On infinitely generated homology of Torelli groups

Abstract

Let $\mathcal{I}_g$ be the Torelli group of an oriented closed surface $S_g$ of genus $g$, that is, the kernel of the action of the mapping class group on the first integral homology group of $S_g$. We prove that the $k$th integral homology group of $\mathcal{I}_g$ contains a free abelian subgroup of infinite rank, provided that $g\ge 3$ and $2g-3\le k\le 3g-6$. Earlier the same property was known only for $k=3g-5$ (Bestvina, Bux, Margalit, 2007) and in the special case $g=k=3$ (Johnson, Millson, 1992). We also show that the hyperelliptic involution acts on the constructed infinite system of linearly independent homology classes in $\mathrm{H}_k(\mathcal{I}_g;\mathbb{Z})$ as multiplication by $-1$, provided that $k+g$ is even, thus solving negatively a problem by Hain. For $k=2g-3$, we show that the group $\mathrm{H}_{2g-3}(\mathcal{I}_g;\mathbb{Z})$ contains a free abelian subgroup of infinite rank generated by abelian cycles and we construct explicitly an infinite system of abelian cycles generating such subgroup. As a consequence of our results, we obtain that an Eilenberg--MacLane CW complex of type $K(\mathcal{I}_g,1)$ cannot have a finite $(2g-3)$-skeleton. The proofs are based on the study of the spectral sequence for the action of $\mathcal{I}_g$ on the complex of cycles constructed by Bestvina, Bux, and Margalit.