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$S^1$-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$S^1$-bundles over a large class of 3-manifolds.