Timeline for Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 11, 2015 at 11:28 | comment | added | Trick1234 | Yes, the gluing map is isotopic to Dehn twist. Thank you very much, again! | |
| Aug 11, 2015 at 11:26 | vote | accept | Trick1234 | ||
| Aug 9, 2015 at 9:54 | history | edited | Futurologist | CC BY-SA 3.0 |
Correcting typos
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| Aug 7, 2015 at 13:29 | comment | added | Futurologist | I forgot to say that instead adding $-dH \wedge d\theta$ to your form, you can take any closed one-form $\omega$ on $E$ and take $-\omega \wedge d\theta$. Then again, the new form is closed and you have a symplectically perturbed map. | |
| Aug 7, 2015 at 13:26 | comment | added | Futurologist | I see, so if the fibers are compact, then everything is ok. This means, as I expected, that the fibration is locally smoothly trivializible, by Ehresmann's theorem. By the way, you can obtain the fibration as a mapping torus of a more general smooth map, not necessarily a symplectomorphism. It requires only constructions from differential topology, without reference to symplectic forms. Of course, if you need your isotopy class to be symplectic, then you use the form. But in general, if I am not wrong, for a generic singularity, I think the gluing map is isotopic to a Dehn twist. | |
| Aug 6, 2015 at 13:24 | comment | added | Trick1234 | In fact, I am reading Michael Usher's paper arXiv:math/0603128. Suppose $F:E_d(\pi) \to D$ is relative Hilbert scheme of $E$, which carries some symplectic form $\tilde{\Omega}$, $Y_d(\pi)=F^{-1}(\partial D)$. I am just curious that can we establish a correspondence between non-degenerate periodic orbit of $Y_{\phi_{\Omega}}$ with degree $d$ and the constant section of $Y_d(\pi)$. Since $(Y_d(\pi),\tilde{\Omega})$ is isomorphic to a mapping torus $Y_{\Phi}$, where $\Phi$ is a $C^0$ perturbation of $Sym^d(\phi_{\Omega}):Sym^d(\Sigma) \to Sym^d(\Sigma)$, so intuitively it seems true. | |
| Aug 6, 2015 at 12:29 | comment | added | Trick1234 | Just as I described above, a Lefschetz fibration $E$ over a disk with one singularity point and fiber $\Sigma$, where $\Sigma$ is a closed surface. The boundary $Y=\partial E$ is a fibration over $S^1$. Given $d>0$, I want to know that whether all the periodic orbit are non-degenerate, your answer basically solved my problem. Big thanks! | |
| Aug 5, 2015 at 10:23 | comment | added | Futurologist | It looks like it. You can also have a one-parameter family, by taking $\varepsilon H$... By the way, since I do not know what kind of fibration you have, what its topology is and whether it is locally trivializable (something like Ehresmann's theorem), you have to make sure that the trajectories of the lifted vector fields make one full circle to return to the fiber $E_1$. If the fibers are compact, you have no troubles. Would you like to share some details about the fibration and what you need it for? There might be alternative ways for working with it... | |
| Aug 4, 2015 at 13:42 | comment | added | Trick1234 | Thank you very much for your answer. Please allow me to ask one more question. Suppose that $R_H$ is the horizontal lift of $\partial \theta$ over $M_1$ in your construction, $\phi_H$ is the flow generated by $R_H$. By directly computation, I show that $R_H$ $C^k$ converges to the original one if $H$ $C^{k+1}$ converges to 0. Does this implies that flow $\phi_H$ $C^k$ converges to the original one, and the periodic points of time-1 map will be close to the original periodic points? | |
| Aug 3, 2015 at 0:05 | history | answered | Futurologist | CC BY-SA 3.0 |