Geometric K-homology with coefficients I

TL;DR

This paper develops a geometric model for K-homology with coefficients in finite cyclic groups, extending to countable abelian groups, and explores its connection to the Freed-Melrose index theorem.

Contribution

It introduces a Baum-Douglas type model for K-homology with coefficients in rac{k}{rac}Z and generalizes it to all countable abelian groups.

Findings

Constructed a geometric model for K-homology with rac{k}{rac}Z coefficients.

Established the relationship with the Freed-Melrose index theorem.

Extended the model to arbitrary countable abelian groups.

Abstract

We construct a Baum-Douglas type model for $K$-homology with coefficients in $\mathbb{Z}/k\mathbb{Z}$. The basic geometric object in a cycle is a $spin^c$ $\mathbb{Z}/k\mathbb{Z}$-manifold. The relationship between these cycles and the topological side of the Freed-Melrose index theorem is discussed in detail. Finally, using inductive limits, we construct geometric models for $K$-homology with coefficients in any countable abelian group.