Timeline for Two embedded symplectic spheres with zero square in a symplectic $4$-manifold
Current License: CC BY-SA 3.0
16 events
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| S Nov 2, 2017 at 21:00 | history | bounty ended | CommunityBot | ||
| S Nov 2, 2017 at 21:00 | history | notice removed | CommunityBot | ||
| Oct 31, 2017 at 9:25 | answer | added | aglearner | timeline score: 2 | |
| S Oct 25, 2017 at 19:15 | history | bounty started | aglearner | ||
| S Oct 25, 2017 at 19:15 | history | notice added | aglearner | Authoritative reference needed | |
| Oct 22, 2017 at 12:05 | answer | added | Tsemo Aristide | timeline score: 0 | |
| Oct 22, 2017 at 0:10 | history | edited | aglearner | CC BY-SA 3.0 |
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| Oct 21, 2017 at 22:44 | comment | added | Marco Golla | No doubt about it! I guess my point was that we weren't aware (back when Paolo and I wrote the paper) of any reference, and we proved it by hand. I'm sure people knew the statement in fibre bundles before McDuff, which probably makes it hard to find it explicitly written. Have you tried Chris Wendl's notes on McDuff's theorem (and surroundings)? | |
| Oct 21, 2017 at 22:38 | comment | added | aglearner | Thanks Marco, I would guess, the result I am looking for was proven before 2000. | |
| Oct 21, 2017 at 22:29 | comment | added | Marco Golla | Paolo Lisca and I proved something very similar on Page 29 of On Stein fillings of contact torus bundles (Bull. LMS 48, 2016), within the proof of Theorem 3.5. I think that there is an argument using adjunction alone as well. (Both arguments use McDuff's theorem, as mentioned by Chris above.) | |
| Oct 21, 2017 at 22:18 | comment | added | aglearner | Thanks Marco, I forgot to say that $\pi_1(M)\ne 0$, it is corrected. And this is still not in the article of McDuff. Do you think you know the reference now? | |
| Oct 21, 2017 at 22:17 | history | edited | aglearner | CC BY-SA 3.0 |
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| Oct 21, 2017 at 22:00 | comment | added | Marco Golla | The reason you can't find the statement is that it's false. $S^2 \times S^2$, with the standard symplectic structure, has two symplectic spheres of square 0 that are not (smoothly) isotopic. | |
| Oct 21, 2017 at 21:35 | comment | added | aglearner | Thanks Chris. I went through all the theorems and lemmas in this article but was not able to find the statement... I suspect that the statement can be in the book of McDuff and Salamon on J-holomorphic curves but was not able to get hold of it. Otherwise I know one place from 2010 where this fact is stated, but of course it should be something much earlier... | |
| Oct 21, 2017 at 21:15 | comment | added | Chris Gerig | By a result of McDuff (The structure of rational and ruled symplectic 4-manifolds), $M$ is a blow-up of either $\mathbb{C}P^2$ or of a ruled manifold (i.e. the total space of an $S^2$-fibration). You might find an answer to your question in her (and Lalonde's) related works, which focus on spheres and isotopies of symplectic forms on these manifolds. | |
| Oct 21, 2017 at 17:13 | history | asked | aglearner | CC BY-SA 3.0 |