Timeline for The classifying space of any topological group is paracompact and locally contractible
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Dec 14, 2023 at 18:25 | history | edited | Martin Sleziak |
edited tags
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| Dec 14, 2023 at 17:50 | answer | added | Tyrone | timeline score: 4 | |
| Dec 14, 2023 at 17:05 | comment | added | Mehmet Onat | @Tyrone If $G$ is a compact group, then why $B_G$ is paracompact. you would present a proof | |
| Dec 14, 2023 at 15:11 | history | edited | David White | CC BY-SA 4.0 |
Fixed typos since it was on the front page anyway
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| Dec 14, 2023 at 14:04 | answer | added | David White | timeline score: 6 | |
| Dec 13, 2023 at 14:13 | comment | added | Mehmet Onat | @DavidWhite Actually, I am interested in pathological situations. I study with non-Lie group actions. | |
| Dec 13, 2023 at 14:09 | comment | added | David White | @MehmetOnat Are you willing to restrict to a compact Lie group $G$? Because in that case everything is much simpler. | |
| Dec 11, 2023 at 12:35 | comment | added | Tyrone | I can supply a proof of paracompactness, but I won't be able to comment on any of your other questions. | |
| Dec 11, 2023 at 6:24 | comment | added | Mehmet Onat | @Tyrone How did you mention the fact that $B_G$ is paracompact if $G$ is compact Hausdorff? | |
| Dec 10, 2023 at 15:19 | comment | added | Thorgott | The nlab is not the right place to go for technical details in topology whatsoever. I don't have a counter-example, but it would be very surprising if their claim was true in this generality. The claim certainly isn't in any of the listed references. A true statement in this direction (which is from Milnor's original paper) is that $EG$ is a $G$-CW-complex if $G$ is a countable CW-group, but that's a lot more restrictive. | |
| Dec 10, 2023 at 8:00 | comment | added | Mehmet Onat | @Thorgott On this web page, ncatlab.org/nlab/show/Milnor+construction, it is said that $E_G$ is a $G$-CW complex. I want to make sure of this | |
| Dec 9, 2023 at 18:42 | comment | added | Tyrone | If $G$ is compact Hausdorff, then Milnor's $BG$ is paracompact and Hausdorff. It seems difficult to get any statement about the local contractibility of $BG$, in general. At least what's true, is that if $G$ is a compact Lie group, then Milnor's $BG$ is locally contractible. | |
| Dec 8, 2023 at 17:54 | comment | added | Mehmet Onat | @Tyrone I need for compact topological groups | |
| Dec 8, 2023 at 14:41 | comment | added | Thorgott | You can write down a closed embedding $G\rightarrow BG$ (like $g\mapsto\left[\frac{1}{2}e+\frac{1}{2}g\right]$, up to explaining notation), so paracompactness of the latter necessitates paracompactness of the former. If $EG$ had a $G$-CW-complex structure, then $BG=EG/G$ would have a CW-complex structure and so be paracompact. A non-paracompact $G$ becomes a counter-example to either. | |
| Dec 8, 2023 at 13:31 | comment | added | Tyrone | If you are asking specifically about the Milnor construction, then this is false. How general do you need your topological groups to be? | |
| Dec 8, 2023 at 12:45 | comment | added | R. van Dobben de Bruyn | Ah ok, that is a little more explicit than what I had in mind, so I understand what the question is. | |
| Dec 8, 2023 at 12:11 | comment | added | Mehmet Onat | @R.vanDobbendeBruyn mathoverflow.net/questions/150786/…. I know the construction here. Unfortunately, I'm not an expert on the subject. | |
| Dec 8, 2023 at 12:04 | comment | added | R. van Dobben de Bruyn | Could you specify what definition of $BG$ you have in mind? To me, it is only well defined up to (weak?) homotopy equivalence, and every weak homotopy class of spaces has a CW representative (which is therefore paracompact and locally contractible). But it's possible that you mean something more specific. | |
| Dec 8, 2023 at 7:45 | history | asked | Mehmet Onat | CC BY-SA 4.0 |