Timeline for What is $\pi_{23}(S^2)$?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Jan 22 at 14:12 | comment | added | Ali Caglayan | 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. | |
| Jan 22 at 14:09 | comment | added | Ali Caglayan | Contrast this with stable homotopy theory, for which we have a far greater understanding about how to compute them. There is recent work by Wang, Xu and Isaksen which is a huge improvement over previous methods, in the sense that it is more efficient and is able to compute higher stems without the computation getting blocked by some unknowns. Crucial to solving these unknowns is "random" bracketing of elements from Toda. Eventually, there will be a new method with a deeper understanding and it will replicate these computations and extend them. | |
| Jan 22 at 14:06 | comment | added | Ali Caglayan | Many people have spent a lot of time trying to understand the patterns, theory and structure behind these observations which is why you won't commonly see a table compiling results outside of Wikipedia. In unstable computations, having technical knowledge of the generators of the groups is more important than knowing the isomorphism class of the group in that sense. | |
| Jan 22 at 14:05 | comment | added | Ali Caglayan | @JoeShipman Being able to compute unstable homotopy groups of spheres requires an intense knowledge of their generators and their various bracketings. Toda was mostly successful in this because he had a very good understanding of how the generators can be combined to produce new elements. This kind of method however doesn't scale particularly well and is somewhat akin to being really good at solving integrals. | |
| Jan 19 at 21:27 | comment | added | Daniel Asimov | This paper arxiv.org/pdf/1506.00952.pdf reports that π_n(S^2) ≠ 0 for all n ≥ 2. | |
| Jan 19 at 19:34 | comment | added | Joe Shipman | Thanks, Ali. That table is great (and shows that we have up to Pi_28(S^2)=Pi_28(S^3)). But why should it be so difficult? You did it yourself, and everything in the unstable range except for the last three columns was done 60 years or more ago so the statement “the past 70 years of outstanding endeavour” seems a bit misleading, the last 60 years didn’t add much. It’s amazing how aversive all the papers in the literature are to giving a simple table like that. If I only want to know the groups themselves, why can’t you extend the table to the other known cases? | |
| Jan 19 at 17:04 | comment | added | Ali Caglayan | 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. | |
| Jan 19 at 17:01 | comment | added | Ali Caglayan | The 2-primary part was computed by Mamoru Mimura in 1965 J. Math. Kyoto. Univ. The p-primary part was computed by Toda "On iterated suspensions I." 1965 J. Math. Kyoto. Univ. This is what I wrote down in my notes a few years ago when I studied Toda's computations. Having a look at the papers now they are quite dense and require careful reading to extract out a table. Typically, you are interested in the description of the elements rather than the isomorphism class of the groups themselves as it let's you compute further groups. | |
| S Jan 19 at 16:18 | history | suggested | Marco Ripà | CC BY-SA 4.0 |
Improving readability with LaTeX
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| Jan 19 at 14:44 | answer | added | Joe Shipman | timeline score: 4 | |
| Jan 19 at 14:18 | review | Suggested edits | |||
| S Jan 19 at 16:18 | |||||
| Jan 19 at 11:00 | comment | added | Tyrone | 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 $\eta\circ\nu'\circ\eta_6\circ\mu_7\circ\sigma_{16}$. | |
| Jan 19 at 7:36 | comment | added | Ryan Budney | Do you have any particular reason to be interested in the $23^{rd}$ homotopy group, or is just because it appears to be the first unknown in the list? | |
| Jan 19 at 7:27 | comment | added | Mark Grant | 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. | |
| Jan 19 at 3:30 | comment | added | mme | 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. | |
| Jan 19 at 2:53 | history | asked | Joe Shipman | CC BY-SA 4.0 |