Universal twist in Equivariant K-theory for proper and discrete actions

TL;DR

This paper develops a universal framework for equivariant K-theory twists using projective unitary stable bundles, extending Atiyah-Segal's work to proper and discrete group actions.

Contribution

It constructs universal equivariant bundles for orbit types and classifies them via third equivariant cohomology, generalizing previous results.

Findings

Constructed universal equivariant projective unitary stable bundles.

Determined the homotopy type of these bundles.

Classified isomorphism classes via third equivariant cohomology.

Abstract

We define equivariant projective unitary stable bundles as the appropriate twists when defining K-theory as sections of bundles with fibers the space of Fredholm operators over a Hilbert space. We construct universal equivariant projective unitary stable bundles for the orbit types, and we use a specific model for these local universal spaces in order to glue them to obtain a universal equivariant projective unitary stable bundle for discrete and proper actions. We determine the homotopy type of the universal equivariant projective unitary stable bundle, and we show that the isomorphism classes of equivariant projective unitary stable bundles are classified by the third equivariant integral cohomology group. The results contained in this paper extend and generalize results of Atiyah-Segal.