^{∗}(X), 490 C_{∗} ⊗ D_{∗}, 28 ch^{G}, 282 ch_{K}^{G}, 282 class(G), 358 class(G)_{cf}, 358 class(G)_{f}, 358 class(G)_{f,cf}, 358 class_{0}(G), 357 class_{0}(G)_{cf}, 357 class_{0}(G)_{f}, 357 class_{0}(G)_{f,cf}, 357 clos, 16 coker, 286 colim_{i∈I} M_{i}, 243 con(G), 357 con(G)_{cf}, 357 con(G)_{f}, 357 cone_{∗}(f_{∗}), 35 consub(H), 281 cyl_{∗}(f_{∗}), 34 D_{max}, 57 def(G), 299 deg(f), 446 det, 127 dF, 127 dim, 242 dim′, 239 dim_{N (G)}, 21, 238, 246 dim_{U(G)}, 329, 330 dom(T), 55 d^{p}, 49 E(G, F), 33 E(G, FIN), 33 e^{−tΔ_{p}}(x, y), 52 E_{λ}^{Δ_{p}}(x, y), 104 F, 297, 372, 410 F_{g}, 300, 422 F_{g}^{d}, 40 𝔽_{p}, 300 F ≃ G, 75 F_{p}(C_{∗}), 81 F_{p}^{Δ}(C_{∗}), 81 F_{p}(M), 104 F_{p}^{Δ}(M), 104 F_{p}(X; N (G)), 97 F^{⊥}, 87 {fin. gen. Hilb. N (G)-mod.}, 247 {fin. gen. Hilb. N (G)-mod.}_{U}, 321 {fin. gen. proj. N (G)-mod. with ⟨ ⟩}, 247 {fin. pres. N (G)-mod.}, 288 full(K), 108 f-rk(M), 226 G_{0}(R), 341 GL(R), 120 (g), 335 ⟨g⟩, 360 H^{k}Ω^{p}(M), 59 H_{dR}^{p}(M), 50 H_{p}^{G}(X; N (G)), 263 H_{p}^{G}(X; U(G)), 331 H_{p}^{lf}(X; ℤ), 491 H_{sing}^{∗}(M; ℂ), 50 H_{p}^{(2)}(C_{∗}), 24 H_{(2)}^{p}(M), 58

