group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $X$ a space of dimension $2k$ and $H^k(X)$ a cohomology group on a space $X$ equipped with H-orientation in degree $k$ with coefficients in some $A$, the intersection pairing on cohomology is the map
given by fiber integration
of the cup product
The signature of a quadratic form of the quadratic form induced by the intersection pairing in integral cohomology is the signature genus.
Over a Riemann surface $X$, the intersection pairing on $H^1(X, \mathbb{Z}_2)$ has a quadratic refinement by the function that sends a Theta characteristic to the mod 2-dimension of its space of sections. See Theta characteristic – Over Riemann surfaces.
For the case that the cohomology in question is ordinary differential cohomology,
a cocycle in degree $k$ is a circle (k-1)-bundle with connection,
the cup product is the Beilinson-Deligne cup product;
and the required notion of orientation is now orientation in differential cohomology: a differential Thom class.
The differentially refined intersection pairing is non-trivial and interesting also on manifolds of dimension less than $2k$, where the integral intersection pairing vanishes: it provides a secondary characteristic class, a secondary intersection pairing.
Notably, the diagonal of the intersection pairing in in dimension $2k-1$ is the action functional of quadratic abelian higher dimensional Chern-Simons theory.
Its quadratic refinement is discussed in (Hopkins-Singer).
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
Discussion of the intersection pairing in ordinary differential cohomology and especially its quadratic refinement is in