Analytical Signatures and Proper Actions

TL;DR

This paper compares two definitions of noncommutative signatures for manifolds with proper group actions and discusses a generalization of the coarse index, bridging geometric and analytical perspectives.

Contribution

It provides a comparison between Mishchenko's topological signature and Higson-Roe's analytical signature, and explores a generalized coarse index in this context.

Findings

Mishchenko's and Higson-Roe's signatures are shown to be comparable.

A generalized coarse index framework is proposed.

The paper bridges topological and analytical approaches to signatures.

Abstract

In this short note we compare Mishchenko's definition of noncommutative signature for a manifold with proper $G$-action of a discrete, countable group $G$ with the (more analytical) counterpart defined by Higson and Roe in the series of articles "Mapping Surgery to Analysis". A generalization of the coarse index is also addressed.