The signature package on Witt spaces

TL;DR

This paper develops a comprehensive analytic framework for the signature operator on Witt spaces, establishing self-adjointness, spectral properties, and topological invariance of higher signatures, extending classical results to singular spaces.

Contribution

It introduces a parametrix construction for the signature operator on Witt spaces, enabling the definition of an analytic signature index class and proving stratified homotopy invariance of higher signatures.

Findings

Signature operator is essentially self-adjoint with discrete spectrum.

Analytic signature index class can be defined in K-theory of C*_r extGamma.

Higher signatures are stratified homotopy invariant under certain conditions.

Abstract

In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction, which is inductive over the `depth' of the singularity, is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index -- the analytic signature of X -- is well-defined. This provides an alternate approach to some well-known results due to Cheeger. We then prove some new results. By coupling this parametrix construction to a C*_r\Gamma-Mishchenko bundle associated to any Galois covering of X with covering group \Gamma, we prove analogues of the same analytic results, from which it follows that one may define an analytic signature index class as an element of the K-theory of C*_r\Gamma. We go on to establish in this setting and for this class the full range of conclusions which sometimes goes by the name of the signature package. In particular, we prove a new and purely topological theorem, asserting the stratified homotopy invariance of the higher signatures of X, defined through the homology L-class of X, whenever the rational assembly map K_* (B\Gamma)\otimes\bbQ \to K_*(C*_r \Gamma)\otimes \bbQ is injective.