# Intersection theory in analytic geometry

This might be a weird/stupid question, but it came to me a couple of times, and I would like to get an answer for that.

In some papers I read, constantly the authors define some analytic subspaces, say $$X$$ and $$Y$$, and then the authors take the intersection product of their cycles $$[X]\cdot [Y]$$ in the homology group, without requiring $$x$$ intersect $$Y$$ transversely.

My question is, it seems that there is an intersection theory applied here, but I don't know how it works. Does Fulton-MacPherson's intersection theory directly apply to analytic setting ?

You don't say what kind of space $$X$$ and $$Y$$ are subspaces of. But if they sit in an oriented manifold there's an easy way to define an intersection product in homology. Namely if $$M$$ is an oriented $$d$$-manifold then there is a Poincaré duality isomorphism $$H_i(M,\mathbf Z) \cong H^{d-i}_c(M,\mathbf Z)$$ between homology and compactly supported cohomology. The cohomology with compact support is a ring (though typically a non-unital ring) for the same reason that the usual cohomology is a ring: use contravariant functoriality for the diagonal morphism. In this way we obtain an intersection product on homology.