Equivariant geometric K-homology for compact Lie group actions

TL;DR

This paper introduces an equivariant geometric K-homology theory for compact Lie group actions on compact G-CW-complexes, establishing isomorphisms with existing KK-theory-based definitions.

Contribution

It defines explicit geometric models for equivariant K-homology and proves their equivalence to KK-theory formulations, enhancing the understanding of equivariant topological invariants.

Findings

Defined equivariant geometric K-homology groups for compact Lie groups.

Constructed natural isomorphisms with KK-theory-based K-homology.

Provided a geometric framework for equivariant K-homology.

Abstract

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K^G_*(X), using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the "official" equivariant K-homology groups) and show that these are isomorphism.