Timeline for Dehn twist generators for mapping class group of a genus zero surface with boundary
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Jan 13, 2018 at 19:09 | vote | accept | braid rep | ||
| Jan 13, 2018 at 15:28 | answer | added | Daniele Zuddas | timeline score: 2 | |
| Jan 13, 2018 at 14:43 | history | edited | Marco Golla |
Added the reference-request and surfaces tags.
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| Jan 13, 2018 at 3:50 | comment | added | braid rep | I edited the problem, to pose the same question for a disc with holes. | |
| Jan 13, 2018 at 3:48 | history | edited | braid rep | CC BY-SA 3.0 |
deleted 14 characters in body; edited title
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| Jan 13, 2018 at 2:57 | comment | added | braid rep | Thanks for your answer, for the last few days I was thinking of punctures and boundary components as the same thing. Now everything makes more sense. But the question remains: What are the Dehn twist generators of a disc with $n$ inner boundary components? | |
| Jan 13, 2018 at 1:53 | comment | added | Daniele Zuddas | Dehn's theorem about Dehn twists generation is for mapping class groups with no punctures nor marked points, for which also half twists are needed. | |
| Jan 13, 2018 at 1:45 | comment | added | Daniele Zuddas | To be precise... by punctures we mean points you remove, and they can be permuted by a mapping class. On the other hand, for a compact genus-0 surface with n boundary components, the mapping class group acts trivially on the boundary, so half twists do not occur, and this mapping class group is not the braid group. | |
| Jan 13, 2018 at 0:18 | comment | added | braid rep | I suspect I did not specify what I meant by a punctured disc. I am trying to find Dehn twist generators for a disc with $n$ holes, i.e. a disc with $n$ inner boundary components, or equivalenty a $2$-sphere with $n+1$ boundary components. | |
| Jan 13, 2018 at 0:09 | comment | added | braid rep | @DanieleZuddas So does that mean that Dehn' theorem, that the mapping class group of a surface is generated by Dehn twists, is only valid for closed surfaces? | |
| Jan 12, 2018 at 23:14 | comment | added | Igor Rivin | All you want to know is contained here: math.stackexchange.com/questions/616981/… | |
| Jan 12, 2018 at 23:11 | comment | added | Daniele Zuddas | It is not generated by Dehn twists. The reason is that Dehn twist act trivially on the puncures, so the induced permutation is the identity. In other words, they are pure braids. On the other hand, a half twist induces a transposition. | |
| Jan 12, 2018 at 23:01 | history | edited | braid rep | CC BY-SA 3.0 |
added 9 characters in body
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| Jan 12, 2018 at 22:45 | history | asked | braid rep | CC BY-SA 3.0 |