H_{p}^{(2)}(X; N (G)), 34 H_{(2)}^{p}(X; N (G)), 34 H_{(2),unr}^{p}(M), 58

Ĥ^{∗}(G), 490 Ĥ^{p}(X), 490 HS(P), 358 ℍ^{n}, 53 h^{(2)}(G), 277 h^{(2)}(X; N (G)), 277 h(G), 496 h_{vol}(M), 496 i_{∗}M, 29, 253 im, 16 Isom(M), 224 I(X), 277 K_{0}(R), 341 K_{1}(R), 121, 343 K_{1}^{inj}(R), 343 K_{1}^{W}(N (G)), 203 (K), 282 (K), 389 (K′), 384 ker, 16 L^{dr}(l^{2}(G)^{n}), 329 L^{dr}(R^{n}), 329 L^{0}, 224 L_{n}^{ϵ}(R), 347 L_{ϵ}^{n}(R), 347 L^{∞}(T^{n}), 15 L^{2}(T^{n}), 15 L^{2}Ω^{p}(M), 51 LHS_{∗}(C_{∗}, D_{∗}, E_{∗}), 141 lim_{i∈I} M_{i}, 243 lim inf_{λ→0+}, 75 lim inf_{i∈I} x_{i}, 464 lim sup_{i∈I} x_{i}, 464 l^{2}C^{∗}(K), 61 l^{2}(G), 14 {l^{2}(G)^{n}}, 248 {l^{2}(G)^{n}}_{U}, 322 ||M||, 490 ||[M]_{ℤ}||_{Δ}^{F}, 499 [M], 45 M_{cp}, 226 M_{Eucl}, 226 M_{ncp}, 226 M(K), 217 M(m, n, N (G)), 173 M_{n}(ℤG), 163 mesh(K), 108 minvol(M), 495 m(X; G), 277 NH, 31 N(K), 217 NZD(R), 402 Or(G), 357 Or(G, FIN), 357 PM, 237 p ∼ q, 337 p ≤ q, 337 (R), 389 (R′), 384 R ∗ G, 398 R[t, t^{−1}]_{ϕ}, 398 resolv(T), 56 rk_{ℂ}(G), 226 rk_{ℂ}(𝔤), 226 SO(p, q), 227 SO(p, q)^{0}, 227 SO(n), 227 sec(M), 226 spec(T), 56 st(σ), 61 T_{f}, 42 T M, 49 T_{min}, 55 TM, 237 T_{dim}M, 331 T_{U}M, 331 T ⊂ S, 55 T = S, 55 T (G), 389 T (R ⊂ S), 387 tr_{ℂG}(A), 194 tr_{ℂG}^{u}, 358 tr_{N (G)}, 15 v_{n}, 493 WH, 31 Wh(G), 121 X[∞], 265 ||y||_{1}, 490 α_{p}(C_{∗}), 81 α_{p}^{Δ}(C_{∗}), 81 α_{p}(X; N (G)), 97 α_{p}^{Δ}(X; N (G)), 97 Δ(K), 218 Δ_{p}, 24, 50, 51 δ^{p}, 51 δ_{r}, 87 θ_{F} (t), 181 θ_{p}(M)(t), 179 Λ(G, C_{r}^{∗})_{fgp}, 413 Λ(G, F)_{all}, 374 Λ(G, F)_{fg}, 374 Λ(G, F)_{fgp}, 374 Λ(G, F)_{fp}, 374 ρ(α), 408 ρ^{(2)}(f: G → G), 305 ρ_{virt}^{(2)}(f: G → G), 310 \overline{ρ}^{(2)}(f: G → G), 311 ρ_{an}(M; V), 126 ρ_{an}(M, ∂_{0}M; V), 190 ρ_{an}^{(2)}(M), 179 ρ_{an}^{(2)}(M, ∂_{0}M), 190 ρ_{top}(M; V), 125 ρ_{top}(M, ∂_{0}M; V), 190

ρ_{top}^{(2)}(M), 177
ρ_{top}^{(2)}(M, ∂_{0}M), 190
ρ(X; V), 123
ρ^{(2)}(X; N (G)), 160
ρ^{(2)}(X̃), 163
ΣC_{∗}, 35
Σ(G), 389
Σ(R ⊂ S), 387
χ^{G}(X), 281
χ′(G), 380
χ^{(2)}(G), 277
χ^{(2)}(X; N (G)), 277
χ_{virt}(X), 280
||ψ||_{∞}, 490
⟨ψ, y⟩, 491
Ω^{p}(M), 49
Ω^{p,q}(M), 429
Ω_{c}^{p}(M), 51
Ω_{d}^{p}(M, ∂_{0}M), 63
Ω_{δ}^{p}(M, ∂_{0}M), 63
Ω_{1}^{p}(M, ∂_{0}M), 63
Ω_{2}^{p}(M, ∂_{0}M), 63
ω^{p} ∧ η^{q}, 49
A, 336
AM, 256
B_{d}, 294
B(H), 15
C, 380
D, 381
D(G), 389
D(R ⊂ S), 387
EAM, 256
FIN (G), 371
G, 461
H_{(2)}^{p}(M), 51
H_{(2)}^{p,q}(M), 430
K(H), 394
L(f, λ), 73
L^{0}(H), 394
L^{1}(H), 394
L^{p}(H), 394
N (G), 15
{N (G)^{n}}, 247
R(G), 389

