Timeline for Simple closed curves on genus 2 surfaces
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 19, 2016 at 15:34 | vote | accept | user98640 | ||
| Sep 19, 2016 at 12:09 | answer | added | Jason Starr | timeline score: 1 | |
| Sep 19, 2016 at 10:24 | comment | added | Alex Degtyarev | @Misha: the question is quite clear and quite standard: the monodromy action on something (e.g., on the isotopy classes of curves). To OP: it seems that the whole $\pi_1$ is generated by a few explicit Picard--Lefschetz transformations. Did you try to compute their images in the mapping class group? I would not be surprised if this were an epimorphism. | |
| Sep 19, 2016 at 1:49 | comment | added | user98640 | Any loop based at $x$ gives a homeomorphism from $C$ to $C$. The isotopy class of this homeomorphism depends only on the class of the loop in $\pi_1(U,x)$. Therefore, there's a map from $\pi_1(U,x)$ to the mapping class group of $C$. The mapping class group of $C$ acts on the set of isotopy classes of simple closed curves of C. Therefore the fundamental group also acts on the set of isotopy classes of simple closed curves of C. | |
| Sep 19, 2016 at 1:45 | comment | added | Misha | @user98640: OK, this is better but has nothing to do with isotopy classes of simple closed curves. Please, revise your question to make things clear. | |
| Sep 19, 2016 at 1:42 | comment | added | user98640 | If $X \rightarrow B$ is a family of smooth projective varieties, the Gauss-Manin connection on the relative de Rham cohomology gives a notion of parallel transport. | |
| Sep 19, 2016 at 1:39 | comment | added | Misha | @user98640: What parallel transport do you mean? In order to make this work you would need a flat bundle over this surface whose structure group is the mapping class group. There is no natural way to make this work. Please, revise your question to make the construction which you have in mind clear. | |
| Sep 19, 2016 at 1:13 | comment | added | user98640 | Parallel transport gives a map from $\pi_1(U,x)$ to the mapping class group. I think this is an inclusion, and gives a subgroup of index either 6 or 12, and if the index is 6, I think the quotient is $S_6$ (something similar if the index is 12). | |
| Sep 19, 2016 at 1:06 | comment | added | Andy Putman | Can you spell put the action you are asking about? | |
| Sep 19, 2016 at 0:03 | review | First posts | |||
| Sep 19, 2016 at 0:08 | |||||
| Sep 19, 2016 at 0:00 | history | asked | user98640 | CC BY-SA 3.0 |