4-Manifolds, Part 1: The intersection form

This is the first part in a series that I’m doing on -manifolds, with an eye toward Donaldson’s Diagonalization Theorem. I’ll retro-edit this post to contain more introduction and discussion, but for now I’ll introduce the intersection form on a -manifold, a central invariant. We’ll go on to discuss the algebraic topology of smooth -manifolds.

The intersection form of a compact, oriented -manifold is the symmetric bilinear form

By Poincare duality, the induced form on is nondegenerate, and we sometimes denote it by .

In general, what can we say about the structure of the homology and cohomology groups of ? Because we are dealing with an oriented manifold, , and . By Hurewicz, we have , and we also have . Moreover, . Finally, by the Universal Coefficient Theorem for cohomology, we have

Imposing the further restriction that be simply connected, i.e. , we have that odd-dimension (co)homology vanishes, and is torsion-free.

We will give two further interpretations of the intersection form in subsequent post: (1) via intersections of embedded surfaces, and (2) via differential forms.