Supposed I have an n-dimensional manifold M with a k-dimensional submanifold that is homologous to zero (or, equivalently, two homologous submanifolds). Can I always construct a k+1-dimensional manifold N and a smooth map $N\to M$ so that the boundary maps diffeomorphically to my submanifold? Can I just take abstract k+1-simplecies and glue them along boundaries to make N, and then somehow smooth it out? If not, is there some understandable obstruction?
I'm most interested in the smooth category, but if it makes more sense in some other category (or there is otherwise a better question I should've asked), do tell me.
Update: As I first asked it, the question was a bit stupid because I forgot about cobordisms. However, in the case I care about, this does not seem to be a problem, since I want the boundary of N to be a union of two submanifolds which are diffeomorphic to each other.