Minimal genus in 4-manifolds with a free circle action

Let N be a closed irreducible 3-manifold and assume N is not a graph manifold. We improve for all but finitely many S1-bundles M over N the adjunction inequality for the minimal complexity of embedded surfaces. This allows us to completely determine the minimal complexity of embedded surfaces in all but finitely many S1-bundles over a large class of 3-manifolds.