Quantum Homology of fibrations over S²

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This paper studies the (small) quantum homology and cohomology of fibrations p: P→ S² whose structural group is the group of Hamiltonian symplectomorphisms of the fiber (M,). It gives a proof that the rational cohomology splits additively as the vector space tensor product H*(M)⊗ H*(S²), 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).