The top homology group of the genus 3 Torelli group

The Torelli group of a genus $g$ oriented surface ${\mathrm{\Sigma }}_g$ is the subgroup ${\mathcal{I}}_g$ of the mapping class group $\mathrm{M}\mathrm{o}\mathrm{d}({\mathrm{\Sigma }}_g)$ consisting of all mapping classes that act trivially on ${\mathrm{H}}_1({\mathrm{\Sigma }}_g,\mathbb{Z})$. The quotient group $\mathrm{M}\mathrm{o}\mathrm{d}({\mathrm{\Sigma }}_g)\mathrm{/}{\mathcal{I}}_g$ is isomorphic to the symplectic group $\mathrm{S}\mathrm{p}(2g,\mathbb{Z})$. 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 $\mathrm{S}\mathrm{p}(6,\mathbb{Z})$-module. We prove an isomorphism $${\mathrm{H}}_4({\mathcal{I}}_3,\mathbb{Z})\cong {\mathrm{I}\mathrm{n}\mathrm{d}}_{S_3⋉\mathrm{S}\mathrm{L}(2,\mathbb{Z}{)}^{\times 3}}^{\mathrm{S}\mathrm{p}(6,\mathbb{Z})}\mathcal{Z},$$ 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 $\mathrm{S}\mathrm{L}(2,\mathbb{Z}{)}^{\times 3}$ is trivial). We also construct an explicit set of generators and relations for the group ${\mathrm{H}}_4({\mathcal{I}}_3,\mathbb{Z})$.