Abstract
This paper studies the (small) quantum homology and cohomology of fibrations
p:P→
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)⊗
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).