Timeline for When can we extend a diffeomorphism from a surface to its neighborhood as identity?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Aug 28 at 12:53 | comment | added | Anubhav Mukherjee | @QiuyuRen that's right. | |
| Aug 28 at 10:07 | comment | added | Qiuyu Ren | Effectively, you are asking whether an exotic diffeomorphism of a closed simply-connected 4-manifold that fixes a surface can be assumed to be supported away from the surface after an isotopy. Is that right? | |
| Jul 10 at 12:24 | comment | added | Anubhav Mukherjee | @SamNead The resultant 4-manifold will not be simply-connected. This is important for my situation. | |
| Jul 10 at 5:18 | comment | added | Sam Nead | What about using a Dehn twist on $S^2 \times S^1$ (the double of your solid torus example) and then crossing with the identity on $S^1$? | |
| Jul 9 at 14:51 | history | edited | Anubhav Mukherjee | CC BY-SA 4.0 |
added 681 characters in body
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| Jul 9 at 0:50 | comment | added | Anubhav Mukherjee | @MoisheKohan yes, this is a very subtle point that I have realized and that is what confuses me a lot. This is the reason I added $f^*=Id$ condition. | |
| Jul 9 at 0:40 | comment | added | Moishe Kohan | I see.......... I think isotopy to the linearization works in this case but it does not help with your question. | |
| Jul 9 at 0:36 | comment | added | Anubhav Mukherjee | @MoisheKohan take a diffeomorphism from solid torus ($S^1\times D^2$) to itself which is obtained by a single Dehn twist. Then the central curve is fixed, but you cannot fix a nbd of this curve other wise we will have an isotopy from id to Dehn twist on a 2 torus. This is where the above mentioned obstruction class is not null homotopic. | |
| Jul 9 at 0:30 | comment | added | Moishe Kohan | What would be a counter-example? | |
| Jul 9 at 0:07 | comment | added | Anubhav Mukherjee | @MoisheKohan sorry, I didn't understand your comment. Could you please elaborate? If I understand it correctly, your argument will work for any diffeomorphism on any n-manifold with some fixed submanifold. But I certainly can construct examples where it is not true | |
| Jul 9 at 0:04 | comment | added | Moishe Kohan | Of course, you would have to slow down the homotopy as you move away from $\Sigma$, so that you get the identity map on a suitable circle bundle inside $\nu$. | |
| Jul 9 at 0:01 | comment | added | Anubhav Mukherjee | @MoisheKohan I don't think it is possible. In a local neighborhood, such homotopy is possible but not globally. | |
| Jul 8 at 23:55 | comment | added | Moishe Kohan | What would be a problem if you try to isotope $f$ to the linearization of $f$ on the total space of the normal bundle $\nu$ to $\Sigma$? I think, the straight-line homotopy (for a suitable Riemannian metric) will be an isotopy on a sufficiently small disk subbundle in $\nu$. | |
| Jul 8 at 23:26 | history | edited | LSpice | CC BY-SA 4.0 |
Typo
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| Jul 8 at 23:10 | comment | added | Chris Gerig | The following (specifically Ruberman's comment) might help but I'm just speculating: mathoverflow.net/questions/270197/… | |
| Jul 8 at 22:46 | history | asked | Anubhav Mukherjee | CC BY-SA 4.0 |