Bordism, rho-invariants and the Baum-Connes conjecture

Paolo Piazza and Thomas Schick

January 21, 2006

Abstract

Let Γ be a finitely generated discrete group. In this paper we establish vanishing results for rhoinvariants associated to

(i) the spin-Dirac operator of a spin manifold with positive scalar curvature and fundamental group Γ (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group Γ.

The invariants we consider are more precisely

• the Atiyah-Patodi-Singer (≡ APS) rho-invariant associated to a pair of finite dimensional unitary representations λ_{1}, λ_{2} : Γ → U(d)

• the L^{2}-rho invariant of Cheeger-Gromov

• the delocalized eta invariant of Lott for a non-trivial conjugacy class of Γ which is finite.

We prove that all these rho-invariants vanish if the group Γ is torsion-free and the Baum-Connes map for the maximal group C^{∗}-algebra is bijective. This condition is satisfied, for example, by torsion-free amenable groups or by torsion-free discrete subgroups of SO(n, 1) and SU(n, 1). For the delocalized invariant we only assume the validity of the Baum-Connes conjecture for the reduced C^{∗}-algebra. In addition to the examples above, this condition is satisfied e.g. by Gromov hyperbolic groups or by cocompact discrete subgroups of SL(3, ℂ).

In particular, the three rho-invariants associated to the signature operator are, for such groups, homotopy invariant. For the APS and the Cheeger-Gromov rho-invariants the latter result had been established by Navin Keswani. Our proof re-establishes this result and also extends it to the delocalized eta-invariant of Lott. The proof exploits in a fundamental way results from bordism theory as well as various generalizations of the APS-index theorem; it also embeds these results in general vanishing phenomena for degree zero higher rho invariants (taking values in A/\overline{[A, A]} for suitable C^{∗}-algebras A). We also obtain precise information about the eta-invariants in question themselves, which are usually much more saddle objects than the rho-invariants.

Contents

1 Introduction and main results 3 1.1 Atiyah-Patodi-Singer rho invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 L^{2}-rho invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Delocalized eta invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 General principle of the proofs of vanishing results . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Plan of the paper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Index theory: statement of results 8

2.1 Atiyah-Singer index theory for arbitrary C^{∗}-algebras . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Geometric set-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 The A/\overline{[A, A]}-valued index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 APS index theory for arbitrary C^{∗}-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 Geometric set-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Trivializing perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.3 Eta invariants for perturbed operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.4 The A/\overline{[A, A]}-valued Atiyah-Patodi-Singer index formula . . . . . . . . . . . . . . . . . 12

3 Discrete groups, twisting bundles and index theorems 13 3.1 The group C^{∗}-algebra and the classical Mishchenko-Fomenko twisting bundle . . . . . . . . . 13

4 Surjectivity of the Baum-Connes map and stable rho-invariants 14 4.1 The ρ-index and its vanishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 The stable rho-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Bordism invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Injectivity of the Baum-Connes map and bordism 15 5.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 Rational homology theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Vanishing of stable ρ-invariants for certain product manifolds 18

7 Vanishing of stable rho-invariants 20

8 Unstable rho invariants 21

9 Special perturbations of the signature operator 22

10 From stable to unstable eta-invariants 27 10.1 Limits of eta invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 10.2 Proof of Lemma 10.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11 Homotopy invariance of unstable rho-invariants 31

12 Vanishing results on spin manifolds with positive scalar curvature 32

13 Further remarks on delocalized invariants 33 13.1 Infinite conjugacy classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 13.2 Higher rho-invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

14 The center valued L^{2}-signature formula for manifolds with boundary 36

15 Examples of non-trivial rho-invariants 37

A Index theory: proofs. 40

A.1 Proof of the A/\overline{[A, A]} valued Atiyah-Singer index formula . . . . . . . . . . . . . . . . . . . . 40

A.1.1 The integral operator index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 A.2 Proof of the A/\overline{[A, A]}-valued APS index formula . . . . . . . . . . . . . . . . . . . . . . . . . 45

A.2.1 The algebraic index perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 A.2.2 The integral operator b-index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A.2.3 Local expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A.2.4 The index formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

B Graded Hermitian complexes and the signature operator 49

C Pseudodifferential operators and morphisms of C^{∗}-algebras 51

D Twists with finite dimensional representations 53 E Classical L^{2}-invariants versus N Γ-invariants 54

E.1 Classical L^{2}-invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 E.2 Twisting with l^{2}Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 E.3 Translation between M̃ and M ×_{Γ} l^{2}Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1 Introduction and main results

Throughout this paper, M is an oriented closed manifold of odd dimension 2n − 1. Let u: M → BΓ be a continuous map, classifying a normal Γ-covering M̃ of M. Let (M′, u′ : M′ → BΓ) be a different oriented Γ-covering. We shall say that (M′, u′ : M′ → BΓ) and (M, u : M → BΓ) are oriented Γ-homotopy equivalent if there exists an oriented homotopy equivalence h : M → M′ such that u′ ∘ h is homotopic to u.

1.1 Atiyah-Patodi-Singer rho invariant

Let λ_{1}, λ_{2} : Γ → U(d) be two finite dimensional unitary representations of Γ of the same dimension. Equivalently, λ_{1} − λ_{2} is a virtual representation in the representation ring of Γ of dimension 0. The representations λ_{1}, λ_{2} induce flat bundles L_{1} and L_{2} on M endowed with flat unitary connections: if V_{j} denotes the representation space of λ_{j}, then L_{j} := M̃ ×_{Γ} V_{j} and the unitary connection is induced from the trivial connection on the product M̃ × L_{j}.

Let D be a Dirac-type operator acting on the sections of a Clifford module E. We shall always assume that the Clifford module is unitary and endowed with a unitary Clifford connection ∇^{E}. The operator D can be twisted with the flat bundles L_{1} and L_{2}. We use the notation D_{L_{j}}, but also D_{λ_{j}}, for the twisted operator, acting on sections of E ⊗ L_{j}.

The usual integral defines the eta-invariant of the twisted operator

η(D_{L_{j}}) = \frac{1}{\sqrt{π}} ∫_{0}^{∞} s^{−1/2} Tr(D_{L_{j}} exp(−(sD_{L_{j}}^{2})) ds.  (1.1)

Convergence near s = 0 is ensured by our assumptions on E and ∇^{E}, see [5]. In particular, if M is equipped with a spin structure and D := D / is the associated Dirac operator, then η(D /_{L_{j}}) is defined. Similarly, if D := D^{sign} is the signature operator of an orientation and Riemannian metric on M, the eta-invariant η(D_{L_{j}}^{sign}) is defined by (1.1).

1.2 Definition. The Atiyah-Patodi-Singer rho-invariant associated to the Dirac-type operator D and the virtual representation λ_{1} − λ_{2} is the difference

ρ(D)_{λ_{1}−λ_{2}} := η(D_{L_{1}}) − η(D_{L_{2}})  (1.3)

The first important result we shall prove in this paper is the following.

1.4 Theorem. Assume that Γ is a torsion-free discrete group and that the Baum-Connes assembly map is an isomorphism

μ_{max}: K_{∗}(BΓ) → K^{∗}(C^{∗}Γ),

where C^{∗}Γ is the maximal C^{∗}-algebra of Γ.

(1) Let D be the Dirac operator of a spin structure, D = D /. If the metric on M has positive scalar curvature then the Atiyah-Patodi-Singer rho-invariant is zero.

(2) Let D be the signature operator, D = D^{sign}, then the Atiyah-Patodi-Singer rho invariant ρ(D^{sign})_{λ_{1}−λ_{2}} only depends on the oriented Γ-homotopy type of (M, u : M → BΓ).

1.5 Remark. The result about the APS-ρ-invariant of the signature operator is due to Navin Keswani [27]. We give an independent proof of Keswani’s result. The first special case of this result is due to Mathai [45], who proved it for Bieberbach groups. We shall also try to present a “philosophical” reason for the theorem.

It should be added Keswani’s argument is most likely to be adapted so as to cover the result on the spin Dirac operator as well. Finally, Shmuel Weinberger [64] proves the homotopy invariance of the APS-rho invariants for the signature operator under the assumption that Γ is torsion free and that the L-theory isomorphism conjecture holds for Γ. Chang [9] has used the same assumption to prove the corresponding result for the L^{2}-rho invariant .

1.6 Remark. Note that for finite groups the ρ-invariants of the spin Dirac operator are effectively used to distinguish different metrics with positive scalar curvature, compare e.g. [6]. We see that this is not possible under our assumption on the group Γ. This might be particularly surprising in view of the genuinely non-local definition of the eta-invariants.

We shall use different C^{∗}-algebras associated to the group Γ.

1.7 Definition. Let Γ be a discrete group. The maximal C^{∗}-algebra C^{∗}Γ is the completion of the group ring ℂ[Γ] with respect to the maximal possible C^{∗}-norm on this ring [13].

The maximal C^{∗}-algebra has the universal property that for each unitary representation ϕ: Γ → U(H) on a Hilbert space H, there is a unique C^{∗}-algebra homomorphism C^{∗}Γ → B(H) extending ϕ.

More frequently used is the reduced C^{∗}-algebra of Γ. It is by definition the closure of ℂ[Γ], considered as subalgebra of B(l^{2}(Γ)) with respect to the right regular representation ((∑_{γ∈Γ} λ_{γ}γ) · g := ∑_{γ∈Γ} λ_{γ}(γg)). Note that by the universal property of C^{∗}Γ, we have a canonical homomorphism C^{∗}Γ → C_{red}^{∗}Γ extending the identity on Γ.

We will also make use of the group von Neumann algebra N Γ. This is by definition the weak closure of the right regular representation of ℂ[Γ] in B(l^{2}Γ). In particular it is also the weak closure of C_{red}^{∗}Γ, i.e. we have an inclusion C_{red}^{∗}Γ ↪ N Γ. By the bicommutant theorem, it is also equal to B(l^{2}(Γ))^{Γ}, i.e. all operators which commute with the right regular representation of Γ on l^{2}(Γ). Note that N Γ is also a C^{∗}-algebra.

1.8 Remark. The Baum-Connes conjecture for a torsion-free group Γ asserts that the assembly map

K_{∗}(BΓ) → K^{∗}(C_{red}^{∗}Γ)

is an isomorphism, where the right hand side is the reduced C^{∗}-algebra of the group Γ. It is a fact that K_{∗}(C^{∗}Γ) and K_{∗}(C_{red}^{∗}Γ) are identical for large classes of groups, e.g. amenable groups or discrete subgroups of SO(n, 1) or SU(n, 1). However, for groups with Kazdan’s property T, they definitely differ, and K_{∗}(BΓ) → K^{∗}(C^{∗}Γ) is known not to be surjective in this case.

1.2 L^{2}-rho invariant

If we look at the Γ-covering M̃ directly, we can lift any differential operator D from M to a Γ-invariant differential operator D̃ on M̃. In particular, this is possible for a Dirac-type operator D. Moreover, by using Schwartz kernels, but integrating only over a fundamental domain for the covering M̃ → M, we get an L^{2}-trace Tr_{(2)}, see [1]. This yields the L^{2}-η-invariant defined using essentially formula (1.1) by

η_{(2)}(D̃) = \frac{1}{\sqrt{π}} ∫_{0}^{∞} s^{−1/2} Tr_{(2)}(D̃ exp(−(sD̃^{2})) ds,

where the L^{2}-trace is defined by

Tr_{(2)}(D̃ exp(−tD̃^{2})) = ∫_{F} tr_{x} k_{t}(x, x).  (1.9)

Here tr_{x} is the fiberwise trace and F is a fundamental domain.

In order to ensure the convergence of the integral near t = 0 we still require the operator D to be associated to a unitary Clifford module endowed with a unitary Clifford connection. The convergence of the integral for large t is discussed in [10] and also in [52].

1.10 Definition. We define the L^{2}-ρ-invariant as the difference

ρ_{(2)}(D) := η_{(2)}(D̃) − η(D)

1.11 Theorem. If Γ is torsion free and the Baum-Connes assembly map for the maximal C^{∗}-algebra of Γ, K_{∗}(BΓ) → K_{∗}(C^{∗}Γ), is an isomorphism, then

(1) the L^{2}-rho invariant of the Dirac operator of a spin manifold with positive scalar curvature vanishes.

(2) the L^{2}-rho invariant of the signature operator depends only on the oriented Γ-homotopy type of (M, u : M → BΓ).

1.12 Remark. For the signature operator, this result is proved by Keswani in [28] using methods different from ours.

The result for the spin Dirac operator was originally obtained (with a similar, but slightly more complicated method as the one presented here) in unpublished work of Nigel Higson and the second author. It should be possible to adapt Keswani’s arguments so as to cover the result on the spin Dirac operator.

Our proof also provides some rather delicate information about the eta-invariants directly (and not just of the less saddle rho-invariants).

1.3 Delocalized eta invariants

We continue with the geometric set-up of Section 1.2, with a Galois Γ-covering Γ ⟶ M̃ ⟶ M classified by a map u : M → BΓ. To construct the L^{2}-eta invariant, we used the integral kernel k_{t}(x, y) of D̃ exp(−tD̃^{2}) on M̃. Fix now a non-trivial conjugacy class < g > of Γ. Define the delocalized trace

Tr_{<g>}(D̃ exp(−tD̃^{2})) := ∑_{g∈<g>} ∫_{F} tr_{x} k_{t}(x, gx).  (1.13)

Note that the fibers at x and gx of the pull back vector bundle on which D̃ acts are canonically identified, so that k_{t}(x, gx) can be considered as an endomorphism of this fiber, and its fiberwise trace tr is defined.

Convergence of this (possibly infinite) sum follows from exponential decay, compare [39].

Finite conjugacy classes If the conjugacy class < g > contains only finitely many elements, Lott proves in [39] that the following integral, defining the delocalized eta invariant, converges.

η_{<g>}(D̃) := \frac{1}{\sqrt{π}} ∫_{0}^{∞} t^{−1/2} Tr_{<g>}(D̃ exp(−tD̃^{2}) dt.  (1.14)

We prove:

1.15 Theorem. Assume that Γ is torsion-free and that the assembly map μ_{red}: K_{∗}(BΓ) → K_{∗}(C_{red}^{∗}Γ) for the reduced group C^{∗}-algebra is an isomorphism. Let < g > be a non-trivial finite conjugacy class in Γ.

(1) If D = D / is the spin-Dirac operator of a spin manifold with positive scalar curvature then

η_{<g>}(D̃ /) = 0.

(2) If D = D^{sign} is the signature operator of an oriented Riemannian manifold then η_{<g>}(D̃^{sign}) depends only on the oriented Γ-homotopy type of (M, u : M → BΓ).

1.16 Remark. Notice that our assumption here involves the reduced group C^{∗}-algebra. There are substantially more groups which satisfy the Baum-Connes conjecture for the reduced C^{∗}-algebra, even some with property T, e.g. all Gromov hyperbolic groups [50] or cocompact discrete subgroups of Sl(3, ℂ) [29].

1.17 Remark. The stated result is only interesting if the group contains elements with finite conjugacy class. Not that this does by no means imply that the corresponding element has finite order.

Evidently, each central element has a finite conjugacy class (consisting only of itself). This means that, starting with an arbitrary group Γ, each central extension with kernel ℤ will be a group with large center, and if H^{2}(Γ, ℤ) ≠ {0}, there are non-trivial such central extensions.

Infinite conjugacy classes. Lott establishes the convergence of the delocalized eta invariant under more general assumptions: it suffices to assume that the Dirac-type operator on the covering has a gap at 0 (with 0 allowed in the spectrum) and that the conjugacy class is of polynomial growth with respect to a word-metric on Γ. Thus, for the spin Dirac operator of a manifold with positive scalar curvature, the delocalized eta invariant η_{<g>}(D̃ /) is well defined, provided < g > is of polynomial growth. If, in addition, Γ is torsion-free and the reduced Baum-Connes map is bijective, then we shall prove that, as in the case of finite conjugacy classes stated above, η_{<g>}(D̃ /) = 0.

1.4 General principle of the proofs of vanishing results

In order to simplify the exposition we shall concentrate on the case where Γ is the fundamental group of our manifold and the Γ-covering is the universal covering.

First observe that the homotopy invariance of the rho invariants for the signature operator reduces to the vanishing of the rho invariant for the disjoint union X ⨿ −X′, if X and X′ are homotopy equivalent, since all rho invariants are certainly additive under disjoint union.

For this reason we shall be only concerned with vanishing results. To establish these results , we apply the following general principle. To avoid undo repetitions, let ρ for the moment stand for any of the ρ-invariants we want to investigate.

(1) We first define a stable variant ρ^{s} of ρ. This will be defined as the invariant of a perturbation of our generalized Dirac operator. Such perturbations do not always exist, we need the vanishing of the index class of the generalized Dirac operator. This very strong assumption is satisfied for geometric reasons if one looks at the Dirac operator of a spin manifold with positive scalar curvature, as well as for the signature operator on the disjoint union X ⨿ −X′ of two homotopy equivalent manifolds.

We study the main properties of ρ^{s}. Most important is that it appears as the correction term in an index theorem for manifolds with boundary for suitably perturbed Dirac operators. We use this fact and the assumed surjectivity of the Baum-Connes map in order to show that the stable rho-invariant is well defined, independent of the chosen perturbation (we are always under the assumption that the index class of our operator is zero). Under these assumptions we also prove the fundamental fact that ρ^{s} is a bordism invariant: suitable index theorems on manifolds with boundary will again play a crucial role here.

(2) Then we use our injectivity assumption on the Baum-Connes map, fundamental results in bordism theory and the bordism invariance established in (1) in order to show that the stable invariant ρ^{s}, whenever it is defined, is equal to the stable invariant of a particularly nice manifold. For this nice manifold we compute the stable invariant and show that it vanishes.

To this point, we have therefore shown that in certain special situations one can define an invariant ρ^{s} which turns out to be zero.

(3) As a last step we show that in the two geometric situations we are studying, the stable invariant ρ^{s} coincides with the unstable invariant ρ. This will be done by constructing very special perturbations (used in the definition of the stable invariant) which make the direct comparison of the stable and unstable invariant possible. For the signature operator on X ⨿ −X′ we use perturbations that are inspired by the work of Hilsum-Skandalis

In fact, our results here are much more precise: they give information directly about the unstable η-invariants. This is quite remarkable because of the non-stable nature of the η-invariants under perturbations and might lead to future applications or calculations of η-invariants.

In the case of a spin manifold with positive scalar curvature, we don’t have to perturb at all, so the last step is trivial.

1.5 Examples

Let Γ be a torsion-free discrete group satisfying our basic assumption, the bijectivity of the Baum-Connes map. One might very well wonder whether the signature rho-invariants considered in this paper are nonzero and whether they can be effectively used in order to distinguish manifolds that are not homotopy equivalent. In the last part of the paper we give a careful treatment of some non-trivial examples. In particular, we construct manifolds with the same cohomology but that are not in the same homotopy class and we distinguish their homotopy type through their rho-invariants. This can be done for all non-trivial groups which satisfy our basic examples. The results might also be obtainable using Blanchfield forms of classical algebraic topology; however, it seems that one has to use some advanced version to cover all the cases covered by our invariant. The advantage of our approach is that it is very easy to carry out the calculations.

We also show, along the way, that the vanishing of the signature index class is not sufficient for establishing the vanishing of the signature-rho-invariants. The mere vanishing of the signature index class does not imply the vanishing of our rho-invariants.

1.6 Plan of the paper.

In Section 2 we gather the index theoretic results that will be needed throughout the paper. We treat the general case of a Dirac operator twisted by a bundle L of finitely generated projective A-modules, with A a C^{∗}-algebra. We state A/\overline{[A, A]}-valued index theorems on closed manifolds and on manifolds with boundary. Proofs are given in Appendix A.

In Section 3 we specialize to the C^{∗}-algebras defined by a discrete group Γ and to the corresponding Mishchenko-Fomenko bundle; we also discuss the corresponding notions when we twist with the group von Neumann algebra N Γ.

In Section 4 we use the surjectivity of the assembly map in order to define the stable rho-invariants and establish their bordism invariance. The stable rho-invariants are C^{∗}-algebraic objects and they are defined under the additional assumption that the index class in K_{1}(C^{∗}Γ) is equal to zero.

In Section 5 we employ the injectivity of the assembly map in order to construct a suitable bordism between d copies of the manifold X = M ∪ (−M′), with M and M′ homotopy equivalent, and a manifold of a special type A similar result is proved if X is a manifold with positive scalar curvature. These bordisms take the classifying maps into account.

In Section 6 we show that the stable rho-invariants of these special manifolds are zero. This fact and the bordism invariance of Section 4 are then used in Section 7 in order to prove that under our assumptions the stable rho-invariants for D = D^{sign} are zero if X = M ∪ (−M′), with M and M′ homotopy equivalent. The corresponding result for X a manifold with positive scalar curvature and D = D / is established in Section 12.

In Section 8 we introduce unstable rho-invariants; these are von Neumann objects producing the three rho-invariants defined in Subsections 1.1, 1.2, 1.3 as special cases. The unstable rho-invariants are always defined.

In Section 9, Section 10 and Section 11 we show that if X = M ∪ (−M′), with M and M′ homotopy equivalent, then the unstable rho-invariants are suitable limits of stable rho-invariants. This step completes the proof of our theorems for the signature operator, since we know that stable rho-invariants are zero under our assumptions. The proof for the spin Dirac operator, which is much easier since there is no need for this limit-argument, is given in Section 12

In Section 13 we gather several remarks on delocalized eta-invariant for infinite conjugacy classes. We also prove vanishing results for higher rho-invariants.

In Section 14 we state a general von Neumann signature formula for manifolds with boundary.

In Section 15 we give examples of closed manifolds with torsion free fundamental group satisfying the Baum-Connes assumption and for which the relevant rho-invariants are non-zero. We also construct manifolds with isomorphic homology but with different homotopy type and we distinguish them by using the L^{2}-rho invariant or the delocalized eta invariant.

In four additional Appendices we recall signatures and the signature operator of Hilbert A-module chain complexes with symmetry, and we give a detailed account of the relationship between spectral invariants on Γ-coverings and certain algebra-valued invariants; we also discuss naturality properties of these algebravalued invariants. All appendices either recall known results or immediate extensions of known results; they are included for the reader’s convenience.

1.7 Acknowledgments

We thank Paul Kirk, Eric Leichtnam, Victor Nistor, George Skandalis and Shmuel Weinberger for helpful discussions and remarks. Special thanks go to Nigel Higson and John Roe for pointing out a gap in an earlier version of the paper. Part of this work was carried out during visits of the authors to Göttingen, Paris and Rome funded by Ministero Istruzione Università Ricerca, Italy (Cofin Spazi di Moduli e Teorie di Lie), Institut de Mathématiques de Jussieu , CNRS, Graduiertenkolleg “Gruppen und Geometrie” (Göttingen).

2 Index theory: statement of results

Many of the results of this paper are based on suitable generalizations of the Atiyah-Singer and Atiyah-Patodi-Singer index theorem. In this section we gather the index theoretic results that we shall need in the rest of the paper. Most of these index theorems are described (sometimes implicitly) in the literature; they are due to Mishchenko-Fomenko in the closed case and Leichtnam and the first author in the boundary case. Nevertheless, we shall give a direct and simplified account of these results, showing in particular that once the index class in K_{∗}(C_{r}^{∗}Γ) is carefully described, it is possible to give heat-kernel-proofs of all needed results following the same steps as for the numeric case (see, for example, [47, Introduction]).

This section is devoted to the statements; Appendix A will contain the proofs. We start with the closed case.

2.1 Atiyah-Singer index theory for arbitrary C^{∗}-algebras

2.1.1 Geometric set-up.

Let M be a closed manifold. We assume that D is a Dirac-type operator acting on a finite dimensional Clifford module E on M, see [5] for the definitions. Our main interest will be the spin-Dirac operator D / of a spin structure, and the signature operator D^{sign} of an orientation and Riemannian metric on M. Let A be a C^{∗}-algebra, and L a bundle of finitely generated projective Hilbert A-modules^{1}, with an A-connection ∇_{L}. We define D_{L} to be the operator D twisted with the bundle with connection L, acting on sections of E ⊗ L. Then D_{L} is an A-linear differential operator,

D_{L} ∈ Diff_{A}^{1}(M; E ⊗ L, E ⊗ L) ⊂ Ψ_{A}^{∗}(M; E ⊗ L, E ⊗ L).

On the right hand side the Mishchenko-Fomenko pseudodifferential calculus appears. D_{L} is elliptic in the sense of Mishchenko-Fomenko with a well defined index class Ind(D_{L}) ∈ K_{dim(M)}(A). We shall recall below the definition of Ind(D_{L}) ∈ K_{dim(M)}(A) when dim(M) is even.

In many situations, instead of working with the differential operator D_{L} itself, we will have to perturb the operator slightly (thereby leaving the world of differential operators). We do this because on the one hand we will have to improve some technical properties of the operator, such as the large time behavior of the associated heat-kernel; on the other hand, when the operator arises as a boundary operator, we shall need invertibility in order to define a Fredholm problem on the manifold with boundary. Of course, each time such a perturbation is introduced, we shall have then to control how invariants such as the index or the eta-invariant behave.

We start with some generalities about perturbations; this material will be needed throughout the paper.

1A comment on notations: objects in the C^{∗}-algebraic context will always be denoted by calligraphic letters

2.1 Definition. Let C ∈ Ψ_{A}^{−∞} be a smoothing operator in the Mishchenko-Fomenko calculus, acting on sections of E ⊗ L. Set

D_{L,C} := D_{L} + C.

This is a (smoothing) perturbation of D_{L}.

The following lemma is a direct consequence of the Mishchenko-Fomenko (≡ MF) pseudodifferential calculus:

2.2 Lemma. D_{L,C} is an elliptic element in the MF-calculus

D_{L,C} ∈ Ψ_{A}^{1}(M; E ⊗ L, E ⊗ L) ⊂ Ψ_{A}^{∗}(M; E ⊗ L, E ⊗ L)  (2.3)

Moreover, the operators D_{L} and D_{L,C} have the same index

Ind(D_{L}) = Ind(D_{L,C}) ∈ K_{dim(M)}(A) .

Fundamental in what follows is the following Mishchenko-Fomenko decomposition theorem.

2.4 Theorem. Let M be even dimensional, so that E = E^{+} ⊕ E^{−}. Let (E ⊗ L)^{±} := E^{±} ⊗ L. There is a Mishchenko-Fomenko decomposition of the space of sections of E ⊗ L with respect to D_{L}, i.e.

C^{∞}(M, (E ⊗ L)^{+}) = I_{+} ⊕ I_{+}^{⊥}, C^{∞}(M, (E ⊗ L)^{−}) = I_{−} ⊕ D_{L}(I_{+}^{⊥}).  (2.5)

By completion, we obtain a decomposition of the Sobolev Hilbert A-modules H^{m}(M, (E ⊗ L)^{±}) for any m ∈ ℕ.

The second decomposition is not, a priori, orthogonal. However, D_{L} induces an isomorphism (in the Fréchet topology) between I_{+}^{⊥} and D_{L}(I_{+}^{⊥}), and I_{+} and I_{−} are finitely generated projective Hilbert A-modules (consisting of smooth section of L^{2}(M, E ⊗ L)). The projections Π_{I_{+}} onto I_{+} (orthogonal) and Π_{I_{−}} onto I_{−} (along D_{L}(I_{+}^{⊥})) are smoothing operators in the Mishchenko-Fomenko calculus. Because I_{±} are already complete finitely generated projective Hilbert A-modules they are unchanged under the completions. The Mishchenko-Fomenko index class is given by

Ind(D_{L}) = [I_{+}] − [I_{−}] ∈ K_{0}(A) .  (2.6)

2.7 Remark. The theorem is ultimately a consequence of the ellipticity of D_{L} and more precisely of the existence of an inverse for D_{L} modulo smoothing operators. Thus, an identical statement remains true for the more general (elliptic) operator D_{L,C}. The existence of the decomposition is implicitly proved in [51]; the structure of the two projections is analyzed in [33, Appendix A].

2.1.2 The A/\overline{[A, A]}-valued index

We want to give a heat kernel proof of a suitable Atiyah-Singer (≡AS) index theorem for the operator D_{L} of 2.1.1. First of all, we introduce the index we want to compute. Consider the index class Ind(D_{L}) expressed through the MF-decomposition theorem (see 2.6): Ind(D_{L}) = [I_{+}] − [I_{−}] ∈ K_{0}(A) . Expressing the finitely generated projective modules I_{±} as the images of idempotent matrices and taking the difference of the traces, we obtain a well defined element in A/\overline{[A, A]}, with [A, A] equal to the ℂ-subspace generated by the commutators [a, b] := ab − ba for a, b ∈ A.

2.8 Definition. Consider the closure \overline{[A, A]} of the subspace [A, A]. We set

A_{ab} := A/\overline{[A, A]}.  (2.9)

where the subscript ab stands for abelianization. It should be noticed that A_{ab} is a commutative Banach algebra.

In this way we define a homomorphism of abelian groups

tr^{alg} : K_{0}(A) ⟶ A_{ab}  (2.10)

Since this is nothing but the zero-degree part of the Karoubi-Chern character [25], we denote tr^{alg}([I_{+}]−[I_{−}]) by

[I_{+}]_{[0]} − [I_{−}]_{[0]} ∈ A_{ab} ,  (2.11)

or, equivalently, by Ind_{[0]}(D_{L}) ∈ A_{ab}.

