Cohomology of topological graphs and Cuntz-Pimsner algebras

TL;DR

This paper computes the sheaf cohomology of groupoids from local homeomorphisms, identifies twists and Brauer groups, and relates C*-algebras of twists to Cuntz-Pimsner algebras, refining previous results.

Contribution

It advances the understanding of the cohomology of groupoids associated with local homeomorphisms and connects their C*-algebras to Cuntz-Pimsner constructions.

Findings

Calculated sheaf cohomology for groupoids from local homeomorphisms.

Identified twists and Brauer groups over these groupoids.

Established isomorphism between C*-algebras of twists and Cuntz-Pimsner algebras.

Abstract

We compute the sheaf cohomology of a groupoid built from a local homeomorphism of a locally compact space $X$. In particular, we identify the twists over this groupoid, and its Brauer group. Our calculations refine those made by Kumjian, Muhly, Renault and Williams in the case $X$ is the path space of a graph, and the local homeomorphism is the shift. We also show how the C*-algebra of a twist may be identified with the Cuntz-Pimsner algebra constructed from a certain C*-correspondence.