Signatures of foliated surface bundles and the symplectomorphism groups of surfaces

For any closed oriented surface Σ_g of genus g?3, we prove the existence of foliatedΣ_g-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 which is an extension of the flux homomorphism from the identity component to the whole group of symplectomorphisms of Σ_g with respect to the symplectic form ω.