Timeline for group extensions and principal bundles
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 6, 2018 at 18:41 | comment | added | Ben McKay | @hänsel: Yes: both are canonically diffeomorphic to $X$. | |
| Feb 6, 2018 at 18:14 | comment | added | hänsel | is $P_G/G=P/H$? (answering construction of map (1.2) on page 548 of paper "Atiyah-Singer III") | |
| Aug 30, 2016 at 10:58 | comment | added | Ben McKay | I didn't notice this: $H$ doesn't act trivially on $G/H$ by left translation, so $H$ doesn't act trivially on $P_{G/H}$. So $P_{G/H}$ is a bundle whose fibers are diffeomorphic to $G/H$, and has a $G$-action, but is not principal, and has no $G/H$-action. | |
| Aug 30, 2016 at 10:09 | comment | added | far | I am assuming that $0 \to H \to G \to G/H \to 0$ is a short exact sequence of groups, so that $G/H$ is a group and the maps in the sequence are group homomorphisms. By functoriality I would expect the $G/H$ bundle to be a principal bundle as well. Am I missing something? | |
| Aug 29, 2016 at 22:51 | comment | added | Qiaochu Yuan | What is a short exact sequence of bundles? Note that even when $G/H$ is a group, the action of $G$ on it is not an action by group homomorphisms, so the associated bundle with fiber $G/H$ is not a bundle of groups. | |
| Aug 29, 2016 at 21:10 | comment | added | far | I mean a short exact sequence. | |
| Aug 29, 2016 at 19:07 | history | answered | Ben McKay | CC BY-SA 3.0 |