Analysis and Topology in Interaction 2011
Conference in Cortona, Italy
June 6-11, 2011
Titles and abstracts, slides (of Banagl, Friedman, Nest, Wockel)
and notes of talks,
as taken by Pierre Albin
For the notes: many thanks to Pierre Albin who did a teriffic job with them.
- ALBIN
Title: The signature operator on stratified pseudomanifolds
Abstract: The signature operator of a Riemannian metric is an
important tool for studying topological questions with analytic
machinery. Though well-understood for smooth metrics on
compact manifolds, there are many open questions when the
metric is allowed to have singularities. We will report on joint work
of Pierre Albin, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza on the
signature operator on stratified pseudomanifolds and some of
its topological applications.
-
Banagl
Title: Singularities and Intersection Spaces: A Topological-Geometric
Panopticon
Abstract:
In many situations, it is homotopy theoretically possible to associate
to a
singular space in a natural way a generalized geometric Poincare
complex,
called its intersection space, whose cohomology turns out to be a new
cohomology theory for singular spaces, not isomorphic in general to
intersection
cohomology or Cheeger-L2-cohomology. An alternative description of the
new theory by a de Rham complex of global differential forms is
available.
The talk will discuss the properties of the new theory, stressing
differences to
intersection cohomology, as well as indicate numerous applications of
these
methods, even outside singularity theory. We will consider the K-theory
of intersection spaces. We will see how, as a by-product, one
obtains results on equivariant cohomology and flat bundles, closely
related
to results of X. Dai and Jörn Müller. The theory also
addresses
questions in type II
string theory and mirror symmetry.
While intersection cohomology is stable under small resolutions, the new
theory is often stable under deformations of singularities. The latter
result is partly joint work with L. Maxim.
An analytic description remains to be found, but
we shall indicate a partial result based on results of
Melrose and Hausel-Hunsicker-Mazzeo.
-
Bismut
Title: Bott-Chern cohomology and Riemann-Roch-Grothendieck
Abstract:
The Bott-Chern cohomology of a complex manifold is a refinement of its ordinarycohomology. In the talk, I will
explain the proof of a theorem of Riemann-Roch-Grothendieck for proper holomorphic submersions with values in the
Bott-Chern cohomology, the considered manifolds not being assumed to be
Kähler. The proof is based on an exotic
hypoelliptic deformation of Hodge theory.
- BUNKE
Title: The topological contents of eta the invariant (U. Bunke)
Abstract: The talk is about the universal bordism invariant obtainable
from eta invariants. I will present its construction and main
properties. The main result is an index theorem which identifies
the analytic construction of the universal eta invariant with a
homotopy theoretic one. I will also discuss intrinsic formulas
for the universal eta invariant and how classical invariants can
be derived as special cases.
-
CARRON
Title: Subelliptic boundary conditions (with Ballmann and
Brüing)
Abstract :
I will introduce a general class of boundary condition for Dirac type
operator
that imply subelliptic estimate. Within this class, a relative/Splitting
index theorem
can be easily proved.
-
Davis
Title: Torus bundles over lens spaces and topological K-theory
(joint with Lück)
Abstract:
For an odd prime p, there is an (p-1)-integral representation of Z/p
which
is free away from the origin, which gives an action of Z/p on the
(p-1)-dimensional torus T with p fixed points. Let M be the quotient of
the
diagonal Z/p-action on S \times T where S is an odd-dimensional sphere
with
a free action. Let \Gamma be the fundamental group of M.
We compute the equivariant K-(co)homology of the torus, and, using the
Baum-Connes and Farrell-Jones Conjectures, the K-theory of the
C^*-algebra
of \Gamma and the structure set of the classifying space for \Gamma. As
geometric consequences we prove the Gromov-Lawson-Rosenberg Conjecture
for
Spin manifolds with fundamental group \Gamma and classify manifolds
having
the homotopy type of M. This is the first computation of a structure
set
for a manifold whose fundamental group is not torsion-free, finite, or
built
from such using amalgamated products of HNN extensions.
