nLab
intersection pairing

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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 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).

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

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

Revised on June 4, 2012 22:34:50 by Urs Schreiber (212.236.23.114)