Detailed proof of cup product equivalent to intersection 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!
 A: Bott and Tu do this completely, in the de Rham theoretic setting of course.
Here's an alternate proof I have used when I teach this material, which I find slightly more clean and direct than using Thom classes in de Rham theory (which require choice of  tubular neighborhood theorem, etc) and works over the integers.
Definition: Given a collection $S = \{W_i\}$ of submanifolds of a manifold $X$, define the smooth chain complex transverse to $S$, denoted ${C^S}_*(X)$, by using the subgroups of the singular chain groups in which the basis chains $\Delta^n \to X$ are smooth and transverse to all of the $W_i$.
Lemma: The inclusion ${C^S}_*(X) \to C_*(X)$ is a quasi-isomorophism, for any such collection $S$.
Now if $W \in S$ then "count of intersection with $W$" gives a perfectly well-defined element $\tau_W$ of
${\rm Hom}(C^S_*(X), A)$ and thus by this quasi-isomorphism a well-defined cocycle if the $W$ is proper and has no boundary.  It is immediate that this cocycle evaluates on cycles which are represented by closed submanifolds through intersection count.
There are two approaches to show that cup product agrees with intersection on cohomology.  Briefly, one is to take $W, V$ over $M$ and consider the special case of $W \times M$ and $M \times V$ over $M \times M$.  There some work with the K"unneth theorem
leads to direct analysis in this case.  But this case is "universal" - cup products in $M$ are pulled back from ``external'' cup products over $M \times M$.  A second proof given in https://arxiv.org/abs/2106.05986 uses a variant of the theory, where one fixes a triangulation or cubulation, and assumes $W, V$ transverse to those.  There we explicitly see that these products do not agree at the cochain level (they can't since intersection is commutative, but non-commutativity of cup product is reflected in Steenrod operations), but Friedman, Medina and I show a vector field flow leads to a cobounding of the difference.
