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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 1

### Inertia and delocalized twisted cohomology

Pages: 129 – 180

DOI: https://dx.doi.org/10.4310/HHA.2008.v10.n1.a6

#### Authors

#### Abstract

Orbispaces are the analog of orbifolds, where the category of manifolds is replaced by topological spaces. We construct the loop orbispace $LX$ of an orbispace $X$ in the language of stacks in topological spaces. Furthermore, to a twist given by a $U(1)$-banded gerbe $G\to X$ we associate a $U(1)^\delta$-principal bundle $\tilde G^\delta\to LX$. We use sheaf theory on topological stacks in order to define the delocalized twisted cohomology by $$H^*_{{\rm deloc}}(X,G):=H^*(G_L,f^*_L\mathcal{L}),$$ where $f_L\colon G_L\to LX$ is the pull-back of the gerbe $G\to X$ via the natural map $LX\to X$, and $\mathcal{L}\in {\tt Sh}_{\tt Ab}\mathbf{LX}$ is the sheaf of sections of the $\mathbb{C}^\delta$-bundle associated to $\tilde G^\delta\to LX$. The same constructions can be applied in the case of orbifolds, and we show that the sheaf theoretic delocalized twisted cohomology is isomorphic to the twisted de Rham cohomology, where the isomorphism depends on the choice of a geometric structure on the gerbe $G\to X$.

#### Keywords

orbispace, delocalized twisted cohomology, inertia stack

#### 2010 Mathematics Subject Classification

55-xx

Published 1 January 2008