Monopole classes and Einstein metrics

Abstract

We introduce the notion of a special monopole class on a four-manifold. This is used to prove restrictions on the smooth structures of Einstein manifolds. As an application we prove that there are Einstein four-manifolds which are simply connected, spin, and satisfy the strict Hitchin--Thorpe inequality, and which are homeomorphic to manifolds without Einstein metrics. In the Appendix, written with J. Wehrheim, we use special monopole classes to discuss the classification of Ricci-flat four-manifolds.

17 pages. Includes an Appendix on ``Ricci-flat four-manifolds'' by D. Kotschick and J. Wehrheim

References in corpus (3)

• Einstein metrics and smooth structures
• Einstein metrics and the number of smooth structures on a four-manifold
• A characterization of the Z^n lattice

Cited by in corpus (6)

• Simply connected symplectic 4-manifolds with $b_2^{+}=1$ and $c_1^2=2$
• Entropies, volumes, and Einstein metrics
• A survey of Einstein metrics on 4-manifolds
• Einstein metrics and the number of smooth structures on a four-manifold
• Monopole classes and Perelman's invariant of four-manifolds
• Minimal volume invariants, topological sphere theorems and biorthogonal curvature on 4-manifolds