Spin Manifolds, Einstein Metrics, and Differential Topology

We show that there exist smooth, simply connected, fourdimensional 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 [3, 9], in conjunction with curvature estimates previously proved by the second author [17]. These methods also allow us to construct infinitely many new 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.