Our aim is to give a formula for Ind_{[0]}(D_{L}) ∈ A_{ab} and to prove it via heat kernel techniques. First we need the existence of the heat-kernel for the Dirac Laplacian D_{L}^{2} and its perturbation D_{L,C}^{2}. This is not completely obvious, given that we are in the C^{∗}-algebraic context. We shall give the precise statement in Lemma A.1 in Appendix A.1 For the time being we shall content ourselves with the statement that the heat-semigroup {e^{−tD_{L,C}^{2}} , t > 0} exists, provides a fundamental solution of the heat-equation and is such that e^{−tD_{L,C}^{2}} ∈ Ψ_{A}^{−∞}.

Since the heat operator is a smoothing operator in the Mishchenko-Fomenko calculus, we can consider its supertrace

STR(e^{−tD_{L}^{2}}) ∈ A_{ab} .

This is defined as follows: we restrict the heat-kernel to the diagonal Δ in M × M, Δ ↔ M, thus obtaining an element in C^{∞}(M, End(E ⊗ L)), we take for each fiber over x the A_{ab}-valued supertrace on End(E ⊗ L)_{x} and integrate

STR(e^{−tD_{L}^{2}}) := ∫_{M} str_{x}^{alg}(e^{−tD_{L}^{2}}(x, x))vol_{M} ∈ A_{ab} .  (2.12)

Our first result concerns the large time behavior of the heat-kernel in the Mishchenko-Fomenko calculus.

2.13 Proposition.

lim_{t→∞} STR(e^{−tD_{L}^{2}}) = [I_{+}]_{[0]} − [I_{−}]_{[0]} ≡ Ind_{[0]}(D_{L}) ∈ A_{ab},

where part of the assertion is that the limit exists.

The proof of this proposition can be found in Appendix A.1. Next we want to connect Ind_{[0]}(D_{L}) ∈ A_{ab} defined using the index class and the algebraic trace tr^{alg} : K_{0}(A) → A_{ab} to the integral-kernel-trace, TR, of the projection operators P_{+} := Π_{I_{+}}, P_{−} := Π_{I_{−}} given in the Mishchenko-Fomenko decomposition. We recall that these are smoothing operators; the trace TR is thus well defined. We state the result here and defer the proof to Appendix A.1.

2.14 Proposition. The algebraic trace

Ind_{[0]}(D_{L}) := [I_{+}]_{[0]} − [I_{−}]_{[0]} of Ind(D_{L}),

i.e. the image under the map tr^{alg} : K_{0}(A) → A_{ab}, of Ind(D_{L}),can be calculated as

[I_{+}]_{[0]} − [I_{−}]_{[0]} = TR(P_{+}) − TR(P_{−})
≡ ∫_{M} tr_{x}^{alg} P_{+}(x, x) − ∫_{M} tr_{x}^{alg} P_{−}(x, x) ∈ A_{ab},

where P_{+} and P_{−} are the projections onto I_{+} and I_{−} as given by the Mishchenko-Fomenko decomposition.

Proceeding now as in the classical case, one proves the following A_{ab}-valued Atiyah-Singer index theorem:

2.15 Theorem. We have

Ind_{[0]}(D_{L}) = TR(P_{+}) − TR(P_{−}) = ∫_{M} AS(D)(x) ∧ ch L(x)_{[dim M]} ∈ A_{ab}

where AS(D)(x) is the local integrand in the Atiyah-Singer formula for D.

In the above formula the differential form

(x ↦ AS(D)(x) ∧ ch(E)(x) ∧ ch L(x)_{[dim M]}) ∈ Ω^{dim M}(M, A_{ab}),

can be calculated as usual using Chern-Weyl theory and the curvature of the connections, see [57].

2.2 APS index theory for arbitrary C^{∗}-algebras

2.2.1 Geometric set-up.

Let W a compact manifold of even dimension with boundary ∂W = M. We assume that D is a Dirac-type operator acting on a finite dimensional Dirac-bundle E on W. Our main interest will be the spin-Dirac operator of a spin structure, and the signature operator of an orientation and Riemannian metric on W. We always assume that all these structures are of product type near the boundary.

Let A be a C^{∗}-algebra, and L a bundle of finitely generated projective Hilbert A-modules, with an A-connection ∇_{L} (again everything has a product structure near the boundary). We define D_{L} to be the operator D twisted with the bundle with connection L. Associated to this are the boundary operators D_{M} ≡ D_{∂W} of D and D_{M,L} ≡ D_{∂W,L} of D_{L}. We can attach an infinite cylinder (−∞, 0] × M to W along its boundary ∂W = M, thus obtaining a manifold with cylindrical ends Ŵ and with product metric dt^{2} +g_{∂W} along the cylinder. The operators D_{L} extends in a natural way to the manifold Ŵ. The change of coordinates t = log x compactifies Ŵ to a manifold with boundary and with product b-metric dx^{2}/x^{2} + g_{∂M} near the boundary. The operator on Ŵ then defines in a natural way a b-differential operator on the compactified manifold. We refer the reader to the book [47] of Melrose for basics about the b-calculus. As we shall mainly work in the framework of the b-calculus, we keep denoting the compactified manifold by W and the resulting b-differential operators by D_{L}.

2.2.2 Trivializing perturbations

Let M be odd dimensional and without boundary, and let us assume that Ind(D_{M,L}) = 0 ∈ K_{1}(A). This will be the case in the following examples:

• D_{M} is the signature operator on M = X ⊔ (−X′) with X and X′ homotopy equivalent

• D_{M} is the signature operator and (M, L) bounds, i.e. there is a manifold W with ∂W = M such that L extends to W

• M is spin with positive scalar curvature and D_{M} is the spin Dirac operator.

For more on these examples we refer the reader to [20], [54] [30] and [55]. According to [65] [37, Theorem 3], we can find a non-commutative spectral section P for D_{M,L}.^{2} Using the projection P one can construct [30, Proposition 2.10] a smoothing operator C_{P} ∈ Ψ_{A}^{−∞}(M; E ⊗ L) such that

D_{M,L} + C_{P} ∈ Ψ_{A}^{1}(M; E ⊗ L) is invertible in Ψ_{A}^{∗}(M; E ⊗ L).

Moreover, C_{P} is symmetric with respect to the A-valued scalar product on C^{∞}(M, E ⊗L). We shall consider in what follows the set of allowable perturbations:

𝔓 := 𝔓_{D_{M,L}} := {C ∈ Ψ_{A}^{−∞}(M; E ⊗ L) | C is symmetric and D_{M,L} + C is invertible in Ψ_{A}^{∗}(M; E ⊗ L)}. (2.16)

Summarizing, we have just seen that

Ind(D_{M,L}) = 0 ⟹ 𝔓 ≠ ∅ .  (2.17)

Let us go back to the case where M = ∂W, with W even dimensional. As mentioned above, by cobordism invariance, the index class of D_{M,L} is zero in K_{1}(A); thus 𝔓 ≠ ∅. Let C ∈ 𝔓 and consider the invertible operator D_{M,L} + C. According to [30, Lemma 6.1] there exists a smoothing operator in the Mishchenko-Fomenko b-calculus

C_{W}^{+} ∈ Ψ_{b,A}^{−∞}(W; E^{+} ⊗ L, E^{−} ⊗ L)

2 Thus, by definition, P is a self-adjoint projection, P ∈ Ψ_{A}^{0} and there exists functions χ_{1}, χ_{2} ∈ C^{∞}(ℝ, [0, 1]) such that χ_{i}(t) = 0 for t << 0, χ_{i}(t) = 1 for t >> 0, χ_{2} ≡ 1 on a neighborhood of the support of χ_{1} and Im χ_{1}(D_{M,L}) ⊂ Im P ⊂ Im χ_{2}(D_{M,L}).

such that D_{L}^{+} + C_{W}^{+} has D_{M,L} + C as boundary operator. Using the invertibility of D_{M,L} + C one proves the invertibility of the indicial family associated to D_{L}^{+} + C_{W}^{+} ; this property can in turn be used to prove that D_{L}^{+} + C_{W}^{+} is A-Fredholm as an operator between suitable Sobolev Hilbert A-modules. Thus there is a well defined index class in K_{0}(A), denoted Ind_{b}(D_{L} + C_{W}). More precisely

Ind_{b}(D_{L} + C_{W}) = [I_{+}] − [I_{−}] ∈ K_{0}(A)  (2.18)

with I_{±} finitely generated projective A-modules and

I_{±} ⊂ x^{ϵ}H_{b}^{∞}(W, (E ⊗ L)^{±}), ϵ > 0.

The construction of C_{W} from C involves choices; the index class, on the other hand, does not depend on these choices. We refer the reader to [30] for the details.

2.19 Remark. The results in [30], [37] are noncommutative generalizations of the results first proved in [48], [49] for families of Dirac operators (i.e. for C(B)-linear operators, with B a compact manifold).

2.20 Notation. We shall often denote by Ind_{b}(D_{L}, C) the index class in (2.18).

2.21 Remark. Notice that the index class Ind_{b}(D_{L}, C) does depend on C ^{3}

2.22 Definition. Let tr^{alg} : K_{0}(A) → A_{ab} be the trace introduced in Subsection 2.1.2. The A_{ab}-valued b-index is by definition tr^{alg}(Ind_{b}(D_{L}, C)) ∈ A_{ab}. As in the closed case we use the notation

tr^{alg}(Ind_{b}(D_{L}, C)) =: Ind_{b,[0]}(D_{L}, C) = [I_{+}]_{[0]} − [I_{−}]_{[0]} ∈ A_{ab} .

Our goal is to prove an index formula for Ind_{b,[0]}(D_{L}, C). First of all, we introduce the boundary correction term that will appear in the formula.

2.2.3 Eta invariants for perturbed operators.

Let C ∈ 𝔓_{D_{M,L}}. Consider the pseudo-differential operator

D_{M,L} + C ∈ Ψ_{A}^{1}(M; E ⊗ L).

We define the A_{ab}-valued eta invariant η_{[0]}(D_{M,L} + C) ∈ A_{ab} by

η_{[0]}(D_{M,L} + C) := \frac{1}{\sqrt{π}} ∫_{0}^{∞} TR ((D_{M,L} + C)e^{−t(D_{M,L}+C)^{2}}) \frac{dt}{\sqrt{t}}.  (2.23)

The integral converges for large t because of the invertibility of D_{M,L} + C. The convergence for t ↓ 0 follows from the local A_{ab}-valued index theorem on closed manifold and the observation that

e^{−t(D_{M,L}+C)^{2}} = e^{−tD_{M,L}^{2}} + tC^{∞}([0, ∞), Ψ_{A}^{−∞}(M, E ⊗ L)) ,

a consequence of Duhamel’s formula and fact that C ∈ Ψ_{A}^{−∞}(M, E ⊗ L).

2.2.4 The A/\overline{[A, A]}-valued Atiyah-Patodi-Singer index formula

In Appendix A.2 we shall recall the precise form of the Mishchenko-Fomenko decomposition theorem in the b-context. From this result we get the two finitely generated projective modules I_{±} entering into the definition of the b-index class Ind_{b,[0]}(D_{L}, C). b-elliptic regularity implies that the projections P_{±} onto these two modules have well defined traces TR(P_{±}) ∈ A_{ab}. Analyzing the large-time behavior of the b-supertrace of the heat-kernel for (D_{L} + C_{W})^{2}, computing the short-time limit and using the commutator formula for the b-trace, one finally proves the following theorem (see Appendix A.2 for proofs and relevant definitions)

3For example if C_{P} ∈ 𝔓 and C_{Q} ∈ 𝔓 are defined by spectral sections P and Q respectively, then (see [35], [32])

Ind_{b}(D_{L}, C_{P}) − Ind_{b}(D_{L}, C_{Q}) = [Q − P] in K_{0}(A) , where on the right hand side the difference class defined by the two projections Q, P appears. This is known to be non-zero in general.

2.24 Theorem.

Ind_{b,[0]}(D_{L}, C) ≡[I_{+}]_{[0]} − [I_{−}]_{[0]}
= lim_{t→+∞} b STR(e^{−t(D_{L}+C_{W})^{2}}) = TR(P_{+}) − Tr(P_{−})
= ∫_{W} AS(D)(x) ∧ ch(E)(x) ∧ ch L(x) dx − \frac{1}{2}η_{[0]}(D_{M,L} + C) ∈ A_{ab}.

where AS(D) is the local integrand in the APS index theorem for D.

3 Discrete groups, twisting bundles and index theorems

In this section we show how we can derive the particular index theorems needed for the results stated in the introduction from the general ones presented above. We shall specialize to the maximal or the reduced C^{∗}-algebra of the group, and also to the group von Neumann algebra.

3.1 The group C^{∗}-algebra and the classical Mishchenko-Fomenko twisting bundle

We now specialize the C^{∗}-algebra A and the Hilbert A-module bundle L in 2.2.1 and 2.1.1 to the cases which lead to essentially all our applications.

Specifically, let Γ be a discrete finitely generated group. Let Γ → M̃ → M be a Galois covering and let u: M → BΓ be the associated classifying map, i.e. M̃ is the pull back of the universal Γ-covering EΓ → BΓ under u. We twist our Dirac type operator D with the flat C^{∗}Γ-bundle L associated to the covering, i.e. with

L := M̃ ×_{Γ} C^{∗}Γ.

3.1 Theorem. For this operator

Ind_{[0]}(D_{M,L}) = (∫_{M} AS(D)) · 1 ∈ (C^{∗}Γ)_{ab}.  (3.2)

Here, AS is the integrand in the classical Atiyah-Singer index theorem and we recall that (C^{∗}Γ)_{ab} := C^{∗}Γ/\overline{[C^{∗}Γ, C^{∗}Γ]}

Proof. We apply Theorem 2.15. Since L is flat, and each fiber is isomorphic to the free C^{∗}Γ-module of rank 1, ch(L)(x) = 1 ∈ (C^{∗}Γ)_{ab}, so that we get the constant 1 in the formula.

We now look at a compact manifold W with boundary M, again with a Γ-covering Γ → W̃ → W, classified by a map (also called u) u: W → BΓ. Note that this induces, by restriction, a Γ-covering M̃ of the boundary.

Let D be a Dirac type operator on W. Then let C ∈ 𝔓_{D_{M,L}} give rise to an allowable perturbation of the boundary operator D_{M,L}.

3.3 Theorem.

Ind_{b,[0]}(D_{L}, C) = (∫_{W} AS(D)) · 1 − \frac{1}{2}η_{[0]}(D_{M,L} + C) ∈ (C^{∗}Γ)_{ab}.  (3.4)

Proof. This is a special case of Theorem 2.24. Note that, since the twisting bundle L is flat and fiberwise isomorphic to the free Hilbert C^{∗}Γ-module of rank 1, again ch L(x) = 1 ∈ (C^{∗}Γ)_{ab}, so that it does only contribute the 1 in the formula.

3.5 Remark. Consider the reduced group C^{∗}-algebra C_{r}^{∗}Γ of Definition 1.7. One can form a Mishchenko-Fomenko twisting bundle M̃ ×_{Γ} C_{r}^{∗}Γ and prove (C_{r}^{∗}Γ)_{ab}-valued index formulas completely analogous to formulas (3.2) and (3.4).

4 Surjectivity of the Baum-Connes map and stable rho-invariants

4.1 The ρ-index and its vanishing

Let Γ be a discrete group. Let Γ → W̃ → W be an even-dimensional Galois covering of a compact manifold with boundary W, classified by u: W → BΓ. Let Γ → M̃ → M be the boundary covering. As before, we consider D := D_{L}, a Dirac-type operator on W twisted with the Mishchenko-Fomenko line bundle L. Since the boundary operator D_{M,L} has trivial index in K_{1}(C^{∗}Γ) we can choose a trivializing perturbation C ∈ 𝔓_{D_{M,L}}. With respect to this perturbation, we Ind_{b}(D, C) ∈ K_{0}(C^{∗}Γ). By applying tr^{alg} : K_{0}(C^{∗}Γ) → (C^{∗}Γ)_{ab} we have finally defined the (C^{∗}Γ)_{ab}-valued b-index Ind_{b,[0]}(D, C) ∈ (C^{∗}Γ)_{ab}.

Since (C^{∗}Γ)_{ab} is a vector space with one dimensional subspace generated by [1] = 1 + \overline{[C^{∗}Γ, C^{∗}Γ]}, we can project the degree zero part of the index onto the quotient (C^{∗}Γ)_{ab}/ < [1] >.

4.1 Definition. The ρ-index associated to D is the image of Ind_{b,[0]}(D, C) in the quotient (C^{∗}Γ)_{ab}/ < [1] >. We shall denote the ρ-index by

ind_{b,[0]}^{ρ}(D, C) ∈ (C^{∗}Γ)_{ab}/ < [1] > .  (4.2)

4.3 Remark. Suppose now that W is closed and let Ind_{[0]}(D) ∈ (C^{∗}Γ)_{ab} be the associated (C^{∗}Γ)_{ab}-valued index. Using the degree-zero Atiyah-Singer index formula 3.1, we see that the ρ-index of D vanishes:

ind_{[0]}^{ρ}(D) = 0 ∈ (C^{∗}Γ)_{ab}/ < [1] > .  (4.4)

This simple observation plays a fundamental role.

4.5 Lemma. If Γ is torsion-free and the Baum-Connes map μ_{max} : K_{0}(BΓ) → K_{0}(C^{∗}Γ) is surjective, then the ρ-index vanishes:

ind_{b}^{ρ}(D, C)_{[0]} = 0 in (C^{∗}Γ)_{ab}/ < [1] > .

Proof. Observe that we can define a homomorphisms

Φ^{ρ}: K_{0}(C^{∗}Γ) → (C^{∗}Γ)_{ab}/ < [1] > ,

by simply composing tr^{alg} : K_{0}(C^{∗}Γ) → (C^{∗}Γ)_{ab} with the quotient map (C^{∗}Γ)_{ab} → (C^{∗}Γ)_{ab}/ < [1] >. Using the above remark, we see that this homomorphism is zero on the image of μ_{max}, since the image of μ_{max} consists precisely of index classes associated to closed manifolds. We use the Baum-Douglas [4] description of the K-homology of BΓ here. By assumption, μ_{max} is surjective, so that Φ^{ρ} = 0 on all of K_{0}(C^{∗}Γ). Thus 0 = Φ^{ρ}(Ind_{b}(D, C)) = ind_{b}^{ρ}(D, C)_{[0]} and the assertion follows.

4.2 The stable rho-invariant

Assume that D is a Dirac type operator on a closed odd-dimensional manifold M, twisted with the Mishchenko-Fomenko line bundle L associated to some classifying map u: M → BΓ. Assume that Ind(D) = 0 ∈ K_{1}(C^{∗}Γ). Pick a trivializing perturbation C ∈ 𝔓_{D}.

4.6 Lemma. If Γ is torsion-free and the Baum-Connes map μ_{max}: K_{0}(BΓ) → K_{0}(C^{∗}Γ) is surjective, then

[η_{[0]}(D + C)] ∈ (C^{∗}Γ)_{ab}/ < [1] >  (4.7)

i.e. the image of η_{[0]}(D +C) ∈ (C^{∗}Γ)_{ab} in the quotient (C^{∗}Γ)_{ab}/ < [1] >, does not depend on the particular perturbation C chosen.

Proof. Let C′ ∈ 𝔓 be a different perturbation. Consider the cylinder [−1, 1] × M. On the boundary of the cylinder we have an invertible operator, obtained by considering the operator D + C on {−1} × M and the operator D+C′ on {1}×M. As already explained, there is a well defined b-index class in K_{0}(C^{∗}Γ), obtained by lifting the two perturbations to the cylinder and defining a b-pseudodifferential operator D_{[−1,1]×M} + C_{[−1,1]×M} with invertible indicial family. We denote this index class by

Ind_{b}(D_{[−1,1]×M} + C_{[−1,1]×M}) .

Using the APS-index formula 3.3 we obtain

ind_{b}^{ρ}(D_{[−1,1]×M} + C_{[−1,1]×M})_{[0]} =

− \frac{1}{2} (ρ_{[0]}(D + C) − ρ_{[0]}(D + C′)) ∈ (C^{∗}Γ)_{ab}/ < [1] >

On the other hand, by the assumed surjectivity of the Baum-Connes map we know that ind_{b}^{ρ}(D_{[−1,1]×M} + C_{[−1,1]×M}) = 0 and the Lemma is proved.

4.8 Definition. Assume that Ind(D) = 0 ∈ K_{1}(C^{∗}Γ) and that the max-Baum-Connes map K_{0}(BΓ) → K_{0}(C^{∗}Γ) is surjective. Then the stable degree-zero ρ-invariant is the class

ρ_{[0]}^{s}(D) := [η_{[0]}(D + C)] ∈ (C^{∗}Γ)_{ab}/ < [1] >  (4.9)

for any perturbation C ∈ 𝔓_{D}, and it is well defined because of Lemmas 4.6.

4.10 Notation. Let r: M → BΓ be a classifying map. We shall frequently use the notation ρ_{[0]}^{s}(M, r) with the understanding that this rho-invariant is associated either to the signature operator on the oriented Riemannian manifold M, or to the spin Dirac operator on the spin manifold M (both twisted by the Mishchenko-Fomenko bundle), depending on the context. Sometime, we shall omit the classifying map from the notation thus writing ρ_{[0]}^{s}(M).

4.3 Bordism invariance

Let (M, g) be odd dimensional endowed with a a bundle of Clifford modules defining a Dirac-type operator D_{M}. Let (M, r: M → BΓ) be a Galois covering and let D := D_{M,L} be the twisted Dirac operator in the Mishchenko-Fomenko calculus. Let (M′, g′) another odd dimensional Riemannian manifold endowed with a Clifford module and let (M′, r′: M′ → BΓ) be a Galois covering on M′; we let D′ := D_{M′,L′}. Assume now the existence of a bordism (W, R : W → BΓ) between (M, r : M → BΓ) and (M′, r′ : M′ → BΓ); we also assume the existence of a Riemannian metric and of a bundle of Clifford modules on W restricting to the given data on M and M′.

4.11 Proposition. Assume Γ is torsion free and such that the Baum-Connes map μ_{max} is surjective. Assume that Ind(D) = 0 in K_{1}(C^{∗}Γ) so that the stable rho-invariant ρ_{[0]}^{s}(D) is well defined. By the bordism invariance of the index class we know that Ind(D′) = 0 so that ρ_{[0]}^{s}(D′) is also well defined. We have

ρ_{[0]}^{s}(D) = ρ_{[0]}^{s}(D′)

Proof. The argument establishing Lemma 4.6 can be repeated here, provided we substitute the cylinder [−1, 1] × M there with the bordism W here.

5 Injectivity of the Baum-Connes map and bordism

5.1 Statement of results

In contrast with the previous section we now consider only the spin Dirac operator D / and the signature operator D^{sign} and not arbitrary generalized Dirac type operators.

5.1 Definition. Consider (M, u: M → BΓ). If d ∈ ℕ \ {0}, we denote by d(M, u : M → BΓ) the disjoint union of d copies of M with obvious induced map du: dM → BΓ, and with obvious induced structure (orientation, etc.).

5.2 Proposition. Assume that M is a closed oriented manifold, u: M → BΓ a classifying map. Let D^{sign} be the associated Mishchenko-Fomenko signature operator. Assume that μ_{max}: K_{∗}(BΓ)⊗ℚ → K_{∗}(C^{∗}Γ)⊗ℚ is injective, and assume that Ind(D^{sign}) = 0 in K_{dim M}(C^{∗}Γ) ⊗ ℚ. Then there exists d ∈ ℕ \ {0} such that d · [M, u: M → BΓ] is bordant to

∪_{j=1}^{k}(A_{j} × B_{j}, r_{j} × 1 : A_{j} × B_{j} → BΓ)

in Ω_{dim M}^{SO}(BΓ), with dim B_{j} = 4b_{j}, π_{1}(B_{j}) = 1 and < L(B_{j}), [B_{j}] >= sgn(B_{j}) = 0. Here r_{j} : A_{j} → BΓ is a continuous map and (r_{j} × 1)(a, b) := r_{j}(a).

5.3 Proposition. Assume that M is a closed spin manifold, u: M → BΓ a classifying map. Let D / := D /_{L} be the associated Mishchenko-Fomenko spin Dirac operator. Assume that μ_{max}: K_{∗}(BΓ) ⊗ ℚ → K_{∗}(C^{∗}Γ) ⊗ ℚ is injective and that Ind(D /) = 0 ∈ K_{∗}(C^{∗}Γ) ⊗ ℚ. Then there is d ∈ ℕ \ {0} such that d[M, u: M → BΓ] is bordant to

⋃_{j=1}^{k}(A_{j} × B_{j}, r_{j} × 1: A_{j} × B_{j} → BΓ)

in Ω_{dim M}^{spin}(BΓ), with dim B_{j} = 4b_{j}, π_{1}(B_{j}) = 1 and < Â(B_{j}), [B_{j}] >= 0. Here r_{j} : A_{j} → BΓ is a continuous map and (r_{j} × 1)(a, b) := r_{j}(a).

5.4 Remark. In Propositions 5.2 and 5.3, the condition that the index of the Mishchenko-Fomenko operator in the K-theory of the maximal C^{∗}-algebra vanishes can be replaced by the more familiar condition that the index in the K-theory of the reduced C^{∗}-algebra vanishes, if we replace the assumption on the rational injectivity of the maximal Baum-Connes map by the analogous one for the reduced C^{∗}-algebra. Moreover, it should be remarked that the rational injectivity of the complex Baum-Connes map is equivalent to the rational injectivity of the real Baum-Connes map. This is a well known result, a proof can be found in [58].

We prove Propositions 5.2 and 5.3 with the same method from algebraic topology, which characterizes rationalized homology theories.

5.2 Rational homology theories

Let h_{∗} and k_{∗} be two generalized homology theories. We will be interested in the examples Ω_{∗}^{spin} of spin bordism, Ω_{∗}^{SO} of oriented bordism, K_{∗} and KO_{∗} of complex or real K-homology (the homology theory dual to K-theory).

Let ȟ_{∗} := h_{∗}(pt) and ǩ_{∗} := k_{∗}(pt) be the coefficients.

It is easy to see that h_{∗} ⊗ ℚ and H_{∗}(X; ℚ) ⊗_{ℚ} (ȟ_{∗} ⊗ ℚ) (graded tensor product, the n-th group is ⊕_{p+q=n}H_{p}(X) ⊗ ȟ_{q}) are again homology theories, compare e.g. [21, 3.18 ff]^{4}.

5.5 Proposition. (1) Every natural transformation T: h_{∗} ⊗ ℚ → k_{∗} ⊗ ℚ is determined by Ť: ȟ_{∗} ⊗ ℚ → ǩ_{∗} ⊗ ℚ.

(2) Every homomorphism Ť: ȟ_{∗} ⊗ℚ → ǩ_{∗} ⊗ℚ has a unique extension to a natural transformation T: h_{∗} ⊗ ℚ → k_{∗} ⊗ ℚ.

(3) There is a unique natural transformation H_{∗}(·; ℚ) ⊗ (ȟ_{∗} ⊗ ℚ) → h_{∗}(·) ⊗ ℚ which is the identity on the point. It is a natural isomorphism of ȟ_{∗} ⊗ ℚ-modules if h_{∗} is a multiplicative homology theory.

Proof. Compare [21, (3.20),(3.21), (3.22)].

Now, we apply this to the examples we want to study. By definition, elements in K_{∗}(X) are represented by triples (M, E, ϕ), where M is a closed oriented Riemannian manifold of dimension congruent to ∗ (mod 2), E is a bundle of Clifford modules on M and ϕ: M → X is continuous. Similarly, KO_{∗}(X) is defined by triples (M, E, ϕ) as above, with the exception that E is a bundle of real Clifford modules. Of course, a suitable equivalence relation has to be factored out. See [4] and [26].

4One remark on the reference [21]: there, only cohomology is considered, but everything works exactly in the same way if cohomology is replaced by homology.

In particular, we get natural transformations

T_{SO}: Ω_{∗}^{SO}(X) → K_{∗}(X);  [M ⟶^{ϕ} X] ↦ [M, Λ^{sign}, ϕ]
T_{spin}: Ω_{∗}^{spin}(X) → KO_{∗}(X);  [M ⟶^{ϕ} X] ↦ [M, S /, ϕ].

Here, Λ^{sign} stands for the complex Clifford module defining the signature operator on the oriented Riemannian manifold M, whereas S / is the spin Clifford module defining the real Dirac operator of the given spin structure^{5} . If X is a point, T_{SO} is just given by the signature (using the isomorphism K_{2n}(pt) = ℤ). Also if X = {pt}, T_{spin} is the α-invariant, which is (up to a multiple) equal to Â(M) for dimensions divisible by 4 (again use the isomorphism KO_{4n}(pt) = ℤ). Note that, in other dimensions, KO_{k}(pt) is finite, and K_{2n+1}(pt) = 0.

Using Proposition 5.5 we get commutative diagrams

\begin{matrix}H_{∗}(X; ℚ) ⊗ (Ω_{∗}^{SO}(pt) ⊗ ℚ) & ⟶^{id⊗T_{SO}^{ℚ}} & H_{∗}(X; ℚ) ⊗ (K_{∗}(pt) ⊗ ℚ) \\ ↓≅ & & ↓≅ \\ Ω_{∗}^{SO}(X) ⊗ ℚ & ⟶^{T_{SO}^{ℚ}} & K_{∗}(X) ⊗ ℚ. \end{matrix}  (5.6)

\begin{matrix}H_{∗}(X; ℚ) ⊗ (Ω_{∗}^{spin}(pt) ⊗ ℚ) & ⟶^{id⊗T_{spin}^{ℚ}} & H_{∗}(X; ℚ) ⊗ (KO_{∗}(pt) ⊗ ℚ) \\ ↓≅ & & ↓≅ \\ Ω_{∗}^{spin}(X) ⊗ ℚ & ⟶^{T_{spin}^{ℚ}} & KO_{∗}(X) ⊗ ℚ. \end{matrix}  (5.7)

It is not really necessary for us to understand completely the vertical isomorphisms. The important point is that Ω_{∗}^{spin}(X) ⊗ ℚ is a free module over Ω_{∗}^{spin}(pt) ⊗ ℚ (free generators given by a basis of H_{∗}(X; ℚ) under the vertical isomorphism). The corresponding statements hold for Ω_{∗}^{SO}(X) ⊗ ℚ and K_{∗}(X) ⊗ ℚ, KO_{∗}(X) ⊗ ℚ. Moreover, we also see that the basis can be chosen in a natural way and such that the transformations T_{SO}^{ℚ} and T_{spin}^{ℚ} are diagonal with respect to them. In particular, ker(T_{SO}^{ℚ}(X)) is a free module over ker(T_{SO}^{ℚ}(pt)) and ker(T_{spin}^{ℚ}(X)) is a free module over ker(T_{spin}^{ℚ}(pt)).

