Torsion at the Threshold for Mapping Class Groups

Abstract

The mapping class group ${\Gamma}_g^ 1$ of a closed orientable surface of genus $g \geq 1$ with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation preserving homeomorphims of the circle. This inclusion pulls back the powers of the discrete universal Euler class producing classes $\text{E}^n \in H^{2n}({\Gamma}_g^1;\mathbb{Z})$ for all $n\geq 1$. In this paper we study the power $n=g,$ and prove: $\text{E}^g$ is a torsion class which generates a cyclic subgroup of $H^{2g}({\Gamma}_g^1;\mathbb{Z})$ whose order is a positive integer multiple of $4g(2g+1)(2g-1)$.