Spin Manifolds, Einstein Metrics, and Differential Topology

2001/07/16 by Masashi Ishida, Claude LeBrun, Ishida, Masashi +1 · 1 citation
Mathematics · #53C25 #57R57 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #math.DG #math.GT #msc:53C25 #msc:57R57

paper · pdf · doi:10.48550/arxiv.math/0107111

17 pages, LaTeX2e; minor errors corrected; references added

arxiv created 2001/10/31 · arxiv updated 2009/11/30

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.

Cited by

• Normal Forms and Einstein 4-manifolds

Related

• 4-Manifolds without Einstein Metrics
• Einstein Metrics, Four-Manifolds, and Differential Topology
• Monopole classes and Einstein metrics
• Seiberg-Witten invariants of non-simple type and Einstein metrics
• Einstein metrics and the number of smooth structures on a four-manifold