Is it true that for even dimensional differentiable manifold $M^{2n}$ all singular homology classes in dimension less than $n$ can be represented by a submanifold?

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    $\begingroup$ Look at the two answers to this duplicate question:mathoverflow.net/questions/21171/… $\endgroup$ – Achim Krause Mar 12 '15 at 0:02
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    $\begingroup$ In particular, there are obstructions for an integral homology class to be represented by a manifold map altogether (you can get some odd multiple in general), but as soon as that part works you can turn such a map into an embedding in the dimensions you're asking for. $\endgroup$ – Achim Krause Mar 12 '15 at 0:04

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