On the Non-vanishing of the Powers of the Euler Class for Mapping Class Groups

Abstract

The mapping class group of an orientable closed surface 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 “discrete universal Euler class” producing a non-zero class in the second integral cohomology of the mapping class group. In this largely expository note, we determine the non-vanishing behavior of the powers of this class. Our argument relies on restricting the cohomology classes to torsion subgroups of the mapping class group.