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A Radius Sphere Theorem

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A Radius Sphere Theorem Invent. math. 112, 577-583 (1993) Inventiones mathematicae Springer-Verlag1993 A radius sphere theorem Karsten Grove 1'* and Peter Petersen 2'** 1 Department of Mathematics, University of Maryland, College Park, MD 20742, USA 2 Department of Mathematics, University of California, Los Angeles, CA 90024-1555, USA Oblatum 9-X-1992 Introduction The purpose of this paper is to present an optimal sphere theorem for metric spaces analogous to the celebrated Rauch-Berger-Klingenberg Sphere Theorem and the Diameter Sphere Theorem in Riemannian geometry. There has lately been considerable interest in studying spaces which are more singular than Riemannian manifolds. A natural reason for doing this is because Gromov-Hausdorff limits of Riemannian manifolds are almost never Riemannian manifolds, but usually only inner metric spaces with various nice properties. The kind of spaces we wish to study here are the so-called Alexandrov spaces. Alexan- drov spaces are finite Hausdorff dimensional inner metric spaces with a lower curvature bound in the distance comparison sense. The structure of Alexandrov spaces was studied in [BGP], [P1] and [P]. We point out that the curvature assumption implies that the Hausdorff dimension is equal to the topological dimension. Moreover, if X is an Alexandrov space and peX then the space of directions Zp at p is an Alexandrov space of one less dimension and with curvature > 1. Furthermore a neighborhood of p in X is homeomorphic to the linear cone over Zv- One of the important implications of this is that the local structure of n-dimensional Alexandrov spaces is determined by the structure of (n - 1)-dimen- sional Alexandrov spaces with curvature > 1. Sphere theorems in this context seem to be particularly interesting. For if one can give geometric characterizations of spheres, then one can also characterize manifold points in Alexandrov spaces (see however, Example 2.1). In [P] there is a generalization of the Diameter Sphere Theorem (see [GS]): If X is an Alexandrov space with curvature > 1 and diameter > g/2 then X is * Supported in part by the NSF ** Supported in part by the NSF, the Alfred P. Sloan Foundation and an NSF Young Investigator Award 578 K. Grove and P. Petersen a suspension. Closed hemispheres or suspensions over projective spaces, however, are examples of suspensions that have curvature > 1 and diameter n, but which are not spheres. So to get a sphere theorem for Alexandrov spaces we therefore need a stronger invariant than the diameter. The radius of a metric space X is defined as: rad X = minp~x maxq~x d(p, q), where d(, ) denotes the distance function on X (see [GP1], [GM] and [SY] for other results using the radius concept). If therefore radX > r then for every point p~X there is qeX such that d(p, q) > r. With this behind us we can now state our main result as. Main theorem. Let X be an n-dimensional Alexandrov space with curvature > 1 and radius > n/2 then X is homeomorphic to the n-sphere S". This theorem is optimal in the sense that the radius condition cannot be relaxed to a condition on diameter or to the condition that radius > n/2. To see this just note that the above mentioned examples have radius = n/2 (see also Sect. 2 for more examples). In [GW] two different optimal sphere theorems for n-dimensional Alexandrov spaces with curvature > 1 are proved. In the first one it is assumed that Pack,+2(X) > n/4. This condition is easily seen to imply our radius condition, so in this case our result is more general. In the second result it is assumed that X has no boundary and that Pack,(X)> n/4. These conditions are, however, neither stronger nor weaker than our assumption. The Main Theorem implies another result about Alexandrov spaces with large 1-systole. The 1-systole sysl (X) for a compact connected Alexandrov space X is the length of the shortest closed non-contractible curve. The existence of such a curve follows from the fact that X is a compact ANR. The 1-systole of real projective space for instance is n, while other space forms have smaller 1-systole. Corollary. Let X be an Alexandrov space with curvature > 1 and 1-systole > n/2. Then the universal cover of X is homeomorphic to a sphere and the fundamental group of X is cyclic of order 2 or 3. Note that if a cyclic group of order k acts freely by isometries