R(R ⊂ S), 388 ; U, 341 ; U(G), 322 ; \{U(G)^{n}\}, 322 ; Z_{d}, 452 ; Z(A), 337 ; 𝔤, 225  𝔨, 225 ; 𝔭, 225 ; \frac{1}{|FIN (G)|}ℤ, 371 ; ∗^{p}, 50 ; ∂, 429 ; \overline{∂}, 429 ; ℜ(s), 178

Index

additive invariant 281

– universal 281 adjoint of the exterior differential 50 adjoint representation 224 affiliated operator 318 affine structure on a manifold 503 algebra

– ∗-algebra 392 algebra with involution 392 amenable discrete homogeneous space

456 amenable exhaustion 469 analytic

– Betti number 52

– L^{2}-Betti number 53

– L^{2}-torsion 179

– Novikov-Shubin invariant 104

– Reidemeister torsion 126

– spectral density function 103 anti-homomorphism of ℂ-algebras 339 aspherical 45 associated matrices 388 asymptotic dimension 444

– finite asymptotic dimension 443 atoroidal

– geometrically 213

– homotopically 214

B-representation 392 based free chain complex 121 Betti number

– analytic 52

– cellular 13 boundary conditions

– Dirichlet boundary conditions 63 – Neumann boundary conditions 63 boundary operator in the long weakly

exact homology sequence 25 bounded cohomology 486 bounded form 422 Burnside group 281 Burnside integrality conditions 283 Burnside ring 281

C^{∗}-algebra

– reduced C^{∗}-algebra of a group 339 Cartan decomposition 224 Cartan involution 224 Cartan subalgebra

– θ-stable Cartan subalgebra 225 category

– ℂ-category 246

– of compactly generated spaces 31 Cayley graph 93 Cayley transform 320 cell

– equivariant closed n-dimensional cell 31

– equivariant open n-dimensional cell 31 cellular

– cochain complex 60

– L^{2}-Betti number 34

– L^{2}-chain complex 33

– L^{2}-cochain complex 33

– L^{2}-cohomology 34

– L^{2}-homology 34

– L^{2}-torsion 160

– map 32

– Novikov-Shubin invariant 96

– permutation structure 32

– – uniqueness 32

– spectral density function 96

– ℤG-basis 32

– ℤG-chain complex 32 center of a ring 337 central carrier of a projection 337 chain complex see Hilbert chain

complex chain contraction 82

– weak 145

characteristic sequence 194 classifying space for a family of

subgroups 33 classifying space for proper G-spaces

33 closed under

– directed unions 256

– extensions 256 closure of a submodule 237 cochain complex of square-summable

cochains 60 cocompact 32 colimit of a directed system 242 colimit topology 31 commensurable groups 390 commutant 336

– double commutant 336 complete orthonormal system see

Hilbert basis complete Riemannian manifold 51 Conjecture

– Amenability and dimension-flatness of N (G) over ℂG 262

– Amenability and the regular representation in G-theory 364

– Approximation Conjecture 453 – Atiyah Conjecture 370

– – strong 369

– Bass Conjecture 358

– – Strong Bass Conjecture for ℂG 358

– – Strong Bass Conjecture for ℤG 358

– – Weak Bass Conjecture 359

– Baum-Connes Conjecture 357

– Bost Conjecture 360

– Determinant Conjecture 454

– Flatness of N (G) over ℂG 263

– Gromov’s Conjecture about positivity of the simplicial volume for non-negative sectional and negative Ricci curvature 501

– Gromovs’s Conjecture on affine structures and L^{2}-Betti numbers 504

– Homotopy invariance of L^{2}-torsion 163

– Hopf Conjecture 417

– Isomorphism Conjecture for K_{0}(ℂG) 357

– Kadison Conjecture 391

– Kaplansky Conjecture 376

– Kneser’s Conjecture 212

– L^{2}-torsion for aspherical manifolds 418

– Poincaré Conjecture 123 – Positivity of Novikov-Shubin

invariants 112

– Positivity of the simplicial volume of locally symmetric spaces of non-compact type 501

