Equivariant K-theory for proper actions of non-compact Lie groups

TL;DR

This paper develops a new equivariant K-theory for proper actions of arbitrary Lie groups, extending Segal's theory to non-compact cases through a classifying space construction.

Contribution

It introduces a generalized equivariant cohomology theory for proper G-CW complexes for any Lie group, including non-compact ones, via a novel classifying space approach.

Findings

Generalizes Segal's equivariant K-theory to non-compact Lie groups

Constructs a classifying space from a Gamma-G-space for proper G-actions

Proves the new theory aligns with classical K-theory when G is compact

Abstract

Generalizing a construction of Wolfgang L\"uck and Bob Oliver, we define a good equivariant cohomology theory on the category of proper G-CW complexes when G is an arbitrary Lie group (possibly non-compact). This is done by constructing an appropriate classifying space that arises from a Gamma-G-space. It is proven that this theory effectively generalizes Segal's equivariant K-theory when G is compact.