Stratified surgery and K-theory invariants of the signature operator

TL;DR

This paper extends the concept of signature invariants and secondary invariants like the rho-class from closed manifolds to stratified spaces called Cheeger spaces, establishing a new surgery exact sequence and exploring geometric applications.

Contribution

It introduces a natural transformation linking surgery exact sequences and K-theory for Cheeger spaces, extending invariants from smooth to stratified spaces.

Findings

Secondary invariants extend to Cheeger spaces.

A new surgery exact sequence for stratified spaces is established.

Connections between invariants and geometric applications are discussed.

Abstract

In work of Higson-Roe the fundamental role of the signature as a homotopy and bordism invariant for oriented manifolds is made manifest in how it and related secondary invariants define a natural transformation between the (Browder-Novikov-Sullivan-Wall) surgery exact sequence and a long exact sequence of C*-algebra K-theory groups. In recent years the (higher) signature invariants have been extended from closed oriented manifolds to a class of stratified spaces known as L-spaces or Cheeger spaces. In this paper we show that secondary invariants, such as the rho-class, also extend from closed manifolds to Cheeger spaces. We revisit a surgery exact sequence for stratified spaces originally introduced by Browder-Quinn and obtain a natural transformation analogous to that of Higson-Roe. We also discuss geometric applications.