TITLE: Intersection product in a manifold, taking values in one factor
QUESTION [7 upvotes]: In a joint paper that I am working on, we are interested in taking the intersection product $[X] \cap [Y]$ of the fundamental classes of two compact, oriented pseudomanifolds $X$ and $Y$ in a compact, oriented manifold $M$.  Now, the usual intersection product takes values in $H_*(M)$, but I need an intersection product that takes values in $H_*(Y)$.  I think that if $X$ and $Y$ are favorably stratified, whatever that means, then such an intersection product should exist.  My question is, what is a good approach for such a construction?  What is a good proof that it's well-defined?  What is a good reference?
The intuitive idea is to wiggle $X$ generically to make it transverse to $Y$, and then take transverse intersections along strata.  But there are various different rigorous frameworks that you could use to prove that this is well-defined.
For example, if $X$, $Y$, and $M$ are all PL, then one way to define the right thing is to restrict the Poincaré dual $[X]^*$ to $H^*(M,M\setminus N) \cong H^*(N,\partial N)$, where $N$ is a regular neighborhood of $Y$, then take the Poincaré dual of that to obtain an element of $H_*(N) \cong H_*(Y)$.  This way to define the intersection is based on very standard ideas.  You can prove that it works is to use the fact that $N$ is unique up to a relevant isotopy.
In the case that I/we need, $M$ is a smooth complex projective variety and $X$ and $Y$ are singular complex subvarieties.  Now, it's easy to suppose that the definition and proof from the PL case still work using the fact that $X$ and $Y$ are Whitney-stratified.  However, I no longer know how standard the argument is.  For all I know, a regular neighborhood argument is not as standard as a direct argument with transverse position.  Or, for all I know, $X$ and $Y$ don't have to be Whitney-stratified; maybe they can be much more general.  Certainly it seems like window dressing to demand that they be complex projective.

REPLY [14 votes]: I don't think you have to get involved with strata or transversality at all. Poincare duality (Edit: and cap products) will do all the work. Let's assume:
$M$ is a compact oriented $n$-manifold. 
$\xi$ is a $p$-dimensional homology class of $M$. (We don't care if it comes as the fundamental class $[X]$ of some pseudomanifold in $M$.)
$Y$ is a subset of $M$ and $[Y]$ is an element of $H_q(Y)$. (We don't care if $Y$ is a pseudomanifold, or $q$-dimensional, or anything.) 
Then duality in $M$ yields an element $\xi^*\in H^{n-p}(M)$. Restrict it to get an element of $H^{n-p}(Y)$. Cap with $[Y]$ to get an element of $H_{p+q-n}(Y)$.
(Final paragraph now simplified after comment from Greg.)