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In a survey paper of Korkmaz it is stated that $H_2(\mathrm{Mod}_3)$ is either $\Bbb Z$ or $\Bbb Z \oplus \Bbb Z_2$, but I was not able to find out a precise computation of this group (resolving the ambiguity). Is this group known? Any reference?

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Did you have a look at Chapter 12 of sciencedirect.com/science/book/9780444824325 (Ivanov's survey on mapping class groups)? I don't have a copy right now, so can not check, but usually the book contains everything one would like to know about the homology of mapping class groups. –  ThiKu Aug 24 at 6:02
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@ThiKu : while I think that Ivanov's survey is great, it is pretty dated at this point (eg it predates Madsen-Weiss). –  Andy Putman Aug 24 at 13:44
    
@ThiKu : I gave a look at Ivanov's survey, but it consider this particular case with rational coefficinets. As in Andy's answer, the integral second homology of $\mathrm{Mod}_3$ has been computed recently by Sakasai. Anyway, thank you for your comment! –  Daniele Zuddas Aug 25 at 15:24

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up vote 11 down vote accepted

In his paper "Lagrangian mapping class groups from a group homological point of view" (available here), Sakasai proves that the desired homology group is $\mathbb{Z} \oplus \mathbb{Z}/2$.

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Sorry, typed this on my phone and will not have access to a computer soon. Can someone fix the arxiv link and the tex? Thanks! –  Andy Putman Aug 24 at 13:46
    
Thanks Kevin!!! –  Andy Putman Aug 24 at 13:51
    
thank you Andy! –  Daniele Zuddas Aug 25 at 11:43

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