Timeline for Is it true that every vector bundle over a non compact smooth manifold is trivial at infinity?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 13, 2022 at 12:09 | answer | added | Jason Starr | timeline score: 5 | |
| Dec 10, 2022 at 16:44 | comment | added | Anton Petrunin | @JasonStarr Could you make an answer from you comment? (to move it from unanswered questions). | |
| May 11, 2018 at 23:39 | comment | added | user95283 | That's right, Jason Starr. | |
| May 11, 2018 at 23:13 | comment | added | Jason Starr | That fails already for rank $1$ vector bundles over $M = \mathbb{R}\times \mathbb{S}^1$. For the tautological rank $1$ vector bundle $F$ on $\mathbb{S}^1=\mathbb{RP}^1$, the pullback $E=\text{pr}_2^*F$ gives a counterexample. For every compact subset $K$ of $\mathbb{R}\times \mathbb{S}^1$, the image $\text{pr}_1(K)$ is a compact subset of $\mathbb{R}$, hence bounded. Thus, there exists $t\in \mathbb{R}\setminus \text{pr}_1(K)$. For the section $\sigma_t:\mathbb{S}^1\to M$ by $\sigma_t(u)=(t,u)$, the pullback $\sigma_t^*E$ equals $F$. | |
| May 11, 2018 at 22:58 | history | asked | user95283 | CC BY-SA 4.0 |