Timeline for What part of the fundamental group is captured by the second homology group?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Aug 5, 2016 at 16:15 | answer | added | Rui Loja Fernandes | timeline score: 7 | |
| Jul 23, 2012 at 10:33 | history | edited | Ronnie Brown | CC BY-SA 3.0 |
corrected Hurewitz to Hurewicz
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| Aug 15, 2011 at 21:53 | answer | added | Sergey Melikhov | timeline score: 9 | |
| Aug 14, 2011 at 20:06 | answer | added | Ronnie Brown | timeline score: 8 | |
| Sep 30, 2010 at 8:30 | comment | added | Matthias Künzer | A bit off the question maybe, but there is a pullback due to Eilenberg-MacLane that describes the second cohomology group in terms of $\pi_1$ and $\pi_2$ (Determination of the second homology and cohomology groups of a space by means of homotopy invariants. Proc. Nat. Acad. Sci. USA 32 (1946), p. 277–280; see also arxiv.org/abs/0911.2864). | |
| Sep 29, 2010 at 20:58 | history | edited | Daniel Moskovich | CC BY-SA 2.5 |
added 45 characters in body
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| Sep 29, 2010 at 20:55 | vote | accept | Daniel Moskovich | ||
| Sep 29, 2010 at 15:33 | answer | added | Primoz | timeline score: 8 | |
| Sep 29, 2010 at 5:42 | answer | added | HJRW | timeline score: 11 | |
| Sep 29, 2010 at 5:39 | comment | added | Daniel Pomerleano | that should be ZxZ... | |
| Sep 29, 2010 at 5:38 | comment | added | Daniel Pomerleano | I guess the comments and answers probably say this already but $\pi_1$ tells you something about $H_2$ and not the other way. An example from Hatcher is that if $pi_1$ is $Z x Z$ you know that $H_2$ is non-vanishing. The basic idea is take your space, attach three cells and higher to build a $K(\pi_1)$ so $H_2(X)$ surjects onto $H_2(\pi_1)$ | |
| Sep 29, 2010 at 2:36 | answer | added | Autumn Kent | timeline score: 15 | |
| Sep 29, 2010 at 2:30 | answer | added | Ryan Budney | timeline score: 15 | |
| Sep 29, 2010 at 2:24 | answer | added | Tom Goodwillie | timeline score: 46 | |
| Sep 29, 2010 at 2:18 | comment | added | Ryan Budney | From the point of view of the Serre spectral sequence for the Postnikov tower it tells you a little bit about how the Postnikov system twists for the $\pi_2 X$ stage over the $\pi_1 X$ stage. I imagine you could clean that up into a very clear statement but off the top of my head I don't know what it is. In particular you'll need more than just $\pi_1 X$ information. If you're only thinking of $\pi_1 X$ as input there is likely very little (if any) information that survives. | |
| Sep 29, 2010 at 1:31 | history | asked | Daniel Moskovich | CC BY-SA 2.5 |