We have the intersection pairing defined on homology class is the following: For a compact oriented manifold $M$ with dimension $m$, denote $[M]$ by its fundamental class. Consider two homology classes of complementary dimensions $[A] \in H_k(M,\mathbb{Z})$, $[B]\in H_{m-k}(M,\mathbb{Z})$, then the intersection pairing is defined to be
$<PD[A] \cup PD[B], [M]> \ \in \mathbb{Z}$
This is a well-defined notion given by the cup product.
However, there seems to be a parallel story of its geometric picture.
For two compact oriented embedded submanifolds $C$ and $D$ of complementary dimension (I'm not sure if the condition "embedding" and "compact" here are necessary, just to phrase it to be safe) that intersect transversally, we count their intersection points with signs (by compactness such points form a finite set), i.e. positive if their orientation together is consistent with the orientation on $M$, negative otherwise, this also gives us an integer.
My question is, how much are the two stories above equivalent?
To be precise, I want to know about the following:
If the two homology classes can be represented by two oriented embedded submanifolds (by representing I mean the pushforward of the fundamental class under the inclusion map is the homology class) that intersect transversally, do the above definition coincide? If yes, is there any way to see why this is true?
If we have the two homology classes can be represented by compact oriented embedded submanifolds, does the transversality comes for free (under diffeomorphisms)? i.e. for example, do we have that there always exist a diffeomorphism $f$ of $M$ s.t. $C$ and $f(D)$ intersect transversally?
However in the general picture, it is not always true that a given homology class can be represented by such a submanifold. In this situation, do we have a geometric interpretation of the intersection pairing?
Any related comments are appreciated.