Timeline for Classifying spaces, Brown representability, and homotopy equivalences
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Sep 17, 2011 at 0:26 | comment | added | Andy Putman | @Fernando Muro : I think the OP is correct that properness is needed. Otherwise, you would get pathological behavior. For instance, $\mathbb{Z}^2$ acts freely on $\mathbb{R}$ with $(1,0)$ acting as translation by $1$ and $(0,1)$ acting as translation by $\pi$. Of course, if $G$ is a compact group then properness is automatic. | |
| Sep 16, 2011 at 22:16 | comment | added | Fernando Muro | Not really, the fact that it is a principal bundle is superfluous. It's enough that EG be contractible and the action of G be free. This is what I usually take as the definition of classifying space. Now I see that you take a different one. | |
| Sep 16, 2011 at 20:47 | comment | added | Will B | @Fernando : Two points. First, I think that the wikipedia article is flawed in that you also have to assume that the action by $G$ is proper (otherwise, the map $EG \rightarrow BG$ might not be a principal $G$-bundle). Second, if you define it like in the wikipedia article, then it is a theorem that if it exists, then it classifies the functor "principal $G$-bundles", and thus it is not completely obvious that the object that the Brown representability theorem gives you actually satisfies the conditions of the theorem. You need some argument like that given by Oscar or Johannes above. | |
| Sep 16, 2011 at 18:07 | comment | added | Fernando Muro | It is, indeed en.wikipedia.org/wiki/Classifying_space | |
| Sep 16, 2011 at 15:52 | comment | added | Will B | I don't think it is part of the definition of classifying spaces that $EG$ is contractible. All we know is that $[X,BG]$ is naturally in bijection with the set of principal $G$ bundles on $X$ via the map that takes $\phi : X \rightarrow BG$ to $\phi^{\ast}(EG)$. Of course, it is true that $EG$ is contractible, but I don't know how to see this except via one of the explicit constructions of $BG$. Do you know how to do this? | |
| Sep 16, 2011 at 15:45 | history | answered | Fernando Muro | CC BY-SA 3.0 |