实Riemann surfaces模群的Poincaré定理
Poincaré's theorem for the modular group of real Riemann surfaces
摘要：
设$Mod_{g}$是亏格$g$的曲面的模群。每个元$[h]\in Mod_{g}$在亏格$g$的曲面的整数同调中诱导出一个辛自同构$H([h])$，Poincaré证明了$H:Mod_{g}\to Sp(2g,\mathbb{Z})$是满态的.实代数曲线理论证明了实黎曼曲面的定义是具有反共形对合$σ$的黎曼曲面$S$。设$(S,σ)$是一个实Riemann曲面，$Mod_{g}$的子群$Mod_{g}^σ$由具有一个表示$h$的元素$[h]\in Mod_{g}$组成，使得$h\circσ=σ\circ h$在实Riemann曲面理论中起模群的作用。在这项工作中，我们描述了图像的$H$的$Mod_{g}^σ$。这种图像依赖于对合$σ$的拓扑类型。
英文摘要：
Let $Mod_{g}$ be the modular group of surfaces of genus $g$. Each element $[h]\in Mod_{g}$ induces in the integer homology of a surface of genus $g$ a symplectic automorphism $H([h])$ and Poincaré shown that $H:Mod_{g}\to Sp(2g,\mathbb{Z})$ is an epimorphism. The theory of real algebraic curves justify the definition of real Riemann surface as a Riemann surface $S$ with an anticonformal involution $σ$. Let $(S,σ)$ be a real Riemann surface, the subgroup $Mod_{g}^σ$ of $Mod_{g}$ that consists of the elements $[h]\in Mod_{g}$ that have a representant $h$ such that $h\circσ=σ\circ h$, plays the rôle of the modular group in the theory of real Riemann surfaces. In this work we describe the image by $H$ of $Mod_{g}^σ$. Such image depends on the topological type of the involution $σ$.
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