Timeline for Milnor's universal bundle as a colimit?
Current License: CC BY-SA 4.0
13 events
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| Jan 8, 2021 at 18:28 | comment | added | Nikhil Sahoo | @Tyrone Thank you very much for your explanation! If you make it into an answer, I'll be happy to accept it. | |
| Jan 8, 2021 at 10:59 | comment | added | Tyrone | In the good news, the reason for introducing the coarse join is due to issues with the continuity of the $G$-action. If $G$ is locally compact, then $G$ acts freely and continuously on $colim\,E_n$ (because $(colim\,E_n)\times G\cong colim\,(E_n\times G)$), so you can construct the universal bundle directly using the colimit. | |
| Jan 8, 2021 at 10:54 | comment | added | Tyrone | Nikhil, fix a group $G$ and let $E_\infty^c$ be the coarse join of countably many copies $G$. The coordinate functions induce an embedding $E^c_\infty\subseteq \prod_\omega CG$ ($CG$ being the coarse cone). Suppose $G$ is a non-discrete compact Lie group. Then $CG$ and hence $\prod_\omega CG$ are metrisable, so therefore so is $E_\infty^c$. On the other hand $colim\,E_n$ will be an infinite-dimensional CW complex and will not be metrisable. (If $G$ is a sphere, then $\prod_\omega CG$ is the Hilbert cube.) | |
| Jan 7, 2021 at 22:37 | history | edited | Nikhil Sahoo | CC BY-SA 4.0 |
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| Jan 7, 2021 at 21:05 | comment | added | Nikhil Sahoo | @Tyrone Thank you for your explanation. I see how compactness stops helping once we are taking an infinite join. But what about for familiar examples, like $S^1$ and $S^3$? Is the coarse topology on $S^\infty$ distinct from the usual colimit topology?! (Feel free to ignore this question if you feel like it's something I should work out myself :P) | |
| Jan 7, 2021 at 21:02 | comment | added | Nikhil Sahoo | @TomGoodwillie I have tried to clarify the definitions of the spaces involved. | |
| Jan 7, 2021 at 21:02 | history | edited | Nikhil Sahoo | CC BY-SA 4.0 |
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| Jan 7, 2021 at 20:56 | history | edited | Nikhil Sahoo | CC BY-SA 4.0 |
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| Jan 7, 2021 at 20:24 | history | edited | Nikhil Sahoo | CC BY-SA 4.0 |
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| Jan 7, 2021 at 16:29 | comment | added | Tyrone | To clear up confusion, $X\ast Y$ with the 'initial topology' is also called the coarse join. It it the weakest topology making the three obvious functions $t:X\ast Y\rightarrow I$, $x:t^{-1}[0,1)\rightarrow X$ and $y:t^{-1}(0,1]\rightarrow Y$ continuous. $1)$ In the coarse topology there is a homoeomorphism $X\ast Y\cong (CX\times Y)\cup (X\times CY)$, so its easy to see that $X\hookrightarrow X\ast Y\hookleftarrow Y$ are closed cofibrations. $2)$ the colimit topology on $colim\, E_n$ is strictly finer than the infinite coarse join topology $E_\infty$. Even for compact Hausdorff groups. | |
| Jan 6, 2021 at 23:44 | comment | added | Tom Goodwillie | What do you mean by the "initial topology" on, say, $G\ast G$? | |
| Jan 6, 2021 at 22:26 | history | edited | Nikhil Sahoo | CC BY-SA 4.0 |
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| Jan 6, 2021 at 22:10 | history | asked | Nikhil Sahoo | CC BY-SA 4.0 |