The signature package on Witt spaces, II. Higher signatures

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.

💡 Research Summary

This paper continues the authors’ investigation of the signature operator on stratified Witt pseudomanifolds, focusing on higher signatures and their invariance properties. Building on the parametrix construction introduced in the first part of the series, the authors first reaffirm that the signature operator (D_{\mathrm{sign}}) on a Witt space is essentially self‑adjoint and admits a well‑behaved Sobolev domain. Consequently, it defines a K‑homology class (