cohomology

# Contents

## Idea

For $X$ a space of dimension $2k$ and ${H}^{k}\left(X\right)$ 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

${H}^{k}\left(X\right)×{H}^{k}\left(X\right)\to A$H^k(X) \times H^k(X) \to A

given by fiber integration

$\left(\lambda ,\omega \right)↦{\int }_{X}\left(\lambda \cup \omega \right)\phantom{\rule{thinmathspace}{0ex}},$(\lambda, \omega) \mapsto \int_X (\lambda \cup \omega) \,,

of the cup product

$\cup :{H}^{k}\left(X\right)×{H}^{k}\left(X\right)\to {H}^{2k}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$\cup : H^k(X) \times H^k(X) \to H^{2k}(X) \,.

## Examples

### In integral cohomology

The signature of a quadratic form of the quadratic form induced by the intersection pairing in integral cohomology is the signature genus.

### In ${ℤ}_{2}$-cohomology

Over a Riemann surface $X$, the intersection pairing on ${H}^{1}\left(X,{ℤ}_{2}\right)$ 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.

### In ordinary differential cohomology: higher abelian Chern-Simons theory

For the case that the cohomology in question is ordinary differential cohomology,

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).

$4k$signature genusintersection pairingintegral Wu structure
$4k+2$Kervaire invariantframing