论文标题
3属Torelli组的顶级同源组
The top homology group of the genus 3 Torelli group
论文作者
论文摘要
Torelli属于$ g $的表面$σ_g$的torelli组是映射类组$ {\ rm mod}（σ_g）$的子组$ \ MATHCAL {i} _g $，由所有映射类组成的所有映射类组成，这些类别在$ {\ rm h} _1 $ _1 $ _1（n）= n = fim = n = n = n n n parter上。商组$ {\ rm mod}（σ_g） / \ mathcal {i} _g $与sympletic group $ {\ rm sp}（2G，\ m athbb {z}）$是同构。组$ \ MATHCAL {i} _g $的共同体学维度等于$ 3G-5 $。本文的主要目的是计算Torelli组的顶级同源组$ g = 3 $ as $ {\ rm sp}（6，\ Mathbb {z}）$ - 模块。我们证明了同构 $$ {\ rm h} _4（\ Mathcal {i} _3，\ Mathbb {z}）\ cong {\ rm ind}^{{\ rm sp}（6，\ m马比布{z}} \ Mathbb {Z}）^{\ times 3}} \ Mathcal {z}，$$其中$ \ Mathcal {z} $是$ \ Mathbb {z}^3 $的商的商，其对角线子组$ \ mathbb {z} $ \ mathbb {z} $ with the自然动作$ s_3 sl}（2，\ mathbb {z}）^{\ times 3} $是微不足道的）。我们还为组$ {\ rm H} _4构建了一组显式的发电机和关系集（\ Mathcal {i} _3，\ Mathbb {Z}）$。
The Torelli group of a genus $g$ oriented surface $Σ_g$ is the subgroup $\mathcal{I}_g$ of the mapping class group ${\rm Mod}(Σ_g)$ consisting of all mapping classes that act trivially on ${\rm H}_1(Σ_g, \mathbb{Z})$. The quotient group ${\rm Mod}(Σ_g) / \mathcal{I}_g$ 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 ${\rm H}_4(\mathcal{I}_3, \mathbb{Z})$.
