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An element $\alpha \in H_k(X; \mathbb{Z})$ is said to be realisable if there is a $k$-dimensional connected, closed, orientable $k$-dimensional submanifold $Y$ such that $\alpha = i_*[Y]$. The Steenrod Problem was whether every class is realisable. Thom showed in 1954 that this is not true in general. However, it is true if $0 \leq k \leq 6$. What is true however is that for any class $\alpha$, there is an integer $N$ such that $N\alpha$ is realisable.

We could ask the analogue of Steenrod's Problem for $\mathbb{Z}_2$ coefficients: is every class in $H_k(X; \mathbb{Z}_2)$ realisable? The answer is yes.

What if we ask the analogue of Steenrod's Problem for $\mathbb{Z}_p$ coefficients, for integer $p>2$?


Edit: Following Mark's comment, I am asking about realizability by smooth maps $i:Y \rightarrow X$.

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    $\begingroup$ Thom's counterexample involves a space with 3-power-torsion cohomology, so it might already be a counterexample to the $p=3$ version. $\endgroup$
    – Will Sawin
    Apr 12 at 0:18
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    $\begingroup$ Rationally, the bottom HQ splits off the Postnikov tower of Q otimes MSO. After 2-completion, the bottom HZ_2 splits off of the Postnikov tower of MSO_2. After p-completion at an odd prime, the HZ_p does not split off. Roughly, this is because the spectrum MSO_p splits as a wedge of BPs. $\endgroup$ Apr 12 at 0:53
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    $\begingroup$ @JeremyHahn Can you explain a bit more for those of us who are interested in this problem but less well versed in spectra? $\endgroup$ Apr 12 at 6:44
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    $\begingroup$ @GregFriedman Akhil Mathew's blog posts here explain some more amathew.wordpress.com/tag/oriented-cobordism $\endgroup$ Apr 12 at 14:46
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    $\begingroup$ In the first paragraph you talk about realizability by embeddings. But then the mod 2 result you quote in the second paragraph is false for embeddings. I suspect that you're really asking about realizability by smooth maps $i:Y\to X$ throughout? $\endgroup$
    – Mark Grant
    Apr 13 at 12:07

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