Timeline for Hyperelliptic loci in Teichmueller spaces
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 13, 2013 at 17:56 | comment | added | Ian Agol | I've added an explanation of why the different lifts of the hyperelliptic universal cover do not intersect. | |
| Jul 26, 2011 at 12:15 | comment | added | algori | Dear JSE -- many thanks for the answer, but I've decided to throw in a couple of dollars. Hope you'll forgive me. | |
| Jul 26, 2011 at 12:10 | history | edited | algori | CC BY-SA 3.0 |
a few more dollars
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| Jul 26, 2011 at 11:49 | history | edited | JSE | CC BY-SA 3.0 |
deleted 3 characters in body
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| Feb 4, 2010 at 2:44 | comment | added | Andy Putman | The category of orbifolds is set up so that you can perform this sort of thing. Without mentioning orbifolds explicitly, you can do the following. Let M_g(L) be the moduli space of curves with level L structures, ie T_g/Mod_g(L) where Mod_g(L) is the subgroup of the mapping class group Mod_g that acts trivially on H_1(S_g;Z/L). For L>=3, the action of Mod_g(L) on T_g is free, so T_g really is the universal cover of M_g(L). Let X(L) be the pullback of the hyperelliptic locus to M_g(L). If X(L) is disconnected, then we are done. Otherwise, we can apply Jordan's argument to X(L). | |
| Feb 4, 2010 at 2:41 | comment | added | JSE | Sorry! I indeed had the orbifold pi_1 in mind for both M_g and H_g. So the Teichmuller space IS the universal cover of the moduli space; not the coarse moduli space but the actual moduli space, which is an orbifold. | |
| Feb 4, 2010 at 2:37 | comment | added | algori | Thanks, JSE. What I don't quite get is the following: the Teichmueller space is not exactly the universal cover of the moduli space. So why does the "$\pi_1$'s map surjectively" criterion still work? | |
| Feb 4, 2010 at 2:29 | comment | added | Andy Putman | This is morally correct, but Teichmuller space is not quite the universal cover of M_g because the action of the mapping class group on Teichmuller space is not free. It is thus a universal cover only in the sense of orbifolds, but everything still goes through like you'd hope. Of course, I'm sure JSE knows this, but this kind of thing can be confusing to a beginner. | |
| Feb 4, 2010 at 2:21 | history | answered | JSE | CC BY-SA 2.5 |