Relative Equivariant Coarse Index Theorem and Relative $L^2$-Index Theorem

TL;DR

This paper introduces a relative equivariant coarse index and a relative $L^2$-index, extending classical index theorems to a relative, equivariant setting for proper group actions.

Contribution

It defines the relative equivariant coarse index and the relative $L^2$-index, and proves their corresponding index theorems, generalizing Roe's and Atiyah's classical results.

Findings

Established a relative equivariant coarse index theorem linking to localized indices.

Proved a relative $L^2$-index theorem as a generalization of Atiyah's $L^2$-index theorem.

Extended classical index theorems to a new relative, equivariant context.

Abstract

In this paper, we give a definition of the relative equivariant coarse index for proper actions and derive a relative equivariant coarse index theorem connecting this index with the localized equivariant coarse indices. This is an equivariant version of Roe's relative coarse index theorem in arXiv:arch-ive/1210.6100. Furthermore, we present a definition of the relative $L^2$-index and prove a relative $L^2$-index theorem which is a relative version of Atiyah's $L^2$-index theorem.