Gromov invariants of \(S ^{2}\)-bundles over 4-manifolds (Q1862056)

Given a symplectic \(S^2\)-fibration \(\pi:W\to M\) and homology class \(A\) such that \(\pi(A) \neq 0\), the author develops a relation between the Gromov-Witten invariants of the total space \(W\) of the fibration and those of the base space \(M\). In particular, the author shows that for suitable generic points and fibers, the Gromov-Witten invariants of a lift \(\hat{A}\in H_2(W,\mathbb{Z})\) of a simple class \(A\) are equal to \(2^g\) times the Gromov-Witten invariants of the class \(A\). As a corollary, the author constructs infinitely many deformation classes of symplectic structures on \(W\) when the base \(M\) is a simply connected minimal elliptic surface.