CONFERENCE TALKS xv
A topological index theorem for manifolds with corners
Victor Nistor
We define a topological and an analytical index for manifolds with corners
M. They both live in the K-theory groups K0(Cb
∗(M))
of the groupoid algebra
associated to our manifold with corners M by integrating the Lie algebra of vector
fields tangent to all faces of M. We prove that the topological and analytic index
coincide. For M smooth (no corners), this is the Atiyah-Singer index theorem. If
all the open faces of M are euclidean spaces, then the index maps are isomophisms,
which gives a way of computing the K-theory of the groups K∗(Cb
∗(M)).
The
proof uses a double-deformation groupoid obtained by integrating a suitable Lie
algebroid. This is a joint work with Bertrand Monthubert.
The signature package on Witt spaces
Paolo Piazza
Let X be an orientable closed compact riemannian manifold with fundamental
group G. Let X be a Galois G-covering and r: X BG a classifying map for X .
The signature package for (X, r : X BG) can be informally stated as follows:
there is a signature index class in the K-theory of the reduced
C∗-algebra
of G
the signature index class is a bordism invariant
the signature index class is equal to the
C∗-algebraic
Mishchenko signa-
ture, also a bordism invariant which is, in addition, a homotopy invariant
there is a K-homology signature class in
K∗(X)
whose Chern character
is, rationally, the Poincare’ dual of the L-Class
if the assembly map in K-theory is rationally injective one deduces from
the above results the homotopy invariance of Novikov higher signatures
The goal of my talk is to discuss the signature package on a class of stratified
psedomanifolds known as Witt spaces. The topological objects involve intersection
homology and Siegel’s Witt bordism groups. The analytic objects involve some
delicate elliptic theory on the regular part of the stratified pseudomanifold. Our
analytic results reestablish (with completely different techniques) and extend results
of Jeff Cheeger. This is joint work, some still in progress, with Pierre Albin, Eric
Leichtnam and Rafe Mazzeo.
Group measure space decomposition of factors and W
∗-superrigidity
Sorin Popa
A free ergodic measure preserving action of a countable group on a probability
space, Γ X, gives rise to a II1 factor,
L∞(X)
Γ, through the group measure
space construction of Murray and von Neumann. In general, much of the initial
data Γ X is “forgotten” by the isomorphism class of
L∞(X)
Γ, for instance all
free ergodic probability measure preserving actions of amenable groups give rise to
isomorphic II1 factors (Connes 1975). But a rich and deep rigidity theory underlies
the non-amenable case. For instance, I have shown in 2005 that any isomorphism
of II1 factors associated with Bernoulli actions Γ X, Λ Y , of Kazhdan
groups Γ, Λ comes from a conjugacy of the actions (W
∗-strong
rigidity). I will
present a recent joint work with Stefaan Vaes, in which we succeeded to prove a
W
∗-superrigidity
result for Bernoulli (+ other) actions Γ X of amalgamated
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