Timeline for Bott & Tu differential forms Example 10.1
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 24 at 12:30 | vote | accept | Jaehwan Kim | ||
| May 23 at 5:11 | comment | added | Jaehwan Kim | @ChrisGerig I think my version is outdated | |
| May 23 at 5:10 | comment | added | Jaehwan Kim | @ChrisSchommer-Pries The sheaf associated with the given presheaf is locally constant, but I do not understand the significance of that sheaf. | |
| May 22 at 5:59 | comment | added | Chris Gerig | As GregFriedman highlights, different editions of the book may be more/less accurate -- I hope your version is older? (My version is "revised third printing" which agrees with Friedman's post.) | |
| May 22 at 5:29 | answer | added | Greg Friedman | timeline score: 9 | |
| May 22 at 5:22 | history | edited | Greg Friedman | CC BY-SA 4.0 |
added 4 characters in body
|
| May 21 at 14:40 | comment | added | Chris Schommer-Pries | This is a good question. As your example shows, the map $\rho_V^U$ need not be an isomorphisms. That seems to be a mistake. I think it is still true that $\mathcal{F}$ is a locally constant presheaf, but this might depend on the definition that Bott and Tu are using. Is it enough for the sheafification to be locally constant? | |
| May 21 at 10:29 | comment | added | Francesco Polizzi | Actually, using the notation in the book for the trivial line bundle on $\mathbb{R}^2$, if $U=D^2$ and $V=D^2-\{(0, \, 0)\}$, I do not see how the pullback ("restriction") map $$H^1( U \times \mathbb{R}) \to H^1(V \times \mathbb{R})$$ can be the identity, since $$H^1(U \times \mathbb{R})= 0, \; H^1(V \times \mathbb{R})= \mathbb{R}$$ | |
| May 21 at 9:01 | comment | added | Francesco Polizzi | If I understand the question correctly, it seems that the OP's doubt is that a contractible open set $U$ can contain a non-contractible open set $V$. For instance, a disk in $\mathbb{R}^2$ contains an open annulus, whose cohomology is non-trivial. | |
| May 21 at 7:48 | review | Close votes | |||
| May 27 at 3:03 | |||||
| May 21 at 7:40 | history | edited | gmvh |
Added top-level tags
|
|
| S May 21 at 7:25 | review | First questions | |||
| May 21 at 7:40 | |||||
| S May 21 at 7:25 | history | asked | Jaehwan Kim | CC BY-SA 4.0 |