2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61856We show that there exist smooth, simply connected, four-dimensional spin manifolds which do not admit Einstein metrics, but nonetheless satisfy the strict Hitchin-Thorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy refinement of the Seiberg-Witten invariant, in conjunction with curvature estimates previously proved by the second author. These methods also easily allow one to construct examples of topological 4-manifolds which admit an Einstein metric for one smooth structure, but which have infinitely many other smooth structures for which no Einstein metric can exist.17 pages, LaTeX2e; minor errors corrected; references addedDifferential GeometryGeometric Topology53C25; 57R57text