-
Deeley
Title: A relative construction in geometric K-homology and R/Z-valued
index theory
Abstract:
Given a unital *-homomorphism between (unital) C*-algebras, we
construct a "relative" K-homology group via geometric cycles defined
using spin^c manifolds with boundary and Hilbert C*-module bundles.
This group fits into a natural six-term exact sequence and has a
natural index mapping to the K-theory of the mapping cone of the
*-homomorphism. A prototypical example occurs when the *-homomorphism
is the inclusion of the complex numbers into a II_1-factor. In this
case, the construction gives a geometric realization of K-homology
with coefficients in R/Z.
-
DEGERATU
Title: Singular Spin Structures
Abstract: Unlike a 3-dimensional manifold, a higher dimensional manifold
need not be spin. On an oriented Riemannian manifold the obstruction to
having a spin structure is given by the
second Stiefel-Whitney class. I will show that even when this
obstruction
does not vanish, it is still possible to define a notion of singular
spin
structure and associated singular Dirac operator. I will conclude with
insights this approach gives into the Positive Mass Theorem for higher
dimensional non-spin-manifolds. This is joint work with Richard Melrose
and
Mark Stern.
-
GOETTE
Title: Generalized Kreck-Stolz invariants and 2-connected homology
7-spheres
Abstract:
We generalise the Kreck-Stolz invariants $s_2$ and $s_3$ by
defining a new
invariant, the $t$-invariant, for quaternionic line bundles over
closed
spin-manifolds $M$ of dimension $4k-1$ with $H^3(M; \Q) = H^4(M;
\Q) = 0$.
The $t$-invariant classifies closed smooth oriented $2$-connected
rational
homology $7$-spheres up to almost-diffeomorphism, that is, up to
connected
sum with an exotic sphere. It also detects exotic homeomorphisms
between
such manifolds.
The $t$-invariant also gives information about quaternionic line
bundles
over a fixed manifold and we use it to give a new proof of a
theorem of
Feder and Gitler about the values of the second Chern classes of
quaternionic line bundles over $\H P^k$. The $t$-invariant for
$S^{4k-1}$
is closely related to the Adams $e$-invariant on the $(4k-5)$-stem.
-
Friedman
Title:
Cup
and cap products and symmetric signatures in intersection (co)homology
(Coauthors: James McClure)
Abstract: Intersection homology was introduced by Goresky and MacPherson in order
to extend Poincare duality and related invariants to singular spaces,
such as algebraic varieties. We'll provide a quick review of the basics
and then discuss recent joint work with McClure extending other
classical algebraic topology techniques such as cup and cap products
to the world of intersection homology. In particular, this work allows us to construct a stratified-homotopy
invariant symmetric signature for Witt spaces.
-
Hambleton
Title: Gauge theory and smooth group actions on 4 manifolds
Abstract: we discuss which groups can act non-trivially on manifolds,
in particular on 4-manifolds, of a given topology. For 4-manifolds, this
involves analysis of the Yang-Mills moduli spaces.
- HANKE
Title: Homotopy groups of the moduli space of metrics of positive scalar curvature.
Abstract: This is joint work with B. Botvinnik, T. Schick and M. Walsh.
We show by explicit examples that in many degrees in a stable range the
homotopy groups of the moduli spaces of Riemannian metrics of positive
scalar curvature on closed smooth manifolds can be non-trivial. This is achieved by further
developing and then applying a family version of the surgery construction of Gromov-Lawson
to certain nonlinear smooth sphere bundles constructed by Hatcher.
- Kahle
Title: T-duality and differential K-theory
Abstract: I will give a precise formulation of T-duality for
Ramond-Ramond fields.
This gives a canonical isomorphism between the "geometrically invariant"
subgroups
of the twisted
differential K-theory of certain principal torus bundles. Our result
combines
topological T-duality
with the Buscher rules found in physics.
