Geometric interpretation of the evaluation of Poincaré dual with a fundamental class

Question

Given oriented, closed submanifolds $X^k$ and $Y^{n-k}$ in an oriented, closed $n$-manifold, is there a nice geometric interpretation of the evaluation $\langle \operatorname{PD}([X]),[Y]\rangle$? I think that this should be the oriented intersection number of $X$ and $Y,$ and I'm sure that it is extremely well known, but I'm having a hard time finding a good reference for this.

Answer 1

Yes it is indeed the oriented interesction number. And it is well-known (it is the basic fact for intersection theory).

In your case, where the submanifolds are complmentary dimensional, it is rather easy to figure out why: just look at the Poincaré Dual of $[X]$ as the restricted Thom class of the normal bundle (under the inclusion $M \to (M,M-X)$, where $M$ is the manifold where the magic happens). And locally you know what happens with the Thom class. Hence you just consider intersection points and look whether they give you (after specifying a convention) the usual preferred (induced) orientation on the normal bundle, where the orientation on the normal bundle (which is at that point up to orientation the tangent space of $Y$) is induced by $M$. In other words you just look if at any intersection point the orientations of $TX,TY$ give you the orientation of $TM$ or not, which is counting the oriented intersection.

But also in general you have the immense useful result: cup product is dual to transverse intersection.