#### Volume 7, issue 2 (2007)

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Sheaf theory for stacks in manifolds and twisted cohomology for $S^1$–gerbes

### Ulrich Bunke, Thomas Schick and Markus Spitzweck

Algebraic & Geometric Topology 7 (2007) 1007–1062
 arXiv: math.KT/0603698
##### Abstract

In this paper we give a sheaf theory interpretation of the twisted cohomology of manifolds. To this end we develop a sheaf theory on smooth stacks. The derived push-forward of the constant sheaf with value $ℝ$ along the structure map of a $U\left(1\right)$ gerbe over a smooth manifold $X$ is an object of the derived category of sheaves on $X$. Our main result shows that it is isomorphic in this derived category to a sheaf of twisted de Rham complexes.

##### Keywords
sheaf theory, stacks, twisted cohomology
Primary: 46M20
Secondary: 14A20