# Is there an analogue of Steenrod's problem for $p>2$?

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 $$i:Y \rightarrow X$$.

• Thom's counterexample involves a space with 3-power-torsion cohomology, so it might already be a counterexample to the $p=3$ version. Apr 12 at 0:18
• 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. Apr 12 at 0:53
• @JeremyHahn Can you explain a bit more for those of us who are interested in this problem but less well versed in spectra? Apr 12 at 6:44
• @GregFriedman Akhil Mathew's blog posts here explain some more amathew.wordpress.com/tag/oriented-cobordism Apr 12 at 14:46
• 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? Apr 13 at 12:07