Timeline for Questions about local triviality
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 12, 2012 at 13:14 | comment | added | Fabio | I added question 3 later. | |
| Nov 12, 2012 at 13:13 | history | edited | Fabio | CC BY-SA 3.0 |
added 290 characters in body
|
| Nov 12, 2012 at 12:15 | vote | accept | Fabio | ||
| Nov 12, 2012 at 9:53 | answer | added | Sergei Ivanov | timeline score: 6 | |
| Nov 12, 2012 at 6:27 | comment | added | Kevin Ventullo | Small remark: if you wanted the fiber bundle to be smooth in question 1, then this fails even for $n=0$. | |
| Nov 11, 2012 at 20:28 | comment | added | Fabio | For question 1, I would like to see an example of a smooth function between smooth manifolds $\pi: E \rightarrow B$ which is surjective and such that $\pi^{-1}(x)$ is diffeomorphic to $\mathbb{R}^{n}$ for every $x \in B$, but which is not a fiber bundle. Anyway, even a continuous example is ok. For 2, I mean a map $\pi: E \rightarrow B$ as in question 1, with a vector space structure on each fiber such that the sum is smooth as a function $E \times_{B} E \rightarrow E$ and the exterior product is smooth as a function $\mathbb{R} \times E \rightarrow E$, but which is not locally trivial. | |
| Nov 11, 2012 at 19:14 | comment | added | Tom Goodwillie | I don't understand question 2. What does "smooth family of vector spaces" mean? I sort of understand question 1, but let me ask: You want an example a map $E\to B$ that is: smooth? continuous with $E$ and $B$ topological manifolds, and such that near each point in $E$ it looks like projection of a product of disks on one factor? continuous with $E$ and $B$ topological manifolds? continuous? and with each fiber homeomorphic to $\mathbb R^n$ but not locally trivial. | |
| Nov 11, 2012 at 18:12 | history | asked | Fabio | CC BY-SA 3.0 |