Timeline for comparing homology of a space and homology of the classifying space of its fundamental group
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 1, 2016 at 23:49 | vote | accept | awivil | ||
| Mar 1, 2016 at 23:44 | comment | added | awivil | Thank you, Igor. Dranishnikov's paper that you referred to is very helpful to me. | |
| Mar 1, 2016 at 21:59 | comment | added | Igor Belegradek | If the rational fundamental class of $X$ survives under the classifying map to $K(\pi_1(X),1)$, the manifold is called rationally essential. Gromov and more recently Dranishnikov studied the interplay between being rationally essential and the the so called "macroscopic dimension". See Dranishnikov's papers at arxiv: arxiv.org/find/grp_math/1/au:+dranishnikov/0/1/0/all/0/1 | |
| Mar 1, 2016 at 21:36 | answer | added | Sebastian Goette | timeline score: 2 | |
| Mar 1, 2016 at 21:17 | answer | added | Jens Reinhold | timeline score: 6 | |
| Feb 26, 2016 at 3:32 | comment | added | Jens Reinhold | It definitely also fails for any manifold of the form $X = N \times S^1$ with $N$ simply-connected, just because $B\mathbb Z = S^1$ has trivial homology above degree 1. | |
| Feb 25, 2016 at 18:14 | comment | added | Qiaochu Yuan | $f_{\ast}$ is an isomorphism for the torus because the torus is already the classifying space of its fundamental group. And note that if $X$ has any nontrivial rational homology then the rational map can't be injective if $G$ is finite (say for $X = \mathbb{RP}^3$). In general, look at the Serre spectral sequence of the fibration $\widetilde{X} \to X \to B \pi_1(X)$. | |
| Feb 25, 2016 at 17:50 | review | First posts | |||
| Feb 25, 2016 at 18:16 | |||||
| Feb 25, 2016 at 17:49 | history | asked | awivil | CC BY-SA 3.0 |