Timeline for Third cohomology of mapping class group
Current License: CC BY-SA 3.0
3 events
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| Apr 3, 2014 at 20:32 | comment | added | Dan Petersen | 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. | |
| Apr 3, 2014 at 19:54 | comment | added | Ryan Budney | 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. | |
| Apr 3, 2014 at 19:41 | history | asked | Christoph Wockel | CC BY-SA 3.0 |