Does there exist a tabulation of the known rational homology of mapping class groups of genus $g$ with $m$ punctures? I'm most interested in the case when punctures can be permuted.
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3 Answers
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There's not a lot known about unstable cohomology. Here are a couple of known facts. I'll restrict myself to the closed case.
1) Of course, it known that the rational homology groups of the genus $1$ mapping class group all vanish.
2) Igusa proved that all the rational cohomology groups of the genus $2$ mapping class group vanish. I'm not entirely sure of the correct reference for this (I'm at home so I can't go look at my notes), but I think this is contained in his paper
MR0114819 (22 #5637) Igusa, Jun-ichi Arithmetic variety of moduli for genus two. Ann. of Math. (2) 72 1960 612–649.
3) The rational cohomology groups of the genus 3 mapping class group were calculated by Looijenga in
MR1234266 (94i:14032) Looijenga, Eduard(NL-UTREM) Cohomology of M3 and M13. (English summary) Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991), 205–228, Contemp. Math., 150, Amer. Math. Soc., Providence, RI, 1993.
A correction to this is in Looijenga's paper "The Hodge polynomial of $\overline{M}_{3,1}$" with Getzler, which is available here.
4) Tommasi has calculated the rational cohomology groups of the genus $4$ mapping class group here :
MR2134272 (2006c:14043) Tommasi, Orsola(NL-RUNJ) Rational cohomology of the moduli space of genus 4 curves. (English summary) Compos. Math. 141 (2005), no. 2, 359–384.
5) The virtual cohomological dimension of the genus $g$ mapping class group is $4g-5$. This implies that $H_k(Mod_g;\mathbb{Q})=0$ for $k > 4g-5$. It is now known that in addition $H_{4g-5}(Mod_g;\mathbb{Q})=0$ for $g$ at least $2$. This was independently discovered by two groups of people (Tom Church, myself, and Benson Farb are one group and Shigeyuki Morita, Takuya Sakasai, and Masaaki Suzuki are the other). Both groups have papers that will (hopefully) appear soon. John Harer has also claimed to have proven this in unpublished work.
6) From Harer-Zagier's computation of the Euler characteristic of the moduli space of curves, it follows that there exists exponentially many unstable classes!
answered Jul 20, 2011 at 2:29
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$\begingroup$ This is very helpful. What about when you have punctures? $\endgroup$ Jul 20, 2011 at 2:52
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1$\begingroup$ In fact, the paper of Morita, Sakasai and Suzuki appeared today: http://arxiv.org/abs/1107.3686. $\endgroup$ Jul 20, 2011 at 6:18
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You should look at
http://www.rasmusvillemoes.dk/thesis/thesis.pdf
and M. Korkmaz' survey http://arxiv.org/abs/math/0307111
answered Jul 20, 2011 at 0:11
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$\begingroup$ That's a very interesting looking thesis, and I'm glad you brought it to my attention. However neither it nor Korkmaz's nice survey have anything like what I want. Korkmaz tells me only first and second homology, even though there are lots of other unstable classes known, while the thesis doesn't even deal with rational coefficients. $\endgroup$ Jul 20, 2011 at 1:59
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A related MO question asks about Betti numbers of $M_{g,n}$ and has some quite complete answers. However, it focuses on the case where the $n$ marked points cannot be permuted.
(Sorry, I know this should have been posted as a comment, but I do not have enough karma yet.)
