On torsion in the homology of the Torelli group

Abstract

Let $S_g$ be a closed, oriented surface of genus $g$, and let $\mathrm{Mod}(S_g)$ denote its mapping class group. The Torelli group ${\mathcal{I}}_g$ is the subgroup of $\mathrm{Mod}(S_g)$ 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 $g\ge 4$.

21 pages, 7 figures. Minor revision. To appear in Algebr. Geom. Topol. journal