Suppose $M$ is an oriented diff. manifold and $X$ and $Y$ are two submanifolds of codimension $m$ and $n$ in $M$. Then under some conditions one can define the intersection of $X$ and $Y$ and this is related to the cup product of the cohomology classes of these submanifolds. This is the rough idea that I know from the relation of cup product and intersection of cycles. Where I can see the rigorous theory of such things? Are there good references which developed the cohomology theory from this intuitive notion of intersection?
As there exists a notion of cup product on the more general class of topological manifolds, Does there exists a good reference for the definition of intersection product without the assumption of differentiability?
Thanks a lot!
In the book of Bredon you'll find a rigorous proof of the connection between the cup product and intersection of submanifolds in the setting of smooth manifolds (with boundary) in Chapter VI.11.