on the standard sphere then the 1-systole of the quotient space is 2n/k. The corollary is therefore optimal, although it still leaves open the question of what exactly X can be (see [W] for related results). The rest of the paper is divided into two sections. In the first section we prove the Main Theorem and the corollary. In the second section we give some general procedures for constructing Alexandrov spaces with curvature ~ 1 and big radius. The abundance of examples we construct will show that it is hard to get a grip on those spaces which have diameter or radius equal to hi2. This is somewhat in contrast with the manifold situation, where all Riemannian manifolds with sec- tional curvature > 1 and diameter = n/2 have been classified (see [GG1]). 1 The antipodal map For the rest of this section we will assume that X is an n-dimensional Alexandrov space with curvature > 1 and radius > ~/2. If we fix a point peX then our radius condition implies that the complement to the open ~/2-ball around p, C(p)= X - B(p, ~/2) is a closed set with non empty interior. Toponogov's Distance (or Angle) Comparison Theorem (see IT, P1, BGP]) then shows that C(p) is convex A radius sphere theorem 579 and that the distance function d(p,. ) is concave on C(p) and strictly concave on the interior of C(p). Thus there is a unique point A(p) at maximal distance from p. Note that the map p ~ A(p) is clearly continuous. We call this map the antipodal map for X, since on the standard sphere A is the antipodal map. It follows from [P] that X is the suspension over 2~a~p) for all p. All we need to do, to show that X is a sphere, is to make sure that A(p) is a manifold point (see also Example 2.1). Because then X itself becomes a manifold, since it is a suspension and spheres are the only manifolds which are also suspensions. We prove that A(p) is a manifold point for some p by contradiction. Denote by S c X the set of non manifold points in X, thus we assume that A : X ---, S. It follows from the topological stratification of X described in [P] that S is a closed subset of dimension < n - 1. Furthermore the dimension can only be n - 1 if X has boundary. Now any Alexandrov space with curvature > 1 and non empty boundary must have radius < n/2 (see e.g. [P-l). Thus our radius condition implies that dim S < n - 2. We will show in Lemma 1 and 2 below that the Alexander-Spanier cohomology group H"(X, 7/2) = 7/2 and that A 2= A.A is homotopic to the identity map id:X--,X. Thus A 2 induces a map on/4"(X, 7/z), which on one hand is the identity map, and on the other hand is zero since A 2 factors through S which has dimension < n. Hence we get a contradiction. Lemma 1 Let X be a compact Alexandrov space without boundary. Then X has a fundamental class in Alexander-Spanier cohomology with 7Z2 coefficients i.e. l~"(X, 7/2) = 7/2. Proof. We use Alexander-Spanier cohomology as it is described in IS]. Denote again by S the set of non manifold points. Since X doesn't have any boundary it follows from the topological stratification results in [P] that dim S < n - 2. Then X - S is a connected n-dimensional manifold. Therefore the top cohomology class with compact support satisfies/~2(X - S, 7/2) = 7/2. Now the advantage of using Alexander-Spanier cohomology is that/~"(X, S, 7/2) =/42(X - S, 712) as long as X and S are compact. Using the long exact sequence for the pair (X, S) now yields: 0 =/~"- I(S, 7/2) -~ ~q"(x, S, 7/2) --'/~"(X, Z2) --, a"(S, 7/2) = 0, because S has dimension < n - 2. Thus we get the desired conclusion. [] Lemma 2 Let X be as in the Main Theorem and A the antipodal map described above. Then A 2 is homotopic to the identity map on X. Proof All of the results on ANR's and decompositions we are going to need for the proof can be found in [D]. The sets C(p)= X- B(p, g/2) are compact and vary continuously in the Hausdorff metric. Furthermore it is proved in [P] that they are Alexandrov spaces which are contractible. Now any Alexandrov space is an ANR since it is finite dimensional and locally contractible. Thus the sets C(p) are cell-like. Consider now the decomposition G on X x X consisting of the sets {(x, y)}, where y(~C(A(x)), and {(x, C(A (x)))}. This is an upper semicontinuous cell-like decomposition, since the sets C(A(x)) are cell-like and vary continuously with x. Let p: X xX~ X x X/G be the natural projection and P2: X x X ~ X the projection onto the second factor.
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