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There are nice results for representing homology classes by submanifolds, in particular for any class in $H_i(X)$ with $i\le 6$, see here. When $X$ is low-dimensional I can start getting explicit, but this uses Poincare duality and appeals to classifying maps and characteristic classes of bundles. What is the right way to approach the problem for relative homology? Does the formulation of (Thom) spectra extend here? While I do have Lefschetz duality, I don't think I can do much with bundles, and I am not comfortable with (Thom) spectra.

Given a submanifold $A\subset X$ of a closed (smooth?) manifold, when can I (not) represent a homology class in $H_i(X,A;\mathbb{Z})$ by some submanifold with appropriate boundary?

Take whatever restrictions on $i$ and $\dim X$, but assume we're in a setting where the Hurewicz map isn't an isomorphism. If $i>\dim A+1$ then $H_i(X)$ surjects onto $H_i(X,A)$ in the LES, so we can take a submanifold which represents a homology class of $X$ to be the submanifold that represents the relative homology class.

Perhaps there is an example of an unrepresentable class in $H_2(X,S^1;\mathbb{Z})$ with $\dim X =3$ and $S^1$ highly knotted. When $X=S^3$ this group is $\mathbb{Z}$ (as seen by the LES), and Seifert surfaces do the trick.

As a toy model, consider the equator in a sphere. Then $H_2(S^2,S^1)\cong\mathbb{Z}^2$, as seen by reduced homology $\widetilde{H}_2(S^2/S^1)\cong H_2(S^2\bigvee S^2)\cong H_2(S^2)\oplus H_2(S^2)\cong\mathbb{Z}^2$. It looks like the classes $(1,0)$ and $(0,1)$ come from the hemispheres, and the relative Hurewicz theorem applies I think.

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I think the arguments are all pretty much the same. Let $X$ be a manifold, and $A$ a subset that's "tame enough" to admit a tubular / regular neighbourhood $V$, then

$$H_i(X,A) \simeq H_i(X,V)$$

by homotopy. And $H_i(X,V) \simeq H_i(X \setminus int(V), \partial V)$ by excision.

But this group is isomorphic to $H^{n-i}(X \setminus int(V))$ by Poincar\'e duality (if you assume orientability). $n=dim(X)$

At this point the standard Serre-Thom representability constructions work very nicely, for example if $i=n-1$ you use that $H^1(X \setminus int(V)) \simeq [X \setminus int(V), S^1]$ and apply transversality.

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  • $\begingroup$ Ah nice, excision allows Lefschetz duality to get rid of the reduced homology. But at the very end, we are applying the Pontryagin-Thom construction... does it really work for manifolds with boundary? It looks like I have to build language to handle cobordisms/framings with corners. $\endgroup$ – Chris Gerig Aug 18 '15 at 22:05
  • $\begingroup$ If you restrict the cohomology class to be one on the boundary, you can get transversality of the map $\partial V \to S^1$ and then extend it transversally (not affecting transversality on the boundary). So your representing manifold might be incident to $A$ with multiplicity. $\endgroup$ – Ryan Budney Aug 18 '15 at 23:52

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