Inertia and delocalized twisted cohomology

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 aWe use sheaf theory on topological stacks in order to define the delocalized twisted cohomology byis the pull-back of the gerbe G → X via the natural map LX → X, and L ∈ Sh Ab LX is the sheaf of sections of the C δ -bundle associated toG 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.

Sheaf theory for stacks in manifolds and twisted cohomology forS1–gerbes

“…In particular, H * (X; G) carries the action of the automorphisms of the gerbe G → X. One can define the map u * without the assumption that u is smooth, but then the argument is more complicated, see [9].…”

“…A parallel theory can be set up in the topological context. Together with applications to T -duality and delocalized cohomology it will be discussed in detail in the subsequent papers [9], and [10]. [3] and Behrend-Xu [5], a different version of sheaf theory and cohomology of stacks is developed.…”

“…X . One can define the map u without the assumption that u is smooth, but then the argument is more complicated, see [9].…”