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~Gδ → LX. We use sheaf theory on topological stacks
in order to define the delocalized twisted cohomology by
where fL: GL → LX
is the pull-back of the gerbe G → X via the natural map
LX → X, and
L' ∈ ShAbLX is the sheaf of sections of the Cδ-bundle associated to~Gδ → 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.
H*deloc(X,G) := H*(GL,fL*L'),|
Homology, Homotopy and Applications,
Vol. 10 (2008), No. 1,