This means that every element in the kernel of T_{SO}^{ℚ} or T_{spin}^{ℚ}, respectively, will after multiplication with a suitable non-zero integer (to clear the denominators) have the form described in Proposition 5.2 or Proposition 5.3, respectively. Here we use the fact that the Ω̌_{∗}-module structure of Ω_{∗}(X) is given by the Cartesian product.

Finally, recall that

Ind(D^{sign}) = μ_{max}(T_{SO}([M ⟶^{u} BΓ])) ∈ K_{dim M}(C^{∗}Γ))

in the situation of Proposition 5.2, and that

Ind(D /_{ℝ}) = μ_{ℝ,max}(T_{spin}([M ⟶^{u} BΓ])) ∈ KO_{dim M}(C_{ℝ}^{∗}Γ)

in the situation of Proposition 5.3. Because we assume that μ_{max} ⊗ ℚ is injective, Proposition 5.2 on the signature operator follows from the descriptions of ker(T_{SO}^{ℚ}) we have just given. Concerning Proposition 5.3, we point out that the rational vanishing of the complex index class associated to D / is equivalent to the rational vanishing of the corresponding real index class for the real spin Dirac operator D /_{ℝ}. Recalling that the rationally injectivity of the real Baum-Connes map μ_{ℝ,max} is equivalent to the rationally injectivity of the complex Baum-Connes map μ_{max} [58], we see that the result stated in Proposition 5.3 is again a consequence of the description of ker(T_{spin}^{ℚ}) we have just given.

5Topologists refer to these transformations as the natural K-orientation associated to an (ordinary) orientation and natural KO-orientation associated to a spin structure on M, respectively.

6 Vanishing of stable ρ-invariants for certain product manifolds

6.1 Theorem. Assume that M = U × V, where V is a simply connected oriented Riemannian manifold of dimension divisible by 4 with vanishing signature, and U is any oriented Riemannian manifold of odd dimension. Let A be a C^{∗}-algebra and L a flat finitely generated projective Hilbert A-module bundle on U (defining a bundle, also called L, on M by pullback). Let D_{L} be the corresponding signature operator.

Then Ind D_{L} = 0 ∈ K_{dim M}(A) and there exists a perturbation C ∈ 𝔓_{D_{L}} such that

η_{[0]}(D_{L} + C) = 0 ∈ A_{ab}.

Proof. For the differential forms on U × V we have the following decomposition:

L^{2}Ω(U × V) = L^{2}Ω(U) ⊗ L^{2}Ω^{+}(V) ⊕ L^{2}Ω(U) ⊗ L^{2}Ω^{−}(V)

where Ω^{±}(V) refers to the grading associated to the signature operator on V. Since the bundle L pulls back from V, we get a corresponding decomposition of the Hilbert A-module of forms with coefficients in L:

L^{2}Ω(U × V; L) = L^{2}Ω(U; L) ⊗ L^{2}Ω^{+}(V) ⊕ L^{2}Ω(U; L) ⊗ L^{2}Ω^{−}(V).

With respect to this decomposition, the signature operator splits as

D_{U×V} = (\begin{matrix}D_{U} ⊗ 1_{V} & 1_{U} ⊗ D_{V}^{−} \\ 1_{U} ⊗ D_{V}^{+} & −D_{U} ⊗ 1_{V} \end{matrix} ) = D_{U} ⊗ τ + 1_{U} ⊗ D_{V} ,

where τ is the grading operator on L^{2}Ω(V). Since L is flat, we get a corresponding splitting of the signature operator twisted with L:

D_{L,U×V} = (\begin{matrix}D_{L,U} ⊗ 1_{V} & 1_{U} ⊗ D_{V}^{−} \\ 1_{U} ⊗ D_{V}^{+} & −D_{L,U} ⊗ 1_{V} \end{matrix} ) = D_{L,U} ⊗ τ + 1_{U} ⊗ D_{V} .  (6.2)

Let us now define the perturbed operator. Choose an isometry Ψ: ker(D_{V}^{+}) → ker(D_{V}^{−}). Since V is compact with vanishing signature, such an isomorphism of finite dimensional vector spaces exists. Then

C := (\begin{matrix}0 & Ψ^{∗} \\ Ψ & 0 \end{matrix}) : L^{2}Ω^{+}(V) ⊕ L^{2}Ω^{−}(V) → L^{2}Ω^{+}(V) ⊕ L^{2}Ω^{−}(V)

is a smoothing operator on V (here “Ψ” stands for the projection onto ker(D_{V}^{+}) follows by the operator Ψ).

The operator D_{L,U×V} + 1_{U} ⊗ C is invertible. Let f(D_{L,U}) be any smoothing function of D_{L,U} which is the identity on a sufficiently large neighborhood of zero in the spectrum of D_{L,U} so that

D_{C} := D_{L,U×V} + f(D_{L,U}) ⊗ C

becomes invertible. This will be our perturbation; notice that since we have found a smoothing perturbation of D_{L,U×V} which is invertible, we conclude that Ind(D_{L,U×V}) = 0 in K_{dim M}(A), as required.

We want to show that the degree 0 eta invariant of this perturbation D_{C} vanishes. Since L^{2}Ω(V) is an ordinary Hilbert space where orthogonal complements exist, we get another orthogonal decomposition into projective Hilbert A-modules

L^{2}Ω(U × V; L) = (L^{2}Ω(U; L) ⊗ ker(D_{V})) ⊕ (L^{2}Ω(U; L) ⊗ ker(D_{V})^{⊥}) .

The operator D_{L,U×V} as well as f(D_{L,U}) ⊗ C preserves the splitting

L^{2}Ω(U; L) ⊗ ker(D_{V}) ⊕ L^{2}Ω(U; L) ⊗ (ker(D_{V})^{⊥}) .

Moreover, the second operator is zero on the second summand. Therefore, the integrand in the definition of η_{[0]}(D_{C}) splits into two summands. We continue by investigating the two summand separately. We will use freely from Equation (6.2): D_{L,U×V} = D_{L,U} ⊗ τ + 1_{U} ⊗ D_{V}. We restrict to L^{2}Ω(U; L) ⊗ ker(D_{V})^{⊥}. We denote the restriction of D_{C} to this space by (D_{C})_{r} and the restrictions of τ and D_{V} to ker(D_{V})^{⊥} by τ_{r} and (D_{V})_{r}. Then

\begin{aligned}(D_{C})_{r} e^{−t(D_{C})_{r}^{2}} & = D_{L,U×V} e^{−tD_{L,U}^{2}} ⊗ e^{−t(D_{V} )_{r}^{2}} \\ & = D_{L,U}e^{−tD_{L,U}^{2}} ⊗ (τ)_{r}e^{−t(D_{V} )_{r}^{2}} + e^{−tD_{L,U}^{2}} ⊗ (D_{V})_{r}e^{−t(D_{V} )_{r}^{2}}. \end{aligned}  (6.3)

We claim that the degree zero trace with values in A_{ab} of both summands vanishes. Note that TR is multiplicative for tensor products. Here, we get TR of the first factor multiplied with the ordinary complex valued trace of the second factor.

For the first summand

Tr((τ)_{r}e^{−t(D_{V} )_{r}^{2}}) = 0

by the McKean-Singer formula (we restrict to the orthogonal complement of the kernel, so the index is zero); therefore

TR(D_{L,U}e^{−tD_{L,U}^{2}} ⊗ (τ)_{r}e^{−t(D_{V} )_{r}^{2}}) = TR(D_{L,U}e^{−tD_{L,U}^{2}} ) Tr((τ)_{r}e^{−t(D_{V} )_{r}^{2}}) = 0.

For the second summand, the same argument applies since TR(e^{−tD_{L,U}^{2}} ) = 0 on the odd dimensional manifold U by the usual symmetry argument.

Now, we restrict D_{C} to L^{2}Ω(U; L) ⊗ ker(D_{V}) (and we again denote the relevant restriction with ( )_{r}). We will finish by showing that this summand also is identically equal to zero, which then implies imme-

diately that η_{[0]}(D_{C}) = 0. To see this, consider the operator

B := 1_{U} ⊗ (τ ∘ C): L^{2}Ω(U; L) ⊗ ker(D_{V}) → L^{2}Ω(U; L) ⊗ ker(D_{V}).

This is an isometry which anticommutes with D_{L,U} ⊗ τ (since C anticommutes and τ commutes with τ) and which anticommutes with 1_{U} ⊗ C (since C commutes and τ anticommutes with C). Similarly, B anticommutes with f(D_{L,U})⊗C. Consequently, B anticommutes with the restriction of D_{C} to L^{2}Ω(U; L)⊗ ker(D_{V}), which is just given by D_{L,U} ⊗ τ + f(D_{L,U}) ⊗ C. Of course, this implies also that B commutes with e^{−t(D_{L,U×V} )_{r}^{2}}. Therefore (everything restricted to L^{2}Ω(U; L) ⊗ ker(D_{V}))

TR((D_{C})_{r}e^{−t(D_{C})_{r}^{2}}) = TR(B^{−1}B(D_{C})_{r}e^{−t(D_{C})_{r}^{2}})
= − TR(B^{−1}(D_{C})_{r}e^{−t(D_{C})_{r}^{2}}B)
= − TR((D_{C})_{r}e^{−t(D_{C})_{r}^{2}}) ∈ A_{ab}.

The last equality follows from the trace property. Consequently, the integrand also vanishes identically. Putting everything together, we get

η_{[0]}(D_{C}) =\frac{1}{\sqrt{π}} ∫_{0}^{∞} t^{−1/2} TR(D_{C}e^{−tD_{C}^{2}} ) = \frac{1}{\sqrt{π}} ∫_{0}^{∞} 0 = 0.

As a special case of the theorem, we obtain the following Corollary.

6.4 Corollary. Assume that M = U × V, where V is a simply connected oriented 4k-dimensional Riemannian manifold such that ⟨L(V), [V]⟩=0, and U is any oriented Riemannian manifold. Let r: U → BΓ be a continuous map. Then there exists a smoothing perturbation C such that D + C is invertible and η_{[0]}(D + C) vanishes, where D is the Mishchenko-Fomenko signature operator associated to r × 1. In particular, if the Baum-Connes map is surjective, then the stable rho-invariant is defined in C^{∗}Γ_{ab}/ < [1] > and equal to zero: ρ_{[0]}^{s}(U × V, r × 1) = 0.

There are similar results in the spin context:

6.5 Theorem. Assume that M = U ×V, where V is a simply connected spin manifold of dimension divisible by 4 with Â(V) = 0, and U is any spin manifold of odd dimension. Let A be a real C^{∗}-algebra and L a flat finitely generated projective Hilbert A-module bundle on U (defining a bundle, also called L, on M by pullback). There is a product metric on U × V such that for the resulting twisted spin Dirac operator D_{L} we get

η_{[0]}(D_{L}) = 0 ∈ A_{ab}.

Proof. By Stephan Stolz’ solution [60] of the Gromov-Lawson-Rosenberg conjecture for simply connected manifolds, V admits a metric with positive scalar curvature. If we shrink this metric appropriately, the corresponding product metric on U × V will still have positive scalar curvature. The Lichnerowicz formula then implies that its spin Dirac operator D_{L} is invertible (here we use that the bundle is flat). In particular, we don’t need a perturbation and can define η_{[0]}(D_{L}).

Secondly, we can use the product structure (of the Dirac operator) and argue in a way similar to the proof of Theorem 6.4 (but with the simplification that the operator on V has no kernel) to conclude that

η_{[0]}(D_{L}) = 0 ∈ (C^{∗}Γ)_{ab}.

7 Vanishing of stable rho-invariants

7.1 Theorem. Let M be an odd dimensional oriented closed compact manifold and let u: M → BΓ be a continuous map, classifying a Galois Γ-covering M̃ → M. If the max-Baum-Connes map μ_{max}: K_{∗}(BΓ) → K_{∗}(C^{∗}Γ) is bijective and the signature index class of (M, u: M → BΓ) vanishes in K_{dim M}(C^{∗}Γ) then

ρ_{[0]}^{s}(M, u) = 0.  (7.2)

Proof. We use the injectivity of the max-Baum-Connes map, the assumption Ind(D) = 0 and Proposition 5.2 in order to conclude that there exists a bordism (W, F : W → BΓ) between (dM, du: M → BΓ) and

∪_{j=1}^{k}(A_{j} × B_{j}, r_{j} × 1 : A_{j} × B_{j} → BΓ)

with π_{1}(B_{j}) = 1 and < L(B_{j}), [B_{j}] >= 0. Denote briefly

∪_{j=1}^{k}(A_{j} × B_{j}, r_{j} × 1 : A_{j} × B_{j} → BΓ)

by (N, v: N → BΓ). By cobordism invariance, the index of the signature operator associated to (N, v: N → BΓ) is zero in K_{dim M}(C^{∗}Γ). From the surjectivity of the max-Baum-Connes map we know that there are well defined stable rho-invariants ρ_{[0]}^{s}(M, u), ρ_{[0]}^{s}(N, v). Fix allowable perturbations C_{M}, C_{N}. Then there exists a well defined signature b-index class, in K_{dim W} (C^{∗}Γ), associated to the bordism (W, F : W → BΓ) and to the perturbations dC_{M}, C_{N}, which give rise to a Mishchenko-Fomenko b-smoothing operator A_{W} on W. We denote this b-index class by Ind_{b}(D_{W} + A_{W}) ∈ K_{dim W} (C^{∗}Γ) . Now we proceed as in the proof of Lemma 4.6. On the one hand, by the surjectivity of μ_{max} we have

Ind_{b}^{ρ}(D_{W} + A_{W}) = 0 ∈ (C^{∗}Γ)_{ab}/ < 1 >;

on the other hand by applying the APS index theorem 3.3 we get

Ind_{b}^{ρ}(D_{W} + A_{W})) = −\frac{1}{2} (dρ_{[0]}^{s}(M, u) − ρ_{[0]}^{s}(N, v)) ∈ (C^{∗}Γ)_{ab}/ < 1 >;

from which we deduce that

ρ_{[0]}^{s}(M, u) = \frac{1}{d}ρ_{[0]}^{s}(N, v) = \frac{1}{d} ∑_{j=1}^{k} ρ_{[0]}^{s}(A_{j} × B_{j}, r_{j} × 1)

We finish the proof by applying Corollary 6.4.

7.3 Theorem. Let M be an odd dimensional closed compact spin manifold and let u: M → BΓ be a continuous classifying map. Let D / := D /_{L} be the associated Mishchenko-Fomenko spin Dirac operator. If the max-Baum-Connes map μ_{max}: K_{∗}(BΓ) → K_{∗}(C^{∗}Γ) is bijective and the index class Ind(D /) = 0 ∈ K_{dim M}(C^{∗}Γ) vanishes, then

ρ_{[0]}^{s}(M, u) = 0.  (7.4)

Proof. The proof follows exactly the same pattern of the proof of Theorem 7.1. More details in Subsection 13.2 (see in particular the proof of Theorem 13.9).

8 Unstable rho invariants

In this section, let A be a von Neumann algebra. Let Z be a commutative von Neumann algebra, and τ: A → Z a positive and normal trace on A with values in Z.

In Section 4 we managed to define, if the index class is zero and the Baum-Connes map is surjective, the stable rho-invariant ρ_{[0]}. In Section 7 we showed that, under the additional assumption of injectivity of the Baum-Connes map, it is zero.

We now introduce unstable rho-invariants which are potentially more interesting, since they are defined under much more general hypothesis.

Assume that M is a closed manifold of odd dimension and let D be a Dirac type operator on M, acting on sections of a bundle E. Let L be a bundle of finitely generated projective Hilbert A-modules on M, with a connection preserving all the structure. Let D_{L} be the corresponding twisted Dirac operator. Each fiber of L is a finitely generated projective module; L_{x} = Im(p_{x}), with p_{x} a projection in M_{k×k}(A) for some k; let τ(L_{x}) := τ(p_{x}) where we extend the trace τ to M_{k×k}(A) in the obvious way. We assume that τ(L_{x}) is constant in x.

8.1 Definition. We define the τ-eta invariant as

η_{τ} (D_{L}) := \frac{1}{\sqrt{π}} ∫_{0}^{∞} τ(TR(D_{L} exp(−tD_{L}^{2}))) \frac{dt}{\sqrt{t}} ∈ Z .  (8.2)

We have to check that this integral converges. For t → 0, this follows from the usual local heat expansion. Since τ is positive and normal, the estimates of Cheeger and Gromov [10] can be used to obtain convergence for large times.

8.3 Definition. Let A_{1} and A_{2} be von Neumann algebras; let L_{1}, L_{2} be two Hilbert A_{j}-module bundles over M; let τ_{j}: A_{j} → Z_{j} be positive normal traces with values in commutative von Neumann algebras Z_{j}. Let β_{j}: Z_{j} → V be homomorphisms to a fixed target space V (a vector space). Assume that β_{1}(τ_{1}((L_{1})_{x}))+ β_{2}(τ_{2}((L_{2})_{x})) = 0 ∈ V. The unstable rho invariant of D_{L} with respect to β_{j}, τ_{j} and L_{j} is defined as

ρ_{(β_{j},τ_{j},L_{j})}(D) := β_{1}(η_{τ_{1}}(D_{L_{1}})) + β_{2}(η_{τ_{2}}(D_{L_{2}})) ∈ V .

The definition can be extended to a finite number of summands in the evident way.

We give examples showing the interest of such a definition. General example. Let M be a closed oriented Riemannian manifold of odd dimension. Assume that Γ is a discrete group; let u: M → BΓ be a map classifying a covering M̃ = u^{∗}EΓ. Let α_{j}: C^{∗}Γ → A_{j} be homomorphisms to unital von Neumann algebras A_{j} (with j = 1, 2), and τ_{j}: A_{j} → Z_{j} positive normal traces with values in abelian von Neumann algebras Z_{j}. Let L_{j} := M̃ ×_{Γ} A_{j} be the associated Hilbert A_{j}-module bundle, where Γ acts on A_{j} via Γ → C^{∗}Γ ⟶^{α_{j}} A_{j}. Then τ_{j}((L_{j})_{x}) = τ_{j}(α_{j}(1)). If β_{j}: Z_{j} → V satisfy

β_{1}(τ_{1}(α_{1}(1))) + β_{2}(τ_{2}(α_{2}(1))) = 0 ∈ V

then the unstable rho-invariant ρ_{(β_{j},τ_{j},L_{j})}(D) ∈ V is well defined and equal to

β_{1}(η_{τ_{1}}(D_{L_{1}})) + β_{2}(η_{τ_{2}}(D_{L_{2}})) ∈ V.  (8.4)

8.5 Example. (Atiyah-Patodi-Singer rho-invariant.) We refer to the general example. The relevant von Neumann algebras here are A_{1} = M_{d}(ℂ) = A_{2}, with two homomorphisms α_{j}: C^{∗}Γ → M_{d}(ℂ) induced by two representations λ_{1}, λ_{2} : Γ → U(d). The relevant trace is (in both cases) the usual trace τ: M_{d}(ℂ) → ℂ on matrices. Then τ(α_{1}(1)) = d = τ(α_{2}(1)), so that, with V = ℂ, we can choose β_{1} = id and β_{2} = − id. By Equation (D.1), the eta-invariants appearing in the definition of the APS-rho invariant and those appearing in formula (8.4) coincide. Thus with these choices ρ_{(β_{j},τ_{j},L_{j})}(D) = ρ_{λ_{1}−λ_{2}}(D).

8.6 Example. (center-valued rho-invariant.) We refer to the general example. Let A_{1} := N Γ be the group von Neumann algebra of Γ and Z its center. Let A_{2} = ℂ. Let α_{1} be induced by the natural map C^{∗}Γ → N Γ and let α_{2} : C^{∗}Γ → ℂ be induced by the trivial representation. Then (by definition) L_{1} = N := M̃ ×_{Γ} N Γ and L_{2} = M × ℂ.

We take V = Z, τ_{1} = τ: N Γ → Z equal to the canonical center valued trace (compare [24, Chapter 8]); τ_{2} : ℂ → Z given by τ_{2}(z) := z · 1. Note that both are positive and normal traces. Here τ_{1}(α_{1}(1)) = 1 = τ_{2}(α_{2}(1)), so that we can choose again β_{1} = id, β_{2} = − id. By Lemma C.8,

ρ_{(β_{j},τ_{j},L_{j})}(D) = η_{τ} (D_{N}) − η(D) · 1 ∈ Z  (8.7)

We define this element in Z to be the center-valued rho invariant of M.

8.8 Example. (L^{2}-rho and delocalized eta invariants.) Let < g > be a finite conjugacy class in Γ. This defines a trace

τ_{<g>}: N Γ → ℂ; ∑_{h∈Γ} λ_{h}h ↦ ∑_{h∈<g>} λ_{h}.  (8.9)

By the universal property of the central valued trace τ, τ_{<g>} = τ_{<g>} ∘ τ, where we use the restriction of τ_{<g>} to the center Z on the right hand side. We now apply τ_{<g>} to both sides of Equation (8.7). If g ≠ 1, τ_{<g>}(1) = 0, and we obtain (using Proposition E.11)

τ_{<g>}(ρ_{(β_{j},τ_{j},L_{j})}(D)) = η_{<g>}(D̃).

If g = 1, then τ_{<g>} =: τ_{Γ} is the canonical trace, and we obtain by Proposition E.11, using τ_{Γ}(1) = 1,

τ_{Γ}(ρ_{(β_{j}τ_{j}L_{j})}(D)) = ρ_{(2)}(D̃).

Our goal is to prove vanishing results for such generalized unstable rho-invariants and, thanks to the examples just given, derive the assertions of the Introduction as special cases.

9 Special perturbations of the signature operator

Let f : M → M′ be a smooth orientation preserving homotopy equivalence between two closed manifolds. Let A be a C^{∗}-algebra and V′ a flat bundle of finitely generated Hilbert A-modules on M′. We consider the signature operator on M′ with values in V′. We denote this operator by D′_{V′}. Next, we consider the signature operator on M with values in the flat bundle V := f^{∗}V′ and we denote it by D_{V}. These two operators come from the de Rham complexes and the Hodge star operator on M′ and M with values in the flat bundles V′ and f^{∗}V′, respectively. We denote these de Rham differentials by d′_{V′} and d_{V}.

In this section we shall construct an explicit trivializing perturbation for the Mishchenko-Fomenko signature operator

D := (\begin{matrix}D_{V} & 0 \\ 0 & −D′_{V′} \end{matrix})  (9.1)

on M ⊔ (−M′).

Recall that, since f is an orientation preserving homotopy equivalence, the index in K_{∗}(A) of D is zero^{6}. Thus, we already know that there exists a smoothing perturbation of the operator which is invertible, see [65] [37, Theorem 3]. In this section we shall sharpen this result, showing that we can construct certain special perturbations which are spectrally concentrated near zero (definition below). We will use these special perturbations in Section 10 in order to show that, in this situation, the stable rho invariant of D in (9.1) coincides with the unstable one, getting a more precise result about their eta-invariants in that case.

9.2 Theorem. Let f : M → M′ be a smooth orientation preserving homotopy equivalence between two closed manifolds, with B dim(M) = dim(M′) odd. Then, for each ϵ > 0 we can find a special self-adjoint smoothing perturbation B_{ϵ} such that D+B_{ϵ} is invertible and with the additional property that B_{ϵ} is ϵ-spectrally concentrated, that is

B_{ϵ} ∘ ϕ(D) = 0 = ϕ(D) ∘ B_{ϵ}

for each function ϕ: ℝ → ℝ with ϕ(t) = 0 for |t| < ϵ

9.3 Remark. We expect that a corresponding result holds if dim(M) is even. In this case we will also need that B_{ϵ} is an odd operator with respect to the signature grading (where we use on M′ the reverse orientation). Since we don’t need the even dimensional case in this paper, we leave this for further investigation.

Proof. In order to prove Theorem 9.2, we must carefully recall the definition of the signature operator for odd dimensional manifolds. This is done in Appendix B.

We denote the Hilbert A-module L^{2}(M, Λ^{∗}M ⊗f^{∗}V′) by L^{2}(M)^{∗}. Similarly we set L^{2}(M′, Λ^{∗}M′⊗V′) =: L^{2}(M′)^{∗}. We would like to compare L^{2}(M)^{∗} with L^{2}(M′)^{∗} using the pull-back map f^{∗}; but this is in general not L^{2}-bounded.. As in Hilsum-Skandalis [20, p. 90], we modify f^{∗} in order to obtain a L^{2}-bounded cochain map between L^{2}(M′)^{∗} and L^{2}(M)^{∗} as follows. From [20, p. 90], for suitably large N, there is a submersion F: D^{N} × M → M′ such that F (0, m) = f(m). Here D^{N} is an open ball in an Euclidean space of dimension N. Fix v ∈ Ω_{c}^{N}(D^{N}) with ∫_{D^{N}} v = 1. Define a bounded cochain map

T : L^{2}(M′)^{∗} ⟶ L^{2}(M)^{∗} , ω → ∫_{D^{N}} v ∧ F^{∗}(ω).

Let D_{V} and D′_{V′} be the signature operators on M and M′, as introduced above. There is a well-defined functional calculus associated to these regular operators on Hilbert modules. In particular, if ϕ: ℝ → ℝ is a rapidly decreasing (i.e. Schwartz) function, then ϕ(D_{V}) ∈ Ψ_{A}^{−∞} (one can prove this e.g. by using the heat kernel and

9.4 Definition. We define

C_{ϕ,f} := ϕ(D_{V}) ∘ T ∘ ϕ(D′_{V′}): L^{2}(M′)^{∗} → L^{2}(M)^{∗}.

If ϕ is an even function, ϕ(D_{V}) is a function of the Laplacian D_{V}^{2} = Δ and therefore preserves the degree of the differential forms. Consequently, the same is true for C_{ϕ,f}.

9.5 Lemma. The operator C_{ϕ,f} is an integral A-linear operator with smooth kernel on M × M′. Moreover, if ϕ is even then d_{V} ∘ C_{ϕ,f} = C_{ϕ,f} ∘ d′_{V′}.

Proof. Since T is bounded and ϕ(D_{V}), ϕ(D′_{V′}) ∈ Ψ_{A}^{−∞} we immediately get that C_{ϕ,f} is a A-linear smoothing operator. We know that d_{V}g^{∗}ω = g^{∗}(d′_{V′}ω) for any smooth map g. Moreover, d_{V} commutes with (d_{V}+d_{V}^{∗})^{2} = D_{V}^{2} and therefore also with ϕ(D_{V}) since ϕ is even; similarly d′_{V′} commutes with ϕ(D′_{V′}). Since

C_{ϕ,f}(ω) = ϕ(D_{V}) ∘ (∫_{D^{N}} v ∧ F^{∗}ϕ(D′_{V′})(ω))

6This result, the homotopy invariance of the signature index class, has been established by several people and with different techniques. Mishchenko and Kasparov prove it by showing its equality with the C^{∗}-algebraic Mishchenko symmetric signature, an a-priori homotopy invariant. Kaminker-Miller give a more analytical treatment, adapting to the noncommutative context the proof that Lustzig gives in the case Γ = ℤ^{k}; Hilsum -Skandalis prove the homotopy invariance in a purely analytical fashion; we shall follow their approach. For a thorough treatment of the homotopy invariance of the signature index class and its connections with surgery theory we refer the reader to the recent papers by Higson and Roe [18]

we also get d_{V} ∘ C_{ϕ,f} = C_{ϕ,f} ∘ d′_{V′} as required, as explained in [20, Proof of Theorem 3.3 on p. 90], where we use that the form v is closed.

9.6 Definition. We use the usual inner product (coming from a fixed Riemannian metric) on the space of differential forms. We use ^{∗} for the adjoint with respect to this inner product.

The involution τ coming from the Hodge-∗ operator (compare Appendix B) and the above inner product define the signature quadratic form on the space of differential forms, we use ^{†} for the adjoint with respect to this quadratic form. Recall that

A^{∗} = τA^{†}τ, A^{†} = τA^{∗}τ.

Let now ϕ_{ϵ} be a smooth even function ϕ_{ϵ}: ℝ → ℝ such that ϕ_{ϵ}(t) = 1 if |t| ≤ ϵ/4, and ϕ_{ϵ}(t) = 0 if |t| ≥ ϵ/2. Recall that f: M → M′ was an orientation preserving homotopy equivalence and write

T_{ϵ} := C_{ϕ_{ϵ},f}: Ω^{∗}(M′, V′) → Ω^{∗}(M, V).

Clearly T_{ϵ} extends to an L^{2}-bounded A-linear operator. Let T_{ϵ}^{∗} be the adjoint of T_{ϵ} and define T_{ϵ}^{†} := τ′T_{ϵ}^{∗}τ, with τ and τ′ denoting the involutions defined by the Hodge ⋆-operators on M and M′ (see Appendix B). Introduce a new differential d by setting dα := i^{|α|}d_{V}α and similarly for d′.

