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Someone asked me if all finite abelian groups arise as homotopy groups of spheres. I strongly doubted it, and I bet ten bucks that $\mathbb{Z}_5$ is not $\pi_k(S^n)$ for any $n,k$. But I don't know how to prove it's not.

Which finite abelian groups are known to not arise as homotopy groups of spheres?

I conjecture that $\mathbb{Z}_5$ is the smallest one. From some tables we can see that all smaller groups do actually arise:

$$ \begin{array}{ccl} \pi_1(S^2) &\cong & 1 \\ \pi_4(S^3) &\cong& \mathbb{Z}_2 \\ \pi_9(S^3) &\cong& \mathbb{Z}_3 \\ \pi_8(S^4) &\cong& \mathbb{Z}_2 \times \mathbb{Z}_2 \\ \pi_{122}(S^{62}) &\cong& \mathbb{Z}_4. \end{array} $$

In fact, I conjecture that for no odd prime $p \gt 3$ is $\mathbb{Z}_p$ isomorphic to $\pi_k(S^n)$ for any $n,k$.

YCor
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John Baez
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    I think every cyclic group appears as a subgroup of a homotopy group of spheres. Indeed, I think this is already true for the image of j. But I think homotopy groups of spheres are typically groups are “small” unless n-k=0 mod 4, in which case the image of j is already quite big, so it does seem likely that most finite groups will not appear. – Theo Johnson-Freyd Nov 26 '20 at 23:09
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    The cyclic group of order 4 appears in the table https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#Table_of_stable_homotopy_groups of stable homotopy groups of spheres reproduced in Wikipedia's entry for "Homotopy groups of spheres". So it seems that this group appears as $\pi_{n+60}(S^n)$ for $n$ sufficiently large. I do not see a 5-element group tabulated anywhere on that page. – Noam D. Elkies Nov 26 '20 at 23:15
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    Are there heuristics (a la Cohen-Lenstra) for p-part of the homotopy groups of spheres mod the image of J? Or maybe in the stable case? – Noah Snyder Nov 26 '20 at 23:57
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    I can't give you any concrete answer, but my inclination is to say that it seems highly likely that you're correct. The 2-primary torsion forms a real thicket, whereas there can be no p-torsion until you are at least at $\pi_{n+2p-3}(S^n)$. This means it's not clear to me whether there are infinitely many different groups of odd order appearing. (But the unstable part is really outside my wheelhouse...) – Tyler Lawson Nov 27 '20 at 05:58
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    Spaces with simple fundamental groups are spaces which only have two covering spaces: the trivial cover and the universal cover. At the same time, every finitely generated group, whether simple or not, is the fundamental group of a $4$-manifold, so this seems to be the same as asking about to Poincare conjecture in $4$-D? – Rachid Atmai Nov 27 '20 at 21:38
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    @CalicoJackRackham I think you mean "only two normal covering spaces" rather than "covering space". After all non-cyclic simple groups have a lot of subgroups. But since the question is only interesting for higher homotopy groups (which are always abelian) the relevance of your comment is not clear to me. – Denis Nardin Dec 11 '20 at 15:22
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    I was under impression that there some $n, k$ for which nothing is known about $\pi_k(S^n)$ (other than that it is an abelian group of known rank)? If so, the question is clearly open. Why all the discussion in the comments? – Igor Belegradek Dec 11 '20 at 19:27
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    The case of the 3-sphere may be useful to look at, since a result of Selick's states that if p is an odd prime, then the p-local homotopy groups of S^3 (above dimension 3) are all killed by p (i.e., are simple p-torsion). But one thing I observed is that for odd p, the group pi_{2p}(S^3) (which contains the first nontrivial p-torsion element in the homotopy of S^3) always seems to have 3-torsion in it. Is this true? – skd Nov 17 '21 at 15:22
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    Is it possible that this is a sort of Skewes's number phenomenon, where $Z_5$ does occur but only for very large n and k? Is much known about the asymptotics of the homotopy groups of spheres? – Harry Wilson Jun 30 '22 at 14:06
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    I think it's "possible" that $\pi_{n+k}(S^n) \cong \mathbb{Z}_5$ only for large $n$ and $k$, given our extensive ignorance of the homotopy groups of spheres except for small $n$ and/or $k$, but I also don't think we have any good reason to expect it. So, I'm conjecturing it never shows up. – John Baez Jul 01 '22 at 21:17
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    What motivates the statement about primes other than the fact no one has yet found one for small-ish values of $n$ ans $k$? – Pedro Dec 30 '22 at 07:39
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    @Pedro: Presumably this is akin to a tease. We know very little about homotopy groups of spheres, in particular how to compute them. My impression is the intention of the post is to trigger an ambitious young homotopy theorist to crank out some computation or theory, and answer the question. – Ryan Budney Dec 30 '22 at 08:59

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