Quantum Homology of fibrations over $S^2$

This paper studies the (small) quantum homology and cohomology of fibrations $p: P\to S^2$ whose structural group is the group of Hamiltonian symplectomorphisms of the fiber $(M,\om)$. It gives a proof that the rational cohomology splits additively as the vector space tensor product $H^*(M)\otimes H^*(S^2)$, 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).