I would like to know the third cohomology with coefficients in $U(1)$ or $\mathbb{C}^\ast$ of the mapping class group of a surface of genus at least one. I found many results on the rational cohomology (in particular of the stable MCG), but no results on the integral cohomology (or even cohomology with the above coefficients) in degree three.
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1$\begingroup$ For a genus 1 surface that's just the cohomology of $GL_2 \mathbb Z$ which I think is fairly classical. For higher genus I would look at papers of Mustafa Korkmaz. I don't recall the current state of the art but I believe his work would be a good reference-point. $\endgroup$– Ryan BudneyCommented Apr 3, 2014 at 19:54
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3$\begingroup$ As Ryan said, $g=1$ is classical (but I don't know it off the top of my head). For $g=2$ see "Mapping class groups of low genus and their cohomology" by Benson and Cohen, they determine in fact all the torsion in $H^\ast(\Gamma_2)$. It seems you don't care about rational cohomology but let me mention that most likely $H^3(\Gamma_g,\mathbf Q)$ always vanishes. When $g=3, 4$ this is due to Looijenga (resp. Tommasi) and when $g \geq 6$ this follows from Harer stability/Madsen-Weiss theorem, so only $g=5$ could fail. $\endgroup$– Dan PetersenCommented Apr 3, 2014 at 20:32
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