– Simplicial volume and L^{2}-invariants 485

– Singer Conjecture 417

– Singer Conjecture for contractible proper cocompact Poincaré G-CW - complexes 432

– Strong Novikov Conjecture 443 – Sullivan’s Conjecture on affine

structures and Euler characteristic 503

– Thurston’s Geometrization Conjecture 213

– Volume Conjecture 218

– Waldhausen’s Conjecture 214

– Zero-in-the-spectrum Conjecture 437

connected

– d-connected 39 convergence

– strong 18

– ultra-weak 18

– weak 18 core 73 crossed product 396

– homomorphism of 397

– universal property 397

d(bounded) form 422 de Rham cochain complex 49 de Rham cohomology 49 deficiency 299 degree of a (proper) map 442 density function 75 det ≥ 1-class 172 det-L^{2}-acyclic

– G-CW -complex 160

– Hilbert chain complex 140 determinant class

– chain map of 141

– Hilbert chain complex of 140

– morphism of Hilbert modules of 127

– weak exact sequence of chain complexes of 141

– weak exact sequence of modules of 141

differential

– exterior 49 dilatationally equivalent 75 dimension

– extended dimension 239

– extended von Neumann dimension of an arbitrary module over N (G) 246

– extended von Neumann dimension of an arbitrary module over U(G) 330

– of a finitely generated projective N (G)-module 238

– of a finitely generated projective U(G)-module 329

– von Neumann dimension of a Hilbert N (G)-module 21

direct limit see colimit of a directed

system directed set 17 directed system 242 Dirichlet boundary conditions 63 division closure 385 domain of an operator 54 dual

– of a Hilbert chain complex 82 d̃(bounded) form 422

Eilenberg-MacLane space of type (π, 1)

45 entropy

– (volume) entropy of a closed connected Riemannian manifold 492

– of a group 492 equivalence of ℂ-categories (with

involution) 247 equivariant smooth triangulation 32,

52 essentially dense 318 Euler characteristic

– L^{2}-Euler characteristic 277

– equivariant Euler characteristic 281 – virtual Euler characteristic 280 exceptional prime 3-manifold 214 extended dimension 239 extended von Neumann dimension of

an arbitrary module over N (G) 246 extended von Neumann dimension of

an arbitrary module over U(G) 330 extension

– minimal closed 55

– of an operator 54 exterior differential 49

Følner exhaustion 476 factor see von Neumann algebra faithfully flat functor 253 family of subgroups 33 fibration 43

– simple 165 field of fractions 407

– universal 407 flag complex 433 flat Riemannian manifold 226 Følner condition 257 formal adjoint of a differential operator

57 Fredholm

– at p 26

– density function 75

– Hilbert chain complex 26

– morphism 26, 73 Fredholm module

– p-summable (A, B)-Fredholm module 392

Fuglede-Kadison determinant 127 fullness of a triangulation 108 function

– essentially bounded 15

– Γ -function 178 functor

– of ℂ-categories (with involution) 247

fundamental domain 53 fundamental form 422 fundamental rank of a symmetric space

226

G-CW -complex 31

– det-L^{2}-acyclic 160

– finite 32

– finite dimensional 32

– of dimension ≤ n 32

– of finite type 32 G-invariant subordinate smooth

partition 61 Γ -function 178 geodesic symmetry 223 geometry on a 3-manifold 213 globally symmetric space see

symmetric space graph 54 Grothendieck group 341 group

– a-T-menable 256

– amenable 256

– commensurable 390

– connected linear reductive Lie group 225

– det ≥ 1-class 172

– elementary amenable 256 – good 257

– Hopfian 499

– Kähler group 434

– lamplighter group 379

– locally finite 287

– measure equivalent 312

– nilpotent 93

– of finite homological type 377 – of polynomial growth 93

– poly-cyclic 360

– residually (P) 457

– residually amenable 458

– residually finite 458

– Thompson’s group 297

– virtually (P) 93

– virtually cyclic 263

– virtually finitely generated abelian 93

– virtually free 93

– virtually nilpotent 93

– virtually poly-cyclic 360

– with the algebraic eigenvalue property 410

group of isometries 224 group ring 14 group von Neumann algebra 15

h-cobordism 122 Haken 3-manifold 212 Hattori-Stallings rank 358 heat kernel 52 Hilbert basis 17 Hilbert chain complex 24

