CURVATURE, DIAMETER AND BOUNDED BETTI NUMBERS Zhongmin Shen and Jyh-Yang Wu Abstract. In this paper, we introduces the notion of bounded Betti numbers, and show that the bounded Betti numbers of a closed Riemannian n-manifold (M,g) with Ric(M) ≥ −(n − 1) and Diam(M) ≤ D are bounded by a number depending on D and n. We also show that there are only finitely many isometric isomorphism types of bounded cohomology ∗ groups (Hˆ (M), k·k∞) among closed Riemannian manifolds (M,g) with K(M) ≥−1 and Diam(M) ≤ D. Key words: Diameter, Ricci curvature, sectional curvature, bounded cohomology and bounded Betti number 2000 Mathematics Subject Classification: 53C21, 53C23 1. Introduction One of the fundamental problems in Riemannian geometry is to understand the relation between the topology and geometry of a Riemannian manifold. Every closed manifold M admits a Riemannian metric g with the following curvature bound (1.1) K(M) ≥−1. Thus the curvature bound in (1.1) alone does not have any implication for the topological structure of the manifold. With this normalized metric, the topology depends on the “size” of the manifold. The diameter is one of the global geometric quantities to measure the manifold. Assume that (1.2) Diam(M) ≤ D. It was proved by M. Gromov [G1] that for any Riemannian n-manifold (M,g) satisfying (1.1) and (1.2), the total Betti number (with respect to any field) is bounded, namely, n X βi(M) ≤ C(n, D). i=0 This paper was written in 1999 and published in Chinese Ann. Math. 27B(2), 2006, 1-10. J.-Y. Wu is partially supported by a Taiwan NSC grant. Typeset by AMS-TEX 2 CURVATURE, DIAMETER AND BOUNDED BETTI NUMBERS 3 In this paper, we are interested in a much weaker curvature bound (1.3) Ric(M) ≥−(n − 1). First of all, according to Sha-Yang’s examples [SY], the Gromov Betti Number Theorem is not true for Riemannian n-manifolds satisfying (1.2) and (1.3). Nevertheless, the first Betti number is still bounded, i.e., (1.4) β1(M) ≤ C(n, D). This is due to M. Gromov [GLP] and T. Gallot [GT]. Besides the ordinary Betti number, what topological invariants are still controlled by the curvature bounds (1.1) or (1.3) ? To answer this question, we consider the bounded ∗ cohomology groups Hˆ (M). The natural inclusion I : Cˆ∗(M) → C∗(M) induces a homo- ∗ ∗ ∗ morphism ι : Hˆ (M) → H∗(M) with image H˜ (M):=ιhHˆ (M)i (see Section 2 below for details). Put i β˜i(M) := dim H˜ (M). Clearly, β˜∗(M) ≤ β∗(M). We call β˜i(M) the ith bounded Betti numbers of M. Below are some important examples: 1 (1)(Gromov) For any closed manifold M, Hˆ (M) = 0, hence β˜1(M)=0; k (2)(Thurston) For any closed manifold of negative curvature, H˜ (M)=Hk(M) for all k ≥ 2, hence β˜k(M)=βk(M) for k ≥ 2; ∗ ˜∗ (3) (Trauber) If π1(M) is amenable, then Hˆ (M) = 0, hence β (M)=0. The bounded Betti numbers behave quite differently from the ordinary Betti numbers. In particular, the Poincar´e duality for β∗(M) does not hold for β˜∗(M). By [Gr3], we ∗ know that the bounded cohomology groups Hˆ (M) are completely determined by π1(M). ˜∗ However, the bounded Betti numbers β (M) depend not only on π1(M), but also on the higher dimensional topological structure of M, when π1(M) is not amenable. In this paper, we shall prove the following Theorem A. There is a constant C(n, D) depending only on n, D such that for closed Riemannian n-manifold (M,g) satisfying Ric(M) ≥−(n−1) and Diam(M) ≤ D, the total bounded Betti number is bounded n (1.5) X β˜i(M) ≤ C(n, D). i=0 Our proof is based on the following conjecture by M. Gromov. For a given n, there is a positive number r = r(n) > 0 such that for any metric ball B(x, ρ) of radius ρ<r 4 ZHONGMIN SHEN AND JYH-YANG WU in a complete Riemannian n-manifold (M,g) with Ric(M) ≥−(n − 1), the image of the natural homomorphism i : π1(B(x, ρ),x) → π1(M,x) is an amenable group. It is proved by Gromov that there is a small number (n) > 0 such that if a closed oriented n-manifold M satisfies (1.3) and (1.6) sup vol(B(p, 1)) <(n), p∈M then there is a map f from M into an (n − 1)-dimensional polyhedron P n−1 such that the pre-image of any star neighborhood is contained in an amenable open subset. Then he concludes that kMk = 0, which is equivalent to that