-
Natsume
Title:The trace formulae for Toeplitz operators and noncommutative geometry.
abstract:
For $C^\infty$-functions $f$ and $g$ on the circle $S^1$, the commutator $[T_f, T_g]$ of the
Toeplitz
operators is a trace class operator, and the trace formula says
$$ Tr([T_f, T_g]) = \frac{1}{2\pi i}\int_{S^1}fdg.$$
This formula describes Helton-Howe's fundamental trace form $W$ in terms of a cyclic cocycle.
We generalize this formula to higher dimensional cases. Suppose that $D$ is a self-adjoint elliptic
differential differential operator of order one on a closed connected manifold $M$ of dimension n.
As
above, denote by $T_f$ the Toeplitz operator with the symbol $f\in C^\infty(M)$. Then
$$W(T_{f_0}, T_{f_1}, \cdots, T_{f_n})= C_n\int_Mf_0 df_1\wedge\cdots\wedge df_n.$$
-
NEST
Title: Formal deformations of gerbes
Abstract: This talk is an overview of the results on the
deformations of gerbes leading to the gerbe version of the
algebraic index theorem
This is a joint work of P. Bressler, A. Gorokhovsky, R. Nest and B. Tsygan
-
MORIYOSHI
Title: Eta cocycles
Abstract: On a complete manifold with cylindrical end there exists a short exact sequence of
C*-algebra associated
to Roe's cyclic 1-cocycle, which is equivalent to the Wiener-Hopf extension
for the group C*-algebra of the real numbers. On such a sequence we construct a relative cyclic cycle
by
exploiting the notion of b-trace due to Melrose and a relative index class for the Dirac operator. We
also
exhibit that the pairing of those elements yields the Atiyah-Patodi-Singer index theorem. Here the eta
invariant is understood as the transgression term.
With this a basic framework we shall generalize the APS theorem to a foliated manifold with boundary.
We establish an index formula for longitudinal Dirac operators on a foliated bundle associated with
the
Godbillon-Vey cyclic cocycle due to Moriyoshi-Natsume, and introduce the Godbillon-Vey eta invariant.
The talk is a joint work with P. Piazza.
-
Reich
Title: The Farrell-Jones Conjecture for SL_n(Z)
(joint work with Arthur Bartels, Wolfgang Lück and Henrik Rüping)
Abstract: The Farrell-Jones conjecture predicts that the algebraic K-theory of a
group ring RG can be assembled from the algebraic K-theory of the
rings RH, where H runs over all virtually cyclic subgroups of G. There
is an analogous conjecture for L- theory. Among the most important
applications of this conjecture is the Borel conjecture, which asserts
that aspherical manifolds with fundamental group G are topologically
rigid. A large part of the talk will give an introduction to the
Farrell-Jones conjecture and its applications.
We prove the conjecture for the groups SLn(Z). In fact we prove both
the K- and L-theory conjecture in the more general version with
coefficients in an additive category (with involutions). Since these
generalized versions have good inheritance properties one can easily
deduce the following statement.
The K- and L-theoretic Farrell-Jones conjecture with coefficients in
an additive category holds for groups that are subgroups or finite
index overgroups of GLn(S), where S is any ring (not necessarily
commutative) whose underlying abelian group is finitely generated.
We would like to remark that the Borel conjecture for torsionfree
discrete subgroups of GLn(R) is covered by work of Farrell and Jones
and that the Baum- Connes conjecture is not known for SLn(Z) if n ≥ 4.
- ROCHON
Title: Pseudodifferential operators on manifolds with foliated
boundaries
Abstract: After describing the pseudodifferential calculus that Mazzeo
and
Melrose introduced on manifolds with fibred boundary, we will indicate
how
to generalize their construction to situations
where the fibration on the boundary is replaced by a foliation. The
operators obtained in
this way have nice mapping properties. We will provide some simple
criteria
to determine when such operators are compact or Fredholm (when acting on
suitable Sobolev spaces).
Time permitting, we will conclude by exhibiting a formula for the index
of
certain
Dirac-type operators arising in this context.
- STEIMLE
title:
K-theoretic obstructions to fibering a manifold
Abstract:
Given a map f: M --> B between compact manifolds, is it homotopic to
the projection map of a fiber bundle whose fibers are compact manifolds?
