The first $22$ homotopy groups of the $2$-sphere were worked out by Toda in 1962, but I cannot find any results extending that to any higher homotopy groups of $S^2$.
Are any more of these groups known?
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4$\begingroup$ The 2-primary part of the groups $\pi_n(S^2)$ for $n \le 55$ are computed in The unstable Adams spectral sequence for $S^3$ by Curtis and Mahowald. This article appears in the collection Algebraic Topology, volume 96 in the AMS series Contemporary Mathematics. Extracting the actual answer would take a more careful read than I'm willing to do. I also don't know about other primes. $\endgroup$– mmeCommented Jan 19 at 3:30
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1$\begingroup$ Note the question mathoverflow.net/q/130922/8103 asking about $\pi_{31}(S^2)\cong\pi_{31}(S^3)$. The references there probably contain what you want, and Tyler Lawson's answer explains how you might go about extrating the full answer. $\endgroup$– Mark GrantCommented Jan 19 at 7:27
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2$\begingroup$ If I'm not mistaken, the group is $\mathbb{Z}/2\oplus\mathbb{Z}/2$ generated by $\eta\circ\nu'\circ\overline\mu_6$ and $\eta\circ\nu'\circ\eta_6\circ\mu_7\circ\sigma_{16}$. $\endgroup$– TyroneCommented Jan 19 at 11:00
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5$\begingroup$ If however like me a few years ago, you are interested in a table past the "Wikipedia range", then I have a WIP table compiled here with a few citations for further reading: docs.google.com/spreadsheets/d/… I should note we know quite a bit more about the unstable (and stable) range than the table would indicate. Its just very difficult to concisely state the past 70 years of outstanding endeavour in this field in a table. $\endgroup$– Ali CaglayanCommented Jan 19 at 17:04
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4$\begingroup$ To finish off this flurry of comments, I think it is unfair to say that the "last 60 years didn't add much" as this is just not true and a disservice to all the people that worked in the field. I think a better way to say what I think you are saying is: "The last 60 years didn't produce an easy to consume expose of the latest results". Which isn't strictly true, but I think most mathematicians would agree that more recent research could definitely be surveyed and communicated better. $\endgroup$– Ali CaglayanCommented Jan 22 at 14:12
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This post shows that Toda and his collaborator Mimura extended the calculations up to $\pi_{25}(S^2)$ a few years later: Mark Grant (https://mathoverflow.net/users/8103/mark-grant), Unstable homotopy groups of spheres beyond Toda's range, URL (version: 2015-01-05): https://mathoverflow.net/q/191070
That is helpful although it does not actually say what the groups ARE, it just cites papers.
All the other answers I have seen in various places on Math Overflow don’t even get this far, they talk about only the $2$-primary component up to $\pi_{32}$ in very complicated ways without simply giving a table, and without data for the odd-prime components.
It seems a rather shocking lack of progress for over $5$ decades.
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12$\begingroup$ It would strike me as a shocking lack of progress if there was a clear reason to be interested in having a table giving the homotopy groups of $S^2$ up to isomorphism. It seems of rather niche interest. I suspect homotopy theorists find this example more interesting as a test case and as a place to look for general patterns, which is why the papers tend to focus on drawing the spectral sequences. $\endgroup$– mmeCommented Jan 19 at 15:49
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2$\begingroup$ There has also been progress on algorithmic computation, although the algorithms and modern computers are still a little behind the pace of Toda. My impression is the algorithms are too memory intensive. $\endgroup$ Commented Jan 19 at 17:04
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7$\begingroup$ Not sure I agree that it's of "niche interest". It my be relatively niche for modern homotopy theorists, but it's also a simply stated problem with a broad appeal. Writing comprehensible answers to problems that (relatively) lay audiences can understand is a worthwhile undertaking. (To clarify: my understanding is that there are known homotopy groups of spheres that aren't written down anywhere: this is what I agree is a sad state of affairs.) $\endgroup$– HJRWCommented Jan 19 at 17:12
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4$\begingroup$ I agree with @HJRW here. The only reason it might appear "niche" is that the information isn't very easy to collect or use, for people who need information about homotopy groups of spheres. The group isomorphism type is one thing, but concrete descriptions of the elements is another thing, and how do homotopy operations act on them, that is another. i.e. suspension, relations to the orthogonal group fibrations, Stiefel manifolds, configuration spaces, Thom constructions, etc. Applications typically need all this, and the literature isn't convenient. $\endgroup$ Commented Jan 19 at 17:23
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$\begingroup$ Regarding the state of the literature, one should keep in mind that the stable range is both more accessible and has more applications than the unstable range, and correspondingly has had much more progress. If you're really interested in unstable homotopy groups, you might look at some of the more recent work using the Goodwillie spectral sequence to calculate them. $\endgroup$ Commented Jan 19 at 22:10