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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.

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Are there any hypotheses? Are the homologous submanifolds assumed disjoint (or null-homologous submanifold embedded)? –  Igor Rivin Feb 11 '11 at 6:34
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3 Answers

up vote 4 down vote accepted

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 $f_*[A]=0$.)

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Thank you, I might look there if I can't come up with some more elementary way of seeing this. –  Ilya Grigoriev Feb 11 '11 at 19:30
I recommend it! The theory of bordism groups is very beautiful, and was developed precisely to answer questions such as yours. –  Mark Grant Feb 12 '11 at 12:29
Also note that my amended answer completely answers your question in the unoriented case, modulo a good understanding of the cohomology of your manifolds and the map induced by the inclusion. For example, if your embedded submanifolds are spheres, the answer to your question is yes. –  Mark Grant Feb 12 '11 at 12:33
Thank you again! You are right, this is probably the way I should be understanding this after all. I'll look at Conner and Floyd. –  Ilya Grigoriev Feb 18 '11 at 20:39
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Presumably, the answer is NO, since every manifold $K$ embeds in $\mathbb{S}^n$ of high dimension (where it is then null-homologous), but not every manifold is null-cobordant, in other words, there are $K$ such that there is no $N$ such that $\partial N = K.$

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You are right, of course, thaks! You were also correct that I want an additional hypothesis - I think I can assume my manifold to be null-cobordant. –  Ilya Grigoriev Feb 11 '11 at 19:30
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First of all, it doesn't matter whether or not the map is smooth. If you find any continuous map, then it will have a smooth approximation.

The other answers so far explain that the cobordism group gives you an obstruction to improving a singular-simplicial chain into a mapped-in manifold. In fact, it is easy to see that this is basically the only obstruction, and that null corbodism directly gives you a way to improve the simplicial chain. I'll work with integer coefficients rather than over $\mathbb{Z}/2$ so that things survive a little longer. Say that you have this $(k+1)$-dimensional cobounding chain. You can manifold-ize a $k$-dimensional face of the chain because various $k$-dimensional sheets meet the face with opposite sign and you can pair them. This is basically using the fact that the reduced 0-corbodism group is trivial. Then turn to the $(k-1)$-faces. Because of what you did to the $k$-faces, the sheets meet the $(k-1)$-faces in a collection of circles. But circles are null cobordant, so you can smooth them. You can continue in this way using the fact that oriented surfaces and 3-manifold are all null-cobordant. But when you try to improve a $(k-3)$-face, the link of the an incoming sheet can be a 4-manifold that is not null-cobordant, like $\mathbb{C}P^2$. Then you're stuck.

The obstruction is fundamental because the original null-homologous $k$-cycle could have been an embedded $\mathbb{C}P^2$, and the original cobounding chain could have been a cone over it.

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Thank you, your answer was very helpful. I'm especially hopeful because of your statement that cobordism is the only problem - I don't think it's a problem for me. –  Ilya Grigoriev Feb 11 '11 at 19:29
In the construction that I describe, you have to be careful to check that all cobordisms that you will need for the entire cobounding chain are available, and not just that your cycle is represented by a null-cobordant manifold. –  Greg Kuperberg Feb 11 '11 at 20:10
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