Torsion at the Threshold for Mapping Class Groups

TL;DR

This paper investigates the torsion properties of certain cohomology classes in the mapping class group of a surface, revealing that a specific class is torsion with an order divisible by a particular integer related to the genus.

Contribution

It proves that the power of the Euler class at the genus level is a torsion class with an order divisible by a specific integer, advancing understanding of the cohomology of mapping class groups.

Findings

The class E^g is torsion in H^{2g} of the mapping class group.

The order of E^g divides a multiple of 4g(2g+1)(2g-1).

This reveals new torsion phenomena in the cohomology of 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 $4g(2g+1)(2g-1)$.