Curvature, diameter, and quotient manifolds
Curvature, diameter, and quotient manifolds
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曲率、直径和商流形
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发表时间：
2002
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通讯作者：
B. Totaro
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作者：
B. Totaro
本文针对 Grove 的一个问题给出了改进的反例（[11], 5.7）。问题是对于每个正整数 n 和实数 D，截面曲率 ≥ -1 且直径 ≤ D 的单连通闭黎曼 n 流形 M 是否仅落入有限多个有理同伦类型。这是由格罗莫夫定理提出的，该定理用 n 和 D 限制了 M 的贝蒂数 [10]。人们知道，在第 7 维中已经可以有无限多个积分同伦类型，这可能是 Aloff 和 Wallach 首先提出的 [2]。 Fang 和 Rong 最近在所有维度≥22 的情况下对 Grove 的问题给出了否定的答案（[7]，定理 B）。我们使用某些双商流形，即齐次流形 G/H 与自由作用的 G 子群的商，来表明该问题在第 6 维中已经有否定答案。我们的例子实际上是非负弯曲的。更准确地说，我们在具有非负截面曲率的单连通闭合黎曼六流形中发现了无限多个有理上同调环。 （当然，我们可以通过缩放来安排这些流形的直径至多为 1。）这里的维度 6 是最优的，这意味着格罗夫的问题在维度 ≤ 5 上有一个肯定的答案。这是从格罗莫夫对贝蒂数的界限得出的，因为维度 ≤ 5 的单连通流形的贝蒂数决定了其有理同伦类型最多有有限种可能性。更准确地说，Paternain 和 Petean（[15]，推论 3.6）认为非负曲率的简单连通流形是整体椭圆的猜想意味着，非负曲率的简单连通 5-流形仅落入 4 个微分同胚类：S5、S3 × S2、S2 上的非平凡 S3 丛以及 Wu 流形 SU(3)/SO(3) [4]。 Fang 和 Rong 的例子的优点是曲率也有上限。也就是说，对于 n ≥ 22，Fang 和 Rong 找到数字 C 和 D，使得在曲率 −1 ≤ K ≤ C 且直径 ≤ D 的单连通闭黎曼 n 流形中存在无穷多个有理上同调环。本文的下一个主要结果是这样的例子已经存在于 7-流形中。这是最优的，因为 Fang 和 Rong [6] 以及 Tuschmann [19] 已经证明，在维度 ≤ 6 的情况下，给定流形类中只有有限多个微分同胚类。最后，在第 9 维中，我们使用双商仅使用非负弯曲流形给出类似的反例。也就是说，对于某些 C 和 D，在曲率 0 ≤ K ≤ C 且直径 ≤ D 的单连通闭 9 流形之间存在无穷多个有理上同调环。总而言之，人们可以问格罗夫问题的哪种替代可能是正确的。对于曲率上界的问题，格罗夫的问题已经有了一个显着的替代品，即 Petrunin-Tuschmann 定理（[16]，推论 0.2）。即，对于每个 n、C 和 D，存在一组有限的维度 ≥ n 的闭合光滑流形 Ei，使得任何单连通闭合黎曼 n 流形
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