: I hope this is not too dumb. I am kind of unclear on some issues: i) I understand that every bilinear map over a fin. dim. V space /F has a matrix representation (depending on the choice of basis ). Still, the intersection form ( in an orientable 4-mfld M.) is a map: f: H^2(M;Z)xH^2(M;Z) -->Z with: f(a,b) =(a\/b)[M] where M is a fundamental (orientation ) class , i.e., M is a generator of H^4(M;Z) ~ Z (since M is assumed orientable). Now: It is not clear that H^2(M;Z) is a vector space. It may be a ring, and therefore may not have a basis ( I think we consider rings as modules over themselves, and see if they are free or not. ). If H^2(M;Z) is not a ring: How do we represent the form above as a quadratic form?. ii) If we do have a representation: how do we use the relation between cap and cup (together with Poincare duality), to show that the cupping of H_2(M;Z)\/H_2(M;Z) is equivalent to the intersection number a.b , where a,b are the Poincare duals to the H^2(M;Z)'s ( I am working under the result that every class H^2(M;Z) can be represented by an embedded submanifold, and that the submanifolds can be disturbed if they do not intersect transversally ) Thanks For any Help.