β˜n(M) = 0. Below is a generalization of Gromov’s result. Theorem B. Let n ∈ Z and X a compact metric space. There is a small constant = (n, X) > 0 such that for any closed Riemannian n-manifold (M,g) satisfying Ric(M) ≥ i −(n − 1),ifdGH (M,X) <, then β˜ (M)=0for i>dim X. Again, our proof is based on the Gromov conjecture. Without the Gromov conjecture being true, the upper bound C(n, D) in (1.5) and the small constant (n, X) in Theorem B would depend on the amenability radius. See Lemmas 3.1 and 3.2 in Section 3 below. The Gromov conjecture was proved by Fukaya-Yamaguchi [6] under the sectional curvature bound K(M) ≥−1. Then it is shown to be affirmative by Cheeger-Colding [2] under the Ricci curvature bound Ric ≥−(n − 1). Cheeger-Colding verifies all listed conditions in [6] for the Gromov conjecture to be true under a lower Ricci curvature bound. At the time when the paper was written, there was no simple direct argument for the Gromov conjecture. ∗ Under a stronger curvature bound, the bounded cohomology groups Hˆ (M)ofM are controlled too. More precisely, Theorem C. There are only finitely many isometric isomorphism types of bounded co- ∗ homology groups (Hˆ (M), k·k∞) among closed Riemannian n-manifolds (M,g) satisfying K(M) ≥−1 and Diam(M) ≤ D. From Theorem C, we conclude that there is a constant C(n, D) such that if a closed Riemannian n-manifold (M,g) satisfying K(M) ≥−1 and Diam(M) ≤ D, then ∗ (1.7) dim Hˆ (M) ≤ C(n, D), ∗ ∗ provided that dim Hˆ (M) < ∞. However, there are closed manifolds M with dim Hˆ (M)= 1 3 1 3 ∞.TakeM = S ×S #S ×S . The fundamental group π1(M)=Z ∗Z. Thus the second 2 bounded cohomology group Hˆ (M) is not finitely generated. This example is given to us by Fuquan Fang. We do not know whether or not there exist closed manifolds with non- i amenable fundamental group such that dim Hˆ (M) < ∞ for some i>1. Recently, M. Gromov told the first author that he is doubt about the existence of such manifolds. CURVATURE, DIAMETER AND BOUNDED BETTI NUMBERS 5 2. Preliminaries In this section, we shall summarize some of Gromov’s results [G3] which will be needed in our proof. Let M be a connected topological space. Denote by Σ the set of all singular simplices σ :∆→ M. The standard pseudo-L∞ norm on C∗(M) is defined by kck∞ := sup |c(σ)|. σ∈Σ Consider the subcomplex Cˆ∗(M) of bounded singular cochains of M. The homology groups ∗ ˆ ˆ∗ b H (M)ofC (M) is called the bounded cohomology of M. Let k·k∞ denote the induced ∗ ∗ ˆ ˆ b norm on H (M). Then (H (M), k·k∞) becomes a normed space. The natural inclusion I : Cˆ∗(M) → C∗(M) induces a homomorphism ∗ (2.1) ι : Hˆ (M) → H∗(M). Put ∗ ∗ H˜ (M):=ιhHˆ (M)i. ∗ ∗ H˜ (M) is called the bounded part of H∗(M) (see [Br]). Cohomology classes in H˜ (M) are called bounded classes of H∗(M). Put b (2.2) kαk∞ := inf kβk∞. β∈I−1(α) ∗ Then (H˜ (M), k·k∞) is a normed space. The most important fact in the bounded cohomology theory is that the normed space ∗ ∗ ˆ b ˆ (H (M), k·k∞) actually depends only on the fundamental group π1(M) ([G3]) and H (M)= 0 for connected closed manifolds with π1(M) amenable. Below, we shall sketch Gromov’s ideas to prove the above fact. One is referred to [Br][I] for different arguments. Although Gromov’s theory is for general topological spaces, we shall focus on closed manifolds. First, Gromov introduced a notion of multipcomplex. A (simplicial) multicomplex is defined as a set K divided into the union of closed affine simplices ∆σ ⊂ K, σ ∈ I such that the intersection of any two simplices ∆σ ∩ ∆τ is a subcomplex in both ∆σ and ∆τ . The set K with the weakest topology which agrees with the decomposition K = ∪σ∈I ∆σ is denoted by |K|. The union of all i-dimensional simplices in K is called the i-skeleton of K, denoted by Ki ⊂ K. Let M be an n-dimensional closed manifold. Denote by Σ the set of all singular simplices σ :∆i → M, i =0, 1, 2, ···, which are injective on the vertices of the standard (oriented) i i i i i-simplex ∆ . Take one copy of ∆ for each σ, denoted by ∆σ, and put K := ∪σ∈Σ∆σ. This union has a natural structure of a multicomplex such that the canonical map: S : |K|→M 6 ZHONGMIN SHEN AND JYH-YANG WU i defined by S| i := σ :∆ → M, is continuous.