  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.

Symplectic Geometry;Geometric Topology