Obstructions in higher algebraic K-theory to fibering the given map
f will be defined. These refine the obstructions recently defined by
Farrell-Lueck-Steimle; the vanishing of these obstructions has a
concrete geometrical meaning: the obstructions are zero if and only if
f fibers stably, i.e. after crossing M with a high-dimensional disk.
The methods also provide a classification of the different ways of
stably fibering f in terms of algebraic K-theory.
-
VAN ERP
Title: Fredholm operators in K-homology: Beyond ellipticity
Abstract:
The isomorphism between topological K-homology (Baum-Douglas) and
analytic K-homology (Atiyah, Kasparov) provides a powerful context in
which to study the index theory of elliptic operators. In recent work
with Paul Baum we showed that the index theory of certain non-elliptic
Fredholm operators is greatly clarified by placing it in the
K-homology framework. We present the computation of a geometric
K-cycle (and Chern character) for a class of hypoelliptic operators on
contact manifolds. This K-cycle has interesting features that are not
seen in the elliptic case.
This is joint work with Paul Baum.
-
WAHL
Title: Higher rho-invariants for the signature operator and the
surgery exact sequence
Abstract: Using their higher Atiyah-Patodi-Singer index theorem
Leichtnam and Piazza constructed a commuting diagram mapping part of
Stolz's positive scalar curvature exact sequence to noncommutative de
Rham homology. The vertical maps are given by the higher index on
manifolds with boundary and the higher rho-invariant for the spin
Dirac operator, respectively. Chang and Weinberger constructed a
similar commutative diagram for the surgery exact sequence using
L^2-signatures and L^2-rho-invariants. We show how to define the
latter diagram in the higher case. It is still an open question
whether the diagram commutes.
- WANG
Title: Stringy product on orbifold K-theory
Abstract: Motivated by orbifold string theory models in physics, Chen and Ruan discovered a new product called
the stringy product on the cohomology for the inertia orbifold of an almost complex orbifold. This cohomology with
the stringy product is now called the Chen-Ruan cohomology, as a classical limit of orbifold quantum cohomology
theory. Later, various versions of stringy product on orbifold K-theory of the inertia orbifold were proposed. In
recent joint work with Hu, we define a stringy product on the orbifold K-theory for the orbifold itself and show
that a modified de-localized Chern character to the Chen-Ruan cohomology is an isomorphism over the complex
coefficient. As an application, we find a new product on the equivariant K-theory of a finite group with the
conjugation product, which is different to the well-known Pontryajin (or fusion) product.
-
Weiss
TITLE: Smooth maps to the plane and Pontryagin classes
(joint work with Rui Reis)
The long-term goal is to prove that the Pontryagin classes for fiber
bundles with
fiber R^n, due to Novikov and Thom, satisfy the standard relations that
we expect
from Pontryagin classes of vector bundles of dimension n. Namely, square
of Euler
class is a Pontryagin class, and as a consequence of that, Pontryagin
class in
degree 4i of a bundle of fiber dimension n vanishes when 4i>2n. Using
smoothing
theory and some functor calculus, we translate this into a hypothesis in
differential topology, concerning spaces of regular (nonsingular) smooth
maps from
D^n times D^2 to D^2 which extend the standard projection on the entire
boundary of
D^n times D^2. The talk is in part about the translation, and then about
ideas from
concordance theory, and again from functor calculus, which lead to a
strategy for
proving the hypothesis on regular maps to D^2.
-
Wockel
Title: A smooth model for the string group
Abstract: There have been various constructions of the string group that are suited for differential geometric applications.
It cannot be a finite-dimensional Lie group, so one of the most natural things to expect would be an
infinite-dimensional Lie group. Surprisingly, such a model did not exist in the past.
In this talk I will explain how to construct such an infinite-dimensional model, based on the construction of a
topological model by Stolz. Moreover, it will be shown how this can be extended to a Lie 2-group model, making
explicit comparisons between ordinary and categorical differential geometry possible.