Extended flux maps on surfaces and the contracted Johnson homomorphism

TL;DR

This paper introduces new constructions of extended flux maps on closed symplectic surfaces of genus two or more, utilizing Jacobian topology and hyperbolic geometry, and relates them to the Johnson homomorphism.

Contribution

It presents novel methods to construct extended flux maps on surfaces of genus two or more, connecting symplectic topology, Jacobian topology, and hyperbolic geometry.

Findings

Constructed extended flux maps using Jacobian topology for genus ≥ 2.

Developed hyperbolic geometry-based extended flux maps for genus ≥ 3.

Linked the extended flux maps to the 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 Symp(Sigma) to the cohomology group H^1(Sigma;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.