A characteristic number of Hamiltonian bundles over S2S2

Each loop ψψ in the group Ham(M) of Hamiltonian diffeomorphisms of a symplectic manifold MM determines a fibration EE on S2S2, whose coupling class [V. Guillemin, L. Lerman, S. Sternberg, Symplectic Fibrations and Multiplicity Diagrams, Cambridge U.P., Cambridge, 1996] is denoted by cc. If VTE is the vertical tangent bundle of EE, we relate the characteristic number ∫Ec1(VTE)cn to the Maslov index of the linearized flow ψt∗ψt∗ and the Chern class c1(TM). We give the value of this characteristic number for loops of Hamiltonian symplectomorphisms of Hirzebruch surfaces.