Timeline for Extending diffeomorphisms
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Jan 31, 2023 at 17:14 | history | edited | Piotr Hajlasz | CC BY-SA 4.0 |
added 102 characters in body
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| Jan 30, 2023 at 2:36 | history | became hot network question | |||
| Jan 30, 2023 at 1:19 | vote | accept | Piotr Hajlasz | ||
| Jan 29, 2023 at 23:12 | answer | added | Moishe Kohan | timeline score: 12 | |
| Jan 29, 2023 at 23:06 | comment | added | Moishe Kohan | @PiotrHajlasz: Soon. | |
| Jan 29, 2023 at 21:28 | comment | added | Igor Belegradek | @PiotrHajlasz: yes, you are right. | |
| Jan 29, 2023 at 21:22 | comment | added | Piotr Hajlasz | @IgorBelegradek I still need to complete reading of Palais' paper, but I don't think the boundary is an issue since Palais' result deals with diffeomorphism up to the bounday so we can assume that one of them is the identity on the closed hemisphere. | |
| Jan 29, 2023 at 21:06 | comment | added | Moishe Kohan | Yes. You are right, I was overthinking it. | |
| Jan 29, 2023 at 20:45 | comment | added | Igor Belegradek | @PiotrHajlasz: I think just like you explain everything follows from Palais. The only technical issue is regularity at the equator. If a diffeomorphism of the upper hemisphere is identity on the equator, then its extension by the identity on the lower hemisphere need not be smooth on the ambient sphere. What one has to do is to first deform to a diffeomorphism of the upper hemisphere that is identity near the boundary. | |
| Jan 29, 2023 at 20:23 | comment | added | Igor Belegradek | Oh, I see: the sequence extends to the right and the next term is $\pi_0(BDiff(D^n, rel\partial))$ which is always trivial (contactibility of $EG$ implies path-connectedness of $BG$). Thus it looks like $r_*$ is onto on $\pi_0$, and hence by the homotopy lifting property one can always extend. | |
| Jan 29, 2023 at 20:08 | comment | added | Piotr Hajlasz | @MoisheKohan I think Palais' theorem says exactly that you can always extend, because (if I correctly understand his paper) you can compose $f$ with a diffeomorphism of $\mathbb{S}^n$ so that $f$ becomes identity. I am confused. | |
| Jan 29, 2023 at 20:04 | comment | added | Igor Belegradek | @MoisheKohan: I think your comment refers to the restriction map $r: Diff(D^n)\to Diff(S^{n-1})$ which is a fibration whose homotopy fiber is $Diff(D^n, rel \partial)$. Look at the $\pi_0$ portion of the homotopy exact sequence of the fibration. We are interested in whether the rightmost arrow $r_*$ is onto. If $n>4$, the kernel of $r$ is the group of homotopy $(n+1)$-spheres. What do homotopy spheres have to do with surjectivity of $r_*$? Could you elaborate? | |
| Jan 29, 2023 at 19:31 | comment | added | Moishe Kohan | By Palais, you can assume that $F$ maps hemisphere to itself. Then in dimension $n<7$ an extension always exists. In higher dimensions, it depends on the number of exotic spheres. | |
| Jan 29, 2023 at 19:19 | comment | added | Igor Belegradek | I think the issue in Schoenflies is whether a smoothly embedded codimension one sphere bounds a ball. Your setting is easier because you have an embedded disk. I think any two smoothly embedded codimension zero disks are equivalent up to ambient diffeomorphism by a result of Palais in "Extending diffeomorphisms", ams.org/journals/proc/1960-011-02/S0002-9939-1960-0117741-0/…. | |
| Jan 29, 2023 at 18:33 | history | asked | Piotr Hajlasz | CC BY-SA 4.0 |