Intersection Theory - Topological Intersection Form

Topological Intersection Form

For a connected oriented manifold M of dimension 2n the intersection form is defined on the nth cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class . Stated precisely, there is a bilinear form

given by

with

This is a symmetric form for n even (so 2n=4k doubly even), in which case the signature of M is defined to be the signature of the form, and an alternating form for n odd (so 2n=4k+2 singly even). These can be referred to uniformly as ε-symmetric forms, where ε = respectively for symmetric and skew-symmetric forms. It is possible in some circumstances to refine this form to an ε-quadratic form, though this requires additional data such as a framing of the tangent bundle. It is possible to drop the orientability condition and work with coefficients instead.

These forms are important topological invariants. For example, a theorem of Michael Freedman states that simply connected compact 4-manifolds are (almost) determined by their intersection forms up to homeomorphism – see intersection form (4-manifold).

By Poincaré duality, it turns out that there is a way to think of this geometrically. If possible, choose representative n-dimensional submanifolds A, B for the Poincaré duals of a and b. Then (a, b) is the oriented intersection number of A and B, which is well-defined because of the dimensions of A and B. This explains the terminology intersection form.