Poincar\'{e}'s theorem for the modular group of real Riemann surfaces
  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 $\sigma$. 
Antonio F. Costa, Sergey Natanzon