TITLE: How to identify cup product with intersection
QUESTION [7 upvotes]: 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?

REPLY [5 votes]: You should look at "Lectures on algebraic topology" by A. Dold chapter VIII section 13.
A more intuitive approach, closer to H. Poincaré's viewpoint and to the intersection product of algebraic geometers, is to use singular manifolds to represent cycles. For example, one can use Kreck's stratifolds in order to get a very geometric definition of the intersection product (transversality holds in this framework), you can look at M. Kreck's book "Differential  Algebraic Topology".