Hi, this might sound like a stupid question (and most likely it is). What do we know about representing homologies between singular chains via cobordisms? let me be more specific, given a smooth closed n-dim manifold $M$, and two homologous chains $\alpha$ and $\alpha'$ of degree $n-1$ which can be both represented by two codim 1 sub manifolds $V,V'$.

Can I claim there is an n-dim mfld W with boundary, together with a map W to M that sends the two boundary components to V and V' and represents the homology between them?