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

2006/02/19 by Antonio F. Costa, Costa, Antonio F., S. M. Natanzon +2
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AG

paper · pdf · doi:10.48550/arxiv.math/0602413

17 pages, LaTex

openalex publication_date 2006/02/19 · arxiv created 2006/09/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Mod_g be the modular group of surfaces of genus g. Each element [h]∈ 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→ Sp(2g,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]∈ Mod_g that have a representant h such that h∘σ=σ∘ 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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