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. 



Can homologous submanifolds be connected by an immersed manifold with boundary?
As Igor has noted, an obvious obstruction is that your embedded submanifold be null-cobordant.


It seems that you are really asking about the kernel of the realization map $MO\_k(M)\to H\_k(M;\mathbb{Z}\_2)$ (assuming that you don't care about orientations). This appears as the edge homomorphism in the unoriented bordism spectral sequence, which collapses since every homology class is Steenrod realizable (as follows from the work of Thom). A basic reference for this fact is the book "Differentiable periodic maps" by Conner and Floyd. It follows that there is a module isomorphism $H\_\*(M; \mathbb{Z}\_2)\otimes MO\_\*\cong MO\_\*(M)$. The bordism class of a map $f\colon A\to M$ is determined by its Stiefel-Whitney numbers (see Section 17 of Conner and Floyd). From this you should be able to piece together what you need.


**More detail:** Let $f\colon A^k\hookrightarrow M$ be the embedding of your submanifold. Every cohomology class $x\in H^\ell(M;\mathbb{Z}\_2)$ and multi-index $(i\_1,\ldots,i\_r)$ with $i\_1+\cdots + i\_r=k-\ell$ gives a *Stiefel-Whitney number of the map $f$*, defined by 
$$\langle w\_{i\_1}(A)\cdots w\_{i\_r}(A)f^\*(x),[A]\rangle\in\mathbb{Z}\_2.$$
 Your map is null-bordant if and only if these are all zero. (Note when $x$ is the unit class we get the S-W numbers of $A$. Also the multi-index $(0)$ gives trivial numbers by your assumption that $f\_\*[A]=0$.)


