On a KK-theoretic counterpart of relative index theorems

TL;DR

This paper extends the concept of relative index theorems to KK-theory, providing a noncommutative geometric framework for comparing homotopy invariants beyond classical indices.

Contribution

It introduces a KK-theoretic relative index theorem that generalizes classical results to noncommutative geometry settings.

Findings

Establishes a KK-theoretic analogue of relative index theorems.

Provides tools for comparing KK-elements in cutting and pasting scenarios.

Enhances understanding of homotopy invariants in noncommutative geometry.

Abstract

Relative index theorems, which deal with what happens with the index of elliptic operators when cutting and pasting, are abundant in the literature. It is desirable to obtain similar theorems for other stable homotopy invariants, not the index alone. In the spirit of noncommutative geometry, we prove a full-fledged "relative index" type theorem that compares certain elements of the Kasparov KK-group KK(A,B).