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 $g : X \to \mathbf{B} SO$ through the spin group fibration $\mathbf{B} Spin \to \mathbf{B} SO$. The obstruction for this to exist is a cohomology class $w_2 \in H^2(X, \mathbb{Z}_2)$ – the second Stiefel-Whitney class: it exists precisely if this class is trivial, $[w_2(g)] = 0$.
Conversely, one can ask for an $SO$-cocycle $g$ with prescribed non-trivial obstruction $[w_2(g)] = \alpha \in H^2(X, \mathbb{Z}_2)$. These may usefully be understood as $\alpha$-twisted $spin$-structures, following the general logic of twisted cohomology.
Let
be the fiber sequence in $\mathbf{H} =$ETop∞Grpd or $\mathbf{H} =$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 $X \in \mathbf{X}$ we have a characteristic class
For $X$ a manifold define the groupoid of twisted spin-structures $SpinStruc_{tw}(X)$ to be the (∞,1)-pullback
where the right vertical morphism picks one cocycle representative in each cohomology class.
The cocycles in $SpinStruc_{tw}(X)$ are twisted Spin-bundles.
The obstruction cocycles in $\mathbf{H}(X, \mathbf{[B}^2 \mathbb{Z}_2)$ are $\mathbf{B}\mathbb{Z}_2$-principal 2-bundles. These may be modeled by $\mathbb{Z}_2$-bundle gerbes. In this incarnation the obstruction cocycles $w_2(g)$ 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
The general abstract discussion given above appears as an example in