group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
An ordinary spin structure on a special orthogonal group-principal bundle is a lift of the corresponding cocycle through the spin group fibration . The obstruction for this to exist is a cohomology class – the second Stiefel-Whitney class: it exists precisely if this class is trivial, .
Conversely, one can ask for an -cocycle with prescribed non-trivial obstruction . These may usefully be understood as -twisted -structures, following the general logic of twisted cohomology.
Let
be the fiber sequence in ETop∞Grpd or Smooth∞Grpd given by the spin group extension of the special orthogonal group (regarded as a topological group or as a Lie group, respectibely). Its delooping defines the second Stiefel-Whitney class
so that for any we have a characteristic class
For a manifold define the groupoid of twisted spin-structures to be the (∞,1)-pullback
where the right vertical morphism picks one cocycle representative in each cohomology class.
The cocycles in are twisted Spin-bundles.
The obstruction cocycles in are -principal 2-bundles. These may be modeled by -bundle gerbes. In this incarnation the obstruction cocycles above have been discussed as spin gerbes in (MurraySinger).
cohomology, twisted cohomology, Whitehead tower, twisted smooth cohomology in string theory
twisted differential c-structure
spin structure, twisted spin structure
A model of twisted spin structures by bundle gerbes is discussed in
Volume 26, Number 4, 355-367, DOI: 10.1023/B:AGAG.0000047514.71785.96 (arXiv:math/0302096)
The general abstract discussion given above appears as an example in
Last revised on February 25, 2020 at 10:32:19. See the history of this page for a list of all contributions to it.