nLab
intersection pairing

Idea

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

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

given by fiber integration

$$(\lambda ,\omega )\mapsto {\int }_X(\lambda \cup \omega ),$$(\lambda, \omega) \mapsto \int_X (\lambda \cup \omega) \,,

of the cup product

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

• 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).

Related concepts

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

line bundle	square root	choice corresponds to
canonical bundle	Theta characteristic	over Riemann surface: spin structure
density bundle	half-density bundle
canonical bundle of Lagrangian submanifold	metalinear structure	metaplectic correction
determinant line bundle	Pfaffian line bundle
quadratic secondary intersection pairing	partition function of self-dual higher gauge theory	integral Wu structure

References

Discussion of the intersection pairing in ordinary differential cohomology and especially its quadratic refinement is in

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory