(i) Topology of embedded surfaces. Let X be a smooth, simply-connected 4-manifold, and ξ a 2-dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly

Abstract. In this paper, the symplectic genus for any 2−dimensional class in a 4−manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular, we completely determine which classes with square at least −1 in such manifolds can be represented by embedded spheres. Moreover, based on a new characterization of the action of the diffeomorphism group on the intersection forms of a rational manifold, we are able to determine the orbits of the diffeomorphism group on the set of classes represented by embedded spheres of square at least −1 in any 4−manifold admitting a symplectic structure. Let M be a smooth, closed oriented 4−manifold. An orientation-compatible symplectic form on M is a closed two−form ω such that ω ∧ω is nowhere vanishing and agrees with the orientation. For any oriented 4−manifold M, its symplectic cone CM is defined as the set of cohomology classes which are represented by orientationcompatible