– contractible 82

– d-dimensional 24

– det-L^{2}-acyclic 140

– dim-finite 140

– finite 24

– finite dimensional 24

– finitely generated 24

– L^{2}-acyclic 140

– positive 24

– weakly acyclic 140 Hilbert module

– finitely generated 16

– Hilbert N (G)-module 16

– map of 16 Hodge star-operator 50 Hodge-de Rham decomposition 51 – of a Hilbert chain complex 24 – for manifolds with boundary 64 homology sequence

– counterexample to weak exactness 26 – long weakly exact 27 homomorphism

– full 406

– left full 406

– local homomorphism of R-fields 407 – right full 406 hyperbolic space 53

idempotent completion 250 identity component 224 incompressible

– incompressible boundary of a 3-manifold 212

– incompressible surface in a 3-manifold 212

index of a differential operator 507 induction

– of group von Neumann algebras 29 – of Hilbert modules 29

– of modules 253

inductive limit see colimit of a

directed system inner product

– on a finitely generated projective N (G)-module 247

– standard inner product on N (G)^{n} 248

inverse limit see limit of an inverse

system inverse system 242 involution

– involution on a ℂ-category 246 irreducible

– 3-manifold 211

– selfhomeomorphism of a surface 307

Kähler hyperbolic manifold 422

Kähler manifold 422 Kähler metric 422 Kähler non-elliptic manifold 424 Killing form 224 knot complement 217 Kronecker product 487

L-groups 347

L^{1}-norm of a singular chain 486 L^{1}-seminorm of a homology class 486 L^{2}-acyclic see Hilbert chain complex L^{2}-Betti number

– analytic 53

– cellular 34

– for arbitrary G-spaces 263, 331 – of a densely defined operator of

Hilbert modules 73

– of a density function 75

– of a group 263

– of a Hilbert N (G)-chain complex 24

L^{2}-character map 282

– global 282 L^{2}-cohomology

– cellular 34 L^{2}-de Rham cochain map 61 L^{2}-de Rham cohomology

– reduced 58

– unreduced 58 L^{2}-Euler characteristic 277 L^{2}-homology

– cellular 34

– of a Hilbert N (G)-chain complex 24

L^{2}-index of a differential operator 507 L^{2}-torsion

– analytic 179

– – of a manifold with boundary 190 – of a det-L^{2}-acyclic free finite

G-CW -complex 160

– of a group automorphism 305

– of a Hilbert chain complex 140, 141 – of a knot 217

– topological 176

– – of a manifold with boundary 190 L^{∞}-norm of a singular cochain 486 L^{∞}-seminorm of a homology class 486 lamplighter group 379 Laplace operator 51

– of a Hilbert N (G)-chain complex 24

Laplace transform 181 lattice 313 Lefschetz invariant

– equivariant 288 lens space 124 Lie algebra

– semisimple 224

– simple 224 limit inferior 75, 460 limit of an inverse system 242 limit property

– for a G-CW -complex 92

– for a density function 92

– for Hilbert chain complex 92 limit superior 460 localization

– Ore localization 324 locally finite homology 487 locally symmetric space 223 lower central series 93

manifold

– closed 40

– exceptional prime 3-manifold 214 – flat Riemannian manifold 226 – Haken 3-manifold 212

– hyperbolic 53, 212

– hyperEuclidean 442

– irreducible 3-manifold 211

– Kähler hyperbolic manifold 422 – Kähler manifold 422

– of analytic determinant class 178 – prime 3-manifold 211

– Seifert manifold 212 mapping cone 35 mapping cylinder 35 mapping torus 42 Markov operator 379 measure equivalent groups 312 mesh of a triangulation 108 minimal volume 491 module

