Timeline for What do the stable homotopy groups of spheres say about the combinatorics of finite sets?
Current License: CC BY-SA 3.0
22 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Feb 10, 2015 at 2:14 | answer | added | Qiaochu Yuan | timeline score: 26 | |
| Feb 28, 2012 at 10:26 | answer | added | Johannes Ebert | timeline score: 33 | |
| Oct 6, 2011 at 14:48 | comment | added | Dai Tamaki | @Mark: I didn't realize your comment when I wrote my answer. Your comment is essentially the same as my answer. Would you like to write it as an answer? Then I will remove my answer. | |
| Oct 6, 2011 at 13:15 | answer | added | Dai Tamaki | timeline score: 21 | |
| Sep 29, 2011 at 20:01 | comment | added | Daniel Litt | @Mark: That is a beautiful paper! Thanks for pointing it out :). @Hans: I guess because the question relates something reasonably well-understood and ubiquitous (combinatorics of finite sets) to something notoriously poorly understood but fascinating (the stable homotopy ring of the sphere spectrum)? But perhaps your question is one for meta.mathoverflow, and not the comment thread to this question. | |
| Sep 29, 2011 at 19:48 | comment | added | Hans-Peter Stricker | How is the astonishingly large number of upvotes for this - rather specific - question (starting with "The Barratt-Priddy-Quillen(-Segal) theorem says...") to be understood? | |
| Sep 29, 2011 at 18:35 | history | edited | Daniel Litt | CC BY-SA 3.0 |
Added a weaker question.
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| Sep 28, 2011 at 6:57 | comment | added | Mark Grant | @Daniel: Have you checked out the work of Jie Wu and collaborators, "Configurations, braids, and homotopy groups" in the JAMS? They give a combinatorial description of the homotopy groups of $S^2$ in terms of Brunnian braids. This might not be quite what you were asking about, but its certainly related. | |
| Sep 28, 2011 at 5:34 | comment | added | Daniel Litt | @Dustin: Indeed I am interested! | |
| Sep 28, 2011 at 5:18 | comment | added | Dustin Clausen | (I meant the complex image of J by the way) | |
| Sep 28, 2011 at 5:12 | comment | added | Dustin Clausen | Daniel - you might be interested in the ``discrete models map'' for the J-homomorphism appearing in papers of Snaith and the book of May-Quinn-Ray-Tornehave. It comes from the forgetful functor from finite-dimensional vector spaces over F_p to finite sets, and gives maps K_n(F_p) --> \pi_n^S[1/p] that capture the image of J away from p. | |
| Sep 28, 2011 at 3:57 | comment | added | Daniel Litt | (cont.)...constructions 2-4 above. Maybe using the $J$-homomorphism? But who knows. And I'd be happy if the homomorphisms went the other way also, of course. Basically, it would be awesome to have some good combinatorial way of building or interpreting stably nontrivial homotopy classes. | |
| Sep 28, 2011 at 3:54 | comment | added | Daniel Litt | @Dylan: Gotcha. Yeah, there's some cool stuff on this and related things in Weibel's $K$ book, though a lot of it is in the exercises. I personally find the book hard to read, though I know some disagree... In any case, the miracle I am hoping for is that one would be able to come up with some combinatorial interpretation of $\pi_*^S$ without having to do so for all the related $K$-theoretic constructions, which seems very hard--or, more weakly, some sequence of groups $G_n$, defined combinatorially, with nontrivial homomorphisms $G_n\to \pi_n^S$, which one could see in one of the... | |
| Sep 28, 2011 at 1:36 | comment | added | Dylan Wilson | @Daniel: Well I knew about the analogy between Quillen's definition of K-theory and what we have here, but I didn't realize the other interpretations of the lower K-groups also had analogs in this case. Very cool! | |
| Sep 28, 2011 at 0:34 | comment | added | Daniel Litt | @Dylan: It's exactly reminiscent, via any of 2-4 above, which should be viewed as exactly analogous to the $+$ or $S^\bullet$ constructions of algebraic $K$-theory. @Jacob Lurie: Another way to see this is to use that Hurewicz is an isomorphism from $\pi_1\to H_1$, and (I think) is a ring homomorphism for ring spectra, and then to use that the generators of $\pi_1$ and $H_1$ square to give the generators of $\pi_2$ and $H_2$. | |
| Sep 28, 2011 at 0:09 | comment | added | Dylan Wilson | This sounds vaguely reminiscent of $K_2$ in algebraic $K$-theory and its realization as the universal central extension of some stable general linear group, or Schur multipler of elementary matrices etc. | |
| Sep 27, 2011 at 23:55 | answer | added | Scott Carter | timeline score: 10 | |
| Sep 27, 2011 at 23:13 | comment | added | Jacob Lurie | The homology equivalence of the identity component of QS^0 with the classifying space of the infinite symmetric group gives a homology equivalence of its universal cover with the classifying space of the infinite alternating group G. Hence $\pi_2 QS^0 = H_2(G)$ is the Schur multiplier of G: that is, the kernel of the universal central extension of G (which is also the universal central extension of A_n as soon as n is reasonably large; I think it starts at n=8.) | |
| Sep 27, 2011 at 20:16 | comment | added | Daniel Litt | @Qiaochu: That would indeed be awesome, but I don't see why it should be true. Do you have any reason to think there might be a relationship? | |
| Sep 27, 2011 at 20:03 | comment | added | Dylan Wilson | I'd love to know an answer to this! +1 | |
| Sep 27, 2011 at 18:18 | history | asked | Daniel Litt | CC BY-SA 3.0 |