Inertia and delocalized twisted cohomology

Ulrich Bunke, Thomas Schick and Markus Spitzweck

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 → X we associate a U(1)^δ-principal bundle~G^δ → LX. We use sheaf theory on topological stacks in order to define the delocalized twisted cohomology by

H^*_deloc(X,G) := H^*(G_L,f_L^*L'),

where f_L: G_L → LX is the pull-back of the gerbe G → X via the natural map LX → X, and L' ∈ Sh_AbLX is the sheaf of sections of the C^δ-bundle associated to~G^δ → 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 → X.

Homology, Homotopy and Applications, Vol. 10 (2008), No. 1, pp.129-180.

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