Timeline for Minimal symmetry of a fibre bundle
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 14, 2021 at 13:26 | comment | added | Neil Strickland | If you try to do this more homotopically, with fibrations rather than fibre bundles, then it will look roughly like this: a symmetry group is an $A_\infty$ group $G$ equipped with maps $\Omega B\to G\to\text{hAut(F)}$ whose composite is the given map $\Omega B\to\text{hAut}(F)$, and the inverse limit of those is $\Omega B$. It's not so clear what modifications are needed for fibre bundles. | |
| Apr 14, 2021 at 13:22 | comment | added | Neil Strickland | That is not a remotely sufficient definition to support the rest of your question. For example, you want to take the limit of all symmetry groups, so you need to give a definition which gives rise to well-defined transition maps between different symmetry groups so that you can form a diagram and take the limit. And you should note that a diagram in the homotopy category is insufficient, you need a strictly commuting diagram of topological groups, or corresponding $\infty$-categorical data. | |
| Apr 14, 2021 at 13:05 | comment | added | Student | Thanks for your comment! I meant that $G$ is a structure group of the bundle. Edited. | |
| Apr 14, 2021 at 13:05 | history | edited | Student | CC BY-SA 4.0 |
clarify "can be characterized as a map"
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| Apr 14, 2021 at 12:55 | comment | added | Neil Strickland | If you don't think that's right, you should spell out in more detail what you mean by "can be characterized as a map $B\to BG$". | |
| Apr 14, 2021 at 12:32 | comment | added | Student | @NeilStrickland but the minimal symmetry group depends not only on the base $B$. $\Omega B$ can't be the answer. | |
| Apr 14, 2021 at 9:41 | comment | added | Neil Strickland | If $B$ is connected then simplicial methods give a topological group $G$ of homotopy type $\Omega B$, and $B\simeq BG$. If you make your question precise in a homotopically meaningful way then you will surely end up with $\Omega B$ as the minimal symmetry group. | |
| Apr 14, 2021 at 2:57 | history | asked | Student | CC BY-SA 4.0 |