Curvature, diameter, and quotient manifolds

Abstract

This paper gives improved counterexamples to a question by Grove ([11], 5.7). The question was whether for each positive integer n and real number D, the simply connected closed Riemannian n-manifolds M with sectional curvature ≥ −1 and diameter ≤ D fall into only finitely many rational homotopy types. This was suggested by Gromov’s theorem which bounds the Betti numbers of M in terms of n and D [10]. It was known that there can be infinitely many integral homotopy types already in dimension 7, perhaps first by Aloff and Wallach [2]. Fang and Rong recently gave a negative answer to Grove’s question in all dimensions ≥ 22 ([7], Theorem B). We use certain biquotient manifolds, that is, quotients of homogeneous manifolds G/H by a subgroup of G which acts freely, to show that the question has a negative answer already in dimension 6. Our examples are in fact nonnegatively curved. More precisely, we find infinitely many rational cohomology rings among simply connected closed Riemannian 6-manifolds with nonnegative sectional curvature. (Of course, we can arrange that these manifolds also have diameter at most 1, by scaling.) The dimension 6 here is optimal, meaning that Grove’s question has a positive answer in dimensions ≤ 5. This follows from Gromov’s bound on the Betti numbers, since the Betti numbers of a simply connected manifold of dimension ≤ 5 determine its rational homotopy type up to finitely many possibilities. More precisely, the conjecture that simply connected manifolds of nonnegative curvature are integrally elliptic would imply, by Paternain and Petean ([15], Corollary 3.6), that simply connected 5-manifolds of nonnegative curvature fall into only 4 diffeomorphism classes: S5, S3 × S2, the nontrivial S3-bundle over S2, and the Wu manifold SU(3)/SO(3) [4]. Fang and Rong’s examples have the merit of also having an upper bound on curvature. That is, for n ≥ 22, Fang and Rong find numbers C and D such that there are infinitely many rational cohomology rings among simply connected closed Riemannian n-manifolds with curvature −1 ≤ K ≤ C and diameter ≤ D. The next main result of this paper is that such examples exist already among 7-manifolds. This is optimal, since Fang and Rong [6], and also Tuschmann [19], have proved that in dimensions ≤ 6 there are only finitely many diffeomorphism classes in the given class of manifolds. Finally, in dimension 9, we use biquotients to give a similar counterexample using only nonnegatively curved manifolds. That is, for some C and D, there are infinitely many rational cohomology rings among simply connected closed 9-manifolds with curvature 0 ≤ K ≤ C and diameter ≤ D. To conclude, one can ask what substitute for Grove’s question might be true. For the problem with an upper curvature bound, there is already a remarkable substitute for Grove’s question, the Petrunin-Tuschmann theorem ([16], Corollary 0.2). Namely, for each n, C, and D, there is a finite set of closed smooth manifolds Ei of dimension ≥ n such that any simply connected closed Riemannian n-manifold