Timeline for How does hyperelliptic involution act on the standard generators of the fundamental group of surfaces of genus g with n punctures?
Current License: CC BY-SA 4.0
25 events
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| Aug 31, 2023 at 7:58 | comment | added | HJRW | @MikhailKatz: Sorry for the very slow reply. I don’t have such a formula, but producing one is, as Sam Nead indicates, a well-posed exercise. (Contrary to your comments, which suggested that it isn’t well-posed.) | |
| Aug 14, 2023 at 11:38 | comment | added | Sam Nead | @MikhailKatz - I've already shown how to do this, for $S_{0, 4}$, in my answer below. I can draw the picture I think you want for $S_{1, 4}$ (and for $S_{g, 2g + 2}$ generally)... but not in the comments thread! I suggest you ask your question as a separate post on MO (giving a link back to this post). | |
| Aug 14, 2023 at 11:15 | comment | added | Mikhail Katz | @SamNead: thanks for your comments. I have a similar question as above. | |
| Aug 14, 2023 at 11:14 | comment | added | Mikhail Katz | @HJRW: thanks for the clarification. You refer to "the resulting formula for the action of the hyperelliptic involution", apparently relative the the presentation mentioned in the question. Are you implying that you have one? | |
| Aug 13, 2023 at 19:54 | comment | added | Sam Nead | I would be happy to discuss this further, but the comments thread is not my preferred place for that. If you like, we can open a chat room? | |
| Aug 13, 2023 at 19:48 | comment | added | Sam Nead | @MikhailKatz - I should have reiterated that the question is only interesting because all of the fixed-points of $J$ have been punctured away. Your concern about the presentation is, I think, misplaced. Choose any base point and ask yourself how $J$ “acts” on it… | |
| Aug 13, 2023 at 19:46 | comment | added | Sam Nead | @HJRW - thank you for explaining! | |
| Aug 13, 2023 at 17:22 | comment | added | HJRW | (cont'd) ... The DNB theorem (specificlly the version for surfaces with punctures) implies that any two ways of presenting the fundamental group like that differ taken by a mapping class. Therefore, the resulting formula for the action of the hyperelliptic involution is well defined up to conjugation. | |
| Aug 13, 2023 at 17:21 | comment | added | HJRW | @MikhailKatz: There seem to be several points of confusion here. Most of all, it's a real shame that you have chosen to hassle a new poster who asked a perfectly well-posed question. Regarding the maths: First, the crosses in the question indicate punctures rather than marked points (note that the given presenttion presents a free group), so there are no fixed points. Second, the Dehn--Nielsen--Baer theorem does resolve the ambiguity you now you say you worry about -- which seems to be different from your initial criticism, because it does not just involve base points... | |
| Aug 13, 2023 at 12:57 | comment | added | Mikhail Katz | @HJRW, one doesn't need the Dehn-Nielsen-Baer theorem since one can just choose a fixed point of $J$ to be the basepoint, but the OP didn't even say that. The main problem with the question is that the one-relator presentation is merely an abstract presentation of the fundamental group, and is not directly related to the figure he drew. Since the generating loops are not specified, the question is not really well-posed. | |
| Aug 13, 2023 at 12:55 | comment | added | Mikhail Katz | @Sam, I am not sure what you mean when you say that the hyperelliptic involution does not fix any points. The number of fixed points of the hyperelliptic involution is easily determined by the Riemann-Roch theorem. For example, for the torus one gets 4, and for the genus-2 surface one gets 6. Besides, the fixed points are clearly indicated by x's in the OP's picture (some of them are deleted as punctures). | |
| Aug 13, 2023 at 9:59 | comment | added | Sam Nead | @MikhailKatz - you are correct that the hyperelliptic involution $J$ does not act on the fundamental group - this is because it does not fix any points, so cannot fix a base-point. However, the mapping class of $J$ gives an outer automorphism of the fundamental group (say with basepoint in front of the dotted line, halfway in-between the first two "x"s). As HJRW points out, this is the content of the (easy direction) of the Dehn-Nielsen-Baer theorem: en.wikipedia.org/wiki/… | |
| Aug 13, 2023 at 2:09 | comment | added | HJRW | @MikhailKatz: I suggest you look at Farb and Margalit’s Primer, especially the section on the Dehn—Nielsen—Baer theorem. This is all quite standard. | |
| Aug 12, 2023 at 19:45 | comment | added | Mikhail Katz | @HJRW, what you say is correct once the action is defined. Before making sure the action is well-defined, it is not clear to me that the question is meaningful at all. Believe me, I have several publications on hyperelliptic surfaces. | |
| Aug 10, 2023 at 20:16 | answer | added | Sam Nead | timeline score: 3 | |
| Aug 10, 2023 at 15:35 | comment | added | HJRW | @MikhailKatz: different choices of base point lead to automorphisms that differ by an inner automorphism. This is proved in the first few lectures of any undergraduate course in algebraic topology. | |
| Aug 10, 2023 at 12:33 | review | Close votes | |||
| Aug 14, 2023 at 11:15 | |||||
| Aug 10, 2023 at 11:45 | comment | added | Mikhail Katz | It certainly does matter, since the action of the involution on the elements of the fundamental group cannot even be defined if you don't deal with the issue of the fixed point. | |
| Aug 10, 2023 at 11:42 | comment | added | HJRW | @MikhailKatz: since the question says “unique up to [inner automorphisms]”, the choice of base point doesn’t matter. | |
| Aug 10, 2023 at 11:37 | history | edited | Rajesh Dey | CC BY-SA 4.0 |
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| Aug 10, 2023 at 10:20 | comment | added | Mikhail Katz | A fundamental group is defined relative to a basepoint. Where is the basepoint in your presentation? | |
| Aug 10, 2023 at 9:39 | history | edited | Rajesh Dey | CC BY-SA 4.0 |
added 3 characters in body; edited title
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| Aug 10, 2023 at 9:32 | history | edited | Max Horn | CC BY-SA 4.0 |
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| S Aug 10, 2023 at 9:27 | review | First questions | |||
| Aug 10, 2023 at 9:41 | |||||
| S Aug 10, 2023 at 9:27 | history | asked | Rajesh Dey | CC BY-SA 4.0 |