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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.

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up vote 14 down vote accepted

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.)

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This answer makes me happy, of course, especially the great amount of generalization. I understand everything except the end: If $Y$ is "at all nice". Since it was a nit-picky question all along, what kind of condition is suitable? –  Greg Kuperberg Jul 17 '10 at 4:42
    
I don't know why I said that. I was still hung up on having something like a regular neighborhood in which to do duality. Of course the cap product can be done in $Y$ itself. I'll edit the answer. –  Tom Goodwillie Jul 17 '10 at 4:44
    
Duh, now I'm thinking that I could have thought of all of this. My mental block was that I forgot about cap products. –  Greg Kuperberg Jul 17 '10 at 4:59
    
Of course, if you need an explicit cycle, you might still get involved with strata and transversality ... –  Tom Goodwillie Jul 17 '10 at 12:11
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