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 F̃lux : SympΣg→H 1(Σg;R) which is an extension of the flux homomorphism Flux : Symp 0 Σg→H 1(Σg;R) from the identity component Symp 0 Σg to the whole group SympΣg of symplectomorphisms of Σg with respect to the symplectic form ω. 1. Statement of the main results Let Σg be a closed oriented surface of genus g. It is a classical result that, for any g ≥ 3, there exist oriented Σg-bundles over closed oriented surfaces such that the signatures of the total spaces are non-zero, see Kodaira [18] and Atiyah [1]. In this paper, we prove the existence of such bundles which, in addition to having non-zero signature, are flat, or foliated. This means that there exist codimension two foliations complementary to the fibers, which is equivalent to the existence of lifts of the holonomy homomorphisms from the mapping class group to the diffeomorphism group of the fiber. We will further show that such lifts can be chosen to preserve a prescribed area form, or equivalently a Date: May 8, 2003; MSC 2000 classification: primary 57R17, 57R30, 57R50; secondary 37E30, 57M99, 58H10.