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Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer $\nu$ (which is allowed to depend on $n, m, [\Sigma],M,$) such that $\nu[\Sigma]$ can be represented by a smoothly embedded compact submanifold (not necessarily connected)? If $[\Sigma]$ is of odd-torsion, then we regard this as formally true.

I'm aware of Thom's work on representing homology classes. In the case of the Steenrod problem, i.e., representing homology classes by maps from manifolds, it is often mentioned that some odd multiple of a class can always be represented, and there are plenty of references.

For representation by embedded submanifolds, when $2n<m,$ a generic perturbation of any realizing map solving the Steenrod problem can yield an embedded submanifold, so some odd multiple suffices. However, I do not know how to deal with the case of $2n\ge m$. Due to my limited knowledge, I do not know if my question follows from Thom's work.

I'm sorry if this is a trivial question, as I work in geometric measure theory, which is very far away from algebraic topology. Many thanks!

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  • $\begingroup$ I don't think you can in general do this. I think you can do this for $n=m-1$, and $m-2$. This is probably also in Thom's paper. The poincare dual of the class $H_{n-1}(M)$ is represented by a map $M\rightarrow S^1$. The preimage of a regular value should be a representing submanifold. For $n=m-2$ you take a map $M\rightarrow CP^\infty$ representing the dual cohomology class. Then use cellular approximation to map $M$ to $CP^N$ for some large $N$, and take the preimage of $CP^{N-1}$ after perturbing the map to meet this submanifold transversally. $\endgroup$
    – Thomas Rot
    Commented Jan 31 at 13:32
  • $\begingroup$ I think Mark Grant answers your question mathoverflow.net/questions/375919/… $\endgroup$
    – Thomas Rot
    Commented Jan 31 at 13:44
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    $\begingroup$ Are you aware of this paper: arxiv.org/pdf/1111.0249.pdf . This shows that there are homology classes which cannot be represented by immersed, let alone embedded, manifolds. This is with Z_2 coefficients, but presumably this can be used in the integer coefficient case as well. $\endgroup$
    – Thomas Rot
    Commented Feb 1 at 17:38
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    $\begingroup$ I don't have the bandwidth to do the calculations right now. Regarding the paper with Szucs that @ThomasRot mentioned: since we were interested in non-realizability by immersions, we were only able to use stable cohomology operations. For embeddings any operation can be used (e.g. $\iota Sq^1\iota + Sq^2 Sq^1\iota$) which makes things a bit easier. $\endgroup$
    – Mark Grant
    Commented Feb 2 at 9:49
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    $\begingroup$ Many thanks for the comments! That looks promising! I'll try to do the exercises. $\endgroup$
    – Zhenhua Liu
    Commented Feb 2 at 17:56

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