The signature package on Witt spaces

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.