Spin Manifolds, Einstein Metrics, and Differential Topology

We show that there exist smooth, simply connected, four-dimensional {\em 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 \cite{baufu,bauer2}, in conjunction with curvature estimates previously proved by the second author \cite{lric}. 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.

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Published 1 January 2002