Quantum homology of fibrations over S

This paper studies the (small) quantum homology and cohomology of fibrations p : P → S whose structural group is the group of Hamiltonian symplectomorphisms of the fiber (M,ω). It gives a proof that the rational cohomology splits additively as the vector space tensor product H(M)⊗H(S), and investigates conditions under which the ring structure also splits, thus generalizing work of Lalonde–McDuff–Polterovich and Seidel. The main tool is a study of certain operations in the quantum homology of the total space P and of the fiber M , whose properties reflect the relations between the Gromov–Witten invariants of P and M . In order to establish these properties we further develop the language introduced in [Mc3] to describe the virtual moduli cycle (defined by Liu–Tian, Fukaya–Ono, Li–Tian, Ruan and Siebert). AMS classification number 53C15; key words: quantum cohomology, symplectic fibration, Hamiltonian fibration, Gromov– Witten invariants Partially supported by NSF grant DMS 9704825.