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Signatures of foliated surface bundles and the symplectomorphism groups of surfaces

D. Kotschick·S. Morita
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Abstract

For any closed oriented surface F of genus at least three, we prove the existence of foliated F-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism from the symplectomophism group of the fiber to its first real cohomology. This crossed homomorphism extends the flux homomorphism defined on the identity component of the symplectomorphism group.

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D. Kotschick, S. Morita. 2003-05-13. Signatures of foliated surface bundles and the symplectomorphism groups of surfaces. https://doi.org/10.1016/j.top.2004.05.002

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