Journal of Symplectic Geometry

Volume 9 (2011)

Number 4

Extended flux maps on surfaces and the contracted Johnson homomorphism

Pages: 445 – 482

DOI: http://dx.doi.org/10.4310/JSG.2011.v9.n4.a3

Author

Matthew B. Day

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.

Full Text (PDF format)