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

Let ModgModg denote the modular group of (closed and orientable) surfaces S of genus g  . Each element [h]∈Modg[h]∈Modg induces a symplectic automorphism H([h])H([h]) of H1(S,Z)H1(S,Z). Poincaré showed that H:Modg→Sp(2g,Z)H:Modg→Sp(2g,Z) is an epimorphism. A real Riemann surface is a Riemann surface S together with an anticonformal involution σ  . Let (S,σ)(S,σ) be a real Riemann surface, Homeogσ be the group of orientation preserving homeomorphisms of S   such that h○σ=σ○hh○σ=σ○h and Homeog,0σ be the subgroup of Homeogσ consisting of those isotopic to the identity by an isotopy in Homeogσ. The group Modgσ=Homeogσ/Homeog,0σ plays the role of the modular group in the theory of real Riemann surfaces. In this work we describe the image by H   of Modgσ. Such image depends on the topological type of the involution σ.