Mapping the surgery exact sequence for topological manifolds to analysis

TL;DR

This paper establishes a natural connection between the topological surgery exact sequence and the analytic surgery sequence, extending previous results to topological manifolds and odd dimensions, with applications to rho classes and index theorems.

Contribution

It generalizes the Higson-Roe surgery sequence mapping to topological manifolds using the Lipschitz signature operator and extends results to odd dimensions and equivariant settings.

Findings

Established a natural map from topological to analytic surgery sequences.

Extended the APS delocalized index theorem to odd-dimensional manifolds.

Provided geometric applications to stability of rho classes.

Abstract

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalised index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.