Equivariant K-theory, groupoids and proper actions

TL;DR

This paper develops a complex equivariant K-theory for Lie groupoid actions using finite-dimensional vector bundles, establishing foundational properties and an analogue of the Atiyah-Segal completion theorem.

Contribution

It introduces a new equivariant K-theory framework for Lie groupoids and proves a completion theorem, expanding the theoretical foundation of equivariant topology.

Findings

Defines complex equivariant K-theory for Lie groupoids

Establishes a periodic cohomology theory for Bredon-compatible groupoids

Proves an analogue of the Atiyah-Segal completion theorem

Abstract

In this paper we define complex equivariant K-theory for actions of Lie groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite equivariant CW-complexes. We also establish an analogue of the completion theorem of Atiyah and Segal. Some examples are discussed.