曲线复形商的有理同调
The rational homology of quotients of the curve complex
浏览论文内容
中文总结 AI 辅助
本文针对几何拓扑学家，详细阐述 Boggi 运用混合 Hodge 理论证明的、当$g+n$远大于$k$时曲线复形商的约化有理同调群为零的定理，该定理在映射类群虚拟第一 Betti 数研究中具重要意义。
中文摘要 AI 辅助
设$\text{Mod}_{g,n}$为亏格$g$、$n$个 punctures 的曲面的映射类群，$\boldsymbol{\textit{C}}_{g,n}$为其曲线复形。对有限指数子群$G < \text{Mod}_{g,n}$，Boggi 证明当$g+n \text{ 远大于 }k$时，$\boldsymbol{\textit{H}}_k(\boldsymbol{\textit{C}}_{g,n}/G;\boldsymbol{\textit{Q}})=0$，该结果在 Putman-Wieland 关于$\text{Mod}_{g,n}$的虚拟第一 Betti 数的研究中起重要作用。Boggi 定理的证明运用混合 Hodge 理论，嵌入在他声称映射类群具有同余子群性质的有缺陷论文中。本文针对几何拓扑学家，给出 Boggi 证明的详细阐述。
英文摘要
Let $\mathop{Mod}_{g,n}$ be the mapping class group of a genus-$g$ surface with $n$ punctures and let $\mathcal{C}_{g,n}$ be its curve complex. For a finite-index subgroup $G < \mathop{Mod}_{g,n}$, Boggi proved that $\widetilde{H}_k(\mathcal{C}_{g,n}/G;\mathbb{Q}) = 0$ for $g+n \gg k$. This plays an important role in the work of Putman-Wieland on the virtual first Betti number of $\mathop{Mod}_{g,n}$. The proof of Boggi's theorem uses mixed Hodge theory and is embedded in his flawed paper purporting to show that the mapping class group has the congruence subgroup property. We give a detailed exposition of Boggi's proof aimed at geometric topologists.
发表机构
- University of Notre Dame（圣母大学）
机构由 AI 辅助整理，请以论文原文为准。
