Spin manifolds, Einstein metrics, and differential topology

We show that there exist smooth, simply connected, four-dimensional spin manifolds which do not admit Einstein metrics, but nevertheless 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 allow one to easily construct many 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.