Twists over \'etale groupoids and twisted vector bundles

TL;DR

This paper investigates conditions under which twists over étale groupoids admit twisted vector bundles, linking these conditions to the Brauer group and classifying space properties, advancing understanding in twisted K-theory.

Contribution

It establishes criteria involving classifying spaces for torsion twists over étale groupoids to admit twisted vector bundles, connecting to the Brauer group.

Findings

Twists admitting twisted vector bundles form a subgroup of the Brauer group.

Conditions involving the classifying space $B ext{G}$ imply the existence of twisted vector bundles for torsion twists.

Provides new insights into the structure of twists in twisted K-theory.

Abstract

Inspired by recent papers on twisted $K$-theory, we consider in this article the question of when a twist $\mathcal{R}$ over a locally compact Hausdorff groupoid $\mathcal{G}$ (with unit space a CW-complex) admits a twisted vector bundle, and we relate this question to the Brauer group of $\mathcal{G}$. We show that the twists which admit twisted vector bundles give rise to a subgroup of the Brauer group of $\mathcal{G}$. When $\mathcal{G}$ is an \'etale groupoid, we establish conditions (involving the classifying space $B\mathcal{G}$ of $\mathcal{G}$) which imply that a torsion twist $\mathcal{R}$ over $\mathcal{G}$ admits a twisted vector bundle.