9.7 Lemma. The bounded operator T_{ϵ} verifies the following properties:

(a) T_{ϵ}(dom(d)) ⊂ dom(d′); T_{ϵ}d′ = dT_{ϵ}

(b) T_{ϵ} induces an isomorphism in cohomology, with inverse induced by T_{ϵ}^{†}

(c) there exists a bounded operator y_{ϵ} of degree −1 on L^{2}(M′)^{∗}, with y_{ϵ}^{†} = −y_{ϵ} and such that y_{ϵ}(dom(d′)) ⊂ dom(d′) and 1 − T_{ϵ}^{†}T_{ϵ} = d′y_{ϵ} + y_{ϵ}d′

(d) y_{ϵ} = y_{1} + y_{2} where y_{1} is ϵ-spectrally concentrated and y_{2} commutes with Δ′.

(e) T_{ϵ} is ϵ-spectrally concentrated.

Proof. The results (a) to (c) for the operator T are proved by Hilsum-Skandalis. We need first of all to extend them to T_{ϵ}. Observe that Lemma 9.5 is precisely assertion (a). Next, we observe that the T of Hilsum-Skandalis and our T_{ϵ} are chain homotopic. Indeed, recall that T_{ϵ} = ϕ_{ϵ}(d_{V} +d_{V}^{∗})∘T ∘ϕ_{ϵ}(d′_{V′} +(d′_{V′})^{∗}). Here, we can replace D_{V} by d_{V} + d_{V}^{∗}, since D_{V}^{2} = (d_{V} + d_{V}^{∗})^{2}, and ϕ is even. Choose a homotopy ϕ_{ϵ}^{t} of even functions with ϕ_{ϵ}^{0} = ϕ_{ϵ}, ϕ_{ϵ}^{1} = 1 and such that ϕ_{ϵ}^{t}(x) = 1 for all t ∈ [0, 1] and |x| < ϵ/4. Then \frac{d}{dt}ϕ_{t}(x) = xg_{t}(x) = g_{t}(x)x with a suitable smooth odd family of functions g_{t}: ℝ → ℝ. Observe that

\begin{matrix}T − T_{ϵ} = & ∫_{0}^{1} \frac{d}{dt} (ϕ_{t}(d_{V} + d_{V}^{∗}) ∘ T ∘ ϕ_{t}(d′_{V} + (d′_{V′})^{∗})) dt \\ = & (d_{V} + d_{V}^{∗}) ∫_{0}^{1} (g_{t}(d_{V} + d_{V}^{∗}) ∘ T ∘ ϕ_{t}(d′_{V′} + (d′_{V′})^{∗})) dt \\ + ∫_{0}^{1} (ϕ_{t}(d_{V} + d_{V}^{∗}) ∘ T ∘ g_{t}(d′_{V′} + (d′_{V′})^{∗})) dt ∘ (d′_{V′} + (d′_{V′})^{∗}) \\ = & d_{V}z + zd′_{V′} = dw + wd′ \end{matrix}  (9.8)

with  z = ∫_{0}^{1} g_{t}(d_{V} + d_{V}^{∗}) ∘ T ∘ ϕ_{t}(d′_{V′} + (d′_{V′})^{∗})dt
+ ∫_{0}^{1} ϕ_{t}(d_{V} + d_{V}^{∗}) ∘ T ∘ g_{t}(d′_{V′} + (d′_{V′})^{∗})dt,
w(ω) =i^{1−|ω|}z(ω).

The last but one equality in (9.8) is true since d_{V}^{∗}g_{t}(d_{V} + d_{V}^{∗}) = g_{t}(d_{V} + d_{V}^{∗})d and d_{V}T = td′_{V}.

Since by [20] T induces an isomorphism in homology, and T_{ϵ} is chain homotopic to T, so does T_{ϵ}, proving (b).

Apply now ·^{†} to Equation (9.8) and use that d^{†} = −d to get

T^{†} − T_{ϵ}^{†} = −w^{†}d′ − dw^{†}.

Let now y be an operator such that 1 − T^{†}T = dy + yd′, as constructed in [20]. Then

\begin{matrix}1 − T_{ϵ}^{†}T_{ϵ} = & 1 − T^{†}T + (T^{†} − T_{ϵ}^{†})T + T_{ϵ}^{†}(T − T_{ϵ}) \\ = & dy + yd′ + (−dw^{†} − w^{†}d′)T + T_{ϵ}^{†}(dw + wd′) \\ = & d(y − w^{†}T + T_{ϵ}^{†}w) + (y − w^{†}T + T_{ϵ}^{†}w)d′, \end{matrix}  (9.9)

using that T and T_{ϵ} are chain maps, and therefore in particular T_{ϵ}^{†}d′ = −T_{ϵ}^{†}(d′)^{†} = −d^{†}T_{ϵ}^{†} = dT_{ϵ}^{†}. Equation (9.9) means that we get (c) with ỹ_{ϵ} = y − w^{†}T + T_{ϵ}^{†}w.

We will now modify ỹ_{ϵ} in such a way that it splits as required in (d). To do this, we make the following general observation. Assume that ψ_{1} : ℝ → ℝ vanishes in a neighborhood of 0 and write ψ_{1}(x) = xψ_{2}(x) with a smooth function ψ_{2}. Define

u := (d′_{V′})^{∗}ψ_{2}(Δ′) = ψ_{2}(Δ′)(d′_{V′})^{∗}.  (9.10)

Then

d′_{V′}u + ud′_{V′} = d′_{V}(d′_{V′})^{∗}ψ_{2}(Δ′) + (d′_{V′})^{∗}d′_{V′}ψ_{2}(Δ′) = ψ_{1}(Δ′).  (9.11)

Choose ψ: ℝ → ℝ with support contained in [ϵ/2, ∞) and such that ψ(x) = 1 for x ≥ ϵ. Define now

y_{1} := (1 − ψ)(Δ′) ∘ ỹ_{ϵ} ∘ (1 − ψ)(Δ′).

By construction, this operator is ϵ-spectrally concentrated. Set y_{2} := ψ(Δ′)ỹ_{ϵ}(1 − ψ)(Δ′), y_{3} := ỹ_{ϵ}ψ(Δ′). We compute (since T_{ϵ}^{†} and T_{ϵ} are ϵ/2-spectrally concentrated)

d′y_{2} + y_{2}d′ = d′ ∘ (ψ(Δ′)ỹ_{ϵ}(1 − ψ)(Δ′)) + (ψ(Δ′)ỹ_{ϵ}(1 − ψ)(Δ′)) ∘ d′
=ψ(Δ′)(d′ỹ_{ϵ} + ỹ_{ϵ}d′)(1 − ψ)Δ′)
=ψ(Δ′) ∘ (1 − ψ)(Δ′)
=d′ ∘ u_{1} + u_{1}d′

with a suitable operator u_{1} defined as in (9.10) and (9.11) which commutes with Δ′. Similarly,

d′y_{3} + y_{3}d′ = d′ ∘ ỹ_{ϵ}ψ(Δ′) + ỹ_{ϵ}ψ(Δ′) ∘ d′ = d′u_{2} + u_{2}d′

with a u_{2} which also commutes with Δ′.

Consequently,

\begin{matrix}1 − T_{ϵ}^{†}T_{ϵ} = & d′ỹ_{ϵ} + ỹ_{ϵ}d′ \\ = & d′(y_{1} + y_{2} + y_{3}) + (y_{1} + y_{2} + y_{3})d′ \\ = & d′(y_{1} + u_{1} + u_{2}) + (y_{1} + u_{1} + u_{2})d′. \end{matrix}

We now set y_{ϵ} := y_{1} + u_{1} + u_{2} and observe that (u_{1} + u_{2}) commutes with Δ′. This proves (d).

(e) is an immediate consequence of the construction of T_{ϵ}.

Following closely [20], consider now the operators on L^{2}(M′)^{∗} ⊕ L^{2}(M)^{∗}:

R_{ϵ} = (\begin{matrix}1 & 0 \\ −T_{ϵ}γ & 1 \end{matrix}) , L_{ϵ,α} = (\begin{matrix}1 − T_{ϵ}^{†}T_{ϵ} & (γ + αy_{ϵ})T_{ϵ}^{†} \\ T_{ϵ}(−γ − αy_{ϵ}) & 1 \end{matrix}) , δ_{ϵ,α} = (\begin{matrix}d′ & αT_{ϵ}^{†} \\ 0 & −d \end{matrix})  (9.12)

with γω := (−1)^{|ω|}ω and α a real number ^{7} . Note that γ anticommutes with d and τ and commutes with T and T^{†}, and that γ^{†} = −γ. It is clear that R_{ϵ} and L_{ϵ,α} are bounded. The crucial relation is L_{ϵ,α}δ_{ϵ,α} = −δ_{ϵ,α}^{†}L_{ϵ,α}.

7The differences in signs between these operators and the ones appearing in [20] are due to the fact that we take τ′ for the grading on M′, whereas Hilsum-Skandalis take −τ′

We notice that R_{ϵ} is invertible and that R_{ϵ}^{†}R_{ϵ} = L_{ϵ,0}. Thus L_{ϵ,α} is invertible for |α| small enough. Let

S_{ϵ,α} := \frac{τ∘L_{ϵ,α}}{|τ∘L_{ϵ,α}|}, with τ = (\begin{matrix}τ′ & 0 \\ 0 & τ \end{matrix}). Then S_{ϵ,α} is an involution. We now endow Ω^{∗}(M′, V′) ⊕ Ω^{∗}(M, V) with

the new inner product

< ω_{1}, ω_{2} >_{ϵ,α}:=< ω_{1}, |τ ∘ L_{ϵ,α}|ω_{2} > .

Notice that |τ ∘ L_{ϵ,α}| is positive and self-adjoint with respect to both scalar-products. Let D̂_{ϵ,α} := −i(δ_{ϵ,α}S_{ϵ,α} + S_{ϵ,α}δ_{ϵ,α}) be the signature operator associated to δ_{ϵ,α} and to the grading S_{ϵ,α}. Using [20, Lemme 2.1], D̂_{ϵ,α} is invertible and self-adjoint with respect to < ·, · >_{ϵ,α}. For the adjoint with respect to the original inner product we therefore get:

(D̂)_{ϵ,α}^{∗} == (|τ ∘ L_{ϵ,α}|)^{−1} ∘ D̂_{ϵ,α} ∘ (|τ ∘ L_{ϵ,α}|) .

9.13 Definition. We define the special perturbed signature operator

D_{ϵ,α} := −i (δ_{ϵ,α}τL_{ϵ,α} + \frac{τL_{ϵ,α}}{|τL_{ϵ,α}|}δ_{ϵ,α} \frac{τL_{ϵ,α}}{|τL_{ϵ,α}|}τL_{ϵ,α})

with

δ_{ϵ,α} = (\begin{matrix}d & αT_{ϵ}^{†} \\ 0 & −d′ \end{matrix}) , L_{ϵ,α} = (\begin{matrix}1 − T_{ϵ}^{†}T_{ϵ} & (γ + αy_{ϵ})T_{ϵ}^{†} \\ T_{ϵ}(−γ − αy_{ϵ}) & 1 \end{matrix})

Then D_{ϵ,α} := D̂_{ϵ,α} ∘ |τ ∘ L_{ϵ,α}| is self-adjoint with respect the original inner product and is invertible for α ≠ 0 sufficiently small (depending on ϵ). We complete the proof of the Theorem with the following

9.14 Lemma. Consider the operators of Definition 9.13. Then δ̃_{ϵ,α} := δ_{ϵ,α} − ( \begin{matrix}d & 0 \\ 0 & −d′ \end{matrix} ) and τL_{ϵ,α} − τ are ϵ-spectrally concentrated. Consequently, \frac{τL_{ϵ,α}}{|τL_{ϵ,α}|} − τ and D_{ϵ,α} − D are ϵ-spectrally concentrated (recall that |τ| = 1).

Moreover, all these differences belong to Ψ_{A}^{−∞}, in particular D_{ϵ,α} − D ∈ Ψ_{A}^{−∞}.

Proof. First of all, using the definition of L_{ϵ,α} and Lemma 9.7 we claim that L_{ϵ,α} = 1+Θ_{ϵ,α} with Θ_{ϵ,α} ∈ Ψ_{A}^{−∞} and ϵ-spectrally concentrated. Indeed

L_{ϵ,α} = 1 + (\begin{matrix}−T_{ϵ}^{†}T_{ϵ} & iγT_{ϵ}^{†} \\ −T_{ϵ}γ & 0 \end{matrix}) + (\begin{matrix}0 & αyT_{ϵ}^{†} \\ −T_{ϵ}αy & 0 \end{matrix})

= 1 + Θ_{ϵ,α}^{1} + (\begin{matrix}0 & αy_{1}T_{ϵ}^{†} \\ −T_{ϵ}αy_{1} & 0 \end{matrix}) + (\begin{matrix}0 & αy_{2}T_{ϵ}^{†} \\ −T_{ϵ}αy_{2} & 0 \end{matrix})

= 1 + Θ_{ϵ,α}^{1} + Θ_{ϵ,α}^{2} + + (\begin{matrix}0 & αy_{2}T_{ϵ}^{†} \\ −T_{ϵ}αy_{2} & 0 \end{matrix})

with Θ_{ϵ,α}^{j}, j = 1, 2 smoothing and ϵ-spectrally concentrated because of lemma 9.7. It remains to be checked that the last term appearing in the above formula is also smoothing and ϵ-spectrally concentrated: but since y_{2} commutes with Δ′, it also commutes with ϕ(D′_{V′}) if ϕ is even; using the very definition of T_{ϵ} we then get that

Θ_{ϵ,α}^{3}  =  (\begin{matrix}0 & αϕ_{ϵ}(D_{V}) ∘ y_{2,ϵ} ∘ T ∘ ϕ_{ϵ}(D′_{V′}) \\ −αϕ_{ϵ}(D_{V})T ∘ ∘y_{2,ϵ} ∘ ϕ_{ϵ}(D′_{V′}) & 0 \end{matrix})

is smoothing and ϵ-spectrally concentrated. Our claim now follows with Θ_{ϵ,α} = Θ_{ϵ,α}^{1} + Θ_{ϵ,α}^{2} + Θ_{ϵ,α}^{3}. Thus τ ∘ L_{ϵ,α} = τ + Φ_{ϵ,α}, with Φ_{ϵ,α} ∈ Ψ_{A}^{−∞} and ϵ-spectrally concentrated. Let us now consider |τ ∘ L_{ϵ,α}| = \sqrt{(τ ∘ L_{ϵ,α})^{2}}. We can write

|τ ∘ L_{ϵ,α}| = \frac{i}{2π} ∫_{C} λ^{\frac{1}{2}} ((τ ∘ L_{ϵ,α})^{2} − λ)^{−1}dλ

with C equal to a circle of radius larger than the norm of (τ ∘ L_{ϵ,α})^{2}. From this integral representation it follows that |τ ∘ L_{ϵ,α}| = 1 + Λ_{ϵ,α} with Λ_{ϵ,α} ∈ Ψ_{A}^{−∞} and ϵ-spectrally concentrated; indeed, (τ ∘ L_{ϵ,α})^{2} − λ is equal to (1 − λ) plus a ϵ-spectrally concentrated smoothing operator. Thus by Lemma A.12, its inverse will be of type (1 − λ)^{−1}(1 + Ξ_{ϵ,α}(λ)) with Ξ_{ϵ,α}(λ) ∈ Ψ_{A}^{−∞} and ϵ-spectrally concentrated; carrying out the integral we obtain immediately that |τ ∘ L_{ϵ,α}| = 1 + F_{ϵ,α} with F_{ϵ,α} ∈ Ψ_{A}^{−∞} and ϵ-spectrally concentrated. It then follows that the same is true for |τ ∘ L_{ϵ,α}|^{−1}. Thus for S_{ϵ,α} we obtain: S_{ϵ,α} = τ + G_{ϵ,α} with G_{ϵ,α} smoothing and ϵ-spectrally concentrated. Consider now D̃; this equals −i(δ_{ϵ,α} ∘ S_{ϵ,α} + S_{ϵ,α} ∘ δ_{ϵ,α})|τ ∘ L_{ϵ,α}|. By its very definition and Lemma 9.7, δ_{ϵ,α} = d + E_{ϵ,α} with E_{ϵ,α} smoothing and ϵ-spectrally concentrated. Thus

D_{ϵ,α} = −i ((d + E_{ϵ,α}) ∘ (τ + G_{ϵ,α}) + (τ + G_{ϵ,α}) ∘ (d + E_{ϵ,α})) ∘ (1 + F_{ϵ,α}),

which is equal to D plus a smoothing operator ϵ-spectrally concentrated.

Theorem 9.2 is proved.

10 From stable to unstable eta-invariants

We continue with the setting of Section 9. In this section, we analyze the τ-η-invariant of the special perturbations D_{ϵ,α} of D, where we use the notation and conventions of Section 8. In particular, A is a von Neumann algebra and τ: A → Z a positive normal trace with values in the commutative von Neumann algebra Z.

10.1 Limits of eta invariants

10.1 Theorem. Let f: M′ → M be a smooth oriented homotopy equivalence between closed Riemannian oriented manifolds as in Section 9. This gives rise to twisted signature operators D_{L} on M and D′_{L′} on M′, with L′ := f^{∗}L. Consider the (unperturbed) τ-eta invariants η_{τ} (D_{L}), η_{τ} (D′_{L′}) ∈ Z, see 8.1.

Assume, in addition to the above, that the von Neumann algebra A admits a positive faithful normal trace τ_{A}: A → ℂ.

There are sequences ϵ_{k} > 0 and α_{k} > 0 such that α_{k} is small enough for D_{ϵ_{k},±α_{k}} to be an invertible perturbation of D and such that

lim_{k→∞} \frac{η_{τ} (D_{ϵ_{k},α_{k}}) + η_{τ} (D_{ϵ_{k},−α_{k}})}{2} = η_{τ} (D).

Here, D_{ϵ,α} is the special smoothing perturbation of the signature operator D on the manifold M ⨿ (−M′) as defined in Definition 9.13. Note that on the left hand side we have a sequences of averaged perturbed eta-invariants, converging to the unperturbed eta-invariant on the right hand side.

Proof. We shall prove that there are sequences ϵ_{k} > 0 and α_{k} > 0 such that

lim_{k→∞} η_{τ} (D_{ϵ_{k},α_{k}}) = η_{τ} (D) + σ , lim_{k→∞} η_{τ} (D_{ϵ_{k},−α_{k}}) = η_{τ} (D) − σ , σ ∈ ℝ

Let p_{ϵ} := χ_{[−ϵ,ϵ]}(Δ), where χ_{Y} is the characteristic function of a set Y.

Since D_{ϵ,α} is an ϵ-perturbation of D, we get

D_{ϵ,α} = p_{ϵ}D_{ϵ,α}p_{ϵ} + (1 − p_{ϵ})D_{ϵ,α}(1 − p_{ϵ}), (1 − p_{ϵ})D_{ϵ,α}(1 − p_{ϵ}) = (1 − p_{ϵ})D(1 − p_{ϵ}).

Consequently, by the spectral theorem and the definition of η_{τ}

η_{τ} (D_{ϵ,α}) = η_{τ} (p_{ϵ}D_{ϵ,α}p_{ϵ}) + η_{τ} ((1 − p_{ϵ})D(1 − p_{ϵ})).  (10.2)

To analyze η_{τ} of the compressed operators, we will make use of the following translation into simple functional calculus.

10.3 Lemma. Let p and B be self-adjoint bounded Hilbert A-module morphisms on a Hilbert A-module H, p a projection, such that τ(p) < ∞ and such that B = pBp. Let q: ℝ → ℝ be defined by q(x) = 1 for x > 0, q(0) = 0 and q(x) = −1 for x < 0. We define

sgn_{τ} (B) := τ(q(B)).  Then η_{τ} (B) = sgn_{τ} (B).  (10.4)

Proof. We simply observe that

q(x) = \frac{1}{\sqrt{π}} ∫_{0}^{∞} x exp(−tx^{2}) \frac{dt}{\sqrt{t}} = \frac{2}{π} ∫_{0}^{∞} x exp(−t^{2}x^{2}) dt,

and |q(x)| ≤ 1, therefore q(B) = \frac{1}{π} ∫_{0}^{∞} B exp(−tB^{2}) dt/\sqrt{t}. Here we observe that ∫_{0}^{T} B exp(−t^{2}B^{2}) dt is defined for each T ≥ 0 since B is bounded and is equal to the functional calculus of B applied to ∫_{0}^{T} x exp(−t^{2}x^{2}) dt. The latter family of functions, depending on the parameter T, converges pointwise to q, therefore the limit ∫_{0}^{∞} B exp(−t^{2}B^{2}) dt exists in the von Neumann algebra of Hilbert A-module morphisms on H and equals q(B). Finally observe that we can throughout write pBp instead of B. By assumption p is of τ-trace class. Since those operators form an ideal, it follows that both pBp and pBp exp(−t^{2}B^{2}) are τ-trace class. Normality implies that for a strongly convergent sequence X_{n} of operators, τ(pX_{n}) ⟶^{n→∞} τ(pX). We can now interchange in the above argument integration, passage to limit and application of τ, and the statement follows.

10.5 Lemma.

lim_{ϵ→0} η_{τ} ((1 − p_{ϵ})D(1 − p_{ϵ})) = η_{τ} (D),

where existence of the limit is part of the statement.

Proof. We have discussed in Section 8 that each of the integrals defining the eta-invariants exists in the above statement. Moreover, for 0 < ϵ < 1,

η_{τ} ((1 − p_{ϵ})D(1 − p_{ϵ})) = η_{τ} ((1 − p_{1})D(1 − p_{1})) + η_{τ} ((p_{1} − p_{ϵ})D(p_{1} − p_{ϵ})).

Finally observe that τ(p_{1}) ≤ τ(5 exp(−D)^{2}) < ∞ and that p_{1}(p_{1} − p_{ϵ})D(p_{1} − p_{ϵ})p_{1} = (p_{1} − p_{ϵ})D(p_{1} − p_{ϵ}), so that Lemma 10.3 implies that η_{τ} ((p_{1} − p_{ϵ})D(p_{1} − p_{ϵ})) = sgn_{τ} ((p_{1} − p_{ϵ})D(p_{1} − p_{ϵ})). Finally, p_{ϵ} strongly converges to p_{0}, so that q((p_{1} − p_{ϵ})D(p_{1} − p_{ϵ})) strongly converges to q((p_{1} − p_{0})D(p_{1} − p_{0})). By normality of τ, therefore

sgn_{τ} ((p_{1} − p_{ϵ})D(p_{1} − p_{ϵ})) = τ(q((p_{1} − p_{ϵ})D(p_{1} − p_{ϵ}))) ⟶^{ϵ→0} sgn_{τ} ((p_{1} − p_{0})D(p_{1} − p_{0}))

Finally, by definition and Lemma 10.3

η_{τ} (D) = η_{τ} ((1 − p_{1})D(1 − p_{1})) + sgn_{τ} ((p_{1} − p_{0})D(p_{1} − p_{0})).

10.6 Lemma. There are sequences ϵ_{k} > 0 and α_{k} > 0 such that α_{k} is small enough for D_{ϵ_{k},α_{k}} to be smoothing invertible perturbations of D and such that

sgn_{τ} (p_{ϵ_{k}}D_{ϵ_{k},±α_{k}}p_{ϵ_{k}}) ⟶^{k→∞} sgn_{τ} (D_{0,±1})

Here

D_{0,α} := −ip_{0} (δ_{0,α}τL_{0} + \frac{τL_{0}}{|τL_{0}|}δ_{0,α} \frac{τL_{0}}{|τL_{0}|}τL_{0}) p_{0}

with

δ_{0,α} = p_{0} (\begin{matrix}0 & αT_{0}^{†} \\ 0 & 0 \end{matrix}) p_{0}, L_{0} = (\begin{matrix}1 − T_{0}^{†}T_{0} & γT_{0}^{†} \\ −T_{0}γ & 1 \end{matrix}) , T_{0} = p_{0}T p_{0}.

In particular, D_{0,α} = αD_{0,1} and therefore

sgn_{τ} (D_{0,1}) = − sgn_{τ} (D_{0,−1}).  (10.7)

Assuming Lemma 10.6, we can now finish the proof of Theorem 10.1. Observe that

η_{τ} (D_{ϵ_{k},±α_{k}}) = \underbrace{η_{τ} ((1 − p_{ϵ_{k}})D_{ϵ_{k},±α_{k}}(1 − p_{ϵ_{k}}))}_{=η_{τ} (1−p_{ϵ_{k}})D(1−p_{ϵ_{k}})⟶^{k→∞}η_{τ} (D)} + \underbrace{η_{τ} (p_{ϵ_{k}}D_{ϵ_{k},±α_{k}}p_{ϵ_{k}})}_{⟶^{k→∞}η_{τ} (D_{0,±1})}

where we use Lemma 10.5 for the first convergence statement, and Lemma 10.6 for the second. Averaging the two resulting equations for α_{k} and −α_{k} and using Equation (10.7) gives the statement of the Theorem.

10.8 Remark. We leave it to the reader to check directly that η_{τ} (D_{0,1}) = 0. Using the same proof as above, this gives the following stronger version of Theorem 10.1:

lim_{k→∞} η_{τ} (D_{ϵ_{k},α_{k}}) = η_{τ} (D).

10.2 Proof of Lemma 10.6

Define for α ≠ 0

C_{ϵ,α} := |α|^{−1} p_{ϵ}D_{ϵ,α}p_{ϵ}

and observe that sgn_{τ} (C_{ϵ,α}) = sgn_{τ} (p_{ϵ}D_{ϵ,α}p_{ϵ}). Set for α ≠ 0

C̃_{ϵ,α} := −i |α|^{−1} p_{ϵ} (δ̃_{ϵ,α}τL_{ϵ,α} + \frac{τL_{ϵ,α}}{|τL_{ϵ,α}|}δ̃_{ϵ,α} \frac{τL_{ϵ,α}}{|τL_{ϵ,α}|}τL_{ϵ,α}) p_{ϵ}

with

δ̃_{ϵ,α} = (\begin{matrix}0 & αT_{ϵ}^{†} \\ 0 & 0 \end{matrix}) , L_{ϵ,α} = (\begin{matrix}1 − T_{ϵ}^{†}T_{ϵ} & (γ + αy_{ϵ})T_{ϵ}^{†} \\ T_{ϵ}(−γ − αy_{ϵ}) & 1 \end{matrix}) .

Note that C̃_{ϵ,α} is in general neither self adjoint nor invertible.

10.9 Lemma. There is a constant K > 0 such that the following holds. For each ϵ > 0 there exists α_{ϵ} > 0 such that

∥α_{ϵ}y_{ϵ}∥ < ϵ,  ∥L_{ϵ,±α_{ϵ}}^{−1} ∥ ≤ K.

Proof. We first observe that there is a uniform bound for ∥T_{ϵ}∥. This implies a uniform bound for ∥L_{ϵ,0}∥ and

for ∥L_{ϵ,0}^{−1}∥, using L_{ϵ,0} = ( \begin{matrix}1 & γT_{ϵ}^{†} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ −T_{ϵ}γ & 1 \end{matrix} ), so that (L_{ϵ,0})^{−1} = ( \begin{matrix}1 & 0 \\ T_{ϵ}γ & 1 \end{matrix} ) ( \begin{matrix}1 & −γT_{ϵ}^{†} \\ 0 & 1 \end{matrix} ). Since each y_{ϵ} is bounded

and L_{ϵ,α} = L_{ϵ,0} + α ( \begin{matrix}0 & y_{ϵ}T_{ϵ}^{†} \\ −T_{ϵ}y_{ϵ} & 0 \end{matrix} ) , we can now choose α_{ϵ} such that all the assertions are fulfilled.

10.10 Lemma. Choose a sequence ϵ_{k} > 0 with ϵ_{k} ⟶^{k→∞} 0, and choose α_{k} := α_{ϵ_{k}} > 0 as given by Sublemma 10.9. Then C̃_{ϵ_{k},±α_{k}} strongly converges to D_{0,±1}.

Proof. For ϵ → 0, p_{ϵ} strongly converges to p_{0}, T_{ϵ} strongly converges to T_{0}, and T_{ϵ}^{†} strongly converges to T_{0}^{†}. ∥α_{ϵ}y_{ϵ}∥ ⟶^{ϵ→0} 0. Since products of strongly convergent sequences strongly converge to the product of the limits, by its construction L_{ϵ_{k},α_{k}} strongly converges to L_{0}. Moreover, since ∥(τL_{ϵ_{k},α_{k}})^{−1}∥ ≤ K independent of k ∈ ℕ, \frac{τL_{ϵ_{k},α_{k}}}{|τL_{ϵ_{k},α_{k}}|} = f(τL_{ϵ_{k},α_{k}}) (and \frac{τL_{0,0}}{|τL_{0,0}|} = f(τL_{0,0})) for any continuous function f with f(x) = −1 for

x ≤ −1/K, f(x) = 1 for x ≥ 1/K. Since bounded continuous functions of strongly convergent sequences strongly converge by [53, Theorem VIII.20], \frac{τL_{ϵ_{k},α_{k}}}{|L_{ϵ_{k},α_{k}}|} strongly converges to \frac{τL_{0,0}}{|τL_{0,0}|}. Putting everything

together, and using in particular that |α|^{−1} δ̃_{ϵ,α} = |α|^{−1} ( \begin{matrix}0 & αT_{ϵ}^{†} \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}0 & ±T_{ϵ}^{†} \\ 0 & 0 \end{matrix} ) , the result follows.

10.11 Lemma. There is a sequence ϵ_{k} > 0, with corresponding α_{k} as in Sublemma 10.10, such that C_{ϵ_{k},±α_{k}} converges in the strong resolvent sense to D_{0,±1}. Here, we will use the positive finite faithful normal trace τ_{A}: A → ℂ.

