group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A differential integral Wu structure in degree $2k$ on an oriented smooth manifold $X$ is a refinement of the Wu class $\nu_{2k} \in H^{2k}(X, \mathbb{Z}_2)$ $\nu_{2k}$ by a cocycle $\phi$ in degree $2k$ ordinary differential cohomology $H^{2k}_{diff}(X)$, hence a circle (2k-1)-bundle with connection $\nabla_{2k-1}$ whose underlying higher Dixmier-Douady class $DD(\nabla_{2k-1})$ equals $\nu_{2k}$ modulo 2-reduction
These are the characteristic elements of the intersection product on ordinary cohomology/ordinary differential cohomology, inducing its quadratic refinements.
manifold dimension | invariant | quadratic form | quadratic refinement |
---|---|---|---|
$4k$ | signature genus | intersection pairing | integral Wu structure |
$4k+2$ | Kervaire invariant | framing |
The following table lists classes of examples of square roots of line bundles
The notion was introduced in def. 2.12 of
motivated by considerations about abelian 7d Chern-Simons theory in
A smooth stack refinement is considered in