On torsion in the homology of the Torelli group

Abstract

Let $S_g$ be a closed, oriented surface of genus $g$, and let $\operatorname{Mod}(S_g)$ denote its mapping class group. The Torelli group $\mathcal{I}_g$ is the subgroup of $\operatorname{Mod}(S_g)$ consisting of mapping classes that act trivially on $H_1(S_g)$. For any collection of pairwise disjoint, separating simple closed curves on $S_g$, the corresponding Dehn twists pairwise commute and determine a homology class in $H_k(\mathcal{I}_g)$, which is called an abelian cycle. We prove that the subgroup of $H_k(\mathcal{I}_g)$ generated by such abelian cycles is a $\mathbb{Z}/2\mathbb{Z}$-vector space for all $k$, and that it is finite-dimensional for $k = 2$ and $g \geq 4$.