Analytic and topological index maps with values in the K-theory of mapping cones

TL;DR

This paper develops index maps valued in the K-theory of mapping cones, establishing an index theorem analogous to Freed-Melrose, using geometric K-homology and explicit isomorphisms.

Contribution

It introduces a new framework for index maps with values in the K-theory of mapping cones, connecting geometric K-homology to KK-theory through explicit isomorphisms.

Findings

Defined index maps in the K-theory of mapping cones

Established an index theorem analogous to Freed-Melrose

Constructed explicit isomorphism between geometric K-homology and KK-theory

Abstract

Index maps taking values in the $K$-theory of a mapping cone are defined and discussed. The resulting index theorem can be viewed in analogy with the Freed-Melrose index theorem. The framework of geometric $K$-homology is used in a fundamental way. In particular, an explicit isomorphism from a geometric model for $K$-homology with coefficients in a mapping cone, $C_{\phi}$, to $KK(C(X),C_{\phi})$ is constructed.