Torsion at the Threshold for Mapping Class Groups

Abstract

The mapping class group $\hat{I}{“}_g^1$ of a closed orientable surface of genus $g\ge 1$ 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 ${\text{E}}^n\in H^{2n}(\hat{I}{“}_g^1;\mathbb{Z})$ for all $n\ge 1$. In this paper we study the power $n=g,$ and prove: ${\text{E}}^g$ is a torsion class which generates a cyclic subgroup of $H^{2g}(\hat{I}{“}_g^1;\mathbb{Z})$ whose order is a positive integer multiple of $4g(2g+1)(2g-1)$.

23 pages, 2 figures. Comments welcome!