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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.