Symplectic structures on fiber bundles

Let π : P→B be a locally trivial fiber bundle over a connected CW complex B with fiber equal to the closed symplectic manifold (M,ω). Then π is said to be a symplectic fiber bundle if its structural group is the group of symplectomorphisms Symp(M,ω), and is called Hamiltonian if this group may be reduced to the group Ham(M,ω) of Hamiltonian symplectomorphisms. In this paper, building on prior work by Seidel and Lalonde, McDuff and Polterovich, we show that these bundles have interesting cohomological properties. In particular, for many bases B (for example when B is a sphere, a coadjoint orbit or a product of complex projective spaces) the rational cohomology of P is the tensor product of the cohomology of B with that of M. As a consequence the natural action of the rational homology Hk(Ham(M)) on H∗(M) is trivial for all M and all k>0.