Einstein Metrics on Rational Homology Spheres

In this paper we prove the existence of Einstein metrics, actually SasakianEinstein metrics, on nontrivial rational homology spheres in all odd dimensions greater than 3. It appears as though little is known about the existence of Einstein metrics on rational homology spheres, and the known ones are typically homogeneous. The are two exception known to the authors. Both involve Sasakian geometry and both occur in dimension 7. In [BGN02] the two authors and M. Nakamaye gave a list of 184 rational homology 7-spheres with Sasakian-Einstein metrics. The result was based on a theorem of Johnson and Kollár proving the existance of Kähler-Einstein metric on cartain Fano 3-folds with orbifold singularities [JK01]. More recently, Grove, Wilking and Ziller constructed infinitely many rational homology 7-spheres with 3-Sasakian metrics of cohomogeneity one under an action of S × S [Zil]. Being 3-Sasakian these metrics are necessarily Sasakian-Einstein. Their construction involves orbifold self-dual Einstein metrics discovered by N. Hitchin [Hit96] a decade ago. Hitchin constructed a family of positive self-dual Einstein metrics, indexed by integers k ≥ 4 which live on S \ RP. They are complete in the orbifold sense, i.e., they can be viewed as metrics defined on compact Riemannian orbifolds M k with a Zk quotient singularities along RP . Any positive self-dual Einstein orbifold O admits a compact 3-Sasakian orbifold V -bundle S−−→O over it [BGM94]. Grove, Wilking and Ziller showed that in the case of Hitchin’s metrics the bundle N k−−→M 4 k is actually a smooth 7-manifold and, moreover, computed H(N k ,Z). When k = 2m− 1 the manifold N 7 k turns out to be a rational homology 7-sphere with H3(N 7 k ,Z) = Zm. In dimension 5, aside from the standard 5-sphere S there is only one known case of an Einstein metric on a simply connected rational homology 5-sphere. It is the homogeneous space SU(3)/SO(3) which is a simply connected non-spin manifold with H2(SU(3)/SO(3),Z) = Z2. On the other hand the 5-manifolds M 5 considered in this paper are all spin, i.e. w2(M) = 0. A well known theorem of Smale [Sma62] says that for any simply connected 5-manifold with spin, the torsion group in H2 is of the form G ⊕ G for some Abelian group G. Until recently [BG02] this was the only simply connected nontrivial rational homology sphere known to admit Riemannian metrics of positive Ricci curvature. In this note we go much further by proving the existence of Sasakian-Einstein metrics on infinitely many simply connected rational homology 5-spheres. Furthermore, the metrics typically depend on parameters, that is, there is non-trivial moduli. We prove the following two theorems.