Proof. We will construct a sequence of monotonously increasing projections Q_{1} ≤ Q_{2} ≤ ⋯ with sup_{k} Q_{k} = p_{1}. The latter property will be ensured showing that for all k Q_{k} ≤ p_{1}, and that τ_{A}(Q_{k}) ⟶^{k→∞} τ_{A}(p_{1}) < ∞, therefore τ_{A}(p_{1} − (sup_{k} Q_{k})) = 0, so that p_{1} = sup_{k} Q_{k}, using faithfulness and normality of τ_{A}.

The main property of the Q_{k} will be that

C̃_{ϵ_{n},±α_{n}}Q_{k} = C_{ϵ_{n},±α_{n}}Q_{k}  for all n ≥ k.  (10.12)

The assertion of the Sublemma follows then from Sublemma 10.10. To see this, we can represent A as bounded operators on a Hilbert space H. Since C̃_{ϵ_{k},±α_{k}}(1 − p_{1}) = C_{ϵ_{k},±α_{k}}(1 − p_{1}) = D_{0,±1}(1 − p_{1}) = 0, it suffices to study the restriction to im(p_{1}). Now all operators in question are bounded, therefore ⋃_{k∈ℕ}(im(Q_{k})) will be a common core for all of them (since sup_{k} Q_{k} = p_{1}, ⋃_{k∈ℕ} im(Q_{k}) is dense in im(p_{1})). A reference for the lattice of projections and properties of it we are using here is [23, 24], in particular Section 2.5. Finally, for each v ∈ ⋃_{k} im(Q_{k}) there is a k ∈ ℕ such that v ∈ im(Q_{k}). Then, for n ≥ k, C_{ϵ_{n},±α_{n}}v = C̃_{ϵ_{n},±α_{n}}v ⟶^{n→∞} D_{0,±1}v. By [53, Theorem VIII.25], C_{ϵ_{n},±α_{n}} converges in the strong resolvent sense to D_{0,±1}..

We now tackle the construction of the projections Q_{k}. Choose ϵ_{k} such that τ_{A}(p_{ϵ_{k}} − p_{0}) < 10^{−k}.

Restrict now all operators in question to im(p_{ϵ_{k}}). Note that A_{k} := τL_{ϵ_{k},α_{ϵ_{k}}} now is an invertible operator,

mapping im(p_{epsilon_{k}}) to itself, and the same is true for B_{k} := \frac{L_{ϵ_{k},αϵ_{k}}}{|L_{ϵ_{k},αϵ_{k}} |}L_{ϵ_{k},α_{ϵ_{k}}}. Choose maximal projections

Q_{A_{k}} and Q_{B_{k}} such that

p_{0}A_{k}Q_{A_{k}} = A_{k}Q_{A_{k}} and p_{0}B_{k}Q_{B_{k}} = B_{k}Q_{B_{k}}.  (10.13)

In other words, Q_{A_{k}} is the projection onto the inverse image under A_{k} of im(p_{0}). Set Q_{k}^{0} := inf{p_{0}, Q_{A_{k}}, Q_{B_{k}}}. Observe that, by construction, δ̃_{ϵ,α}p_{0} = δ_{ϵ,α}p_{0}. Consequently,

C̃_{ϵ_{k},α_{k}}Q_{k}^{0} = C_{ϵ_{k},α_{k}}Q_{k}^{0}  ∀k ∈ ℕ.  (10.14)

Finally, we set Q_{k} := (inf_{l≥k} Q_{l}^{0}) + p_{1} − p_{ϵ_{k}}. Observe that (10.12) immediately follows from (10.14) and the fact that C̃_{ϵ_{k},α_{k}}(p_{1} − p_{ϵ_{k}}) = C_{ϵ_{k},α_{k}}(p_{1} − p_{ϵ_{k}}) = 0. Note that τ_{A}(Q_{A_{k}}) = τ_{A}(p_{0}) = τ_{A}(Q_{B_{k}}). This follows since instead of Q_{A_{k}} we can first construct the idempotent e_{k} := A_{k}^{−1}p_{0}A_{k}; it satisfies the relation (10.13) and τ_{A}(e_{k}) = τ_{A}(p_{0}). Use then the standard construction of a selfadjoint projection Q_{A_{k}} with e_{k}Q_{A_{k}} = Q_{A_{k}} and Q_{A_{k}}e_{k} = e_{k} (compare e.g. [7, Theorem 2.1]), then τ_{A}(Q_{A_{k}}) = τ_{A}(e_{k}Q_{A_{k}}) = τ_{A}(Q_{A_{k}}e_{k}) = τ_{A}(e_{k}), and p_{0}A_{k}Q_{A_{k}} = p_{0}A_{k}e_{k}Q_{A_{k}} = A_{k}e_{k}Q_{A_{k}} = A_{k}Q_{A_{k}}.

Note that each of the projections Q_{A_{k}}, Q_{B_{k}}, p_{0} are bounded above by p_{ϵ_{k}}, with τ_{A}(Q_{A_{k}}) = τ_{A}(Q_{B_{k}}) = τ_{A}(p_{0}), τ_{A}(p_{ϵ_{k}}) < τ(p_{0}) + 10^{−k}. The Kaplansky formula [24, Theorem 6.1.7]

p_{0} − inf{Q_{A_{k}}, p_{0}} = sup{p_{0}, Q_{A_{k}}} − Q_{A_{k}} ≤ p_{ϵ_{k}} − Q_{A_{k}},

linearity and positivity of τ_{A} imply τ_{A}(Q_{k}^{0}) ≥ τ_{A}(p_{0}) − 2 · 10^{−k}, and similarly,

τ_{A}(inf_{l≥k} Q_{l}^{0}) ≥ τ_{A}(p_{0}) − 2 · 10^{−k} ∑_{l=0}^{∞} 10^{−l} ≥ τ_{A}(p_{0}) − 4 · 10^{−k}.

It follows that

τ_{A}(Q_{k}) ≥ τ_{A}(p_{0}) − 4 · 10^{−k} + τ_{A}(p_{1} − p_{ϵ_{k}}) ⟶^{k→∞} τ_{A}(p_{0}) + τ_{A}(p_{1} − p_{0}) = τ_{A}(p_{1}).

Since Q_{k} ≤ p_{1}, the properties of Q_{k} and therefore Sublemma 10.11 follow.

10.15 Sublemma. Assume that self adjoint operators A_{n} converge in the strong resolvent sense to the self adjoint operator A. Assume that χ_{{0}}(A_{n}) strongly converges to χ_{{0}}(A). Then q(A_{n}) strongly converges to q(A), where q = χ_{(0,∞)} − χ_{(−∞,0)}.

Proof. By assumption, χ_{{0}}(A_{n}) + A_{n} converges in the strong resolvent sense to χ_{{0}}(A) + A (this is best seen using the Trotter criterion [53, Theorem VIII.21]). Thus by [53, Theorem VIII.24] we know that χ_{[0,∞)}(χ_{{0}}(A_{n}) + A_{n}) strongly converges to χ_{[0,∞)}(χ_{{0}}(A) + A) and similarly for χ_{(−∞,0]}. Hence χ_{(0,∞)}(A_{n}) + χ_{{0}}(A_{n}) converges strongly to χ_{(0,∞)}(A) + χ_{{0}}(A) which implies that χ_{(0,∞)}(A_{n}) converges strongly to χ_{(0,∞)}(A). Similarly, χ_{(−∞,0)}(A_{n}) converges strongly to χ_{(−∞,0)}(A). The proof of the sublemma is complete.

With all the Sublemmas in place, it is now easy to finish the proof of Lemma 10.6. We can restrict to im(p_{1}) (by multiplication with p_{1} from the left), since all operators in question commute with p_{1} and vanish on the complement of im(p_{1}) (i.e. when multiplied with (1 − p_{1})). Observe that τ, restricted to im(p_{1}), is strongly continuous (because τ(p_{1}) < ∞).

Since D_{ϵ,α} is invertible and ϵ-spectrally concentrated near zero, χ_{{0}}(C_{ϵ,±α}p_{1}) = p_{1} − p_{ϵ}. In the same way, when restricted to im(p_{0}), D_{0,α} is invertible and therefore χ_{{0}}(D_{0,α}p_{1}) = p_{1} − p_{0}. Consequently, χ_{{0}}(C_{ϵ_{k},±α_{k}}) converges strongly to χ_{{0}}(D_{0,±1}). Sublemmas 10.11 and 10.15 imply that q(C_{ϵ_{k},±α_{k}}) converge strongly to q(D_{0,1}), so that finally, using Lemma 10.3,

η_{τ} (p_{ϵ_{k}}D_{ϵ_{k},±α_{k}}p_{ϵ_{k}}) = τ(q(C_{ϵ_{k},±α_{k}})) ⟶^{k→∞} τ(q(D_{0,±1})) = η_{τ} (D_{0,±1}).

10.16 Remark. We want to observe that, if 0 is isolated in the spectrum of D, the proof of Lemma 10.6 is almost trivial. By construction, for sufficiently small ϵ in this case D_{ϵ,α} is equal to D_{0,α}.

11 Homotopy invariance of unstable rho-invariants

We are now in the position to prove the statements concerning homotopy invariance of the introduction, along with some more general results.

11.1 Theorem. Let M be a closed oriented Riemannian manifold of odd dimension. Assume that Γ is a torsion free discrete group such that the maximal Baum-Connes map

μ_{max}: K_{∗}(BΓ) → K^{∗}(C^{∗}Γ)

is an isomorphism. Let u: M → BΓ be a map classifying a covering M̃ = u^{∗}EΓ. Let A_{j} be unital von Neumann algebras admitting positive finite faithful normal traces τ_{j}: A_{j} → ℂ. Let α_{j}: C^{∗}Γ → A_{j} be homomorphisms (with j = 1, …, r), and τ_{1}, …, τ_{r}: A_{j} → Z_{j} positive normal traces with values in abelian von Neumann algebras Z_{j}. Assume that β_{1}, …, β_{r}: Z_{j} → V are continuous homomorphisms to a fixed topological vector space V such that

∑_{j=1}^{r} β_{j}τ_{j}(α_{j}(1)) = 0 ∈ V.  (11.2)

Let L_{j} := M̃ ×_{Γ} A_{j} be the associated Hilbert A-module bundle, where Γ acts on A_{j} via Γ → C^{∗}Γ ⟶^{α_{j}} A_{j}. Then ρ_{(τ_{j},β_{j},L_{j})}(M) ∈ V is a homotopy invariant.

11.3 Remark. We leave it to the reader to remove the hypothesis that the A_{j} admit a finite faithful trace in Theorem 11.1.

Proof. Let f: M′ → M be a homotopy equivalence, L′_{j} := f^{∗}L_{j}. Let V := M̃ ×_{Γ} C^{∗}Γ be the Mishchenko-Fomenko line bundle associated to u on M. Then f^{∗}V =: V′ is the Mishchenko-Fomenko line bundle associated to f ∘ u on M′.

Let D_{ϵ_{k},±α_{k}} be the smoothing perturbation of D_{V} ⨿−D′_{V′} on M ⨿(−M′) as in Theorem 10.1. Associated to these perturbation we define the stable rho-invariant

ρ_{[0]}^{s}(M ⨿ −M′) = [η_{[0]}(D_{ϵ_{k},±α_{k}})] ∈ (C^{∗}Γ)_{ab}/ < [1] > ,

where we recall that (C^{∗}Γ)_{ab} := (C^{∗}Γ)/\overline{[C^{∗}Γ, C^{∗}Γ]}. By Theorem 7.1, ρ^{s} does not depend on the particular perturbation and vanishes identically.

From Lemma C.8 we conclude that

α_{j}(η_{[0]}(D_{ϵ_{k},±α_{k}})) = η_{[0]}(D_{ϵ_{k},±α_{k}}^{A_{j}}) ∈ A_{j}/\overline{[A_{j}, A_{j}]},  (11.4)

where D_{ϵ_{k},±α_{k}}^{A_{j}} is the perturbation of the A_{j}-twisted signature operator D_{L_{j}} ⨿ D′_{L′_{j}} on M ⨿ (−M′) given by the corresponding recipe. On the other hand, by Theorem 10.1

\frac{η_{τ_{j}}(D_{ϵ_{k},α_{k}}^{A_{j}}) + η_{τ_{j}}(D_{ϵ_{k},−α_{k}}^{A_{j}})}{2} ⟶^{k→∞} η_{τ_{j}}(D^{A_{j}}) = η_{τ_{j}}(D_{L_{j}}) − η_{τ_{j}}(D′_{L′_{j}}).

Therefore

∑_{j=1}^{r} β_{j} \frac{η_{τ_{j}}(D_{ϵ_{k},α_{k}}^{A_{j}}) + η_{τ_{j}}(D_{ϵ_{k},−α_{k}}^{A_{j}})}{2} ⟶^{k→∞} ∑_{j=1}^{r} β_{j}η_{τ} (D_{L_{j}}) − ∑_{j=1}^{r} β_{j}η_{τ} (D′_{L′_{j}}).  (11.5)

By Equation (11.4), for the left hand side we can write

∑_{j=1}^{r} β_{j}η_{τ_{j}}(D_{ϵ_{k},±α_{k}}^{A_{j}}) = (∑_{j=1}^{r} β_{j}τ_{j}α_{j}) (η_{[0]}(D_{ϵ_{k},±α_{k}}))

The maps τ_{j}α_{j}: C^{∗}Γ → Z_{j} are traces. Therefore, they factorize through (C^{∗}Γ)_{ab}. After composing with β_{j} and summing, we get that ∑_{j=1}^{r} β_{j}τ_{j}α_{j}: C^{∗}Γ → V maps 1 to 0, therefore this map factorizes through C_{ab}^{∗}/ < [1] >.

But the projection of η_{[0]}(D_{ϵ_{k},±α_{k}}) to (C^{∗}Γ)_{ab}/ < [1] > is the stable rho-invariant of M ⨿ (−M′) which is identically zero.

The assertion of the theorem follows now from (11.5).

11.6 Corollary. The statements about the signature operator in Theorem 1.4, Theorem 1.11 and Theorem 1.15 are true.

Proof. It suffices to specialize the result of our main theorem to the Examples 8.5 and 8.8. Notice that in this way we have established the result about the delocalized eta invariant only under the Baum-Connes assumption for the maximal C^{∗}-algebra. In order to sharpen this result to the reduced C^{∗}-algebra we simply observe that if in the statement of Theorem 11.1 we take j = 1, A = N Γ, α : C_{r}^{∗}Γ → N Γ the natural map, Z = ℂ, τ = τ_{<g>} and β_{1} = 1, then the basic condition (11.2) is satisfied, ρ_{(τ,β,L)}(M) = η_{<g>}(D̃^{sign}) and the proof carries over.

11.7 Remark. Let M be a closed odd-dimensional manifold. We want to stress here that the mere vanishing of the signature index class in K_{1}(C^{∗}Γ) does not imply the vanishing of the corresponding L^{2}-rho invariant: we wish to clarify this point. There certainly exist allowable perturbations to define the corresponding stable rho-invariant (by [37])); under our standard assumptions on the fundamental group and the Baum-Connes map this stable rho invariant will be equal to zero. However, in this generality it cannot be guaranteed that stable=unstable. Indeed, there are examples where the index class is trivial but where the rho invariants are non-trivial (even for a fundamental groups as simple as ℤ); we shall construct them in Section 15. From this point of view, the case M = X ⊔ (−X′), with X and X′ homotopy equivalent, is indeed very special.

12 Vanishing results on spin manifolds with positive scalar curvature

Let (M, g) be a Riemannian manifold. We assume that M is spin and that dim M = m = 2k − 1, k ≥ 1. We fix a spin structure and we let D / be the associated Dirac operator.

12.1 Theorem. Assume that M has positive scalar curvature. Under the same assumptions on Γ, A_{j}, α_{j}, τ_{j} and β_{j} as in Theorem 11.1 we have

ρ_{(τ_{j},β_{j},L_{j})}(M) = 0 ∈ Z .  (12.2)

Proof. The proof is parallel to the proof of Theorem 11.1, but much easier since under the assumption of positive scalar curvature the unstable rho-invariant coincides by definition with the stable one.

12.3 Corollary. The statements about the spin Dirac operator in Theorem 1.4, Theorem 1.11 and Theorem 1.15 are true.

Proof. The proof is word by word parallel to the proof of Corollary 11.6, with the same remark applying in order to pass from C^{∗}Γ to C_{r}^{∗}Γ for the delocalized eta invariant.

12.4 Remark. Let M be a closed oriented odd dimensional manifold and let Γ → M̃ → M be a Galois covering, with Γ torsion-free. Let us assume that the signature operator on M̃ is L^{2}-invertible; thanks to the work of Farber-Weinberger [15] and Higson-Roe-Schick [19] we know that there are plenty of examples of such coverings. Then our arguments imply that

ρ_{λ_{1}−λ_{2}}(D^{sign}) = 0 and ρ_{(2)}(D̃^{sign}) = 0

provided the max-Baum-Connes map is bijective and that

η_{<g>}(D̃^{sign}) = 0

provided < g > is of polynomial growth and the reduced-Baum-Connes map is bijective. Conclusion: for such L^{2}-invisible coverings our signature-rho-invariants are zero, precisely as the spin rho-invariants in the presence of positive scalar curvature.

13 Further remarks on delocalized invariants

13.1 Infinite conjugacy classes

Let < g > be a non-trivial conjugacy class in Γ, not necessarily of finite cardinality.

13.1 Definition. If the limit

lim_{T →+∞} \frac{1}{\sqrt{π}} ∫_{0}^{T} t^{−1/2} Tr_{<g>}(D̃ exp(−tD̃^{2})dt.  (13.2)

exists, then its value is, by definition, the delocalized eta invariant η_{<g>}(D̃) associated to D̃ and < g >.

Lott [39] [41] establishes the convergence if the operator D̃ on M̃ has a gap near zero in its spectrum (i.e. zero is isolated in the spectrum) and if the trace τ_{<g>} : ℂΓ → ℂ associated to < g > has a certain extension-property. Let us recall his results. Let B_{Γ}^{∞}, ℂΓ ⊂ B_{Γ}^{∞} ⊂ C_{r}^{∗}Γ, be the Connes-Moscovici algebra [12] and let D^{∞} denote the Dirac operator on M twisted by the flat bundle M̃ ×_{Γ} B^{∞}. First of all, if D̃ has a gap in its spectrum, then the integral defining the 0-degree eta-invariant of D^{∞}

η_{[0]}(D^{∞}) := \frac{1}{\sqrt{π}} ∫_{0}^{∞} t^{−1/2} TR(D^{∞} exp(−t(D^{∞})^{2})dt ∈ B_{Γ}^{∞}/\overline{[B_{Γ}^{∞}, B_{Γ}^{∞}]}  (13.3)

converges in the natural Fréchet topology induced by B_{Γ}^{∞}. We set as usual (B_{Γ}^{∞})_{ab} := B_{Γ}^{∞}/\overline{[B_{Γ}^{∞}, B_{Γ}^{∞}]}. Next, if the trace τ_{<g>} : ℂΓ → ℂ extends from ℂΓ to a continuous trace on B_{Γ}^{∞}, then one can prove (see Remark E.12 in Appendix E.2) that

τ_{<g>}(TR(D^{∞} exp(−t(D^{∞})^{2}))) = Tr_{<g>}(D̃ exp(−tD̃^{2})) .

It is now clear that if D̃ has a gap in its spectrum and if τ_{<g>} extends, then the integral in (13.2) converges and the delocalized eta invariant η_{<g>}(D̃) is well defined. Notice that under our two assumptions

η_{<g>}(D̃) = τ_{<g>}(η_{[0]}(D^{∞}))  (13.4)

For example, groups of polynomial growth have the property that τ_{<g>} extends for each conjugacy class < g >. More generally:

13.5 Proposition. If < g > is of polynomial growth with respect to a word metric, then τ_{<g>} has the extension property.

Proof. This follows from [12, Lemma 6.4] and the Hölder inequality. Compare also [22]

13.6 Remark. It is in general very hard to check when a conjugacy class has polynomial growth. In particular, it seems likely that a group which has a exponential growth will have a conjugacy class whose growth is not polynomial. On the other hand, the conjugacy classes of commutators in a metabelian group are in most cases of polynomial growth

13.7 Theorem. Assume that Γ is torsion-free, μ_{red}: K_{∗}(BΓ) → K_{∗}(C_{red}^{∗}Γ) is an isomorphism and that < g > is a non-trivial conjugacy class in Γ of polynomial growth. If D = D / is the spin Dirac operator on a spin manifold with positive scalar curvature, then η_{<g>}(D̃ /) is defined and η_{<g>}(D̃ /) = 0.

Proof. We shall freely use a (B_{Γ}^{∞})_{ab}-valued APS-index theory. This is the zero-degree part of the higher theory developed in [30] ^{8}. Since we are in the presence of positive scalar curvature, the fact that η_{<g>}(D̃ /) is well defined follows from Lott’s results and Proposition 13.5. The vanishing of η_{<g>}(D̃ /) follows from (13.4) and from the proof of Theorem 7.3 and Theorem 13.9 below, which show that [η_{[0]}(D /^{∞})] ∈ (B_{Γ}^{∞})_{ab}/ < [1] > vanishes. Notice that τ_{<g>} : (B_{Γ}^{∞})_{ab} → ℂ factors through (B_{Γ}^{∞})_{ab}/ < [1] >.

13.8 Remark. For the signature operator one can prove the following statement: If D = D^{sign} is the signature operator of an oriented Riemannian manifold, and if 0 is isolated in the L^{2}-spectrum of D̃^{sign}, then η_{<g>}(D̃^{sign}) is defined for each pair (M′, u′ : M′ → BΓ) Γ-homotopy equivalent to M and it is an oriented Γ-homotopy invariant. That the delocalized eta invariants are well defined follows from Lott’s results, form Proposition 13.5 and from the homotopy invariance of the Novikov-Shubin numbers, see Gromov-Shubin [17]. The proof of the homotopy invariance given in the previous sections can be modified so as to cover this case as well. We omit the details. In fact, under the gap assumption one can argue in the following alternative and more general way.

Because of the gap assumption the signature index class associated to (M, u : M → BΓ) is zero in K_{1}(B_{Γ}^{∞}) = K_{1}(C_{r}^{∗}Γ). Since the Baum-Connes assembly map is by assumption bijective, we conclude from Theorem 7.1 that the stable rho-invariant ρ_{[0]}^{s}(M, u) is equal to zero in (B_{Γ}^{∞})_{ab}/ < [1] >. However, in this case there is a very natural perturbation available for the definition of the stable rho-invariant, namely the projection Π_{Ker(D^{∞})} onto the null space of D^{∞}, which is a finitely generated projective B_{Γ}^{∞}-module because of the gap assumption. Thus ρ_{[0]}^{s}(M, u) = [η_{[0]}(D^{∞} + Π)] = 0. On the other hand, in general, it can be proved [30] that

η_{[0]}(D^{∞} + Π) = η_{[0]}(D^{∞}) + [Ker(D^{∞})]_{[0]} ≡ η_{[0]}(D^{∞}) + TR(Π_{Ker(D^{∞})}) in B_{Γ}^{∞} .

We conclude that under our three assumptions (gap + Baum-Connes + polynomial growth of < g >) the following formula holds:

η_{<g>}(D̃) = τ_{<g>}(TR(Π_{Ker(D^{∞})}))

8The B_{Γ}^{∞} higher APS-index theory developed in [30] assumes the group to be of polynomial growth. Arbitrary groups are treated in [34] in the invertible case, based on results of Lott [40]. In order to extend the general B_{Γ}^{∞}-theory of [30] from groups of polynomial growth to arbitrary groups one simple needs to prove that if Ind(D^{∞}) = 0 in K_{1}(B_{Γ}^{∞}) = K_{1}(C_{r}^{∗}Γ) then there exist trivializing perturbations in the B_{Γ}^{∞}-Mishchenko-Fomenko calculus. However, this is the consequence of a simple density argument. We thank Victor Nistor for pointing this out.

On the other hand, because of the gap assumption, one can establish a Hodge theorem, proving that the whole null space Ker(D^{∞}) is in this case a homotopy invariant, being isomorphic to the cohomology of M with values in the local system M̃ ×_{Γ} B_{Γ}^{∞}. Thus, under our assumption, the homotopy invariance of η_{<g>}(D̃^{sign}) is a consequence of a much more general result.

13.2 Higher rho-invariants.

Let us consider (M, u: M → BΓ) with M spin and with a metric with positive scalar curvature. In this section M is not necessarily of odd dimension. Let D / be the Dirac operator twisted by the Mishchenko-Fomenko bundle u^{∗}EΓ ×_{Γ} B_{Γ}^{∞}. In this case, Lott’s higher eta invariant

η̃(D /) ∈ (Ω_{∗}(B_{Γ}^{∞}))_{ab} := Ω_{∗}(B_{Γ}^{∞})/\overline{[Ω_{∗}(B_{Γ}^{∞}), Ω_{∗}(B_{Γ}^{∞})]}

is well defined, see [40], [30, Appendix]. Higher rho-invariants are obtained by pairing this noncommutative differential form with suitable closed graded traces on Ω_{∗}(B_{Γ}^{∞}). Let us describe these traces. We start with a closed graded trace Φ on Ω_{∗}(ℂΓ); we assume that Φ is identically zero on the noncommutative differential forms concentrated at the identity conjugacy class, i.e. on elements of the form

∑_{γ_{0},...,γ_{k};γ_{0}···γ_{k}=1} ω_{γ_{0},γ_{1},...,γ_{k}}γ_{0}dγ_{1} ⋯ dγ_{k} .

There are examples of such traces, see [38, p. 209]. We briefly refer to Φ as a delocalized closed graded trace. We assume that Φ extends to a closed graded trace Φ_{∞} on Ω_{∗}(B_{Γ}^{∞}). The higher rho-invariant associated to D / and Φ is by definition the complex number

ρ̃_{Φ}(D /) :=< η̃(D /), Φ_{∞} > .

It is clear that the delocalized eta invariant of Lott is a special case of this construction. We shall also use the notation ρ̃_{Φ}(M, u) := ρ̃_{Φ}(D /).

It is proved in [36, Proposition 4.2] that the higher rho invariants defines maps ρ̃_{Φ} : Pos_{n}^{spin}(BΓ) → ℂ. If Γ has torsion, the latter information is used in [36] in order to distinguish metrics of positive scalar curvature (under suitable assumptions on Γ). In the torsion-free case, on the other hand, we can prove the following

13.9 Theorem. Assume that Γ is torsion-free and that the assembly map μ_{red}: K_{∗}(BΓ) → K_{∗}(C_{red}^{∗}Γ) is an isomorphism. Let Φ: Ω_{∗}(ℂΓ) → ℂ be a delocalized closed graded trace. Assume that Φ extends to a closed graded trace on Ω_{∗}(B_{Γ}^{∞}). If M is a spin manifold with positive scalar curvature and u: M → BΓ is a classifying map, then for the associated Dirac operator D /, the higher rho invariant vanish:

ρ̃_{Φ}(M, u) := ρ̃_{Φ}(D /) = 0.

Proof. Since M has positive scalar curvature the index class of D / is equal to zero. Thus, using Proposition 5.3 and the injectivity of μ_{red}, we conclude that for some d ∈ ℕ \ {0}, d(M, u: M → BΓ) is bordant to

⋃_{j=1}^{k}(A_{j} × B_{j}, r_{j} × 1 : A_{j} × B_{j} → BΓ)

with dim B_{j} = 4b_{j}, π_{1}(B_{j}) = 1 and < Â(B_{j}), [B_{j}] >= 0; we denote the latter manifold with classifying map by (N, v: N → BΓ). We can and we shall endow N with a metric of positive scalar curvature. Let (W, F : W → BΓ) be the manifold with boundary realizing the bordism. Since there is a metric of positive scalar curvature on ∂W the b-index class Ind_{b}(D /_{W}) is well defined in K_{∗}(C_{r}^{∗}Γ). By the surjectivity of μ_{red} we know that Ind_{b}(D /_{W}) = Ind(D /_{X}) with X a closed spin manifold with classifying map r: X → BΓ. In particular, using Lott’s treatment of the Connes-Moscovici higher index theorem [12], [38], we see that the Karoubi-Chern character of Ind_{b}(D /_{W}) is concentrated in the trivial conjugacy class. This means that < Ch(Ind_{b}(D /_{W})), Φ_{∞} >= 0. On the other hand, by the higher APS-index theorem in [33] we know that

< Ch(Ind_{b}(D /_{W})), Φ_{∞} >= dρ̃_{Φ}(M, u) − ∑_{j=1}^{k} ρ̃_{Φ}(A_{j} × B_{j}, r_{j} × 1),

since the local part in the index formula is concentrated in the trivial conjugacy class and it is thus sent to zero by Φ_{∞}. Hence using the bijectivity of μ_{red} we have proved that

ρ̃_{Φ}(M, u) = \frac{1}{d} ∑_{j=1}^{k} ρ̃_{Φ}(A_{j} × B_{j}, r_{j} × 1).

Using now the product formula for higher eta-invariants proved in [36, Proposition 2.1] we see that ρ̃_{Φ}(A_{j} × B_{j}, r_{j} × 1) = 0 ∀j. Thus ρ̃_{Φ}(M, u) = 0 and the Theorem is proved.

14 The center valued L^{2}-signature formula for manifolds with boundary

In this section, we relate the eta- and rho invariants of the signature operator, which show up in the APSindex theorem for the signature operator, to the signature of the manifold with boundary (the signature of some possibly degenerate intersection form).

Note that, even in the compact case, the ordinary signature formula for manifolds with boundary does not immediately follow from the Atiyah-Patodi-Singer index theorem for the signature operator. It is a non-trivial result of [2] to connect the APS-index of the signature operator to the signature of the manifold with boundary. This is much more complicated in the L^{2}-case, since in [2] eigenvalue decompositions of the space of L^{2}-sections on the boundary are used, which are not available in our setting. This is overcome in [44] together with [62] for the numerical L^{2}-signature. We now explain how this is done and how it generalizes to the situation we are considering in Section 15.

