The top homology group of the genus 3 Torelli group

TL;DR

This paper computes the top homology group of the genus 3 Torelli group, revealing its structure as an ${ m Sp}(6, bZ)$-module and providing explicit generators and relations.

Contribution

It provides the first explicit description of the top homology group of the genus 3 Torelli group, including its isomorphism class and generators.

Findings

${ m H}_4( ext{Torelli}_3, bZ)$ is isomorphic to an induced module from a subgroup.

Explicit generators and relations for ${ m H}_4( ext{Torelli}_3, bZ)$ are constructed.

The module structure involves the symmetric group $S_3$ and ${ m SL}(2, bZ)$.

Abstract

The Torelli group of a genus $g$ oriented surface $\Sigma_g$ is the subgroup $\mathcal{I}_g$ of the mapping class group ${\rm Mod}(\Sigma_g)$ consisting of all mapping classes that act trivially on ${\rm H}_1(\Sigma_g, \mathbb{Z})$. The quotient group ${\rm Mod}(\Sigma_g) / \mathcal{I}_g$ is isomorphic to the symplectic group ${\rm Sp}(2g, \mathbb{Z})$. The cohomological dimension of the group $\mathcal{I}_g$ equals to $3g-5$. The main goal of the present paper is to compute the top homology group of the Torelli group in the case $g = 3$ as ${\rm Sp}(6, \mathbb{Z})$-module. We prove an isomorphism $${\rm H}_4(\mathcal{I}_3, \mathbb{Z}) \cong {\rm Ind}^{{\rm Sp}(6, \mathbb{Z})}_{S_3 \ltimes {\rm SL}(2, \mathbb{Z})^{\times 3}} \mathcal{Z},$$ where $\mathcal{Z}$ is the quotient of $\mathbb{Z}^3$ by its diagonal subgroup $\mathbb{Z}$ with the natural action of the permutation group $S_3$ (the action of ${\rm SL}(2, \mathbb{Z})^{\times 3}$ is trivial). We also construct an explicit set of generators and relations for the group ${\rm H}_4(\mathcal{I}_3, \mathbb{Z})$.