– semisimple 326 Morita isomorphism 343 multiplicatively closed 324

neighborhood

– normal neighborhood 223 net 17

– convergence of 18 Neumann boundary conditions 63 normalizer 31 Novikov-Shubin invariant

– analytic 104

– cellular 96

– of a Hilbert chain complex 81 – of a density function 75

– of a manifold with boundary 114 – of map of Hilbert modules 76 NS-L^{2}-acyclic 202

operator

– adjoint 55

– affiliated 318

– closable 55

– closed 54

– compact 392

– essentially selfadjoint 55

– G-operator 317

– maximal closure 57

– minimal closure 55 – of finite rank 392

– of trace class 392

– positive 16

– positive unbounded densely defined 55

– selfadjoint 55

– symmetric 55 Ore condition 324 Ore localization 324 orientation covering of a manifold 48

partial isometry 55 partition see G-invariant subordinate

smooth partition Poincaré model 53 Poincaré ℤG-chain homotopy equiva-

lence 39 Polar Decomposition 16

– for unbounded densely defined operators 55

prime 3-manifold 211 product

– ∧-product 49 projection 337

– abelian 337

– central carrier of 337

– equivalent 337

– finite 337

– infinite 337

– properly infinite 337 projective class group 341 projective limit see limit of an inverse

system projective rank function 405

– faithful 405 proper 31

Question

– Approximating Fuglede-Kadison determinants by determinants 478

– Approximating L^{2}-torsion by torsion 483

– L^{2}-torsion of groups and quasiisometry and measure equivalence 313

– Minimal volume and L^{2}-torsion for aspherical closed manifolds 502

– Novikov-Shubin invariants and quasi-isometry 313

– S^{1}-foliations and L^{2}-torsion for closed aspherical manifolds 202

– Simplicial volume and normal infinite amenable subgroups 498 – Simplicial volume and S^{1}-foliations

502

– Vanishing of L^{2}-Betti numbers of groups and epimorphism of groups 297

– Vanishing of the simplicial volume for affine manifolds 503

Vanishing of L^{2}-Betti numbers of the base and the total space of a fibration 67

R-field 407

– epic 407

– Hughes-free 408

– universal 407 rank

– complex rank of a semisimple Lie algebra 226

– complex rank of a semisimple Lie group 226

– fundamental rank of a symmetric space 226

rational closure 385 Ray-Singer torsion 126 recurrency probability of the natural

random walk on a group 94 reduced C*-algebra of a group 339 regular representation

– left 339, 340

– right 339, 340 Reidemeister torsion 123

– analytic 126

– – of a manifold with boundary 190 – topological 125

– – of a manifold with boundary 190 resolvent set of an operator 55 Riemannian metric

– product near the boundary 190 ring

– ∗-regular 333

– ∗-ring 326

– Artinian 397

– division closed 385

– hereditary 405

– Noetherian 397

– rationally closed 385

– regular 325

– regular coherent 380

– semifir 407

– semihereditary 239

– semiprime 397

– semisimple 326

– von Neumann regular 325

– with involution 326 Rothenberg sequence 348

S-inverting 323

– universal 323 Schatten ideal 392 Seifert fibered 3-manifold see Seifert

manifold Seifert manifold 212 selfadjoint

– essentially selfadjoint 55

– operator 55

– selfadjoint morphism in a ℂ-category with involution 247

semisimple

– Lie algebra 224

– Lie group 226

– module 326

– ring 326 Σ-inverting 385

– universal 385 simple homotopy equivalence 122 simple structure 164 simplicial volume

