MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
Did you have a look at Chapter 12 of (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 '14 at 6:02
@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 '14 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 '14 at 15:24
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 $\mathbb{Z} \oplus \mathbb{Z}/2$.

share|cite|improve this answer
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 '14 at 13:46
Thanks Kevin!!! – Andy Putman Aug 24 '14 at 13:51
thank you Andy! – Daniele Zuddas Aug 25 '14 at 11:43

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.