Timeline for Which finite abelian groups aren't homotopy groups of spheres?
Current License: CC BY-SA 4.0
18 events
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| Sep 20, 2023 at 2:12 | history | protected | Noah Schweber | ||
| Dec 30, 2022 at 8:59 | comment | added | Ryan Budney | @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. | |
| Dec 30, 2022 at 7:39 | comment | added | Pedro | What motivates the statement about primes other than the fact no one has yet found one for small-ish values of $n$ ans $k$? | |
| Jul 1, 2022 at 21:17 | comment | added | John Baez | 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. | |
| Jun 30, 2022 at 14:06 | comment | added | Harry Wilson | 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? | |
| Nov 17, 2021 at 15:22 | comment | added | skd | 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? | |
| Nov 16, 2021 at 9:26 | history | edited | YCor | CC BY-SA 4.0 |
added assumption in title
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| Nov 16, 2021 at 3:26 | history | edited | John Baez | CC BY-SA 4.0 |
listed groups smaller than Z/5 and why they do arise as homotopy groups of spheres
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| Dec 11, 2020 at 19:27 | comment | added | Igor Belegradek | 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? | |
| Dec 11, 2020 at 15:22 | comment | added | Denis Nardin | @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. | |
| Dec 11, 2020 at 14:56 | history | edited | YCor |
edited tags
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| Dec 11, 2020 at 14:53 | history | edited | John Baez | CC BY-SA 4.0 |
added conjecture
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| Nov 27, 2020 at 21:38 | comment | added | Rachid Atmai | 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? | |
| Nov 27, 2020 at 5:58 | comment | added | Tyler Lawson | 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...) | |
| Nov 26, 2020 at 23:57 | comment | added | Noah Snyder | 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? | |
| Nov 26, 2020 at 23:15 | comment | added | Noam D. Elkies | The cyclic group of order 4 appears in the table <en.wikipedia.org/wiki/…> 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. | |
| Nov 26, 2020 at 23:09 | comment | added | Theo Johnson-Freyd | 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. | |
| Nov 26, 2020 at 22:40 | history | asked | John Baez | CC BY-SA 4.0 |