2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/96675We consider product 4--manifolds S^1 X M, where M is a closed, connected and oriented 3-manifold. We prove that if S^1 X M admits a complex structure or a Lefschetz or Seifert fibration, then the following statement is true: S^1 X M admits a symplectic structure if and only if M fibers over S^1, under the additional assumption that M has no fake 3-cells. We also discuss the relationship between the geometry of M and complex structures and Seifert fibrations on S^1 X M.Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-24.abs.htmlSymplectic GeometryGeometric Topology57M50, 57R17, 57R57, 53C15, 32Q55text