  We 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.
