# Degree 3 Cohomology: Sheaves of Groupoids

## Abstract

We develop the theory of Dixmier-Douady sheaves of groupoids and relate it to degree 3 cohomology with integer coefficients. In §1 we explain the theory of descent for sheaves, based on local homeomorphisms. In §2, we introduce sheaves of groupoids (also called stacks) and gerbes. We relate gerbes on *X* with band a sheaf of abelian groups *A* with the cohomology group *H* ^{2}(*X, A*). A gerbe with band ℂ _{X}* is called a Dixmier-Douady sheaf of groupoids. In §3 we introduce the notion of connective structure and curving for such sheaves of groupoids; we obtain the 3-curvature Ω, which is a closed 3-form such that the cohomology class of \(\frac{\Omega }{{2\pi \sqrt { - 1} }}\) is integral. We prove that any 3-form with these properties is the 3-curvature of some sheaf of groupoids, and relate this to the constructions of Chapter 4. In §4 we use the path-fibration to define a canonical sheaf of groupoids over a compact Lie group. In §5 we give other examples of sheaves of groupoids connected with Lie group actions on a smooth manifold and with sheaves of twisted differential operators.

## Keywords

Exact Sequence Line Bundle Cohomology Class Connective Structure Holomorphic Line Bundle## Preview

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