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Let $MCG_g$ be the mapping class group of genus $g$ closed surface. (Say $MCG_1=SL(2,Z)$). I would like to know what is the group cohomology of $MCG_g$ with coefficients in Z, such as $H^2(MCG_g,Z)$and $H^3(MCG_g,Z)$.

http://arxiv.org/abs/math/9503230 contains a result $\bar H^n(SL(2,Z),Z)=Z_{12}$ for $n=$even, and $\bar H^n(SL(2,Z),Z)=0$ for $n=$odd. But I do not know what $\bar H^n$ means (Torsion?)

Also, I like to know what is $H^2(SL(n,Z),Z)$. Thanks!

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    $\begingroup$ In general, the cohomology groups of mapping class groups are very much not known. Much more is known about stable mapping class groups, i.e. if you consider the mapping class groups of surfaces of genus $g$ with one boundary curve for $g$ large (see, e.g. arxiv.org/pdf/math/0401168v1.pdf). $\endgroup$
    – Lennart Meier
    Commented Apr 15, 2014 at 15:13
  • $\begingroup$ And I guess, $\bar H^n(SL(2,Z),Z)$ refers to the reduced cohomology, i.e. we leave out the $\mathbb{Z}$ in degree $0$. By the way, there are several further sources for "unstable" homology of mapping class groups: msp.org/gtm/2008/14/gtm-2008-14-001s.pdf and hss.ulb.uni-bonn.de/2011/2610/2610.pdf from the Bödigheimer school come to my mind, which were written after Korkmaz' survey. $\endgroup$
    – Lennart Meier
    Commented Apr 15, 2014 at 15:21
  • $\begingroup$ @Lennart Meier: Thanks. The newer refs are for mapping class group with a boundary. I wonder do you know new refs for the case without boundary. $\endgroup$
    – Xiao-Gang Wen
    Commented Apr 15, 2014 at 23:09
  • $\begingroup$ I'm guessing that you are interested only in certain low-degree cohomology groups for MCGs. If that is indeed the case, I suggest that you modify your question: you'll get substantially more informative answers if you say explicitly which degrees you care about. $\endgroup$
    – André Henriques
    Commented May 16, 2014 at 0:38
  • $\begingroup$ mathoverflow.net/questions/162383/… $\endgroup$
    – Ryan Budney
    Commented May 16, 2014 at 2:40

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See M. Korkmaz' paper For the mapping class group. For the $SL(n, \mathbb{Z})$ see this question.

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