Poincaré's theorem for the modular group of real Riemann surfaces

Details Poincaré's theorem for the modular group of real Riemann surfaces Poincaré's theorem for the modular group of real Riemann surfaces

2006-02-19

arxiv.org/abs/math/0602413v2

Mathematics

Algebraic Geometry

17 pages, LaTex

Scientific paper

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 Poincar\'{e} 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 $\sigma$. Let $(S,\sigma)$ be a real Riemann surface, the subgroup $Mod_{g}^{\sigma}$ of $Mod_{g}$ that consists of the elements $[h]\in Mod_{g}$ that have a representant $h$ such that $h\circ\sigma=\sigma\circ h$, plays the r\^{o}le of the modular group in the theory of real Riemann surfaces. In this work we describe the image by $H$ of $Mod_{g}^{\sigma}$. Such image depends on the topological type of the involution $\sigma$.

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