Contents

Defintion

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

$DD(\nabla_{2k-1}) mod 2 = \nu_{2k} \in H^{2k}(X, \mathbb{Z}_2) \,.$

These are the characteristic elements of the intersection product on ordinary cohomology/ordinary differential cohomology, inducing its quadratic refinements.

manifold dimensioninvariantquadratic formquadratic refinement
$4k$signature genusintersection pairingintegral Wu structure
$4k+2$Kervaire invariantframing

The following table lists classes of examples of square roots of line bundles

References

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

Revised on September 10, 2014 04:33:32 by Urs Schreiber (185.26.182.29)