Timeline for Proofs of Bott periodicity
Current License: CC BY-SA 3.0
14 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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| S Oct 3, 2014 at 22:12 | history | suggested | Robert Bruner | CC BY-SA 3.0 |
Changed guess from "choosing a basepoint" to "adding a disjoint basepoint".
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| Oct 3, 2014 at 20:50 | review | Suggested edits | |||
| S Oct 3, 2014 at 22:12 | |||||
| Dec 22, 2009 at 6:41 | vote | accept | Eric Peterson | ||
| Dec 18, 2009 at 1:48 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
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| Dec 17, 2009 at 22:59 | comment | added | Eric Peterson | Also: step #3 wasn't to formally desuspend both sides, but to reduce the tautological line bundle. This should only introduce a $\Sigma^{-2}$ on one side of the map above, and is part of what made it smell Bott-y. ...Hmm, that may be the map used to build the localized CP^infty -- something worth checking, just to justify why this seemed sane from the start! // I'll probably give this question another few days in case anyone else has ideas, then accept this answer (more accurately this comment thread). "This is another way to view the bottom of Snaith's theorem" is an OK conclusion, if true. | |
| Dec 17, 2009 at 19:04 | comment | added | Eric Peterson | Now that it's not midnight: both Snaith (bottom of 2.11) and Gepner-Snaith (prop 4.3) use the invertibility of the Bott element to assert that their map $\Sigma^\infty_+ CP^\infty \to BU$ descends to the localization. (Snaith also uses periodicity to show that his map is even a map of ring spectra; Gepner uses some space-of-units tech instead.) A next step then would be for a proof of Snaith's theorem without using periodicity, but to do so you'd need to use statements about the localization other than its universal property -- probably to much to ask. Certainly not clearer than Harris' proof! | |
| Dec 17, 2009 at 7:37 | comment | added | Greg Kuperberg | By the way, the key issue remaining is addressed in Snaith's paper, "Localized stable homotopy of some classifying spaces". journals.cambridge.org/action/… | |
| Dec 17, 2009 at 7:28 | comment | added | Greg Kuperberg | You are more than welcome! Lest people think that I already knew all of this stuff, I certainly didn't. These reviews are a great way for me to learn the material. (I did just last month attend a talk by Mike Freedman that discussed Bott's proof. At the time, it went in one ear and out the other.) | |
| Dec 17, 2009 at 7:20 | comment | added | Eric Peterson | The map $\Sigma^\infty_+ CP^\infty[\beta^{-1}] \to BU(\infty)$ is producible without periodicity. To show that it is an equivalence, Gepner (with a more conceptual proof) does use periodicity, but I don't recall if Snaith (with a more computational proof) uses periodicity or not. I suspect he does; I'll look when I have time tomorrow afternoon. Certainly knowing that this map is a homotopy equivalence implies the periodicity of $BU(\infty)$, since it specifically names an invertible degree 2 element in its homotopy. Also, I meant to say thanks in the previous comment! This is all helpful. | |
| Dec 17, 2009 at 6:42 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
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| Dec 17, 2009 at 6:35 | comment | added | Greg Kuperberg | Well maybe my last paragraph is just my mathematical ignorance. In this Snaith result, I wouldn't know what is being proved from what. That is, I wouldn't know whether it could be a proof of Bott periodicity or a theorem that uses Bott periodicity to prove something else. | |
| Dec 17, 2009 at 6:10 | comment | added | Eric Peterson | This is very much along the lines of what I was thinking. However, Snaith (and now Gepner, and probably other people) have a more cheerful view of $\Sigma^\infty CP^\infty$ -- namely, they show that $\Sigma^\infty_+ CP^\infty[\beta^{-1}]$ is homotopy equivalent to $BU(\infty)$, where $\beta$ is the element of $\pi_2 \Sigma^\infty_+ CP^\infty$ classifying the reduced tautological bundle $(L - 1)$ over $CP^1 = S^1$. That is, $\Sigma^\infty_+ CP^\infty$ is a lot like $BU(\infty)$, up to invertibility of the Bott element. That said, stone soup is probably still applicable. | |
| Dec 17, 2009 at 4:29 | history | answered | Greg Kuperberg | CC BY-SA 2.5 |