Timeline for Classification of $O(2)$-bundles in terms of characteristic classes
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 31, 2019 at 8:36 | comment | added | Overflowian | Hi Denis, any update? | |
| Aug 24, 2019 at 16:58 | comment | added | Denis Nardin | @WarlockofFiretopMountain I might be making a subtle mistake. Let me stew on it for a while, I'll try to answer later. | |
| Aug 24, 2019 at 16:51 | comment | added | Overflowian | Thanks for the reply. Sorry but I'm still missing something. Why in this answer mathoverflow.net/questions/335537/… Matthias says that the additional invariant lives in a quotient of $H^4(X,\mathbb{Z})$? Even in this case we are interested in classifying bundles. | |
| Aug 24, 2019 at 14:12 | comment | added | Overflowian | Denis, I was wondering, is it possible that two distinct elements in $H^{n+1}(X;\pi_n G)$ give the same $G$-bundle? I mean, when you say that the choices of the lift are parametrized by a class in $H^{n+1}(X;\pi_n G)$ I thought it comes from the Puppe exact sequence $[X,K(\pi_n G, n+1)]\overset{\alpha}{\to} [X,P_{n+1}BG] \to [X,P_{n}BG]$ but $\alpha $ can also have nontrivial kernel, am I right? | |
| Apr 14, 2019 at 15:23 | history | edited | Denis Nardin | CC BY-SA 4.0 |
Screwed up the indexing, of course.
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| Apr 27, 2016 at 12:43 | comment | added | Denis Nardin | (cont.) When I say "twisted by $\alpha$" I mean in the sense of cohomology with local coefficients (that is $\pi_nG$ is secretly a local system on $X$ via the $\pi_1X$-action). Also beware that the groups where the higher classes lie are not canonically identified (that is you'll have to fix a G-bundle with first class $\alpha$ and use it to compare the other G-bundles) | |
| Apr 27, 2016 at 12:43 | comment | added | Denis Nardin | @Bilateral I don't have a good reference for this particular story, but being familiar with Hatcher's book on algebraic topology will certainly help to see many similar constructions, so that this is not too surprising. I learnt the general structure of the Postnikov tower from Blanc, Dwyer, Goerss The realization space of a Pi-algebra but that might be a bit advanced. (cont.) | |
| Apr 27, 2016 at 7:26 | comment | added | Bilateral | What does exactly mean "coefficients twisted by $\alpha$"? | |
| Apr 27, 2016 at 7:07 | comment | added | Bilateral | Thanks for the detailed answer. Could you recommend a reference to further explore what you are explaining? | |
| Mar 26, 2016 at 17:48 | comment | added | Denis Nardin | As an added remark, the classes of maps $[X,P_n(BG)]$ correspond to the failure of supporting some additional structure. For example if $G=O(k)$ we have that the map $X\to P_1(BO(k))$ is nullhomotopic iff the bundle is orientable, while the map $X\to P_2(BO(k))$ is nullhomotopic iff the bundle is spin etc.. | |
| Mar 26, 2016 at 17:39 | history | answered | Denis Nardin | CC BY-SA 3.0 |