Recently I have been thinking and inquiring about how "cup products are dual to intersection of submanifolds", and wanted to verify whether the following is accurate (and to find a source for this material).
Let $M$ be a closed, oriented manifold, and let $N, K$ be embedded submanifolds of dimension $k$, $(n-k)$ respectively. By genericity we can assume they intersect transversally, in which case they meet at a finite number of points. $N$ and $K$ determine homology classes $[N] \in H_k(M)$, $[K] \in H_{n-k}(M)$.
By Poincaré duality, we can identify $H_{n-k}(M)$ with $H^k(M)$, so we can pair the image of $[K]$ in $H^k(M)$ as a $k$-cochain with the $k$-chain $[N]$ using the evaluation map to get a number in $\mathbb{Z}$; is this simply the signed intersection number of $N$ and $K$ in $M$?
I would appreciate a reference to this and an explanation as to how the cup product is "dual" to intersection of submanifolds; I have not been able to find one which goes about it this way. Thank you!