Timeline for Classification of bundles, Postnikov towers, obstruction theory, local coefficients
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 14, 2019 at 15:55 | history | became hot network question | |||
| Apr 14, 2019 at 15:48 | comment | added | Denis Nardin | @WarlockofFiretopMountain Obstruction theory tells you absolutely how many lifts there are! In fact it tells you the homotopy type of the space of lifts :). I'll try to write something, without repeating too much what Charles has already written. BTW I think I screwed up the indexing in my old answer (what's new, heh? :)), I think I fixed it now, so you might want to double check. | |
| Apr 14, 2019 at 15:47 | comment | added | Overflowian | @DenisNardin Thanks, many of the above questions came out trying to understand your previous answer. Sorry if they are messy, so is my understanding of the subject. In particular question 2) asks about why you state in your answer that "the possible choices are parametrized by a class in $H^{n+3}(X,\pi_{n+1}G)$". As far as I know (very little) obstruction theory tells us if we can lift but -gives us a cohomological obstruction to lifting- but it does not tell how many possible lifts we have. | |
| Apr 14, 2019 at 15:38 | history | edited | Overflowian | CC BY-SA 4.0 |
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| Apr 14, 2019 at 15:38 | answer | added | Charles Rezk | timeline score: 10 | |
| Apr 14, 2019 at 15:19 | comment | added | Denis Nardin | Also, the reason why the homotopy quotient is there is precisely that this is not a principal fibration, so local coefficients appear | |
| Apr 14, 2019 at 15:13 | comment | added | Denis Nardin | The tower in Mark Grant's answer is not the Whitehead tower, but the Postnikov tower (which is fairly degenerate, since $BO(2)$ is a 2-type so that $P_2(BO(2))=BO(2)$ and $P_1(BO(2))=B\mathbb{Z}/2=BO(1)$). I'll see if I can write an answer, but I'd have to say that I don't understand most of your questions | |
| Apr 14, 2019 at 14:52 | comment | added | Charles Rezk | In the second diagram, the bottom right corner should be $K(\pi_2G, 3)_{h\pi_0G}$, so that the fiber of the vertical map over it is equivalent to $K(\pi_2 G, 2)$. | |
| Apr 14, 2019 at 12:23 | history | asked | Overflowian | CC BY-SA 4.0 |