– of a closed orientable manifold 486 – of a manifold without boundary

487 singular homology with coefficients

263, 331 skeleton 31 Sobolev space of p-forms 58 space

– compactly generated see category – of L^{2}-integrable harmonic smooth

p-forms 51

– of L^{2}-integrable harmonic smooth p-forms on a manifold with boundary 63

– of (p, q)-forms 425

– of harmonic L^{2}-integrable (p, q)-forms 426

– of smooth p-forms 49

– of smooth p-forms with compact support 51

specialization 407 spectral density function

– analytic 103, 114

– cellular 96

– of a Hilbert chain complex 81

– of a map of Hilbert modules 73 spectral family 56

– of a selfadjoint operator 56

– of an essentially selfadjoint operator 56 spectrum of an operator 55 star

– closed 61

– open 61 support

– of an element in l^{2}(G) 259

survey

– on 3-manifolds 211

– on G-CW -Complexes 31

– on amenable groups 256

– on crossed products 396

– on deficiency of groups 299

– on division and rational closure 385 – on Ore localization 323

– on positive results about the Approximation Conjecture and the Determinant Conjecture 454

– on positive results about the Atiyah Conjecture 378

– on positive results about the Singer Conjecture 418

– on positive results about the zero-in-the-Spectrum Conjecture 440

– on residually finite groups 458

– on simplicial volume and bounded cohomology 485

– on symmetric spaces 223

– on the Approximation Conjecture and the Determinant Conjecture 454

– on the Atiyah Conjecture 370

– on the Bass Conjecture and the Hattori-Stallings rank 357

– on the Isomorphism Conjecture for K_{0} of complex group rings 357

– on torsion invariants 120

– on unbounded operators and spectral families 54

– on universal localizations 385

– on von Neumann algebras 336

– on von Neumann Regular rings 325 suspension 35 symmetric space 224

– dual 227

– of compact type 225

– of Euclidean type 225

– of non-compact type 225

Theorem

– Analytic L^{2}-torsion of hyperbolic manifolds 187

– Analytic and combinatorial Novikov-Shubin invariants 106 – Analytic and topological L^{2}-torsion

186

– Betti numbers and simplicial volume 494

– Cellular L^{2}-torsion 161

– Cellular L^{2}-torsion for universal coverings 163

– Cheeger’s finiteness theorem 491 – Combinatorial computation of

L^{2}-invariants 194

– Counterexample to the strong Atiyah Conjecture 380

– De Rham Isomorphism Theorem 50 – Detecting elements in Wh(G) 354 – Dimension and colimits 243

– Dimension and limits 244

– Dimension function for arbitrary N (G)-modules 239

– Dimension function for arbitrary U(G)-modules 330

– Dimension-flatness of N (G) over ℂG for amenable G 259

– Double Commutant Theorem 336 – Flat bundles and simplicial volume

503

– Glueing formula for L^{2}-torsion 192 – Goldie’s Theorem 399

– Gromov’s estimate on the simplicial volume 492

– Gromov’s mapping theorem for bounded cohomology 489

– Hadamard-Cartan Theorem 45

– Hodge-de Rham Theorem 51

– Hodge-de Rham Theorem for manifolds with boundary 63

– Hopf-Rinow Theorem 51

– Inverse Mapping Theorem 16

– K-groups of U 345

– K_{0} of finite von Neumann algebras 342

– K_{1} of von Neumann algebras of type I_{f} 344

– K_{1} of von Neumann algebras of type II_{1} 344

– Kadison-Fuglede determinant 128 – L-groups of von Neumann algebras

349

– L^{2}-Betti numbers 37

– L^{2}-Betti numbers and aspherical CW -complexes 48

– L^{2}-Betti number and fibrations 272 – L^{2}-Betti numbers and minimal

volume 502 – L^{2}-Betti numbers and Novikov-Shubin invariants of Kähler hyperbolic manifolds 423

