Twisted equivariant K-theory, groupoids and proper actions

TL;DR

This paper develops a new framework for twisted equivariant K-theory applicable to Lie groupoid actions, providing classification, a completion theorem, and applications to proper group actions, advancing the understanding of equivariant topology.

Contribution

It introduces twisted equivariant K-theory for Lie groupoids, including classification of bundles and a completion theorem, extending the theory to proper group actions.

Findings

Defined twisted equivariant K-theory for Lie groupoid actions.

Classified equivariant stable projective bundles.

Established a completion theorem for the theory.

Abstract

In this paper we define twisted equivariant K-theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite CW-complexes with equivariant stable projective bundles. A classification of these bundles is shown. We also obtain a completion theorem and apply these results to proper actions of groups.