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

Abstract

Let Mod g denote the modular group of (closed and orientable) surfaces S of genus g. Each element [ h ] ∈ Mod g induces a symplectic automorphism H ( [ h ] ) of H 1 ( S , Z ) . Poincaré showed that H : Mod g → Sp ( 2 g , Z ) is an epimorphism. A real Riemann surface is a Riemann surface S together with an anticonformal involution σ. Let ( S , σ ) be a real Riemann surface, Homeo g σ be the group of orientation preserving homeomorphisms of S such that h ○ σ = σ ○ h and Homeo g , 0 σ be the subgroup of Homeo g σ consisting of those isotopic to the identity by an isotopy in Homeo g σ . The group Mod g σ = Homeo g σ / Homeo g , 0 σ plays the role 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.