carries at most finitely many smooth structures up to diffeomorphism when n 4= 4 (cf. [-KS] for n > 5). Combining Theorem A with the Bonnet-Myers theorem ECE] ...
Invent. math. 99,205 213(1990)
Inventiones mathematicae 9 Springer-Verlag 1990
Geometric finiteness theorems via controlled topology 4' Karsten Grove x,., Peter Petersen V 2,., and Jyh-Yang Wu 3 1 Department of Mathematics, University of Maryland, College Park, Maryland 20742, USA 2 Department of Mathematics, Princeton University,Princeton, New Jersey 08540, USA 3 Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA
Although motivated by Riemannian geometry, this paper is purely topological in nature. As a consequence of our main result we obtain two generalizations of Cheeger's finiteness theorem, I-C, P]. In dimensions > 5 these results were announced in [GPW].
Theorem A. For any real number k and positive numbers D, v, the class of closed Riemannian n-manifolds M with sectional curvature KM >=k, diameter d M v contains at most finitely many homeomorphism types when n 4: 3, and only finitely many diffeomorphism types if in addition n 4=4. A h o m o t o p y version of this theorem was proved in FGP 1] for all dimensions. Here therefore, we need only consider dimensions __>4. Moreover, the last claim of the theorem is implied by the first, since every closed topological n-manifold carries at most finitely m a n y smooth structures up to diffeomorphism when n 4=4 (cf. [-KS] for n > 5). Combining Theorem A with the Bonnet-Myers theorem ECE] and Hamilton's work on manifolds with positive Ricci curvature [H], yields a generalization and sharpening of Weinstein's finiteness theorem [W].
Corollary B. For any positive numbers k, v there are at most finitely many homeomorphism classes of complete Riemannian n-manifolds M, with KM > k and VM > v. In particular, there are at most finitely many diffeomorphism types of such manifold if n+-4. The other application of our main theorem is the following isoembolic finiteness theorem, which is a curvature free generalization of Cheeger's finiteness theorem. A h o m o t o p y version of this was proved in EY].
Theorem C. For any positive numbers i, V, the class of closed Riemannian nmanifolds M, with injectivity radius i(M)>=i and VM i d (p is linear for the classes in Theorems A and C). Here by definition a metric space X is LGC(p) provided every z-ball in X is contractible in the concentric p(e)-ball in X, when e < R. If a metric space X is also a topological manifold we simply refer to it as a metric manifold. With this terminology our main result can be stated as: Main Theorem. Let p: E0, R]--* [0, oo) be a concave function with p(e)>e for all 0 < e < R, and p (0) = O. Then any Gromov-Hausdorff precompact class of closed LGC(p) metric n-manifolds contains at most finitely many homeomorphism types when n + 3. Extending the basic ideas developed in [ G P 1], a homotopy version of this result for all dimensions is a special case of the main result in [-PV]. In particular here we need only consider dimensions > 4. If as in Theorem A and C the class of metric manifolds are all smooth, then there are only finitely many diffeomorphism types when n4= 3, 4. In the generality of the above theorem this is clearly wrong for n = 4 since there are topological 4-manifolds (actually complex algebraic surfaces) which carry infinitely many distinct diffeomorphism types of smooth structures (cf. [FM]). The 3-dimensional case may well rely on the Poincar6 conjecture. The strategy in the proof is to show that for any Gromov-Hausdorff convergent sequence {Mi} of closed LGC(p) metric n-manifolds, all but finitely many are homeomorphic. The point of departure here is that if two LGC(p) manifolds are Gromov-Hausdorff close together then they are controlled h o m o t o p y equivalent. This is used to show that the limit space X = lim Mi has a manifold resolution in the sense of Quinn [-Q 1, 2]. Using results of Davermann [-D 1], Edwards [E], Chapman and Ferry [CF, F 1], we then complete the proof via the controlled h-cobordism theorem in spirit as one proves the uniqueness of resolutions
[Q3]. Acknowledgements.We would like to thank severalpeoplefor their interest and suggestionsconcerning this work, in particular K. Corlette, S. Ferry, M. Rothenherg and F. Quinn.
1. The Gromov-Hausdorff distance and controlled maps A map f: X --* Y between metric spaces is called an ~-map, if diam f - l(y) < e for all y e Y. Two maps fo, f l : X ~ Y are said to be e-homotopic, provided there is a h o m o t o p y H: X x [0, 1] ~ Y from fo to f l such that each path H(x, [0, 1]), x e X , is contained in an ~-ball in Y. Finally, f: X --* u is an z-equivalence with inverse g: Y ~ X , if f o g is e-homotopic to id~ and g o f is homotopic to idx by a homotopy H: X x [0, 1] ~ X , such that for all x e X , the p a t h f ( H ( x , [0, 1])) lies in an z-ball in Y.
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Recall also that the Gromov-Hausdorff distance, dH(X, Y), between compact metric spaces X, Y is less than e > 0 if and only if there is a metric on the disjoint union XAI_Y, extending the metrics on X and Y in which B(X, e) = {z e X ALY [dist (z, X) < e} = Y and B(Y, e) = X, i.e., the classical Hausdorff distance between X and Y in XALY is less than e. If dH(X, Y) 0 then X and Y are isometric. In this way we view the class of compact metric spaces as a metric space. Convergence in this space is related to the classical Hausdorff distance via the following useful observation [G 1]. Lemma 1.1. Suppose the sequence {Xi} converges to X in the Gromov-Hausdorff metric. Then there exists a metric o n Y = X - [ L i X i extending the metrics on X and the Xi's such that {XI} converges to X relative to the classical Hausdorff distance on compact subsets of Y. F r o m now on we fix a contractibility function p: [-0, R-]-~[0, ~ ) as in the introduction. The following result has been proved in [PV]. Theorem 1.2. The class of compact LGC(p) metric spaces of dimension
