Extended flux maps on surfaces and the contracted Johnson homomorphism. (English) Zbl 1338.53117

Summary: 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 \(\text{Symp}(\Sigma)\) 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.


53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
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