Timeline for Elegant proof that mapping class groups are generated by Dehn twists?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 17, 2013 at 4:17 | comment | added | Sam Nead | You need to say "orientation preserving" at the right moment... | |
| May 31, 2011 at 3:13 | vote | accept | Daniel Moskovich | ||
| May 20, 2011 at 5:45 | comment | added | Andy Putman | @Daniel : I don't know, how many proofs do you know that $SL_n(\mathbb{Z})$ is generated by elementary matrices? I think of the two facts as being analogous. Anyway, I'll give an answer with some extended comments below. | |
| May 20, 2011 at 5:32 | answer | added | Andy Putman | timeline score: 12 | |
| May 20, 2011 at 0:54 | comment | added | Daniel Moskovich | That's interesting! I learnt this proof from Massuyeau's survey, and it's really the only one I have gone through in any detail. But I hope there are other ways- maybe something slicker and easier to teach. I'd feel enriched by any fundamentally different proof, in fact- important facts should have many findamentally different proofs. | |
| May 19, 2011 at 21:10 | comment | added | Andy Putman | By the way, the proof you describe above is related to Lickorish's proof, but is not identical to it. Indeed, Lickorish's paper predates both the Birman exact sequence (which was Birman's thesis and was published in 1969) and the curve complex (which was first defined by W. Harvey in the late '70's). I believe that the proof you describe is due to Ivanov in his 1998 survey on mapping class groups -- he describes it as a simplification of an argument of Birman. | |
| May 19, 2011 at 15:35 | comment | added | Andy Putman | You can avoid using the Birman exact sequence by proving that the arc complex (consisting of isotopy classes of arcs joining two fixed points on two boundary components) is connected -- the base case would then be a once-punctured torus. However, I rather doubt that you are going to find a proof that is significantly simpler than the one you describe, which strikes me as a pretty elegant proof. | |
| May 19, 2011 at 13:50 | answer | added | Sam Nead | timeline score: 3 | |
| May 19, 2011 at 13:15 | history | edited | Daniel Moskovich | CC BY-SA 3.0 |
added 17 characters in body; added 8 characters in body
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| May 19, 2011 at 13:07 | history | asked | Daniel Moskovich | CC BY-SA 3.0 |