Timeline for Classifying spaces of topological groups whose underlying spaces are homotopy equivalent
Current License: CC BY-SA 3.0
14 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Dec 9, 2015 at 5:45 | vote | accept | user46652 | ||
| Mar 31, 2015 at 21:29 | comment | added | David Roberts♦ | @DanRamras The conditions do imply they are good simplicial spaces. | |
| Mar 31, 2015 at 20:04 | answer | added | archipelago | timeline score: 6 | |
| Mar 30, 2015 at 23:24 | comment | added | archipelago | I somehow disagree. Life isn't always as beautiful as the homotopy type of a CW complex especially if one is not interested in algebraic topological questions for its own sake. | |
| Mar 29, 2015 at 3:31 | comment | added | Qiaochu Yuan | $BG$ is fundamentally an object which only makes sense up to weak homotopy equivalence; asking questions like this means getting bogged down in the technicalities of spaces not having the homotopy type of a CW complex and there's just no reason to torture yourself like that. | |
| Mar 29, 2015 at 1:49 | comment | added | John Klein | @user46652: you need to specify which classifying space functor $B$ you are using for the question to make any sense. | |
| Mar 29, 2015 at 1:43 | comment | added | Denis Nardin | @user46652 If I understand what you're asking, yes if $G$ has the homotopy type of a CW complex so does $BG$ (because after all it can be constructed using the bar construction). | |
| Mar 29, 2015 at 0:48 | comment | added | Dan Ramras | (cont'd) I think your conditions on $G$ and $H$ ought to imply that these nerves are good simplicial spaces. But I'm not certain if that's ensure that a level-wise homotopy equivalence induces a homotopy equivalence of realizations. Goerss and Jardine's book would be one place to look. | |
| Mar 29, 2015 at 0:46 | comment | added | Dan Ramras | You might try looking at Segal's paper Classifying Spaces and Spectral Sequences, where he explains that Milnor's infinite join construction of $BG$ can be built as the classifying space of a certain topological category $G_N$. Then induced functor $G_N \to H_N$ will induce a level-wise homotopy equivalence between the nerves of these cateogies (whose realizations give $BG$ and $BH$)... | |
| Mar 29, 2015 at 0:43 | comment | added | user46652 | Thanks for the comment! This means in order for my very last statement to work, $G$ would have to be a lot nicer, so I guess showing $BG$ has the homotopy type of a CW complex is not the right way to show that $Bf$ is a homotopy equivalence. Nonetheless, do you think the converse to your comment hold somehow? | |
| Mar 29, 2015 at 0:01 | comment | added | Denis Nardin | If $BG$ has the homotopy type of a CW complex then $G=\Omega BG$ has it too. | |
| Mar 28, 2015 at 23:32 | review | First posts | |||
| Mar 28, 2015 at 23:56 | |||||
| Mar 28, 2015 at 23:32 | history | asked | user46652 | CC BY-SA 3.0 |