Suppose that $\omega$ is a differential form on a smooth compact manifold $X$, and let $V$ be a compact submanifold of the same dimension to the form.
Then we can define a fundamental class $[V]$*, and cap it with $\omega$ to an element of $H_0(X;\mathbb{Z})$, which we can associate with a number $\mathbb{Z}$, essentially by counting the number of points.
(*I think just class associated to a triangulation of $V$ -- I'm working with complex varieties, which can be triangulated anyway.)
Alternatively, we can restrict $\omega$ to $V$ and then integrate.
I guess these numbers should be the same, but I don't understand the cap product well enough to prove it. Maybe someone can suggest a hint or a reference?
Moreover, if $[V]$ is just some homology class of $X$, can it be represented by objects on which it is possible to integrate the restriction of $\omega$? (Not necessarily a manifold, but for example a union of manifolds that might overlap, or something...) If so, same question in this case.