When we restrict to the torsion-free part of the cohomology of a manifold, the intersection pairing is nondegenerate. In dimension 2n, this gives a bilinear form on the free part of H^{n} (symmetric if n is even, skew-symmetric of n is odd). Supposedly, this classifies simply-connected 4-manifolds pretty well; more precisely, the map from homeomorphism classes of manifolds is at most 2-to-1, and whenever a bilinear form has two preimages, at least one of them isn't smooth.

Broadly, I'm wondering what else is known about this pairing. (Of course, if anything I've said so far is wrong, please correct me!) For example, can we obtain the bilinear form associated to a connected-sum (or a product, or whatever other natural ways we can get manifolds) in some nice way? It seems like in the connected-sum case, Mayer-Vietoris should tell us something -- I'd imagine the isomorphism it gives (when n>1) between H^{n}(connected-sum)=H^{n}(M_{1})+H(M_{2}) is probably natural, but I don't know if we can tell anything about the cohomology ring structure from it. As for the product of manifolds, I'm pretty sure that the Kunneth formula gives an isomorphism of algebras, so that *should* take care of it unless I'm missing something... Also, is this at all related to the Kirby-Siebemann invariant? (All I really know is that it's related to smoothness/smoothability, but perhaps, for example, we can somehow read off the possible values of the K-S invariant from the bilinear form?)