nLab
intersection pairing

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher geometry

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 XX a space of dimension 2k2k and H k(X)H^k(X) a cohomology group on a space XX equipped with H-orientation in degree kk with coefficients in some AA, the intersection pairing on cohomology is the map

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

given by fiber integration

(λ,ω) X(λω), (\lambda, \omega) \mapsto \int_X (\lambda \cup \omega) \,,

of the cup product

:H k(X)×H k(X)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 dimension 4, see also:

In 2\mathbb{Z}_2-cohomology

Over a Riemann surface XX, the intersection pairing on H 1(X, 2)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 2k2k, 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 2k12k-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.

manifold dimensioninvariantquadratic formquadratic refinement
4k4ksignature genusintersection pairingintegral Wu structure
4k+24k+2Kervaire invariantframing

The following table lists classes of examples of square roots of line bundles

line bundlesquare rootchoice corresponds to
canonical bundleTheta characteristicover Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundlehalf-density bundle
canonical bundle of Lagrangian submanifoldmetalinear structuremetaplectic correction
determinant line bundlePfaffian line bundle
quadratic secondary intersection pairingpartition function of self-dual higher gauge theoryintegral Wu structure

References

Introductions and surveys include

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

Revised on September 2, 2014 18:58:28 by Urs Schreiber (82.136.246.44)