Topology Seminar (Main Talk): Cohomology of mapping class groups near their virtual cohomological dimension

Seminar | April 4 | 4-5 p.m. | 3 Evans Hall

 Søren Galatius, Stanford/Copenhagen

 Department of Mathematics

The mapping class group $\rm {Mod}_g$ of a genus $g$ closed oriented $2$-manifold has virtual cohomological dimension $4g-5$, by a theorem of John Harer, and therefore its cohomology groups $H^i(\rm {Mod}_g;\mathbb Q)$ vanish for $i > 4g-5$. For $i=4g-5$ it also vanishes, by work of Morita-Sakasai-Suzuki, Church-Farb-Putman, and unpublished results of Harer. The highest remaining interesting cohomological degree is therefore $i = 4g-6$. I will discuss joint work with Melody Chan and Sam Payne, in which we prove that the cohomology in this degree is usually non-zero and in fact its dimension grows quite fast as a function of $g$.