On torsion in the homology of the Torelli group

TL;DR

This paper investigates the torsion phenomena in the homology of the Torelli group, showing that certain subgroups generated by abelian cycles are 2-torsion and finite-dimensional in specific cases.

Contribution

It proves that the subgroup generated by abelian cycles in the Torelli group's homology is a 2-torsion vector space and finite-dimensional for genus at least 4 when k=2.

Findings

Subgroup generated by abelian cycles is a Z/2Z-vector space.

This subgroup is finite-dimensional for k=2 and g≥4.

Homology classes from disjoint separating curves exhibit 2-torsion properties.

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 $g \geq 4$.