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 foliated Σg-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 : Symp Σg→H1(Σg;R) which is an extension of the flux homomorphism Flux : Symp0 Σg→H1(Σg;R) from the identity component Symp0 Σg to the whole group Symp Σg of symplectomorphisms of Σg with respect to the symplectic form ω.