发表机构
San Jos\' e State University, One Washington Square, San Jose, CA, 95112
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文类比曲线图的着色结果，研究$M_r$球面图的子图，证明其唯一$r-1$可着色，进而得到Torelli子群到对称群的非平凡同态，并给出映射类群的新生成集。
AI 中文摘要
曲面的曲线图是一个图，其顶点为曲面上本质闭曲线的同痕类，边连接具有不相交代表元的顶点，该曲线图在曲面的映射类群理论中占据核心地位。Gaster、Greene和Vlamis已证明，由单个本原同调类中的曲线诱导的曲线图子图是唯一且有限可着色的。在曲面映射类群理论与自由群外自同构理论的类比中，$M_r$（$r$个$S_1 \times S_2$的连通和）的球面图在该理论中扮演着类似角色。受Gaster、Greene和Vlamis结果的启发，我们研究$\boldsymbol{\frak{S}}_v(M_r)$，即由单个本原同调类$[v] \boldsymbol{\frak{}} H_2(M_r;\boldsymbol{\frak{Z}})$中的球面诱导的$M_r$球面图的子图。我们证明该图是唯一的$r-1$可着色图。作为推论，当$r \boldsymbol{\frak{}} 3$时，我们得到从Torelli子群$IO_r \boldsymbol{\frak{}} \boldsymbol{\frak{ut}}(F_r)$到$r-1$个符号的对称群的非平凡同态；这与Gaster、Greene和Vlamis的着色结果与Chillingworth同态之间的联系相平行。在证明过程中，我们找到了删除球面后的$M_r$映射类群的新的更小生成集。
英文摘要
The curve graph of a surface is the graph whose vertices are isotopy classes of essential, closed curves on the surface. Edges join vertices with disjoint representatives. The curve graph plays a central role in the theory of mapping class groups of surfaces. Gaster, Greene, and Vlamis have shown that the subgraph of the curve graph induced by curves in a single primitive homology class is uniquely and finitely colorable. In the analogy between the theory of mapping class groups of surfaces and outer automorphisms of free groups, the sphere graph of $M_r$, a connect sum of $r$ copies of $S_1 \times S_2$ plays an analogous role in the theory. Motivated by Gaster, Green, and Vlamis' result, we investigate $\mathcal{S}_v(M_r)$, the subgraph of the sphere graph of $M_r$ induced by spheres representing a single primitive homology class $[v] \in H_2(M_r;\mathbb{Z})$. We show that this graph is uniquely $r-1$ colorable. As a corollary, with $r\ge 3$ we obtain a nontrivial homomorphism from the Torelli subgroup $IO_r \le \operatorname{Out}(F_r)$ to the symmetric group on $r-1$ symbols; this parallels Gaster, Greene, and Vlamis' connection between their coloring and the Chillingworth homomorphism. In the course of our proof we find new smaller generating sets for mapping class groups of $M_r$ with spheres deleted.
Comments18 pages, 6 figures
