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 homeomorphisms of the circle. This inclusion pulls back the powers of the discrete universal Euler class producing classes $E^n \in H^{2n}(\Gamma _g^1;\mathbb {Z})$ for all $n\geq 1$. In this paper we study the power $n=g$, and prove: $E^g$ 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 $4g(2g+1)(2g-1)$.