Therefore, let W be a compact oriented Riemannian manifold of dimension 4k with boundary ∂W = M. Let A be a von Neumann algebra and τ: A → Z a finite positive normal trace with values in an abelian von Neumann algebra Z. We are thinking here in particular of the von Neumann algebra of a discrete group Γ with its canonical trace or with its center valued trace.

Let L be a flat bundle of finitely generated projective Hilbert A-modules on W (giving rise to a local coefficient system of such modules). Recall that L is given by a representation of Γ in a finitely generated projective Hilbert A module (which we call L_{x} here). We assume that everything involved (Riemannian metric, bundle, connection) is of product type near the boundary.

We can now define three kinds of intersection forms on W, using the twisting bundle L, and with a signature in K_{0}(A).

The most computable one is obtained combinatorially: we consider a triangulation (or more generally a CW-decomposition) of W. This defines a cellular chain complex C_{∗}(W; L) of finitely generated free Hilbert A-modules, with coefficients in the local coefficient system L. There is a Poincaré duality chain homotopy equivalence to the relative cochain complex C^{4k−∗}(W, ∂W; L). From there we can restrict to W to get a map to C^{4k−∗}(W; L). Since the second map is not a chain homotopy equivalence in general, neither is the composition. But it is self-dual (note that the cochain complex is dual to the chain complex).

Now one can pass to the projective part of the homology and cohomology of these Hilbert A-module (co)chain complexes. This passage to the projective part involves some additional consideration in homological algebra —special to finite von Neumann algebras—, developed in different languages and independently by Farber [14] and Lück [42]. We will later only look at the special example where the homology and cohomology in middle degree is itself a finitely generated free Hilbert A-module (in fact equal to the chainand cochain module). Then the projective part in middle degree is equal to the whole (co)homology. The Poincaré duality chain homotopy equivalence composed with restriction to the boundary will then define a self dual map

H_{2k}(W; L) → H^{2k}(W; L) → Hom_{A}(H_{2k}(W; L), A).

In the special situation we are going to consider this will be given as follows: There is a free finitely generated ℤΓ-module V ≅ (ℤΓ)^{l} with (possibly singular) self dual map σ: V → V^{∗} := Hom_{ℤΓ}(V, ℤΓ) of the form σ = ψ + (−1)^{k}ψ^{∗} (i.e. σ has a quadratic refinement). Identifying V^{∗} with V using the given basis, σ is represented by a matrix B = A + A^{∗} with A ∈ M_{n}(ℤΓ). The self dual map H_{2k}(W; L) → H_{2k}(W; L)^{∗} is then obtained by tensoring σ: V → V^{∗} with the Γ-representation L_{x}, i.e. H_{2k}(W; L) ≅ V ⊗_{ℤΓ} L_{x} ≅ L_{x}^{l}, and the map is given as B′ := B ⊗ id_{L_{x}}.

The combinatorial signature sgn(W, L) ∈ K_{0}(A) is then defined by

sgn(W, L) := sgn_{A}(B′) := χ_{(0,∞)}(B′) − χ_{(−∞,0)}(B′) ∈ K_{0}(A).

Note that, for manifolds with boundary, this can only be defined for a von Neumann algebra, where measurable functional calculus is available, since 0 might well be contained in the spectrum of B′ (in contrast to the closed case, where the topology implies that an intersection form is necessarily invertible).

Using the trace τ: A → Z, we can then define the Z-valued combinatorial signature

sgn_{τ} (W, L) := sgn_{τ} (B′) := τ(sgn_{A}(B′)).

The second version of A-signature is obtained by replacing the combinatorial chain complex by the chain complex of differential forms with values in the flat bundle L. Here, the cup product of forms together with the A-valued inner product in the fibers of L induces an A-valued intersection pairing on the de Rham cohomology in middle degree with coefficients in L. This intersection pairing is again defined by a self adjoint operator B_{dR}, and then sgn_{A}(B_{dR}) := χ_{(0,∞)}(B_{dR}) − χ_{(−∞,0)}(B_{dR}) ∈ K_{0}(A) defines the de Rham signature of W with coefficients in L.

Finally, we can attach infinite cylinders to the boundary of W, and then carry out the construction as above, but now with square integrable differential forms (with values in L) on the enlarged non-compact manifold. We get a third signature invariant in K_{0}(A).

Application of τ gives three signatures in Z.

One can now use the arguments of [43] in order to show that these three invariants in fact always coincide. In [43], the corresponding result is proved for the ordinary L^{2}-signature, i.e. where L = N is the Mishchenko-Fomenko line bundle, and τ: N Γ → ℂ is the canonical trace. The techniques carry over. The relevant properties are that τ is a positive and normal trace. Note that in [43], one works with A-Hilbert spaces instead of the Hilbert A-modules used here. However, one can translate between these two settings as explained in [57, Section 8.6].

Let D_{L} be the signature operator on W twisted by L. It has boundary operator D_{M,L}. Then we can express the three equal higher signature of the bordism W which were defined above using the signature operator.

14.1 Theorem.

sgn_{τ} (W; L) = (∫_{W} AS(D) · τ(L_{x})) − \frac{η_{τ} (D_{M,L})}{2} ∈ Z.

The Atiyah-Singer integrand AS(D) for the signature operator is of course given by the Hirzebruch L-form of the Riemannian manifold W. τ(L_{x}) is the trace of the projection onto the finitely generated projective fiber L_{x}, this is a locally constant A-valued function on W.

Proof. This is proved for the “ordinary” cylindrical end L^{2}-signature by Vaillant [62], using the L^{2}-Atiyah-Patodi-Singer index formula. His proof only uses the formal properties of the canonical L^{2}-trace of being positive and normal and therefore carries over to prove the asserted equality. As explained above, from the work in [43] it follows that the formula also holds for the combinatorially defined signature.

15 Examples of non-trivial rho-invariants

In this section, we show that there are many examples where the rho-invariants considered in this paper (know to be homotopy invariants) show that certain manifolds are not homotopy equivalent. The fundamental group can be as simple as ℤ. Similar explicit calculations (without using the notion of ρ-invariant) have been carried out in [11, Section 5]. In the latter paper, these invariants are used to detect certain knots which do not have the slice property (which implies that certain types of bordisms can not exist).

We use the surgery construction of [44] employed there to construct manifolds with boundary with very general intersection form.

Recall from Section 14 that the combinatorially defined L^{2}-signatures of a triangulated manifold M (with or without boundary) is obtained as follows. Assume that Γ = π_{1}(M). We have an “intersection form” on the combinatorial chain complex. This can be understood as a matrix B with entries in ℤΓ. Actually, because of the symmetry properties of the intersection form, B = A + A^{∗} for A ∈ M_{n}(ℤΓ). We now use A = N Γ and L = N = M̃ ×_{Γ} N Γ. In this situation, since ℤΓ is a subset of N Γ, we understand the matrices A and B to be matrices also over N Γ (i.e. we write B instead of B′ in the notation of Section 14).

Then sgn(M, N) = sgn_{N Γ}(B) = χ_{(0,∞)}(B) − χ_{(−∞,0)}(B).

Applying the center-valued trace τ: N Γ → Z, we get then sgn(M, N , τ) = τ(sgn(M, N)). Set sgn_{τ} (B) := τ(sgn_{N Γ}(B)). Applying the canonical trace, from this we get sgn_{(2)}(B) ∈ ℂ, and applying the delocalized traces corresponding to a finite conjugacy class < g >, we get sgn_{<g>}(B) ∈ ℂ.

Moreover, if λ: Γ → U(d) is a finite dimensional representation of Γ, set λ(B) ∈ M_{n}(M_{d}(ℂ)) = M_{nd}(ℂ), the matrix obtained by applying λ entry-wise. Then sgn(λ(B)) is the λ-twisted signature of M. Recall that λ(B) is a possibly indefinite self-adjoint matrix. sgn(λ(B)) is the difference of the number of positive and negative eigenvalues, the eigenvalue 0 is ignored.

We will use the following Proposition 1.1 of [43].

15.1 Proposition. Fix a dimension 4k ≥ 6 and a finitely presented group Γ. Let X be a closed (4k − 1)- dimensional manifold with fundamental group Γ and with Morse decomposition without a k-handle. Choose A ∈ M_{n}(ℤΓ). Let B = A + A^{∗}.

Then there is a compact manifold with boundary (W; X, Y) of dimension 4k with boundary ∂W = X ⨿Y and with fundamental group Γ, such that the Morse chain complex C_{∗}(W̃) of the universal covering W̃ is isomorphic to C_{∗}(X̃) ⊕ V, where V is considered as trivial chain complex concentrated in the middle dimension k, and with inverse Poincaré duality homomorphism

C_{4k−∗}(W̃) → C_{4k−∗}(W̃ , ∂W̃) ⟶^{P D^{−1}} C^{∗}(W̃)

which in the middle dimension is given by B := A + A^{∗}, as explained in Section 14. Here P D^{−1} is a chain homotopy inverse to the cup product with [W, ∂W].

In particular, sgn_{τ} (W, N) = τ(χ_{(0,∞)}(B))−τ((−∞, 0)(B)), where τ: N Γ → Z is the center valued trace.

Recall that, on the other hand, for the constructed manifold W the ordinary signature is the signature of ρ(B), where ρ: Γ → {1} is the trivial representation.

Using the L^{2}-signature theorem 14.1, we therefore get in principle a large number of examples of manifolds with interesting difference of ρ-invariants.

15.2 Corollary. Given any A ∈ M_{n}(ℤΓ), set B := A + A^{∗}. Then there exist manifolds X, Y with π_{1}(X) = π_{1}(Y) = Γ such that

ρ_{(2)}(X) − ρ_{(2)}(Y) = sgn_{(2)}(B) − sgn(ρ(B)).

If λ_{1}, λ_{2} : Γ → U(d) are two representations, then

ρ_{λ_{1}−λ_{2}}(X) − ρ_{λ_{1}−λ_{2}}(Y) = sgn(λ_{1}(B)) − sgn(λ_{2}(B)).

If g ∈ Γ has finite conjugacy class < g >, then

ρ_{<g>}(X) − ρ_{<g>}(Y) = sgn_{<g>}(B).

15.3 Remark. Note that X ⨿ −Y with the reference map to Bℤ is a boundary and therefore has signature class 0 ∈ K_{0}(C^{∗}ℤ). Nevertheless, our construction shows that the rho-invariants of X ⨿ −Y are non-zero. In particular, for such a manifold the stable and the unstable rho-invariant can differ (we know that the stable one vanishes in such an example).

This theorem implies that we have a great freedom at constructing manifolds X and Y such that the various ρ-invariants differ. There is of course a problem in explicitly calculating these invariants for a given matrix B. Let us therefore recall how this can be done in the easiest case, i.e. if Γ = ℤ.

In this case N ℤ ≅ L^{∞}(S^{1}) using Fourier transform. The subring ℤ[ℤ] ⊂ N ℤ corresponds to Laurant polynomials on S^{1}.

Therefore, B = A + A^{∗} ∈ M_{n}(ℤΓ) can be understood as an n × n-matrix with entries in Laurant polynomials on S^{1}, or alternatively as a function (a Laurant polynomial) B(z) on S^{1} with values in M_{n}(ℂ). To compute sgn_{N ℤ}(B), one then computes pointwise sgn(B(z)), getting an integer valued function on S^{1}. Considered as an element of L^{∞}(S^{1}) = N ℤ = Z, this is exactly sgn_{τ} (B) (since N ℤ is abelian, the center valued trace is the identity).

If λ: ℤ → U(1) is a 1-dimensional representation, sending the generator z to λ ∈ U(1) = S^{1}, then sgn_{λ}(B) = sgn(B(λ)), i.e. we have to evaluate the function sgn(B(z)) at the point λ. In particular, for the trivial representation, sgn_{1}(B) = sgn(B(1)).

If, on the other hand, g = z^{n} ∈ ℤ, then sgn_{<g>}(B) is the Fourier coefficient of g for the function sgn(B(z)), i.e.

sgn_{<g>}(B) = ∫_{S^{1}} sgn(B(z))z^{−n} dz, in particular sgn_{(2)}(B) = ∫_{S^{1}} sgn(B(z)) dz.

Lets look at a particular example: if A = z + z^{−1} + 1 (a 1 × 1-matrix), then B(z) = 2(z + z^{−1} + 1), and sgn(B(z)) = 0 for z = z_{1,2} = exp(±2πi/3), sgn(B(z)) = 1 if z is contained in the connected component of 1 of S^{1} \ {z_{1}, z_{2}}, and sgn(B(z)) = −1 for the remaining points of z.

Observe that this is the general pattern: sgn(B(z)) jumps (at most) at those points on S^{1} where an eigenvalues of B(z) crosses 0, i.e. where the rank of B(z) is lower than the maximal rank.

It follows e.g. that sgn_{(2)}(B) = 1/3, but sgn_{1}(B) = 1, and sgn_{<z>}(B) = 2(exp(2πi/3) − exp(−2πi/3)) = 2i\sqrt{3}.

In particular, since sgn_{(2)}(B) − sgn_{1}(B) ≠ 0, (or since sgn_{<z>}(B) ≠ 0), if we use the matrix A in the construction of Proposition 15.1, the resulting closed manifolds X and Y satisfy

ρ_{(2)}(Y) − ρ_{(2)}(X) = sgn_{(2)}(B) − sgn_{1}(B) = −2/3,

and are therefore not homotopy equivalent.

Note that we can apply Theorems 1.4, 1.11 or 1.15 since ℤ is torsion-free and satisfies the Baum-Connes conjecture for the maximal C^{∗}-algebra.

Unfortunately, this conclusion has one flaw: by [63, Lemma 2.2] and the proof of [63, Theorem 5.8],

H_{k}(Y; ℤΓ) ≅ ker(B) ⊕ H_{k}(X; ℤΓ) and H_{k−1}(Y; ℤΓ) ≅ coker(B) ⊕ H_{k−1}(X; ℤΓ),  (15.4)

whereas the homology of X and Y in all other degrees coincides. This implies that we might as well distinguish X and Y using their homology, which is of course much easier to compute that the ρ-invariants.

One can easily construct more interesting examples as follows: consider a diagonal matrix A with entries A_{1}(z), …, A_{m}(z), and a second diagonal matrix A′ with entries ϵ_{1}A_{1}(a), …, ϵ_{n}A_{n}(z), with ϵ_{i} ∈ {−1, 1}. Starting with a manifold X as in Proposition 15.1, we then get two manifolds Y and Y′. By [63, Lemma 2.2] and the proof of [63, Theorem 5.8], the homology of Y and Y′ is isomorphic in all degree and with arbitrary coefficients.

However, the signatures and therefore the difference of ρ-invariants changes sign if the matrix A changes sign. This means that we can easily arrange that certain rho-invariants of Y and Y′ do not coincide, we have even enough freedom to make sure that there are examples where they neither coincide nor are negative of each other.

The conclusion is that although the homology of Y and Y′ is isomorphic in all degree and with arbitrary coefficients, there is no homotopy equivalence between Y and Y′ (not even one which reverses the orientations). This result has been obtained using rho-invariants, it can be recast in terms of Blanchfield forms (also called linking forms), i.e. in terms of classical methods of advanced algebraic topology.

It is evident that with other matrices A(z), we can get all kinds of further interesting examples.

This was all done for a group as simple as ℤ. One is often interested to get examples for more complicated groups, compare also the problem of constructing knots of e.g. [11]. We can “induce up” these examples by embedding ℤ in an arbitrary torsion-free group Γ and then use the fact that the signature calculations happen entirely in the subgroup ℤ and therefore are unchanged, as long as the L^{2}-signature and the ordinary signature are involved. The relevant result is stated and proved e.g. in [11, Proposition 5.13]. The following is happening: when we have an inclusion i: ℤ ↪ Γ, then this induces inclusions l^{2}(ℤ) ↪ l^{2}(Γ) and i: N ℤ ↪ N Γ. The latter one being an inclusion of von Neumann algebras, it is compatible with functional calculus. If we therefore start with B = A^{∗} + A ∈ M_{n}(ℤΓ), then χ_{(0,∞)}(i(B)) = i(χ_{(0,∞)}(B)). It is not to us clear how the classical methods of algebraic topology mentioned above could to be used in this general setting.

Finally, if we want to compute a delocalized trace of an element i(x), we observe that by definition for g ∈ Γ with |< g >| < ∞.

τ_{<g>}(i(x)) = ∑_{h∈<g>}⟨e · i(x), h⟩_{l^{2}(Γ)}.

But since e ∈ l^{2}(ℤ), also e · i(x) ∈ i(l^{2}(ℤ)), and therefore the inner product is zero if h ∉ i(l^{2}(ℤ)), and

τ_{<g>}(i(x)) = ∑_{h∈<g>∩i(ℤ)}⟨e · i(x), h⟩_{l^{2}(Γ)} = ∑_{h∈<g>∩i(ℤ)} τ_{i^{−1}(h)}(x).

In particular, if g = e then tr_{(2)}(i(x)) = tr_{(2)}(x), and if g ≠ e, one has to analyze which conjugates of g lie in i(ℤ). If < g > is finite, then g^{−1} can be the only power of g contained in < g >. It follows that if g generates i(ℤ), we understand < g > ∩i(ℤ) completely.

For example, if one embeds ℤ into the center of Γ, all delocalized invariants with respect to elements of this embedded ℤ will be the same, independent of the question whether the corresponding matrix is considered as a matrix over ℤ or over Γ. With a little extra care one can use the same “induction process” to obtain examples of non-trivial delocalized rho-invariants for finite conjugacy classes with more than one element.

A Index theory: proofs.

This first appendix is devoted to the proof of the two index theorems stated in Section 2.

A.1 Proof of the A/\overline{[A, A]} valued Atiyah-Singer index formula

First of all we need to deal with the existence of the heat kernel for the Dirac Laplacian D_{L}^{2} and its perturbation D_{L,C}^{2} ≡ (D_{L} + C)^{2}.

A.1 Lemma. For each t > 0 there exists a well-defined operator e^{−tD_{L,C}^{2}} ∈ Ψ_{A}^{−∞} . If v ∈ C^{∞}(M, E ⊗ L), then u(t, ·) := e^{−tD_{L,C}^{2}}v is a solution of the heat equation (∂_{t} + D_{L,C}^{2})u = 0 with initial condition v at t = 0. The operators e^{−tD_{L,C}^{2}} form a semigroup.

Proof. We begin by the Dirac Laplacian D_{L}^{2}. The best way to prove the above lemma is by use of the heat space of Melrose, see [47] Chapter VII. First we make a general comment. Microlocal analysis can be viewed as a geometrization of operator theory, the operators of interest being certain distributions on M × M with precise singularities on the diagonal (conormal distributions). Smoothing operators, for example, are given as smooth functions on M × M, or, more generally, as smooth sections of suitable homomorphism bundles on M ×M. It is not difficult to understand that the Mishchenko-Fomenko pseudodifferential calculus can be easily developed by simply considering A-valued conormal distributions. For example, smoothing operators in the Mishchenko-Fomenko calculus, acting for simplicity on the trivial bundle ℂ^{ℓ} ⊗ A are nothing but C^{∞} functions on M × M with values in M_{ℓ×ℓ}(A). From the microlocal point of view the presence of the C^{∗}-algebra A does not affect in a significant way all the usual arguments culminating in the existence of the pseudodifferential calculus. (Needless to say, operators in the Mishchenko-Fomenko calculus act on A-Hilbert modules and not on Hilbert spaces, it is at this point that much more care is needed.) This general philosophy will be applied here to the construction of the heat semigroup e^{−tD_{L}^{2}}. The advantage of the treatment given by Melrose through the heat space (a certain blow-up of M × M × [0, ∞)_{t}) is that it is as geometric as it can be, thus generalizing without any difficulty to operators acting on sections of bundles of A-modules (such as E ⊗ L). This general principle has been already applied in [33] in the case A = C_{r}^{∗}Γ but it is clear that it extends readily to an arbitrary unital C^{∗}-algebra A. Summarizing: by following closely the treatment given by Melrose for ordinary Dirac operator we can prove the lemma for the heat kernel associated to D_{L}^{2}. A standard argument involving a Volterra series can be applied in order to obtain the lemma for the perturbed operator D_{L,C}^{2}. See [5].

The problem we encounter with the heat kernel in the C^{∗}-algebraic context, is that it doesn’t behave well for t → ∞, since our operators are in general not invertible in the Mishchenko-Fomenko calculus. For this reason, we introduce suitable perturbation which make them invertible.

A.2 Definition. Instead of M consider M_{+}, the disjoint union of the manifold with an additional point ∗. The Dirac type operator on the additional point is by definition the 0 operator on ℂ. Recall the finitely generated projective modules I_{+}, I_{−} appearing in the Mishchenko-Fomenko decomposition theorem. We define L_{+}, the twisting bundle on M_{+} to be L ⨿ (I_{+} ⊕ I_{−}), where we view I_{±} as an abstract finitely generated projective Hilbert A-module (therefore a bundle over the point).

Following [38], Section VI, we now define for α ∈ ℝ the perturbed operators (of D_{L}, but we suppress the L in the notation)

D_{α}^{+} := (\begin{matrix}D_{L}^{+}|_{I_{+}^{⊥}} & 0 & 0 \\ 0 & D_{L}^{+}|_{I_{+}} & α \\ 0 & α & 0 \end{matrix}) ,

and D_{α}^{−} as the (formal) adjoint of D_{α}^{+}. The description is with respect to the Mishchenko-Fomenko decomposition of Theorem 2.4

C^{∞}(M_{+}, (E ⊗ L)_{+}^{+}) = I_{+}^{⊥} ⊕ I_{+} ⊕ I_{−} C^{∞}(M_{+}, (E ⊗ L)_{+}^{−}) = D_{L}(I_{+}^{⊥}) ⊕ I_{−} ⊕ I_{+}

Note that I_{±} has two roles here: first as subset of the section of E ⊗ L on M, secondly as possible values of the sections at the additional point ∗, where the fiber is I_{−} ⊕ I_{+}.

A.3 Lemma. The operator e^{−tD_{α}^{2}} is defined for each t > 0 and each α ∈ ℝ, and is a smoothing operator in the Mishchenko-Fomenko calculus on M_{+}. In particular STR(e^{−tD_{α}^{2}}) ∈ A_{ab} is defined.

Proof. The heat operator e^{−tD_{α}^{2}} is defined by Duhamel’s expansion, using the fact that D_{α} − D_{0} is finite rank as A-linear operator, see [38, Section VI]. Since the heat-operator of D_{0}^{2} is smoothing on M_{+}, the lemma follows. In fact, by using the information that the orthogonal projection onto I_{+} and the projection onto I_{−} along D_{L}(I_{+}^{⊥}) are smoothing operators, it is possible to check that D_{α} − D_{0} is smoothing on M_{+}, i.e. an element in Ψ_{A}^{−∞}(M_{+}, (E ⊗ L)_{+}, (E ⊗ L)_{+}).

The following lemma is clear:

A.4 Lemma. Let α = 0, then

STR(e^{−tD_{0}^{2}}) = STR_{M}(e^{−tD_{L}^{2}}) + [I_{−}]_{[0]} − [I_{+}]_{[0]} ∈ A_{ab} .

A.5 Lemma. For α sufficiently large, D_{α} is invertible in the MF-calculus, and therefore

STR(e^{−tD_{α}^{2}}) ⟶^{t→∞} 0 ∈ A_{ab}.  (A.6)

Proof. The invertibility is explained in [38] Section VI, after formula (107). One can proceed as in [47] and show that the heat kernel defined through the heat space is indeed expressible in terms of the usual integral involving the resolvent. Using the invertibility of D_{α}^{2} one then gets (A.6).

A.7 Lemma. For each α ∈ ℝ and t > 0

STR(e^{−tD_{α}^{2}}) − STR(e^{−tD_{0}^{2}}) = 0 ∈ A_{ab}.

Proof. Use Duhamel’s formula to compute the derivative with respect to α of STR(e^{−tD_{α}^{2}}). The usual calculations show that this is the supertrace of a supercommutator and therefore vanishes in A_{ab}. Details are as in [5], Chapter 3.

As a Corollary we obtain Proposition 2.13:

A.8 Corollary.

lim_{t→∞} STR_{M}(e^{−tD_{L}^{2}}) = [I_{+}]_{[0]} − [I_{−}]_{[0]} ≡ Ind_{[0]}(D_{L}) ∈ A_{ab},

where part of the assertion is that the limit exists.

Proof. By lemmas A.4 and A.7 we have in A_{ab}, for each α ∈ ℝ and t > 0:

STR_{M}(e^{−tD_{L}^{2}}) = STR(e^{−tD_{0}^{2}}) − [I_{−}]_{[0]} + [I_{+}]_{[0]} = STR(e^{−tD_{α}^{2}}) − [I_{−}]_{[0]} + [I_{+}]_{[0]} .

Taking α large enough we get the Corollary by applying lemma A.5.

A.1.1 The integral operator index

Next we tackle the problem of connecting the index Ind_{[0]}(D_{L}) ∈ A_{ab} defined using the index class and the algebraic trace tr^{alg} : K_{0}(A) → A_{ab} to the integral-kernel-trace, TR, of the projection operators given by the Mishchenko-Fomenko decomposition.

A.9 Definition. Define a second smoothing perturbation of D by

B_{α}^{+} := D_{L}^{+} − αP_{−}D_{L}^{+}P_{+} ,

for α ∈ ℝ, with B_{α}^{−} the (formal) adjoint of B_{α}^{+} and with P_{+} := Π_{I_{+}}, P_{−} := Π_{I_{−}}.

A.10 Remark. Observe that

B_{α} = (\begin{matrix}0 & B_{α}^{−} \\ B_{α}^{+} & 0 \end{matrix})

is a smoothing perturbation of D_{L}. Note that P_{+} is self adjoint, but P_{−} is not necessarily. Thus we will also use its adjoint P_{−}^{∗}. Note, finally, that im(1 − P_{−}) = D_{L}^{+}(I_{+}^{⊥}).

A.11 Lemma. We have decompositions

C^{∞}(M, (E ⊗ L)^{−}) = im(P_{−}) ⊕ im(1 − P_{−}) = im(P_{−}^{∗}) ⊕ im(1 − P_{−}).

The second decomposition is an orthogonal decomposition, since

im(1 − P_{−})^{⊥} = ker(1 − P_{−}^{∗}) = im(P_{−}^{∗}).

Let pr be the orthogonal projection onto im(P_{−}^{∗}). Then pr is also a smoothing operator.

Proof. P_{−}^{∗}P_{−} + (1 − P_{−}) = 1 + (P_{−}^{∗}P_{−} − P_{−}) is an isomorphism of C^{∞}(M, E ⊗ L)^{−}) which is diagonal with respect to the two decompositions (it is an isomorphism, since P_{−}^{∗} has kernel im(P_{−})^{⊥} and is surjective, so that P_{−}^{∗}: im(P_{−}) → im(P_{−}^{∗}) is an isomorphism by the open mapping theorem). Then

pr = (1 + (P_{−}^{∗}P_{−} − P_{−}))^{−1}P_{−}(1 + (P_{−}^{∗}P_{−} − P_{−}))

is smoothing by Lemma A.12 below.

A.12 Lemma. If P is smoothing and 1 + P is invertible in the sense of Hilbert-A-module morphisms (on the completed space of sections), then

(1 + P)^{−1} = 1 + Q with Q smoothing.

Proof. Write (1 + P)^{−1} = 1 + Q; then Q satisfies Q = −P − P^{2} − P QP and it is therefore smoothing. Here we are using the fact that smoothing operators in L^{2} are a semi-ideal (a subring R of a ring R is a semi-ideal if i, j ∈ I, r ∈ R ⇒ irj ∈ I, for all i, j ∈ I, for all r ∈ R).

A.13 Lemma. In this lemma we shall suppress the L subscript in the notation of D_{L}. For α = 0, B_{α} = D_{L}; for α = 1, we have

B_{1}^{+} = (\begin{matrix}D|_{I_{+}^{⊥}} & 0 \\ 0 & 0 \end{matrix})

with respect to the decomposition

C^{∞}(M, (E ⊗ L)^{+}) = I_{+}^{⊥} ⊕ I_{+} C^{∞}(M, (E ⊗ L)^{−}) = im(1 − P_{−}) ⊕ im(P_{−}^{∗}).

Observe that this operator decomposes as an invertible operator plus (direct sum) the zero operator between two finitely generated projective modules.

Since the decompositions of domain and range are both orthogonal, the adjoint B_{1}^{∗} decomposes (with respect to the same decomposition) as

B_{1}^{−} = (\begin{matrix}D^{∗}|_{I_{+}^{⊥}} & 0 \\ 0 & 0 \end{matrix}) ,

and again the left upper corner is an isomorphism. From this

B_{1}^{2} = (\begin{matrix}D_{I_{+}^{⊥}}^{∗}D_{I_{+}^{⊥}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & D_{I_{+}^{⊥}}D_{I_{+}^{⊥}}^{∗} \\ 0 & 0 & 0 & 0 \end{matrix}) ,

where we use the same decomposition as before.

It follows that

e^{−tB_{1}^{2}} ⟶^{t→∞} P_{+} ⊕ pr,

where P_{+} is the even part of the operator, and pr the odd part. For the supertraces, this implies

STR(e^{−tB_{1}^{2}}) ⟶^{t→∞} TR(P_{+}) − TR(pr) ∈ A_{ab}.

Here, the trace is always taken in the sense of integration over the diagonal. Note that pr is a smoothing operator by Lemma A.11.

A.14 Lemma.

