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

Let Modg be the modular group of surfaces of genus g. Each element [h] ∈ Modg induces in the integer homology of a surface of genus g a symplectic automorphism H([h]) and Poincaré shown that H : Modg → 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 subgroupModg of Modg that consists of the elements [h] ∈Modg 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 Modg . Such image depends on the topological type of the involution σ.