What's the standard generalization and reference for the following statement:
If two oriented submanifolds $L$, $L'$ of an oriented compact manifold $M$ intersect transversally, then the Poincare dual of the fundamental class $[L\cap L']$ equals to the cup product of the Poincare duals of the fundamental classes $[L]$ and $[L']$.
It seems, in algebraic geometry $L$, $L'$ and $L\cap L'$ don't have to be smooth, it is only required that at a generic point of $L\cap L'$ we have that $L$ and $L'$ are smooth and intersect transversally. In the case of non-transversal intersection sometimes more can be said (e.g. multiplicities appear).
I am asking for a topological version of this.
To exclude proofs with differential forms, let's say we want homology/cohomology with integer or arbitrary coefficients.
In fact, I thought of a sketch of a theory along the following lines (in style of Hatcher's book and adopting an approach from Voisin's book):
First define inductively a class of "good subspaces" by saying that a closed subspace $L\subset M$ in a manifold $M$ is a good subspace of dimension $d$ if there is open $U\subset L$ such that $U$ is a submanifold of $M$ and $L\setminus U$ is a good subspace of dimension $d-1$. Let us call manifolds $U\subset L$ for which $L\setminus U$ have smaller dimension generic.
An orientation on a good space is simply an orientation on some generic subset.
Suppose $M$ is oriented and $L$ is an oriented good subspace of $M$ of codimension $n$. We probably can prove that $H^i(M, M-L)=0$ for $i<n$ and that there is at most one element $c_L\in H^n(M, M-L)$ which restricts to the correct orientation class on $H^n(M-(L-U), U)$. We call this class the cycle class if it exists.
If $L$ has a cycle class in a manifold $M$ and $L$ is compact, its fundamental class in $M$ is defined by the cap product $[L]=c_L\cap [M]$ of the cycle class of $L$ with the fundamental class $[M]\in H_N(M,M-L)$ ($N$ is the dimension of $M$, $N=d+n$). In case $L$ has a fundamental class in $H_d(L)$, e.g. if $L$ is a manifold, its pushforward to $M$ coincides with $[L]$.
Theorem. Suppose cycle classes exist for good oriented spaces $L$, $L'$ in an oriented manifold $M$. Suppose the intersection $K=L\cap L'$ is a good space, suppose there exist generic sets $U\subset L$, $U'\subset L'$, such that the intersection of $U$ and $U'$ is transverse and generic in $K$. Then cycle class of $K$ exists and $c_K=c_L\cup c_{L'}$.
In the case of compact $M$, the Poincare dual class of $L$ is the image of $c_L$ under the natural map $H^n(M,M-L)\to H^n(M)$, so the statement for Poincare dual classes follows the Theorem.
Do you see any problem with this approach, and has something like this been worked out in details in the literature?
