Gromov invariants of S2-bundles over 4-manifolds

We study the Gromov invariants of the total space W of a symplectic fibration pi : W --> M, where (M, w) is a symplectic 4-manifold and the fiber is equal to S-2. We find a relation between the Gromov invariants of W and those of M, for the homology classes (A) over cap such that pi((A) over cap) not equal 0. As an application we construct infinitely many symplectic structures on W for M = E(n), the simply connected minimal elliptic surface. (C) 2001 Elsevier Science B.V. All rights reserved.