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.

36 pages, both the result and the proof are the same as in the previous version but the paper is rewritten considerably

Cited by in corpus (2)

• On the top homology group of Johnson kernel
• On spectral sequence for the action of genus 3 Torelli group on the complex of cycles