Timeline for Cross sections in bundles and principal G-bundles
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 27, 2012 at 20:44 | vote | accept | Shlomi A | ||
| Jun 26, 2012 at 17:56 | answer | added | Oldřich Spáčil | timeline score: 7 | |
| Jun 26, 2012 at 8:47 | comment | added | Shlomi A | The word "principal" in the question is intentional. Another way of putting it is whether applying the forgetful functor (from principal $G$-bundles over $B$ to fiber-bundles over $B$) to a non-trivial principal bundle might yield a fiber bundle with a global cross section. I tend to agree with Claudio's answer. Are sections in both categories indeed the same? | |
| Jun 25, 2012 at 18:33 | comment | added | Dan Ramras | Maybe the question should have been phrased without the word "principal"? In other words, one could ask "Are there non-trivial fiber bundles whose fibers are homeomorphic to a fixed topological group $G$, but which do admit sections?" The answer is certainly "yes". For instance, every (real, say) vector bundle has a zero section, and the fibers are topological groups. Of course vector bundles are not principal bundles, since the transition functions are linear maps, whereas a principal bundle for the additive group $\mathbb{R}^n$ would have translations as its transition maps. | |
| Jun 25, 2012 at 13:11 | comment | added | Andreas Blass | I agree with Claudio. Specifically, I see no difference between the notions of a section of a principal bundle and a section of the same bundle considered as a mere fiber bundle. In either case, a section is just a map from the base to the total space which, followed by the projection, yields the identity map of the base. | |
| Jun 25, 2012 at 11:46 | comment | added | Shlomi A | Yes, locally trivial with fiber G. | |
| Jun 25, 2012 at 11:36 | comment | added | Claudio Gorodski | By "fiber bundle", you mean locally trivial bundle with typical fiber $G$? I think you do not get more cross-sections just by ignoring the free right action of $G$ on the fiber bundle. | |
| Jun 25, 2012 at 11:28 | history | asked | Shlomi A | CC BY-SA 3.0 |