2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63046In this paper we show that the Seiberg--Witten invariant is zero for all smooth 4--manifolds with $b_+{>}1$ which admit circle actions that have at least one fixed point. Furthermore, we show that all symplectic 4--manifolds which admit circle actions with fixed points are rational or ruled, and thus admit a symplectic circle action.Includes new theorem classifying symplectic 4-manifolds with circle actions having fixed points. Minor updates to existing proofs to reflect the new theorem. 7 pages, 2 figuresGeometric TopologySymplectic Geometry57R57; 57M60text