Higher signatures on Witt spaces

Zhizhang Xie  (Texas A&M)

9:30 am to 10:30 am, August 14th, 2015   Science Building A1510

Abstract:

The signature is a fundamental homotopy invariant for topological manifolds. However, for spaces with singularities, this usual notion of signature ceases to exist, since, in general, spaces with singularities fail the usual Poincar\'{e} duality. A generalized Poincare duality theorem for spaces with singularities was proven by Goresky and MacPherson using intersection homology. The classical signature was then extended to Witt spaces by Siegel using this generalized Poincar\'{e} duality. Witt spaces are a natural class of spaces with singularities. For example, all complex algebraic varieties are Witt spaces. In this talk, I will describe a noncommutative geometric approach to higher signatures of Witt spaces.

This is based on joint work with Nigel Higson.