Timeline for $p$-adic Bott periodicity?
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 19, 2020 at 8:47 | vote | accept | Dominik | ||
| May 10, 2016 at 20:29 | answer | added | John Rognes | timeline score: 18 | |
| May 10, 2016 at 11:42 | history | edited | Dominik | CC BY-SA 3.0 |
edited body
|
| May 10, 2016 at 11:23 | comment | added | user51223 |
@MatthiasWendt I should have been more precise, and have said looks like Milnor conjecure' rather that is Milnor conjecture'. Still, the statement made above is much stronger than the original conjecture.
|
|
| May 10, 2016 at 9:31 | comment | added | Matthias Wendt | @user51223: the Friedlander-Milnor conjecture is a statement about algebraically closed fields (and of course finite coefficients as pointed out by Ben Wieland) or $\mathbb{R}$. For $\mathbb{R}$ and $\mathbb{C}$, the map $BG^\delta\to BG$ fails to be a homotopy equivalence because the fundamental groups are non-isomorphic. | |
| May 10, 2016 at 9:26 | comment | added | Matthias Wendt | @QiaochuYuan: it should be noted that the algebraic K-theory spectrum of $\mathbb{Q}_p$ is not periodic which follows from the localization sequence and the finite field case. | |
| May 8, 2016 at 14:49 | comment | added | Ben Wieland | People generally call it the Friedlander-Milnor conjecture to distinguish it from other conjectures of Milnor. It is only for homology with finite coefficients and is false for rational coefficients. Suslin proved the relevant cases $G=U(\infty),O(\infty),Sp(\infty)$ using his rigidity theorem. | |
| May 7, 2016 at 8:49 | comment | added | user51223 | If one had Milnor conjecture, as Amrani has proven, then together with Whitehead Theorem in presence of a nice CW- structure, the above statement on the map $BG^\delta\to BG$ being s homotopy equivalence would be correct. | |
| May 7, 2016 at 8:47 | comment | added | user51223 | @QiaochuYuan Dominik writes ``The classifying space of the topological group should have the same homotopy groups as the classifying space constructed from the underlying discrete group, if I'm not mistaken.'' I presume he is suggesting that the map $BG^\delta\to BG$ is a homotopy equivalence. That the map above induces an isomorphism in homology was Milnor conjecture, and the suggestion of Dominik is much stronger I suppose; I don't know how a CW-complex structure for $BG^\delta$ can be derived if one for $BG$ is known so that one can apply Whitehead Theorem to Milnor's conjecture. | |
| May 7, 2016 at 8:05 | comment | added | Qiaochu Yuan | @user51223: I'm being a bit imprecise; when I want to think of a spectrum as a space I take its zeroth space. I don't understand what your comment about Milnor's conjecture is referring to. | |
| May 7, 2016 at 7:42 | comment | added | user51223 | I have never tried to do the proof of the periodicity theorems on my own, but if something like what you look for holds, then I suppose one can detect new finite families of elements in stable homotopy of spheres; you can use Bott periodicity to prove Adams Hopf invariant one result... | |
| May 7, 2016 at 7:38 | comment | added | user51223 | Just to add something on the notation to the comment of @QiaochuYuan. I think as $\Omega$ sees only based maps, then you have $\Omega^8BO=\Omega^8(BO\times\mathbb{Z})=BO\times\mathbb{Z}$. The equality $KO=\mathbb{Z}\times BO$ seems to equate a spectrum with a space which I don't understand. The part of your comment on the isomorphism of homologies of $BG$ and $BG^\delta$ where $G^\delta$ is $G$ when equipped with discrete topology, is Milnor's conjecture, and I am not sure to what extent is has been verified. Amrani has a recent work on it available at arxiv.org/pdf/1511.06101v2.pdf. | |
| May 7, 2016 at 3:37 | comment | added | Allen Knutson | Also, I think the $\Omega_8$ statement is much cooler as two $\Omega_4$ statements, going between $KO$ and $KSp$. | |
| May 6, 2016 at 16:02 | comment | added | Qiaochu Yuan | Your statements of Bott periodicity are slightly off; you should replace $BU$ and $BO$ with $KU = \mathbb{Z} \times BU$ and $KO = \mathbb{Z} \times BO$ respectively. These are the complex and real K-theory spectra, and so one possible $p$-adic replacement for them is the algebraic K-theory spectrum of $\mathbb{Q}_p$ (which completely ignores the $p$-adic topology). These are used by Dustin Clausen (arxiv.org/abs/1110.5851), for example, to give a $p$-adic J-homomorphism. | |
| May 6, 2016 at 15:42 | history | asked | Dominik | CC BY-SA 3.0 |