Consider a smooth, closed, compact finite-dim manifold. We have Poincare Duality to relate the cocycles and cycles.

I would like to know where I can find a reference for a proof that the cup product of the Cohomology Ring is given by the intersection of the corresponding cycles.

Griffiths and Harris talk about intersection number, and discuss this result in chapter 0, Hatcher's book doesn't mention this explicitly as far as I can tell, Katz' little book on enumerative geometry alludes to this, Fulton's book on Young Tableaux dodges this, etc.

I am preparing to give a talk on Schubert Cells and Schubert calculus, and I realized that I have not checked the details of this proof.

Thanks in advance!