derived smooth geometry
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.
given by fiber integration
of the cup product
In dimension 4, see also:
Over a Riemann surface , the intersection pairing on 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.
For the case that the cohomology in question is ordinary differential cohomology,
the cup product is the Beilinson-Deligne cup product;
The differentially refined intersection pairing is non-trivial and interesting also on manifolds of dimension less than , where the integral intersection pairing vanishes: it provides a secondary characteristic class, a secondary intersection pairing.
|manifold dimension||invariant||quadratic form||quadratic refinement|
|signature genus||intersection pairing||integral Wu structure|
|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|
Introductions and surveys include