∫_{M} tr_{x}^{alg} pr(x, x) = ∫_{M} tr_{x}^{alg} P_{−}(x, x) ∈ A_{ab}.

Proof. By the proof of Lemma A.11 (using Lemma A.12),

pr = (1 − P)^{−1}P_{−}(1 − P) = (1 − Q)P_{−}(1 − P),

where P and Q are smoothing operators. The assertion follows from the trace property for the integral trace for smoothing operators.

A.15 Lemma. STR(e^{−tB_{α}^{2}}) is independent of α.

Proof. The independence follows, as before, from Duhamel’s formula.

We are now in the position of proving Proposition 2.14, which is stated once again here for the convenience of the reader:

A.16 Corollary. The algebraic trace [I_{+}]_{[0]} − [I_{−}]_{[0]} of [I_{+}] − [I_{−}] ≡ Ind(D_{L}), i.e. the image under the induced map tr^{alg} : K_{0}(A) → A_{ab} of the index class Ind(D_{L}), can be calculated as

[I_{+}]_{[0]} − [I_{−}]_{[0]} = ∫_{M} tr^{alg} P_{+}(x, x) − ∫_{M} tr P_{−}(x, x) ≡ TR P_{+} − TR P_{−} ∈ A_{ab},

where P_{+} and P_{−} are the projections onto I_{+} and I_{−} as given by the Mishchenko-Fomenko decomposition.

Proof. Both expressions are limits for t → ∞ of STR(e^{−tD_{L}^{2}}) by Lemma A.15 and Lemma A.7.

A.17 Remark. Note that the proof shows that [I_{+}]_{[0]} − [I_{−}]_{[0]} can also be expressed with many of the other integral operators we have used throughout the proof. In fact it might be useful to observe that the following general proposition holds:

A.18 Proposition. Let D_{L} as above and let Q ∈ Ψ_{A}^{−1}(M, E^{−} ⊗ L, E^{+} ⊗ L) be a parametrix for D_{L}^{+} with remainders S_{±} ∈ Ψ_{A}^{−∞}. Then

ind_{[0]}(D_{L}) = TR(S_{+}) − TR(S_{−}) .

The proof of the proposition is standard.

Having analyzed the large-time behavior of the A_{ab}-valued supertrace of the heat-kernel, we now turn our attention to the short-time behavior.

A.19 Lemma. The local supertrace

str_{x}^{alg}(e^{−tD_{L}^{2}}(x, x)) vol(x)

has a limit for t → 0 which is precisely the differential form

(x ↦ AS(D)(x) ∧ ch(E)(x) ∧ ch L(x)_{[dim M]}) ∈ Ω^{dim M}(M, A_{ab}),

where the Chern forms are defined as usual using Chern-Weyl theory and the curvature of the connections.

Proof. We use Getzer’s proof, as in the book by Berline, Getzler and Vergne [5].

A.20 Remark. In fact, the same statement holds for the smoothing perturbation D_{L,C} since, as already remarked, for the heat kernel

e^{−t(D_{L}+C)^{2}} = e^{−tD_{M,L}^{2}} + tC^{∞}([0, ∞), Ψ_{A}^{−∞}(M, E ⊗ L, E ⊗ L))

(a consequence of Duhamel’s formula together with the fact that C ∈ Ψ_{A}^{−∞}(M, E ⊗ L, E ⊗ L)).

A.21 Lemma.

\frac{d}{dt} STR(e^{−tD_{L}^{2}}) = 0 in A_{ab}.

Proof. This is, once again, a consequence of Duhamel’s formula.

We can finally give a complete proof of the A_{ab}-valued Atiyah-Singer index theorem 2.15:

Proof. Only the index formula itself remains to be established. We integrate from 0 < ϵ < 1 to 1/ϵ the derivative \frac{d}{dt} STR(e^{−tD_{L}^{2}}); we apply the fundamental theorem of calculus, we let ϵ ↓ 0 and use the large and short time behavior of STR(e^{−tD_{L}^{2}}) as in A.19 and in A.8.

A.2 Proof of the A/\overline{[A, A]}-valued APS index formula

In this subsection we shall recall and complement (a special case of) the Atiyah-Patodi-Singer index theory developed by Leichtnam and the first author in [30] [37]. In order to simplify the exposition we shall only consider even dimensional manifolds with boundary. Fundamental in our treatment is the extension of Melrose’ b-calculus to the C^{∗}-algebraic setting. This is developed in [33], [30] when the C^{∗}-algebra A is equal to the reduced C^{∗}-algebra of a discrete group; exactly the same arguments work when A is an arbitrary (unital) C^{∗}-algebra. We shall not enter into the precise definition of the Mishchenko-Fomenko b-calculus with bounds Ψ_{b,A}^{∗,ϵ}, ϵ > 0; we only recall that operators in Ψ_{b,A}^{∗,ϵ} are characterized by their behavior, as distributions, on W × W or, more precisely, by the behavior of their lifts on the so-called b-stretched product W ×_{b} W (the manifold with corners obtained by blowing up ∂W × ∂W in W × W); the b-calculus with bounds is the sum of three spaces of operators

Ψ_{b,A}^{∗,ϵ} = Ψ_{b,A}^{∗} + Ψ_{b,A}^{−∞,ϵ} + Ψ_{A}^{−∞,ϵ} .

The first space on the right hand side is the small b-calculus; it is an algebra and contains as a sub-algebra the space of b-differential operators. Elements in the second space are called (smoothing) boundary terms whereas the operators in the third space are called residual and are directly characterized on W × W. The residual elements play the role of the smoothing operators in the closed case. There are natural composition rules for elements in the b-calculus with bounds. Finally, elements in the b-calculus are bounded on suitable b-Sobolev Hilbert modules:

P ∈ Ψ_{b,A}^{m,ϵ}(W, E ⊗ L, F ⊗ L) ⇒ P: H_{b}^{ℓ}(W, E ⊗ L) → H_{b}^{ℓ−m}(W, F ⊗ L) is bounded ,

where the subscript b in the Sobolev modules indicates that these modules are defined using a b-metric and b-differential operators. We set H_{b}^{∞} := ∩_{k∈ℕ}H_{b}^{k}. It is also important to consider weighted b-Sobolev modules x^{δ}H_{b}^{m}, with x a boundary defining function; in fact the inclusion x^{δ}H_{b}^{ℓ+ϵ} ↪ H_{b}^{ℓ} is an A-compact operator ∀ϵ > 0, ∀δ > 0 .

A short guide to the literature: the basic reference for the b-calculus and its applications to index theory on manifolds with boundary is the book by Melrose [47]. Short but rather complete introductions to the theory are given in the surveys [46] and [16] and also in the Appendix of [48]. The existence of spectral sections, and thus of trivializing perturbations, for a A-linear Dirac-type operator D_{M,L} on a closed manifold M with vanishing index class in K_{dim M}(A) (see 2.17) is established in [65] [37], building on work of Melrose-Piazza [48]. The b-calculus in the C^{∗}-algebraic context (including b-Sobolev modules) is studied in [33]. If M = ∂W and C is a trivializing perturbation for D_{∂W,L}, then the lifted perturbation C_{W} ∈ Ψ_{b,A}^{−∞} is defined in [48] for families and in [30] in the C^{∗}-context. Using the b-calculus and the invertibility of D_{∂W,L} + C one proves that the operator D_{W,L} + C_{W} is invertible modulo residual operators; since a residual operator extends to a A-compact operator on b-Sobolev modules we see that D_{W,L} + C_{W} has a well defined index class in K_{0}(A) (see below for more on this point).

Our main interest is thus in the perturbed b-operator D_{L} + C_{W} ∈ Ψ_{b,A}^{1}(M, E ⊗ L, E ⊗ L). We begin by the existence of the heat kernel.

A.22 Lemma. The operator H(t) := e^{−t(D_{L}+C_{W})^{2}} is a smoothing operator in the (small) Mishchenko-Fomenko b-calculus on W; e^{−t(D_{L}+C_{W})^{2}} ∈ Ψ_{b,A}^{−∞}. The heat operator H(·) satisfies the heat equation with initial condition lim_{t↓0} H(t) = Id.

Proof. The result for the (unperturbed) b-differential operator D_{L}^{2} is obtained by employing the b-heat space as in [47] and applying the same reasoning as in the proof of Lemma A.1. For the perturbed operator (D_{L} + C_{W})^{2} we apply the same arguments used in the proof of Proposition 8 in [48].

The following result is fundamental in developing a higher APS-index theory

A.23 Lemma. There is a Mishchenko-Fomenko decomposition of the space of sections of E ⊗L with respect to D_{L}^{+} + C_{W}^{+} , i.e.

H_{b}^{∞}(W, (E ⊗ L)^{+}) = I_{+} ⊕ I_{+}^{⊥}, H_{b}^{∞}(W, (E ⊗ L)^{−}) = I_{−} ⊕ (D_{L}^{+} + C_{W}^{+} )(I_{+}^{⊥}).

In this decomposition, I_{±} ⊂ x^{ϵ}H_{b}^{∞}, with ϵ > 0. By completion, this decomposition gives a decomposition of the Hilbert A-modules H_{b}^{m}(W, (E ⊗ L)^{±}), m ∈ ℕ.

The second decomposition is not a priori orthogonal. However, D_{L}^{+} +C_{W}^{+} induces an isomorphism (in the Fréchet topology) between I_{+}^{⊥} and (D_{L}^{+} + C_{W}^{+} )(I_{+}^{⊥}). Moreover, I_{+} and I_{−} are finitely generated projective Hilbert A-modules; the projections Π_{I_{+}} onto I_{+} (orthogonal) and Π_{I_{−}} onto I_{−} (along (D_{L}^{+} + C_{W}^{+} )(I_{+}^{⊥})) are residual, i.e. belong to Ψ_{A}^{−∞,ϵ}(W; E ⊗L, E ⊗L). I_{±} are already complete finitely generated projective Hilbert A-modules, i.e. unchanged when passing to any completion H_{b}^{m}(W, E ⊗ L). The index class associated to D_{L}^{+} + C_{W}^{+} , denoted Ind_{b}(D_{L}, C), is by definition

Ind_{b}(D_{L}, C) := [I_{+}] − [I_{−}] ∈ K_{0}(A)  (A.24)

Proof. The Lemma is proved in [33, Appendix B] for an operator D_{L} with invertible boundary operator. As explained in [30], Theorems 6.2 and 6.5, the same proof applies to perturbed operators, such as D_{L} + C_{W}, with invertible indicial family.

A.25 Remark. Suppose now that C is defined from a spectral section P associated to D_{∂W,L} : thus C ≡ C_{P} for some P. Then

Ind_{b}(D_{L}, C_{P}) = Ind_{APS}(D_{L}, P) in K_{0}(A)

where on the right-hand side a suitable generalization of the Atiyah-Patodi-Singer boundary condition appears. See [65], [37]

Recall that our goal is to prove an index formula for

tr^{alg}(Ind_{b}(D_{L}, C)) := Ind_{b,[0]}(D_{L}, C) ≡ [I_{+}]_{[0]} − [I_{−}]_{[0]} ∈ A_{ab} .

A.2.1 The algebraic index perturbation

A.26 Definition. Instead of W consider W_{+}, the disjoint union of the manifold with an additional point. The Dirac type operator on the additional point is by definition the 0 operator on ℂ. Note that the boundary of W_{+} is still M. We define L_{+}, the twisting bundle on W_{+} to be L ⨿ (I_{+} ⊕ I_{−}), where we view I_{±} as an abstract finitely generated projective Hilbert A-module (therefore a bundle over the point).

Now we define for α ∈ ℝ the perturbed operators (of D_{L}^{+} + C_{W}^{+} , but we suppress L and C_{W} in the notation of the perturbation)

D_{α}^{+} := (\begin{matrix}(D_{L}^{+} + C_{W}^{+} )|_{I_{+}^{⊥}} & 0 & 0 \\ 0 & (D_{L}^{+} + C_{W}^{+} )|_{I_{+}} & α \\ 0 & α & 0 \end{matrix}) ,

and D_{α}^{−} as the (formal) adjoint of D_{α}^{+}. The description is with respect to the b-Mishchenko-Fomenko decomposition of Lemma A.23,

H_{b}^{∞}(W_{+}, (E ⊗ L)_{+}^{+}) =I_{+}^{⊥} ⊕ I_{+} ⊕ I_{−}
H_{b}^{∞}(W_{+}, (E ⊗ L)_{+}^{−}) =(D_{L}^{+} + C_{W}^{+} )(I_{+}^{⊥}) ⊕ I_{−} ⊕ I_{+}

Note that I_{±} has two roles here: first as subset of the section of E ⊗ L on W, secondly as possible values of the sections at the additional point ∗, where the fiber is I_{−} ⊕ I_{+}.

A.27 Lemma. For each α ∈ ℝ, D_{α} is a bounded perturbation of D_{0}. More precisely D_{α} − D_{0} belongs to the residual space Ψ_{A}^{−∞,ϵ}(M_{+}, (E ⊗ L)_{+}, (E ⊗ L)_{+}), for some ϵ > 0.

It follows by Duhamel expansion that e^{−tD_{α}^{2}} is defined for each t > 0 and each α ∈ ℝ, and is an element in Ψ_{b,A}^{−∞,ϵ}(M_{+}, (E ⊗ L)_{+}, (E ⊗ L)_{+}). In particular the b-supertrace bSTR(e^{−tD_{α}^{2}}) ∈ A_{ab} is defined. Let us recall the definition of the b-supertrace, from [47]. First we recall the definition of the regularized integral on the manifold W endowed with a product b-metric. Let vol_{b,W} be the associated volume form. Let us fix once and for all a trivialization ν ∈ C^{∞}(∂W, N_{+}∂W) of the inward pointing normal bundle to the boundary and let x ∈ C^{∞}(W) a boundary defining function for ∂W such that dx · ν = 1 on ∂W. For any function f ∈ C^{∞}(W) we set

ν∫_{W} f := lim_{ϵ→0^{+}}[∫_{x>ϵ} f vol_{b,W} + log ϵ ∫_{∂W} f|_{∂W} vol_{∂W}]

Consider now an element K ∈ Ψ_{b,A}^{−∞,ϵ} and its restriction to the lifted diagonal Δ_{b} ⊂ M ×_{b} M. Using the identification Δ_{b} ≡ W and taking the A_{ab}-valued supertrace of the endomorphism of (E ⊗ L)_{x}, denoted str_{x}, we define

bSTR_{ν}(K) := ν∫_{W} str_{x}^{alg} K|_{Δ_{b}}(x) .

One can easily compute the effect of changing the trivialization ν; from now on we shall suppress the subscript ν in the notation of the b-supertrace. Needless to say, in the ungraded case we can similarly define the b-trace. The b-trace does not vanish on commutators; however, there is an explicit formula due to Melrose, see [47], for computing the defect. The formula involves the indicial families of the two operators:

A.28 Proposition. For any K, K′ belonging to Ψ_{b,A}^{−∞,ϵ} one has:

bSTR[K, K′] = \frac{\sqrt{−1}}{2π} ∫_{ℝ} STR( \frac{∂|}{∂λ}I(K, λ) ∘ I(K′, λ) )dλ.  (A.29)

If we replace K by a differential operator in Diff_{b,A}^{∗} and K′ by the composition of K′ with an element of the calculus with bounds Ψ_{b,A}^{∗,ϵ} then the same commutator formula is valid.

A.30 Lemma. The following formula is clear from the definition:

bSTR(e^{−tD_{0}^{2}}) = bSTR_{W} (e^{−t(D_{L}+C_{W})^{2}}) + [I_{−}]_{[0]} − [I_{+}]_{[0]} ∈ A_{ab} .

A.31 Lemma. For α sufficiently large, D_{α} is invertible in the MF b-calculus with bounds, and therefore

bSTR(e^{−tD_{α}^{2}}) ⟶^{t→∞} 0.

Proof. Directly from the b-Mishchenko-Fomenko decomposition theorem we prove the invertibility of D_{α} , α large, exactly as in the closed case, see [38], Section VI.

A.32 Lemma.

bSTR(e^{−tD_{α}^{2}}) − bSTR(e^{−tD_{0}^{2}}) = F (α, t) (= 0),

where for each fixed α and t, F (α, t) = 0 in A_{ab}.

Proof. Use Duhamel’s formula to compute the derivative with respect to α of bSTR(e^{−tD_{α}^{2}}). The usual calculations show that this is the b-supertrace of a supercommutator. Using A.29 we see that this is an explicit term F (α, t), localized on the boundary. As explained in [33], formula (14.16), one can show that F (α, t) = 0 in A_{ab}; this is a consequence of the particular structure of the two projections onto I_{±}, namely, that they are residual.

A.33 Corollary.

lim_{t→∞} bSTR_{W} (e^{−t(D_{L}+C_{W})^{2}}) = [I_{+}]_{[0]} − [I_{−}]_{[0]} ≡ Ind_{b,[0]}(D_{L}, C) ,

where part of the assertion is that the limit exists.

A.2.2 The integral operator b-index

As in the closed case, we wish to connect the index Ind_{b,[0]}(D_{L}, C) ∈ A_{ab} defined using the index class and the algebraic trace tr^{alg} : K_{0}(A) → A_{ab} to the integral-kernel-trace, TR, of the projection operators given by the b-Mishchenko-Fomenko decomposition.

A.34 Definition. We set P_{+} := Π_{I_{+}}, P_{−} := Π_{I_{−}}. We define a second smoothing perturbation of (D_{L} + C_{W})^{+} by

B_{α}^{+} := (D_{L}^{+} + C_{W}^{+} ) − αP_{−}(D_{L}^{+} + C^{+})P_{+} ,

for α ∈ ℝ; define B_{α}^{−} as the (formal) adjoint of B_{α}^{+}.

A.35 Remark. Observe that

B_{α} = (\begin{matrix}0 & B_{α}^{−} \\ B_{α}^{+} & 0 \end{matrix})

is a residual perturbation of D_{L}+C_{W} in the b-calculus. We shall also use pr := orthogonal projection onto im P_{−}^{∗}.

A.36 Lemma. STR(e^{−tB_{α}^{2}}) is independent of α.

Proof. The result follows from Duhamel’s formula, the formula for the b-supertrace of a commutator and the fact that the perturbation involved in the definition of B_{α} is residual.

Proceeding exactly as in the closed case one proves the analog of lemma A.11 and, in particular, that pr is a residual operator; the analog of lemma A.12 is established using the semi-ideal property of the residual operators (see [47], formula (5.23) and Proposition 5.38). Then, as in the closed case,

bSTR(e^{−tB_{1}^{2}}) ⟶^{t→∞} bTR(P_{+}) − bTR(pr) = TR(P_{+}) − TR(pr) ∈ A_{ab},

with the last equality following once again from the fact that P_{+} and pr are residual. Finally, from the trace property we get TR(pr) = TR(P_{−}), exactly as in the closed case. Using lemma A.36 we finally get

A.37 Corollary. The A_{ab}-valued b-index Ind_{b,[0]}(D_{L}, C), i.e. the image under the induced map tr^{alg} : K_{0}(A) → A_{ab} of the index class Ind_{b}(D_{L}, C), can be calculated as

∫_{M} tr^{alg} P_{+}(x, x) − ∫_{M} tr^{alg} P_{−}(x, x) ∈ A_{ab},

where P_{+} and P_{−} are the projections onto I_{+} and I_{−} as given by the b-Mishchenko-Fomenko decomposition.

Proof. Both expressions are limits for t → ∞ of bSTR(e^{−t(D_{L}+C_{W})^{2}}) by Lemma A.36 and Corollary A.33.

A.2.3 Local expansion

A.38 Lemma. The local supertrace str^{alg}(e^{−t(D_{L}+C_{W})^{2}}(x, x))vol_{b}(x) has a limit for t → 0 which is exactly the differential form

(x ↦ AS(D)(x) ∧ ch(E)(x) ∧ ch L(x)_{[dim w]}) ∈ Ω^{dim W} (W, A_{ab}),

where the Chern forms are defined as usual using Chern-Weyl theory and the curvature of the connections.

Proof. The formula holds for str(e^{−tD_{L}^{2}}(x, x))vol_{b}(x), with proof employing the rescaled b-heat calculus. See [47], Chapter VIII. For the perturbed operator we simply observe that

e^{−t(D_{L}+C_{W})^{2}} = e^{−tD_{L}^{2}} + tC^{∞}([0, ∞), Ψ_{b,A}^{−∞}) ,

See [48], Proposition 8.

A.2.4 The index formula

The index formula, as stated in subsection 2.2, follows from the large and short time limits for bSTR(e^{−t(D_{L}+C_{W})^{2}}) together with the commutator formula A.28, with details as in [48], [30]:

A.39 Proposition.

lim_{t→∞} bSTR(e^{−t(D_{L}+C_{W})^{2}}) − lim_{t→0} bSTR(e^{−t(D_{L}+C_{W})^{2}})

= ∫_{0}^{∞} \frac{d}{dt} bSTR(e^{−t(D_{L}+C_{W})^{2}}) = −\frac{1}{2}η_{[0]}(D_{M,L} + C) ∈ A_{ab}.

Notice that the derivative with respect to t of bSTR(e^{−t(D_{L}+C_{W})^{2}}), as given by the commutator formula, is not precisely the eta integrand

−\frac{1}{2} \frac{1}{\sqrt{π}} t^{−1/2} TR ((D_{M,L} + C)e^{−t(D_{M,L}+C)^{2}}) .

However, as explained in [48], its integral from 0 to +∞ is indeed equal to

− ∫_{0}^{∞} \frac{1}{2} \frac{1}{\sqrt{π}} t^{−1/2} TR ((D_{M,L} + C)e^{−t(D_{M,L}+C)^{2}}) dt ≡ −\frac{1}{2}η_{[0]}(D_{M,L} + C) ∈ A_{ab}

The A_{ab}-valued APS index theorem, as stated in 2.24, now follows immediately from the above results.

B Graded Hermitian complexes and the signature operator

In this Appendix we shall make precise our conventions for the signature operator We follow [20, Section 3.1] and also [31], giving the general definition of graded regular n-dimensional Hermitian complex and its associated signature operator. The only difference between our conventions and those of [20] is that we deal with left modules, whereas [20] deals with right modules. The following material is taken directly from [31].

Let A be a C^{∗}-algebra with unit.

B.1 Definition. A graded regular n-dimensional Hermitian complex consists of the following data

• A ℤ-graded cochain complex (E^{∗}, d) of finitely-generated projective left A-modules,

• A nondegenerate quadratic form Q : E^{∗} × E^{n−∗} → A and

• An operator τ ∈ Hom_{A} (E^{∗}, E^{n−∗}) such that Q(bx, y) = bQ(x, y), Q(x, y)^{∗} = Q(y, x), Q(dx, y) + Q(x, dy) = 0, τ^{2} = I, and < x, y >:= Q(x, τy) defines a Hermitian metric on E.

Let M be a closed oriented n-dimensional Riemannian manifold. Let V be a flat A-vector bundle on M. We assume that the fibers of V have A-valued Hermitian inner products which are compatible with the flat structure.

Let Ω^{∗}(M; V) denote the vector space of smooth differential forms with coefficients in V. If n = dim(M) > 0 then Ω^{∗}(M; V) is not finitely-generated over A, but we wish to show that it still has all of the formal properties of a graded regular n-dimensional Hermitian complex. If α ∈ Ω^{∗}(M; V) is homogeneous, denote its degree by |α|. In what follows, α and β will sometimes implicitly denote homogeneous elements of Ω^{∗}(M; V). Given m ∈ M and (λ_{1} ⊗ e_{1}), (λ_{2} ⊗ e_{2}) ∈ Λ^{∗}(T_{m}^{∗}M) ⊗ V_{m}, we define (λ_{1} ⊗ e_{1}) ∧ (λ_{2} ⊗ e_{2})^{∗} ∈ Λ^{∗}(T_{m}^{∗}M) ⊗ A by

(λ_{1} ⊗ e_{1}) ∧ (λ_{2} ⊗ e_{2})^{∗} = (λ_{1} ∧ \overline{λ_{2}})⊗ < e_{1}, e_{2} > .

Extending by linearity (and antilinearity), given ω_{1}, ω_{2} ∈ Λ^{∗}(T_{m}^{∗}M)⊗V_{m}, we can define ω_{1}∧ω_{2}^{∗} ∈ Λ^{∗}(T_{m}^{∗}M)⊗ A.

Define an A-valued quadratic form Q on Ω^{∗}(M; V) by

Q(α, β) = i^{−|α|(n−|α|)} ∫_{M} α(m) ∧ β(m)^{∗}.

It satisfies Q(β, α) = Q(α, β)^{∗}. Using the Hodge duality operator ∗, define τ : Ω^{p}(M; V) → Ω^{n−p}(M; V) by

τ(α) = i^{−|α|(n−|α|)} ∗ α.

Then τ^{2} = 1, and the inner product < ·, · > on Ω^{∗}(M; V) is given by < α, β >= Q(α, τβ). Let d_{V} be the de Rham differential with values in the flat bundle V; define d : Ω^{∗}(M; V) → Ω^{∗+1}(M; V) by

dα = i^{|α|}d_{V}α.  (B.2)

It satisfies d^{2} = 0. Its dual d^{†} with respect to Q, i.e. the operator d^{†} such that Q(α, dβ) = Q(d^{†}α, β), is given by D^{†} = −D. The formal adjoint of D with respect to < ·, · > is D^{∗} = τD^{†}τ = −τDτ.

B.3 Definition. If n is even, the signature operator is

D^{sign} := d + d^{∗} = d − τdτ.  (B.4)

It is formally self-adjoint and anticommutes with the ℤ_{2}-grading operator τ. If we denote by Ω^{±}(M, V) the ±1-eigenspaces of τ then

D^{sign} = (\begin{matrix}0 & D_{−}^{sign} \\ D_{+}^{sign} & 0 \end{matrix})

The index class of the signature operator in K_{0}(A) is, by definition, the index class of the elliptic operator D_{+}^{sign}. If n is odd, the signature operator is

D^{sign} = −i(dτ + τd).  (B.5)

It is formally self-adjoint and defines an index class in K_{1}(A).

Now suppose that M is a compact oriented manifold of dimension n = 2m with boundary ∂M. We fix a non-negative boundary defining function x ∈ C^{∞}(M) for ∂M and a Riemannian metric on M which is isometrically a product in an (open) collar neighborhood U ≡ (0, 2)_{x} × ∂M of ∂M. The signature operator D^{sign} is a well-defined differential operator; it is associated to the graded regular n-dimensional Hermitian complex defined by Ω_{c}^{∗}(int(M); V) and the Riemannian metric on M. Let V_{0} denote the pullback of V from M to ∂M; there is a natural isomorphism

V|_{U} ≅ (0, 2) × V_{0} .

Let Q_{∂M}, τ_{∂M}, d_{∂M} and D^{sign}(∂M) denote the expressions defined above on Ω^{∗}(∂M; V_{0}^{∞}). One can decompose Q, τ, d and D^{sign}, when restricted to compactly-supported forms on (0, 2) × ∂M, in terms of Q_{∂M}, τ_{∂M}, D_{∂M} and D^{sign}(∂M). This computation is given in great detail in [31]; one proves in particular that with the definitions given above the operator D^{sign}(∂M) is the boundary operator of D_{+}^{sign} in the sense of Atiyah-Patodi-Singer [APS, (3.1)].

B.6 Remark. Let (X, g) be closed oriented and of dimension 2m − 1. Assume that V = X × ℂ is the trivial complex line bundle. We wish to compare the eta-invariant of the signature operator defined with our sign-conventions, denoted D^{sign}, and the eta-invariant of the operator B appearing in the work of Atiyah-Patodi-Singer, see [3]. In order to simplify the notation we set D := D^{sign}

Recall that Bϕ = (\sqrt{−1})^{m}(−1)^{p+1}(ϵ ⋆ d − d⋆)ϕ with ϵ = 1 if ϕ ∈ Ω^{2p}(X) and ϵ = −1 if ϕ ∈ Ω^{2p−1}(X). The operator D is given instead by Dϕ = (−1)^{m}((\sqrt{−1}⋆d−d⋆))ϕ if ϕ ∈ Ω^{2p}(X) and Dϕ = −⋆dϕ−\sqrt{−1}d⋆ϕ if ϕ ∈ Ω^{2p−1}(X). Each operator preserves the parity of the form-degree. Moreover, both commute with the following self-adjoint odd involution: Ξ := ηΘ with Θ = (−1)^{p} ⋆ on both Ω^{2p} and Ω^{2p−1} and η = 1 if m = 2k; η = \sqrt{−1} if m = 2k − 1. Thus B = B^{even} ⊕ B^{odd}, D = D^{even} ⊕ D^{odd}, with B^{even} = (Ξ)^{−1}B^{odd}Ξ and similarly for D ^{9}. The Hodge theorem implies the following orthogonal decomposition of the space of differential forms on X, where H stands for the space of harmonic forms:

Ω^{∗} = Ω^{0} ⊕ Ω^{1} ⊕ ⋯ dΩ^{m−2} ⊕ H^{m−1} ⊕ d^{∗}Ω^{m} ⊕ dΩ^{m−1} ⊕ H^{m} ⊕ d^{∗}Ω^{m+1} ⊕ ⋯ ⊕ Ω^{2m−1}

Consider now the following two subspaces of Ω^{∗}:

V = H^{m−1} ⊕ d^{∗}Ω^{m} ⊕ dΩ^{m−1} ⊕ H^{m};
W = Ω^{0} ⊕ Ω^{1} ⊕ ⋯ dΩ^{m−2} ⊕ d^{∗}Ω^{m+1} ⊕ ⋯ ⊕ Ω^{2m−1}

Both operators B and D are block diagonal with respect to the orthogonal decomposition Ω^{∗} = V ⊕ W; we denote by D_{V} , B_{V} and D_{W} , B_{W} their restrictions. Notice that both V and W are invariant under the action of Ξ so that, as above, B_{V} = B_{V}^{even} ⊕ B_{V}^{odd}, B_{W} = B_{W}^{even} ⊕ B_{W}^{odd} and similarly for D_{V} and D_{W}. It’s easy to check from the explicit expression given above that B_{V}^{even} = −D_{V}^{even}. We will show that

η(B_{W}) = 0 = η(D_{W})  (B.7)

which implies immediately that

η(B) = −η(D)  (B.8)

In order to establish (B.7) we argue as follows. Define

Ω^{<} = Ω^{0} ⊕ Ω^{1} ⊕ ⋯ Ω^{m−2} ⊕ dΩ^{m−2} ; Ω^{>} = d^{∗}Ω^{m+1} ⊕ Ω^{m+1} ⋯ ⊕ Ω^{2m−1}

so that W = Ω^{<} ⊕ Ω^{>}. There is a natural involution α on W defined as follows:

α = Id on Ω^{<} α = −Id on Ω^{>}

It is immediate from the structure of D and B that

D_{W} ∘ α + α ∘ D_{W} = 0 , B_{W} ∘ α + α ∘ B_{W} = 0

In other words α gives a grading to W, W^{+} = Ω^{<}, W^{−} = Ω^{>} and both D_{W} and B_{W} are odd with respect to such a grading; thus η(B_{W}) = 0 = η(D_{W}) as required. Conclusion: the odd-signature operator considered here and the odd signature operator considered in the work of Atiyah-Patodi-Singer are different. However, their eta-invariants are equal up to a sign.