– L^{2}-Betti numbers and S^{1}-actions 43, 271

– L^{2}-Betti numbers and volume 501 – L^{2}-Betti numbers for arbitrary

spaces 265

– L^{2}-Betti numbers of 3-manifolds 214

– L^{2}-Betti numbers of Kähler non-elliptic manifolds 424

– L^{2}-Betti numbers of Thompson’s group 298

– L^{2}-Euler characteristic 277

– L^{2}-Hodge decomposition for Kähler manifolds 426

– L^{2}-Hodge-de Rham Theorem 52 – L^{2}-Hodge-de Rham Theorem for manifolds with boundary 64

– L^{2}-invariants of symmetric spaces 228

– L^{2}-Lefschetz Theorem 426

– L^{2}-torsion for manifolds with boundary 190

– L^{2}-torsion and aspherical CW - complexes 172

– L^{2}-torsion and fibrations 166

– L^{2}-torsion and S^{1}-actions 168

– L^{2}-torsion of 3-manifolds 216

– L^{2}-torsion of Hilbert chain complexes 142

– L^{2}-torsion of hyperbolic manifolds with boundary 191

– L^{2}-torsion of mapping tori 169

– Linnell’s Theorem 378

– Logarithmic estimate for spectral density functions 456

– Measure equivalence and L^{2}-Betti numbers 312

– Modules over N (G) and Hilbert N (G)-modules 249

– Moody’s induction Theorem 402 – Novikov Shubin invariants and aspherical CW -complexes 103

– Novikov-Shubin invariants 97

– Novikov-Shubin invariants and S^{1}-actions 102

– Novikov-Shubin invariants of 3-manifolds 215

– Proportionality principle for L^{2}-invariants 201 – Proportionality principle for the

simplicial volume 496

– S-cobordism Theorem 122

– Simplicial volume of hyperbolic manifolds 490

– Simplicial volume of low-dimensional manifolds 490

– Strong version of Linnell’s Theorem 387

– Survey on residually finite groups 458

– The algebra of affiliated operators 327

– Thurston’s estimate on the simplicial volume 491

– Torus Theorem 214

– Vanishing of L^{2}-Betti numbers of mapping tori 42, 270

– von Neumann dimension 21

– von Neumann trace 18

– Weakly exact long L^{2}-homology sequence 27

– Wedderburn’s Theorem 400 topological L^{2}-torsion 176 topology

– norm 336

– strong 336

– ultra-weak 336

– weak 336 toral splitting of an irreducible

3-manifold 213

– geometric 213 totally geodesic 226 trace

– center valued 338

– faithful 338

– finite 338

– normal 338

– of an endomorphism of a Hilbert module 17

– over the complex group ring 193 – universal 338

– von Neumann trace 15 triangulation see equivariant smooth

triangulation twisted Laurent ring 396

uniformly contractible 437 unitary

– unitary morphism in a ℂ-category with involution 247 – unitary natural equivalence 247 universal R-field 407 universal additive invariant 281 universal R-field of fractions 407 universal S-inverting 323 universal Σ-inverting 385

virtually homotopy-finite CW -complex

280 von Neumann algebra 336

– center 337

– countably composable 339

– factor 337

– finite 337

– infinite 337

– of a group 15

– of type I 337

– of type I_{f} 337

– of type I_{∞} 337

– of type II 337

– of type II_{1} 337

– of type II_{∞} 337

– of type III 337

– properly infinite 337 von Neumann dimension

– center valued 341

– extended von Neumann dimension of an arbitrary module over N (G) 246

– extended von Neumann dimension of an arbitrary module over U(G) 330

– of a finitely generated projective N (G)-module 238

– of a finitely generated projective U(G)-module 329

– of a Hilbert N (G)-module 21

– properties 21 von Neumann trace 15

weak chain contraction 145 weak chain isomorphism 144 weak homology equivalence 141 weak homotopy equivalence 32 weak isomorphism 16 weakly acyclic see Hilbert chain

complex weakly exact sequence 16, 238 Weyl group 31 Whitehead group 121 Whitehead torsion 122 Whitney map 61 Witt group 347