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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.

MSC:

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