The Torelli group of a genus $g$ oriented surface ${\mathrm{\Sigma }}_g$ is the subgroup ${\mathcal{I}}_g$ of the mapping class group $\mathrm{Mod}({\mathrm{\Sigma }}_g)$ consisting of all mapping classes that act trivially on ${\mathrm{H}}_1({\mathrm{\Sigma }}_g,\mathbb{Z})$. The quotient group $\mathrm{Mod}({\mathrm{\Sigma }}_g)/{\mathcal{I}}_g$ is isomorphic to the symplectic group $\mathrm{Sp}(2g,\mathbb{Z})$. 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 $\mathrm{Sp}(6,\mathbb{Z})$-module. We prove an isomorphism

中文翻译：

---

属 3 Torelli 群的顶级同源群

托雷利属组$G$定向表面${\mathrm{\Sigma }}_G$是子群${\mathcal{我}}_G$映射类组的$\mathrm{模组}（{\mathrm{\Sigma }}_G）$包含所有作用不大的映射类${\mathrm{H}}_1（{\mathrm{\Sigma }}_G,\mathbb{Z}）$。商群$\mathrm{模组}（{\mathrm{\Sigma }}_G）/{\mathcal{我}}_G$与辛群同构$\mathrm{斯普}（2G,\mathbb{Z}）$。群的上同调维数${\mathcal{我}}_G$等于$3G-5$。本文的主要目标是计算该情况下 Torelli 群的顶级同调群$G=3$作为$\mathrm{斯普}（6,\mathbb{Z}）$-模块。我们证明同构