Abstract
We show that there exist smooth, simply connected, fourdimensional 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 re nement of the Seiberg-Witten invariant [4, 3], in conjunction with curvature estimates previously proved by the second author [17]. 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 in nitely many other smooth structures for which no Einstein metric can exist.
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