Mathematical Research Letters
Volume 3 (1996)
The classification of ruled symplectic $4$-manifolds
Pages: 769 – 778
Let $M$ be an oriented $S^2$-bundle over a compact Riemann surface $\Sigma$. We show that up to diffeomorphism there is at most one symplectic form on $M$ in each cohomology class. Since the possible cohomology classes of symplectic forms on $M$ are known, this completes the classification of symplectic forms on these manifolds. Our proof relies on a simplification of our previous arguments and on the equivalence between Gromov and Seiberg-Witten invariants that we apply twice.