Geometric $K$-homology with coefficients II

TL;DR

This paper explores the analytic aspects of geometric K-homology with coefficients, establishing an isomorphism with its analytic model for finite CW-complexes and relating it to the Freed-Melrose index theorem.

Contribution

It constructs a map from geometric to analytic K-homology with coefficients and proves it is an isomorphism for finite CW-complexes, extending Baum-Douglas results.

Findings

Constructed a map from geometric to analytic K-homology with coefficients.

Proved the map is an isomorphism for finite CW-complexes.

Connected the results to the Freed-Melrose index theorem.

Abstract

We discuss the analytic aspects of the geometric model for $K$-homology with coefficients in $\mathbb{Z}/k\mathbb{Z}$ constructed in "Geometric K-homology with coefficients I". In particular, using results of Rosenberg and Schochet, we construct a map from this geometric model to its analytic counterpart. Moreover, we show that this map is an isomorphism in the case of a finite CW-complex. The relationship between this map and the Freed-Melrose index theorem is also discussed. Many of these results are analogous to those of Baum and Douglas in the case of $spin^c$ manifolds, geometric K-homology, and Atiyah-Singer index theorem.