Paolo Piazza(Rome La Sapienza)

"The signature operator on Witt spaces"

"The signature operator on Witt spaces"

Let X be an orientable closed compact riemannian manifold with fundamental group G. Let X' be a Galois G-covering and r: X\to BG a classifying map for X'. The signature package for (X,r:X\to 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 signature, 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 pseudomanifolds 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.

In this talk I will concentrate on the geometric and analytic aspects of this projects

This is joint work, some still in progress, with Pierre Albin, Eric Leichtnam and Rafe Mazzeo. Part of our results is available on arXiv