Timeline for Is a $G$-cell complex always a $G$-CW complex?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 27, 2019 at 11:56 | history | edited | Denis Nardin | CC BY-SA 4.0 |
added 69 characters in body
|
| Jan 27, 2019 at 11:02 | comment | added | Denis Nardin | @jdc I added a reference for the compact Lie group case. It is a bit subtle, but it can be deduced from the equivariant version of HELP (homotopy extension and lifting property) | |
| Jan 27, 2019 at 11:01 | history | edited | Denis Nardin | CC BY-SA 4.0 |
Added words for the compact Lie group case
|
| Jan 27, 2019 at 0:34 | comment | added | jdc | Well, thanks for taking the question seriously. | |
| Jan 27, 2019 at 0:34 | vote | accept | jdc | ||
| Jan 25, 2019 at 23:11 | comment | added | Denis Nardin | @jdc I don't know. I'm not too familiar about how people construct CW structures on the nose, even non-equivariantly. | |
| Jan 25, 2019 at 23:10 | comment | added | jdc | Thanks! Is the difficulty of the second problem alleviated if one instead demands a $G$-invariant CW/cell structure? My intuition is that this would be closer to the difficulty of the non-equivariant problem (which I still think isn't solvable in general). | |
| Jan 25, 2019 at 23:07 | comment | added | Denis Nardin | @jdc I never think about compact Lie groups, but basically you need to pay attention at the connectivity of the inclusion of the skeleton of the $n$-fixed points inside the fixed points of the $n$-skeleton (this is for the proof of the lemma). I'll see if I can manage to generalize the proof tomorrow. Constructing a $G$-CW-structure on a space on the nose and not just up to homotopy is a much more difficult problem, and I'm not sure I can say anything intelligent about it. | |
| Jan 25, 2019 at 23:03 | comment | added | jdc | Not explicit was another question about how strong this result is. It seems clear from your answer that this only produces a homotopy-equivalent $G$-CW-complex, and I am expecting this can't be hoped to preserve the space up to homeomorphism. I encountered a proposition in a preprint seemingly claiming a certain space has a $G$-CW structure but only proving a $G$-cell structure, and was trying to see if the result as stated could be rescued. | |
| Jan 25, 2019 at 23:01 | comment | added | jdc | Thanks for this. I was really just hoping to a pointer to somewhere in the literature, so this is more than I needed in some sense, but it is also much easier than I had imagined. I did want (I thought this was clear, sorry), though, a compact Lie group; what else does one need to do in that case? | |
| Jan 25, 2019 at 22:28 | history | answered | Denis Nardin | CC BY-SA 4.0 |