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Let $MCG_g$ be the mapping class group of closed genus $g$ Riemannian surface. What is the group cohomology $H^n(MSG_g,Z)$ for $n=2$ (and other values).

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I do not know if this will be helpful:

http://www.northeastern.edu/jimenez_rolland/rita_images/RepStabPMCG_AGT.pdf

which discussed the case for the mapping class group of Riemann surface with $n$ points removed. But most of the work cannot be applied to the current case, because Rita's work showed the (rational) cohomology is dependent on the number of distinguished points on the surface. And I assume you are working with the full mapping class group. If I am not mistaken an automorphism $M_{g}\rightarrow M_{g}$ can happen without any fixed points.

Others you can look up are her advisor Benson Farb, Tom Church (https://math.stackexchange.com/users/639/tom-church ?), and Jordan Ellenberg. Also I think this question is quite relevant:

What can Lefschetz fixed point theorem tell us about the group of automorphisms of a compact Riemann surface?

A proof for the $H^{2}$ case with $k$ fixed points can be found at her PhD thesis (page 33):

http://www.northeastern.edu/jimenez_rolland/rita_images/thesis.pdf

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