C Pseudodifferential operators and morphisms of C^{∗}-algebras

Let A and B be unital C^{∗}-algebras. We assume that there exists a morphism of unital C^{∗}-algebras λ : A → B. Let E_{A} → M be a bundle of finitely generated projective Hilbert A-modules, with fibers isomorphic to a fixed finitely generated projective Hilbert A-module V. In particular, there exists N ∈ ℕ and W, a Hilbert A-module, such that V ⊕W = A^{N} as Hilbert A-modules; moreover V is the image of a self-adjoint projection p ∈ M(N × N, A), p = (p(ij)), i, j = 1, …, N. The morphism λ induces in a natural way a morphism of matrix algebras M(N × N, A) → M(N × N, B) that will be still denoted by λ. If we define p_{λ} := (p_{λ}(ij)) with p_{λ}(ij) := λ(p(ij)), briefly p_{λ} = λ ∘ p, then p_{λ} is still a self-adjoint projection in M(N × N, B) so that λ(V) := Im p_{λ} is a finitely generated projective Hilbert B-module. Applying this reasoning to a more global situation, we see that λ induces a bundle of finitely generated projective Hilbert B-modules, E_{B}^{λ}, with fibers diffeomorphic to λ(V): in fact, if E_{A} is defined by p ∈ C^{∞}(M, M_{N×N}(A)) with p = p^{∗} = p^{2}, then E_{B}^{λ} is simply defined by p_{λ} := λ ∘ p ∈ C^{∞}(M, M_{N×N}(B)). Alternatively, if {U_{α}} is a trivializing covering for E_{A} → M, with transition function g_{α,β}^{A} : Uα ∩ U_{β} → Iso_{A}(V) then E_{B}^{λ} → M is defined by g_{α,β}^{B} := λ ∘ g_{α,β}^{A} : U_{α} ∩ U_{β} → Iso_{B}(λ(V)).

9Notice that there is a sign-mistake in [35]: the Θ given there in Section 1 must be replaced with the Ξ given here

If E_{A} → M is endowed with a connection ∇^{A}, then E_{B} → M inherits in a natural way a connection ∇^{B,λ} defined as follows. If ∇^{A} is equal to p ∘ d ∘ p then ∇^{B,λ} is simply equal to p_{λ} ∘ d ∘ p_{λ}; if, on the other hand, ∇^{A} is arbitrary, then ∇^{A} = p ∘ d ∘ p + ω, with ω ∈ C^{∞}(M, End_{A}(E_{A})) and we then define

∇^{B,λ} = p_{λ} ∘ d ∘ p_{λ} + ω_{λ}

with ω_{λ} ∈ C^{∞}(M, End_{B}(E_{B}^{λ})) the 1-form induced by ω and λ. Alternatively, if ∇^{A} is defined by a collection of local 1-forms {ω_{α}} associated to the trivializing cover {U_{α}}, with

ω_{α} ∈ Ω^{1}(U_{α}, End_{A}(V)) , ω_{β} = g_{α,β}^{−1}dg_{α,β} + g_{α,β}^{−1}ω_{β}g_{α,β}

then we define the connection ∇^{B,λ} through the local 1-forms {ω_{α}^{λ}} where, once again, ω_{α}^{λ} ∈ Ω^{1}(U, End_{B}(λ(V))) is defined in a natural way by ω_{α} and the extension of λ to a morphism M(N × N, A) → M(N × N, B).

Consider now the graded vector space of pseudodifferential Hilbert A-module bundle operators Ψ_{A}^{∗}(M, E_{A}, F_{A}), with F_{A} a second bundle of finitely generated projective Hilbert A-modules. This space, defined for the first time in [51], is nothing but

Ψ^{∗}(M) ⊗_{C^{∞}(M×M)} C^{∞}(M × M, HOM_{A}(F_{A}, E_{A}))  (C.1)

where we are considering the bundle HOM_{A}(F_{A}, E_{A})) ⟶ M×M with fiber at (x, y) equal to Hom_{A}((E_{A})_{y}, (F_{A})_{x}). Here we are using the C^{∞}(M × M)-module structure of both Ψ^{∗}(M) and C^{∞}(M × M, HOM_{A}(F_{A}, E_{A})). The morphism λ induces in a natural way a morphism of graded vector spaces λ_{♮} : Ψ_{A}^{∗}(M, E_{A}, F_{A}) → Ψ_{B}^{∗}(M, E_{B}^{λ}, F_{B}^{λ}) which is simply induced by the natural map

C^{∞}(M × M, HOM_{A}(F_{A}, E_{A})) ⟶ C^{∞}(M × M, HOM_{B}(F_{B}^{λ}, E_{B}^{λ})) .

If F_{A} = E_{A} then λ_{♮} is in fact a morphism of graded algebras; we shall often denote Ψ_{A}^{∗}(M, E_{A}, E_{A}) by Ψ_{A}^{∗}(M, E_{A}) or, more simply, by Ψ_{A}^{∗}. Thus the following simple conclusion is true.

C.2 Corollary. If Q ∈ Ψ_{A}^{∗} is invertible as an element in Ψ_{A}^{∗}, then λ_{♮}(Q) is invertible in Ψ_{B}^{∗}.

As an important example we shall consider twisted Dirac operators. Thus E_{A} = E ⊗ V_{A} with E a vector bundle over M and V_{A} a line bundle with typical fiber A and endowed with a flat connection ∇^{A}. We can then choose trivializations of V_{A} with locally constant transition functions {a_{α,β} ∈ U_{1}(A) ⊂ A}. Let D ∈ Diff^{1}(M, E) be a Dirac-type operator acting on the sections of E and consider D_{V_{A}} ∈ Diff_{A}^{1}(M, E_{A}), the twisted Dirac operator defined by the connection ∇^{A}. Then λ_{♮}(D_{V_{A}}) = D_{V_{B}^{λ}} with V_{B}^{λ} the flat bundle induced by λ. Notice that the transition functions of the flat bundle V_{B}^{λ} will only involve the image λ(A).

Let now χ ∈ C_{c}^{∞}(ℝ, ℝ). The operator D_{V_{A}} defines a self-adjoint unbounded regular operator on the (full) Hilbert A-module L_{A}^{2}(M, E_{A}). It is well-known that there is a continuous functional calculus for regular operators on (full) Hilbert A-modules; in particular χ(D_{V_{A}}) is a well defined bounded A-linear endomorphism on L_{A}^{2}(M, E_{A}). As observed in [30], following the arguments in [61, page 300], one can show that χ(D_{V_{A}}) is in fact a smoothing operator: χ(D_{V_{A}}) ∈ Ψ_{A}^{−∞}(M, E_{A}). Since χ(D_{V_{A}}) is defined through a functional integral, we see immediately that

λ_{♮}(χ(D_{V_{A}}) = χ(λ_{♮}(D_{V_{A}})) = χ(D_{V_{B}^{λ}}) .  (C.3)

The same is true for other functions and other operators, for the same reason. E.g. if C is a smoothing operator, then

λ_{♯} ((D_{V_{A}} + C) exp(t(D_{V_{A}} + C)^{2})) = (D_{V_{B}^{λ}} + λ_{♯}C) exp(t(D_{V_{B}^{♯}} + λ_{♯}C)^{2}).  (C.4)

C.5 Remark. Let ϕ ∈ C_{c}^{∞}(ℝ, [0, 1]) be any real function equal to 1 on [−ϵ, ϵ] and equal to 0 on (−∞, −2ϵ]∪ [2ϵ, ∞). Let C_{A} ∈ Ψ_{A}^{−∞} and let C_{B}^{λ} := λ_{♮}(C_{A}). Then the following implication holds:

if (Id − ϕ(D_{V_{A}})) ∘ C_{A} = 0 then (Id − ϕ(D_{V_{B}^{λ}})) ∘ C_{B}^{λ} = 0 .  (C.6)

Notice that if (Id − ϕ(D_{V_{A}})) ∘ C_{A} = 0 then, taking adjoints, C_{A} ∘ (Id − ϕ(D_{V_{A}})) = 0 .

Let now K ∈ Ψ_{A}^{−∞}(M, E_{A}) be a smoothing operator, and let us consider TR(K) ∈ A_{ab}. As an immediate consequence of the definitions,

λ(TR(K)) = TR(λ_{♮}K) ∈ B_{ab}.  (C.7)

If now C_{B}^{λ} = λ_{♮}(C_{A}), then

C.8 Lemma.

λ(η_{[0]}(D_{V_{A}} + C_{A})) = η_{[0]}(D_{V_{B}^{λ}} + C_{B}^{λ})) in B_{ab}  (C.9)

Proof. This simply follows from the definition of η_{[0]} as a convergent integral. Then, using for the second equality (C.7),

λη_{[0]}(D_{V_{A}} + C_{A}) = \frac{1}{\sqrt{π}} λ ∫_{0}^{∞} TR ((D_{V_{A}} + C_{A}) exp(t(D_{V_{A}} + C_{A})^{2})) \frac{dt}{\sqrt{t}}

= \frac{1}{\sqrt{π}} ∫_{0}^{∞} TR (λ_{♯} ((D_{V_{A}} + C_{A}) exp(t(D_{V_{A}} + C_{A})^{2}))) \frac{dt}{\sqrt{t}}

= \frac{1}{\sqrt{π}} ∫_{0}^{∞} TR ((D_{V_{B}^{λ}} + C_{B}^{λ}) exp(t(D_{V_{B}^{λ}} + C_{B}^{λ})^{2})) \frac{dt}{\sqrt{t}}

= η_{[0]}(D_{V_{B}^{λ}} + C_{B}^{λ}).

D Twists with finite dimensional representations

As a particular case of the principles explained in the previous subsection, we consider a discrete group Γ and a finite dimensional unitary representation λ: Γ → U(d) ⊂ M(d × d, ℂ) . This induces a morphism of C^{∗}-algebras (we also call it λ) λ: C^{∗}Γ → M(d × d, ℂ). Here, C^{∗}Γ is the maximal C^{∗}-algebra of Γ, and the extension of λ follows from the universal property of C^{∗}Γ.

Given, as in Section 3.1, a classifying map u: M → BΓ defined on a closed manifold M, we obtain the corresponding Mishchenko-Fomenko bundle L and the twisted Dirac operator D_{L}; thus in this case

A = C^{∗}Γ , V_{A} ≡ L := M̃ ×_{Γ} C^{∗}Γ , B = M(d × d, ℂ)

Let V_{λ} be the flat vector bundle M̃ ×_{λ} ℂ^{d} and let D_{λ} be the Dirac operator twisted by V_{λ}. Then it is easy to see that the bundle V_{B}^{λ} introduced in the previous subsection is equal to the direct sum of d-copies of V_{λ} and that D_{V_{B}^{λ}} is the diagonal operator D_{λ} ⊗ I_{d}, with I_{d} the d × d-identity matrix. If C ∈ Ψ_{C^{∗}Γ}^{−∞} is a trivializing perturbation, then λ_{♮}C = C_{λ} ⊗ I_{d}, with C_{λ} ∈ Ψ^{−∞} and D_{λ} + C_{λ} invertible. Let tr: M(d × d, ℂ) → ℂ be the usual trace and consider tr_{d} := d^{−1} tr; let tr_{λ} := tr_{d} ∘λ: C^{∗}Γ_{ab} → ℂ. Then, by Lemma C.8,

tr_{λ}(η_{[0]}(D_{L} + C) = η(D_{λ} + C_{λ})  (D.1)

Let us go back to closed manifolds and let X = M ⊔ (−M′) with M and M′ homotopy equivalent. Let us fix a unitary representation λ: Γ → U(d) and let D_{X,λ}^{sign} be signature operator twisted by the flat vector bundle associated to λ. Let 0 < ϵ such that spec(D_{λ,X}^{sign}) ∩ (−ϵ, ϵ) ⊆ {0}. Let ϕ ∈ C_{c}^{∞}(ℝ, [0, 1]) be a real function equal to 1 on [−ϵ, ϵ] and equal to 0 on (−∞, −2ϵ] ∪ [2ϵ, ∞). Let D_{X}^{sign} be the Mishchenko-Fomenko signature operator on X. We know from Section 10 that in this situation we can construct a trivializing perturbation C of D_{X}^{sign} such that (Id − ϕ(D_{X}^{sign})) ∘ C = 0. Thus, using (C.6), we see that C_{λ} is a smoothing operator such that (Id − ϕ(D_{λ,X}^{sign})) ∘ C_{λ} = 0; by Remark 10.16 we can now conclude as in Section 10.1, but without making use of the limiting procedure of Section 10.2, that η(D_{λ}) = (η(D_{λ} + C_{λ}) + η(D_{λ} − C_{λ}))/2. Using Remark 10.8 we can even conclude that the stable and unstable APS-rho invariants coincide, without making use of the limiting procedure of Section 10.2.

E Classical L^{2}-invariants versus N Γ-invariants

E.1 Classical L^{2}-invariants

In Section 1 we introduced (delocalized) L^{2}-invariants by working directly on a normal covering M̃ of a closed manifold M. However, all the main arguments of this paper are given in terms of A/\overline{[A, A]}-valued invariants for Dirac operators twisted by Hilbert A-module bundles. We briefly called these invariants degree zero invariants. In this appendix, we will explain how the classical L^{2}-invariants can be derived from these degree zero invariants, where the C^{∗}-algebra A in question is N Γ, and the bundle to twist with is the Mishchenko-Fomenko line bundle N = M̃ ×_{Γ} N Γ. This procedure is well known to the experts. In this appendix, we will give a detailed description of it, since we are not aware of a place in the literature where this would be covered, in particular when dealing with eta invariants (the focus of interest here).

E.2 Twisting with l^{2}Γ

The passage from the covering situation to the twist with the N Γ bundle is achieved by using an intermediate step: instead of twisting with N := M̃ ×_{Γ} N Γ we twist with H := M̃ ×_{Γ} l^{2}Γ.

The typical fiber of this bundle is isomorphic to l^{2}Γ, which we consider as an N Γ-Hilbert space (in the notation of [57]).

We start with giving the basic definitions, repeated from [57, Section 8].

E.1 Definition. A finitely generated projective N Γ-Hilbert space V is a Hilbert space together with a right action of N Γ such that V embeds isometrically preserving the N Γ-module structure as a direct summand into l^{2}(Γ)^{n} for some n.

A (general) N Γ-Hilbert space V satisfies the same conditions a finitely generated projective N Γ-Hilbert space does, with the exception that l^{2}(Γ)^{n} is replaced by H ⊗ l^{2}(Γ) for some Hilbert space H with trivial N Γ-action (the tensor product has to be completed).

E.2 Definition. An N Γ-Hilbert space morphism is a bounded N Γ-linear map between two N Γ-Hilbert spaces. If it is an isometry for the Hilbert space structure, it is called N Γ-Hilbert space isometry. A N Γ-Hilbert space bundle H on a space X is a locally trivial bundle of N Γ-Hilbert spaces, the transition functions being N Γ-Hilbert space isometries. A smooth structure is given by a trivializing atlas where all the transition functions are smooth.

If the fibers are finitely generated projective N Γ-Hilbert space, the bundle is called a finitely generated projective N Γ-Hilbert space bundle.

E.3 Lemma. The L^{2}-sections of a N Γ-Hilbert space bundle W on a Riemannian manifold X form themselves a N Γ-Hilbert space.

Proof. Compare [57, Lemma 8.10]

E.4 Example. Essentially the only example we will be dealing with is the N Γ-Hilbert space l^{2}(Γ). This is obviously finitely generated projective. Since the left regular representation acts by N Γ-Hilbert space isometries, H := M̃ ×_{Γ} l^{2}(Γ) is a finitely generated projective N Γ-Hilbert space bundle on M.

The trivial connection on M̃ × l^{2}(Γ) descends to a canonical flat connection on M̃ ×_{Γ} l^{2}(Γ).

E.5 Remark. Note that N Γ is canonically a subset of l^{2}(Γ) (the inclusion given by b ↦ 1 · b), and l^{2}(Γ) is the completion of N Γ with respect to the inner product ⟨b_{1}, b_{2}⟩ := tr_{Γ}(b_{1}^{∗}b_{2}). In the same way, M̃ ×_{Γ} l^{2}(Γ) is the fiberwise completion of M̃ ×_{Γ} N Γ. For all of this, compare [57, Section 8.6]

Given a Dirac type operator D: C^{∞}(M, E) → C^{∞}(M, E) on M, we can now also form the twisted Dirac operator D_{H} as usual.

If M is closed, this is an elliptic differential operator of order 1 on N Γ-Hilbert space bundles in the sense of [8]. In any case, it extends to an unbounded operator on L^{2}(E ⊗ H). The main point, as observed in [57, Section 8] is now the following:

E.6 Proposition. The operator D_{H} maps the sections of the subbundle E ⊗ N of E ⊗ H to sections of E ⊗ N, and the restriction is exactly the operator D_{N}. This holds for smooth sections and all kinds of completions. Moreover, all the functions of the operator D_{H} we are considering (like D_{H} exp(−tD_{H}^{2})) restrict to the corresponding functions of D_{N}.

Vice versa, we get D_{H} by applying the procedure of completion of [57, Section 8] to D_{N}.

The above properties together imply that there is a canonical bijection of algebras

{ψ(D_{N}) | ψ is a Schwartz function} and {ψ(D_{H}) | ψ is a Schwartz function}.

From now on, making use of this identification, we will write D_{H} and D_{N} interchangeably. Put it differently, we now have defined the invariants η_{τ} (D_{H}) := η_{τ} (D_{N}), η_{τ_{<g>}}(D_{H}) := η_{τ_{<g>}}(D_{N}).

E.3 Translation between M̃ and M ×_{Γ} l^{2}Γ

In this subsection, we are going to explain why it is equivalent to consider D̃ acting on Ẽ → M̃ and D_{H}, H = M̃ ×_{Γ} l^{2}(Γ), acting on E ⊗ H → M. In fact, there is a dictionary that allows to pass back and forth between objects on the covering and twisted objects. For the sake of completeness we indicate the constructions. Other accounts (where certain aspects are explained in more detail) can be found e.g. in [38, Section III], [59, Section 3.1] and [56, Example 3.39].

The translation is summarized in the following table.

no.  M̃  · ⊗ H
1  L^{2}(M̃, Ẽ)  L^{2}(M, E ⊗ H)
2 {s ∈ C^{∞}(M̃, Ẽ) | ∑_{γ∈Γ} |s(γx)|^{2} < ∞ ∀x ∈ M̃} C^{∞}(M, E ⊗ H)
3  D̃  D_{H} (= D_{N})
4  D̃ exp(−tD̃^{2})  D_{H} exp (−tD_{H}^{2})
5  Tr_{<g>}  τ_{<g>} ∘ TR
6  Tr_{(2)}  τ_{Γ} ∘ TR
7  η_{<g>}(D̃)  η_{τ_{<g>}}(D_{H}).

Recall that for an operator Ã given by the smooth integral kernel k̃, by definition

Tr_{<g>}(Ã) = ∑_{γ∈[g]} ∫_{M̃/Γ} tr_{x} k(x, γx) dx,  (E.7)

and in particular Tr_{(2)}(Ã) = ∫_{M̃/Γ} tr_{x} k̃(x, x) dx.

We give some explanation how to translate between the two sides.

(1) A section s of Ẽ corresponds to the section ŝ of E ⊗ H with ŝ(x) = ∑_{γ∈Γ} s(γx̃) ⊗ [x̃, γ], where x̃ ∈ M̃ is an arbitrary lift of x ∈ M along the covering projection. We identify the fibers E_{x} and Ẽ_{γx̃}. Moreover, by definition H_{x} = Γx̃ ×_{Γ} l^{2}(Γ). This construction is well defined by the very definition of the twisted bundle H, with fiber identified with l^{2}(Γ) using the chosen lift x̃.

(2) The above identification defines an isometry of the spaces of L^{2}-sections. Moreover, it is compatible with the Γ-actions. L^{2}(M̃, Ẽ) is well known to be an N Γ-Hilbert space, and the identification is an identification of N Γ-Hilbert spaces. In addition, it preserves smoothness and continuity, where the condition as given in the table is used to really get a section of E ⊗ H.

(3) The operators D̃ and D_{H} are conjugated to each other under the isomorphism of the section spaces. This follows from their local definition as follows. For a small connected neighborhood U of x ∈ M, we can choose a lift Ũ, a connected neighborhood of a lift x̃, such that there is a unique section U → Ũ of the restriction of the covering M̃ → M to U, and then y ↦ [ỹ, γ] is a by definition of the connection a flat section of H|_{U} for each γ ∈ Γ. Consequently, using the identification of (2) and this flatness, we see that D̃(s̃) corresponds on the set U to

∑_{γ∈Γ} D̃s̃(γx̃) ⊗ [x̃, γ] = D_{H}(∑_{γ∈Γ} s(γx̃) ⊗ [x, γ]),

i.e. to D_{H} applied to the section corresponding to s̃.

(4) Since the self-adjoint unbounded operators D̃ and D_{H} are unitarily equivalent, the same is true for all bounded measurable functions of them, using functional calculus. In particular, this is the case for D̃ exp(−tD̃^{2}), but also for any other bounded measurable function ϕ: ℝ → ℝ.

(5) Choose a subset U ⊂ M such that M \ U has measure zero and such that the restriction of the covering M̃ → M to U is trivial. If we choose an appropriate lift of U then M̃|U ≅ U ×Γ. This induces a trivialization H|_{U} ≅ U × l^{2}(Γ). Using this, we identify L^{2}(U, E ⊗ H|_{U}) = L^{2}(U, E|_{U}) ⊗ l^{2}(Γ).

On the other hand, using the corresponding trivialization of the covering M̃|U ≅ U × Γ we get the identification L^{2}(Ũ, Ẽ|U) ≅ L^{2}(U, E|U) ⊗ l^{2}(Γ), and our unitary isomorphism defined above becomes the identity under these identifications.

So, in this picture, also the operators D̃ and D_{H} are identical. It remains to show that the two ways to define the trace are equal. We take traces of operators like S = D̃e^{−tD̃^{2}}, which have integral kernels k̃(x̃, ỹ) on M̃ × M̃. Note that it is well known that these operators are of Γ-trace class, by the results of [1]. They are also well known to be smoothing operators in the N Γ-Mishchenko-Fomenko calculus, and therefore TR(S) is defined.

Using the identification Ũ = U × Γ, k̃(u, g, v, h) has the property that for a section s of E|_{U} and u ∈ U, g, γ ∈ Γ

S(s(·) ⊗ γ)(u, g) = ∫_{U} k̃(u, g, x, γ)s(x) dx.

On the other hand, if we look at the integral kernel k_{H} on M × M of S, now considered as obtained from D_{H}, using again the identification L^{2}(U, E|_{U} ⊗ H) = L^{2}(U, E|_{U}) ⊗ l^{2}(Γ), we get for a section s of E|_{U}

S(s(·) ⊗ γ)(u) = ∫_{U} k_{H}(u, x)(s(u) ⊗ γ) dx = ∑_{g∈Γ} ∫_{U} k_{H}(u, x)(γ, g)(s(u) ⊗ γ) ⊗ g dx.

where we write the homomorphism k_{H}(u, x) from E_{x} ⊗ l^{2}(Γ) to E_{u} ⊗ l^{2}(Γ) as a matrix (k_{H}(u, x)(γ, g))_{γ,g∈Γ} of homomorphisms from E_{x} to E_{u}, using the orthonormal basis Γ of l^{2}(Γ).

Since these are two representations of the same operator S, it follows that

k_{H}(u, x)(γ, g) = k̃(u, γ, x, g).  (E.8)

Now, by definition of Tr_{<g>}, we get

Tr_{<g>}(S) = ∑_{h∈<g>} ∫_{U} tr_{x} k̃(x, 1, x, h) dx.  (E.9)

On the other hand, by the definition of τ_{<g>} in terms of a matrix decomposition of elements of N Γ ⊂ B(l^{2}Γ), using the basis Γ of l^{2}(Γ),

τ_{<g>} TR(S) = ∫_{U} τ_{<g>} tr_{x}^{alg}(k_{H}(x, x)) dx = ∫_{U} ∑_{h∈<g>} tr_{x}(k_{H}(x, x)(1, h)) dx.  (E.10)

From Equation (E.8), the right hand sides of Equations (E.9) and (E.10) are equal. Consequently,

Tr_{<g>}(S) = τ_{<g>} ∘ TR(S),

as claimed. (6)Since Tr_{(2)} = Tr_{<1>}, this is a special case of (5).

(7) This is a direct consequence of (4) and of (5), using the definition of the η-invariants.

For more details, in particular with respect to the delocalized trace, compare also [39, Section 4] and [38, Section 3]

We want to single out the relevant proposition which is used in the body of this paper, using the results of the above table and of Section :

E.11 Proposition. Let M be a compact manifold with Γ-covering M̃, and D a Dirac type operator on M. Let < g > be a finite conjugacy class in Γ. If D̃ is the lift of D to M̃, and D_{N} is the twist of D with the Mishchenko-Fomenko line bundle N = M̃ ×_{Γ} N Γ, then

η_{<g>}(D̃) = η_{τ_{<g>}}(D_{N}).

In particular, for g = e we compute the classical L^{2}-eta invariant

η_{(2)}(D̃) = η_{τ_{Γ}}(D_{N}).

E.12 Remark. There is an alternative route to Proposition E.11, due to John Lott, and based on particular properties of the heat-kernel. We briefly explain the main ideas, referring to [38] for the details.

There is a sequence of inclusions of algebras

B_{Γ}^{ω} ⊂ B_{Γ}^{∞} ⊂ C_{r}^{∗}Γ ⊂ N Γ ,  (E.13)

where B_{Γ}^{ω} is the algebra of functions on Γ which are exponentially rapidly decreasing and where B_{Γ}^{∞} is the Connes-Moscovici algebra. A generalized Dirac operator can be twisted with the corresponding flat bundles

V^{ω} := M̃ ×_{Γ} B_{Γ}^{ω} , V^{∞} := M̃ ×_{Γ} B_{Γ}^{∞} , V := M̃ ×_{Γ} C_{r}^{∗}Γ , N := M̃ ×_{Γ} N Γ.  (E.14)

producing D^{ω}, D^{∞}, D := D_{V}, D_{N} There are obvious compatibility conditions for these operators, coming from the inclusions of vector spaces

C^{∞}(M, E ⊗ V^{ω}) ⊂ C^{∞}(M, E ⊗ V^{∞}) ⊂ C^{∞}(M, E ⊗ V) ⊂ C^{∞}(M, E ⊗ N)

We also have natural inclusions for the corresponding Mishchenko-Fomenko calculi

Ψ_{B_{Γ}^{ω}}^{∗} ⟶^{j_{ω}} Ψ_{B_{Γ}^{∞}}^{∗} ⟶^{j_{∞}} Ψ_{C_{r}^{∗}Γ}^{∗} ⟶^{j_{r}} Ψ_{N Γ}^{∗}

For the smoothing operators we have natural traces and a commutative diagram

Ψ_{B_{Γ}^{ω}}^{−∞} ⟶^{j_{ω}} Ψ_{B_{Γ}^{∞}}^{−∞} ⟶^{j_{∞}} Ψ_{C_{r}^{∗}Γ}^{−∞} ⟶^{j_{r}} Ψ_{N Γ}^{−∞}
TR_{ω} ↓  ^{TR}_{∞}↓  ^{TR}↓  ^{TR}_{N Γ}↓
(B_{Γ}^{ω})_{ab} ⟶ (B_{Γ}^{∞})_{ab} ⟶ (C_{r}^{∗}Γ)_{ab} ⟶ (N Γ)_{ab}

Using Proposition 6 and Proposition 7 in [38] we have that

τ_{<g>} TR_{ω}(D^{ω}e^{−t(D^{ω})^{2}}) = Tr_{<g>}(D̃e^{−t(D̃)^{2}}) .

Since from the commutative diagram and the definition of τ_{<g>} we clearly have

τ_{<g>} TR_{ω}(D^{ω}e^{−t(D^{ω})^{2}}) = τ_{<g>} TR_{N Γ}(D_{N} e^{−tD_{N}^{2}} ),

we see that Proposition E.11 is proved once again. Notice that we have also established that

τ_{<g>} TR_{∞}(D^{∞}e^{−t(D^{∞})^{2}}) = Tr_{<g>}(D̃e^{−t(D̃)^{2}}) .

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