In topology, Poincaré duality implies that given a connected and oriented compact manifold M of dimension 4k, the cup product gives rise to an integral non-degenerate symmetric bilinear form on the "middle" cohomology group $H^{2k}(M,\mathbf Z)$.

This gives rise to the definition of the signature, which will maybe be of interest to you. A variant for 4k+2 dimensional manifolds gives rise to the famous Kervaire invariant.

See https://en.wikipedia.org/wiki/Signature_%28topology%29 for example.

A Good reference is the book by Milnor and Husemoller.