Timeline for What do the stable homotopy groups of spheres say about the combinatorics of finite sets?
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17 events
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| Nov 6, 2018 at 5:48 | comment | added | mme | I'm really glad you linked this elsewhere. I learned a lot from reading this. | |
| Sep 22, 2017 at 1:26 | comment | added | Qiaochu Yuan | @Saal: yes. The "etc." means the $n$-morphisms are framed $n$-cobordisms between framed $(n-1)$-cobordisms. | |
| Sep 21, 2017 at 20:12 | comment | added | Saal Hardali | @QiaochuYuan So... $Bord$ here is some kind of $(\infty,\infty)$-category? Otherwise I don't understand "it has objects framed points, morphisms framed 1-cobordisms between points etc.." What does the "etc" mean here? | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Feb 11, 2015 at 1:14 | comment | added | Qiaochu Yuan | @Daniel: modulo the fact that I don't know an explicit $3$-cocycle describing this extension, this construction is called the twisted Drinfel'd double and it's made explicit, for example, here: arxiv.org/abs/math/0503266 But I asked an MO question about these central extensions and they are probably not too interesting, unfortunately. | |
| Feb 11, 2015 at 0:25 | comment | added | Daniel Litt | Your last comment is extremely interesting. Can one make it explicit? | |
| Feb 10, 2015 at 6:11 | comment | added | Qiaochu Yuan | @Daniel: hmm, actually I can knock the explicit category number down a bit more. The universal $2$-group extension of $\widetilde{A}_n$ should give a distinguished central extension of the adjoint quotient $\widetilde{A}_n / \widetilde{A}_n$ (a groupoid, not a group, but made up of groups), and in particular it should give a distinguished family of central extensions of the centralizers of the conjugacy classes of $\widetilde{A}_n$ by $\mathbb{Z}_{24}$. | |
| Feb 10, 2015 at 4:59 | comment | added | Daniel Litt | Let us continue this discussion in chat. | |
| Feb 10, 2015 at 4:57 | comment | added | Qiaochu Yuan | @Daniel: the lesson I take from the cobordism hypothesis, etc. is that $\pi_n(\mathbb{S})$ is about an $n$-groupoid, and so in particular if you want to understand $\pi_3(\mathbb{S})$ you will have to, implicitly or explicitly, think about a $3$-groupoid somehow. The description above is about as low as I can get the category number while there's still a $3$-groupoid somewhere in the background, namely the $3$-groupoid of $2$-groups. If you'd like to ask a separate question about "combinatorial" examples of topological field theories, I'd be happy to attempt to answer that. | |
| Feb 10, 2015 at 4:48 | comment | added | Qiaochu Yuan | @Daniel: so it seemed like you were happy with $\pi_2(\mathbb{S}) \cong H_2(BA_{\infty}, \mathbb{Z})$. What about $\pi_3(\mathbb{S}) \cong H_3(B \widetilde{A}_{\infty}, \mathbb{Z})$? This can be interpreted as claiming the existence of a "universal $2$-group extension" of the universal central extension $\widetilde{A}_n$ (for $n$ sufficiently large, but I don't know enough about these stabilization results to guess how large) by $B \mathbb{Z}_{24}$. | |
| Feb 10, 2015 at 4:44 | comment | added | Daniel Litt | My complaint is mostly about your comment that "this question doesn't respect the symmetries of the situation." But in the spirit of your comment, I do think that this answer is like saying that "representation theory is a good way to study groups because of the Tannakian formalism." That's true, but once you have the Tannakian formalism in hand, you have to actually do something with it. My question, reinterpreting it in these terms, is: can one do anything very, very concrete? And so far the answer seems to be: not for $\pi_n^{st}$ with $n>2$. | |
| Feb 10, 2015 at 4:37 | comment | added | Qiaochu Yuan | @Daniel: well, roughly speaking there are two ways to try to understand a group: by looking at its structure theory or looking at its representation theory. This answer describes in particular a natural class of representations, in a suitable sense, of the groups $\pi_n(\mathbb{S})$... is the sense in which this is unsatisfactory that I haven't told you how to compute these representations more explicitly? | |
| Feb 10, 2015 at 4:08 | comment | added | Daniel Litt | Of course, I do take your point that topological field theories are a good way of understanding stable homotopy groups of spheres; if you gave me a "combinatorial" example of one, where things ended up being somehow computable, I'd probably be happier. | |
| Feb 10, 2015 at 4:07 | comment | added | Daniel Litt | OK, I've voted you up because I like your explanation of $\pi_2$ a lot. But I think much of what you've written is philosophically backwards--the question is not about saying something about computing $\pi_n(\mathbb{S})$, but rather about understanding it. Breaking symmetries (e.g. writing down a basis of a vector space) is a common way to do that...In fact, this is exactly the answer I was trying to avoid by discouraging a discussion of the sphere spectrum as a categorification of the integers. But I do greatly appreciate your attempt at making that philosophy more concrete. | |
| Feb 10, 2015 at 2:40 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Feb 10, 2015 at 2:29 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Feb 10, 2015 at 2:14 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |