Signatures of foliated surface bundles and the symplectomorphism groups of surfaces

Abstract

For any closed oriented surface Σ g of genus g⩾3, we prove the existence of foliated Σ 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 Flux : Symp Σ g→H 1(Σ g; R) which is an extension of the flux homomorphism Flux : Symp 0 Σ g→H 1(Σ g; R) from the identity component Symp 0 Σ g to the whole group Symp Σ g of symplectomorphisms of Σ g with respect to the symplectic form ω.