Timeline for Elegant proof that mapping class groups are generated by Dehn twists?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 27, 2014 at 23:39 | comment | added | QcH | @AndyPutman: Thanks for your answer! | |
| Jan 26, 2014 at 21:10 | comment | added | Andy Putman | @QcH : Notice that I said they proved "equivalent" results; they did not explicitly prove that the mapping class group is finitely presentable. Here's what I meant. Ignoring stacky issues, the mapping class group is the fundamental group of the moduli space of curves. Quasiprojective varieties are homotopy equivalent to compact CW-complexes, and in particular have finitely presentable fundamental groups. The desired result thus follows from the fact that the moduli space of curves is quasiprojective, which was originally proven by Baily in 1960. | |
| Jan 26, 2014 at 17:28 | comment | added | QcH | @AndyPutman: Could you give a reference where the algebraic geometers proved the finite-presentability of mapping class group? Thanks. | |
| May 31, 2011 at 3:13 | vote | accept | Daniel Moskovich | ||
| May 24, 2011 at 21:57 | comment | added | Andy Putman | @Dave : I've not read Dehn's original proof that the mapping class group is finitely generated in detail, but my impression is that the germ of the idea that the curve complex is connected is contained in there as well. In any case, I definitely agree that it's lurking in the background of Lickorish's paper. | |
| May 24, 2011 at 21:05 | comment | added | Dave Futer | @Andy: I believe the standard combinatorial proof of connectivity is due to ... Lickorish. Of course, as you said in a previous comment, his work predates Harvey's definition of the curve complex. But it's abundantly clear in hindsight that connectivity of the curve complex is exactly what he's proving. An analogous phenomenon: Schreier's normal form for 3-braids (published in 1924) sorts conjugacy classes into 5 families. In hindsight, it's clear that one family is reducible, three are periodic, and one is pseudo-Anosov. But this predates Thurston's work by a half-century! | |
| May 20, 2011 at 5:40 | comment | added | Andy Putman | As far as how to present it in a single lecture, it's perfectly doable if you don't insist on giving every detail. Just give the big picture (assuming the connectivity of the complex of curves and the Birman exact sequence). This could easily be done in 30 minutes if the students already know what a Dehn twist is or an hour if they don't. If you have time remaining, sketch one of the two assumed facts. | |
| May 20, 2011 at 5:32 | history | answered | Andy Putman | CC BY-SA 3.0 |