nLab intersection pairing

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 correspond 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 dimension 4, see also:

• Wikipedia, Intersection form (4-manifold).

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), 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.

Related concepts

• functional cup product

• homotopy Whitehead integral formula

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 and Hermitian manifold (e.g.Kähler manifold): 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

Introductions and surveys include

• Akhil Mathew, Intersection theory on surfaces, 2013 (web)

• Wikipedia, Intersection theory

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