Contents

cohomology

# Contents

## Idea

On a space of suitable even dimension the cup product on suitable mid-dimensional cohomology is often called the intersection product – this, or its evaluation on the fundamental class of the whole space.

Under Poincaré duality these cohomology classes may corrrespond to cycles and then under suitable conditions or in a suitable sense, the cup product dually counts (or otherwise detects) literally the intersection points of the two subspaces, whence the name. It is the topic of intersection theory to make this statement precise, classical results to this extent include Bézout's theorem and its refinement to the Serre intersection formula.

If here cohomology is replaced by differential cohomology then quadratic refinements of the intersection product provide the Lagrangians for higher dimensional Chern-Simons theory and govern the structure of self-dual higher gauge theory. See there for more.

In a little more detail: 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

$H^k(X) \times H^k(X) \to A$

given by fiber integration

$(\lambda, \omega) \mapsto \int_X (\lambda \cup \omega) \,,$

of the cup product

$\cup : H^k(X) \times H^k(X) \to H^{2k}(X) \,.$

## 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 $\mathbb{Z}_2$-cohomology

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.

### 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), induced from/modeled on characteristic elements given by integral Wu structures.

### On framed manifolds

On a framed manifold the intersection pairing has a canonical quadratic refinement, leading to the Kervaire invariant.

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