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Spin Manifolds, Einstein Metrics, and Differential Topology

Masashi Ishida·Claude LeBrun
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Abstract

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.

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Masashi Ishida, Claude LeBrun. 2001-10-31. Spin Manifolds, Einstein Metrics, and Differential Topology. https://arxiv.org/abs/math/0107111

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