Twists over étale groupoids and twisted vector bundles

Farsi, Carla; Gillaspy, Elizabeth

doi:10.1090/proc/13165

Twists over étale groupoids and twisted vector bundles
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by Carla Farsi and Elizabeth Gillaspy PDF

Proc. Amer. Math. Soc. 144 (2016), 3767-3779 Request permission

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 étale 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.

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