The signature package on Witt spaces, II. Higher signatures

TL;DR

This paper extends the study of higher signatures on Witt spaces, demonstrating their invariance under stratified homotopies and Witt bordisms, and linking analytic and topological signatures through K-theory and assembly maps.

Contribution

It proves the invariance of higher signatures on Witt spaces under stratified homotopies and bordisms, and relates analytic and topological signatures via the assembly map.

Findings

Signature index class is invariant under rational Witt bordisms.

Analytic signature class matches the topological Mischenko symmetric signature.

Witt-Novikov higher signatures are stratified homotopy invariants under certain conditions.

Abstract

This is a sequel to the paper "The signature package on Witt spaces, I. Index classes" by the same authors. In the first part we investigated, via a parametrix construction, the regularity properties of the signature operator on a stratified Witt pseudomanifold, proving, in particular, that one can define a K-homology signature class. We also established the existence of an analytic index class for the signature operator twisted by a C^*_r\Gamma Mischenko bundle and proved that the K-homology signature class is mapped to the signature index class by the assembly map. In this paper we continue our study, showing that the signature index class is invariant under rational Witt bordisms and stratified homotopies. We are also able to identify this analytic class with the topological analogue of the Mischenko symmetric signature recently defined by Banagl. Finally, we define Witt-Novikov higher signatures and show that our analytic results imply a purely topological theorem, namely that the Witt-Novikov higher signatures are stratified homotopy invariants if the assembly map in K-theory is rationally injective.