Signatures of foliated surface bundles and the symplectomorphism groups of surfaces

For any closed oriented surface Σg of genus g⩾3, we prove the existence of foliatedΣg-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 ω on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism Flux:SympΣg→H1(Σg;R) which is an extension of the flux homomorphism Flux:Symp0Σg→H1(Σg;R) from the identity component Symp0Σg to the whole group SympΣg of symplectomorphisms of Σg with respect to the symplectic form ω.