Twists over étale groupoids and twisted vector bundles

Abstract

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