Relative-partitioned index theorem

TL;DR

This paper develops a new theorem combining relative and partitioned index theorems within the framework of coarse geometry and $K$-theory, advancing the understanding of index theory on non-compact spaces.

Contribution

It introduces a novel partitioned-relative index theorem that unifies two important concepts in the index theory of non-compact spaces.

Findings

Formulation of a combined partitioned-relative index theorem.

Extension of index theory to broader classes of non-compact spaces.

Provides a new tool for analyzing operators on coarse spaces.

Abstract

It seems that the index theory for non-compact spaces has found its ultimate formulation in realm of coarse spaces and $K$-theory of related operator algebras. Relative and partitioned index theorems may be mentioned as two important and interesting examples of this program. In this paper we formulate a combination of these two theorems and establish a partitioned-relative index theorem.