A classifying space for Phillips' equivariant K-theory

TL;DR

This paper constructs a classifying space for Phillips' equivariant K-theory, establishing an isomorphism with a new, more general equivariant cohomology theory for Lie groups, extending prior work and resolving a longstanding issue.

Contribution

It introduces a classifying space for Phillips' equivariant K-theory and demonstrates an isomorphism with a new equivariant cohomology theory for Lie groups.

Findings

Constructed a classifying space for Phillips' equivariant K-theory.

Established an isomorphism between the new theory and Phillips' theory under certain conditions.

Extended the theory to proper G-CW-complexes for Lie groups.

Abstract

In a previous paper, we have constructed, for an arbitrary Lie group G and any of the fields F=R or C, a good equivariant cohomology theory KF_G^*(-) on the category of proper $G$-CW-complex and have justified why it deserved the label ``equivariant K-theory". It was shown in particular how this theory was a logical extension of the construction of L\"uck and Oliver for discrete groups and coincided with Segal's classical K-theory when G is a compact group and only finite G-CW-complexes are considered. Here, we compare our new equivariant K-theory with that of N.C. Phillips: it is shown how a natural transformation from ours to his may be constructed which gives rises to an isomorphism when G is second-countable and only finite proper G-CW-complexes are considered. This solves the long-standing issue of the existence of a classifying space for Phillips' equivariant K-theory.