Extended flux maps on surfaces and the contracted Johnson homomorphism

Abstract

On a closed symplectic surface $\Sigma$ of genus two or more, we give a new construction of an extended flux map (a crossed homomorphism from the symplectomorphism group $\operatorname{Symp}(\Sigma)$ to the cohomology group $H^1(\Sigma;\mathbb{R})$ that extends the flux homomorphism). This construction uses the topology of the Jacobian of the surface and a correction factor related to the Johnson homomorphism. For surfaces of genus three or more, we give another new construction of an extended flux map using hyperbolic geometry.