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. Define the maps i: X ~ X x X as i(x) = (x, x) and f: X ~ X x X as f(x) = (x, A2(x)). From the construction of A we see that x, A2(x)e C(A(x)). Thus the two maps p .f and p. i are identical. But then also f and i are homotopic since 580 K. Grove and P. Petersen their domain is a finite dimensional ANR and p: X x X ~ X • X/G is a cell-like map, and hence a (fine) homotopy equivalence. Thus also A 2 = P2 .fand id = P2" i must be homotopic which is what we wanted to prove. []

Now that we know X is a sphere we can use this to get more information about the geometry of X, just as was done in [GPI] for Riemannian manifolds with sectional curvature > 1 and radius > n/2. The important results are summarized in:

Theorem 3 Let X be an n-dimensional Alexandrov space with curvature > 1 and radius > n/2, then the following statements are true: (1) The "'antipodal map" A: X--* X is onto. (2) Any ball B(p, r) where r < radX, is contractible. (3) The n-dimensional Hausdorff measure vol X > c, > 0, where c, is a constant depending only on n. (4) There is an e, > O, such that if in addition rad X > n- ~, then X is bi- Lipschitz equivalent to the standard sphere S".

Proof. (1) Suppose we have identified X with the standard sphere by some homeo- morphism. Since the map A doesn't have any fixed points it must be homotopic to the standard antipodal map. In particular A has nonzero degree and must therefore be onto. (2) If p~X then we can find xeX such that p = A(x). Hence the distance function from p has only p and A(p) as critical points. This implies that all balls B(p, r), where r < radX < d(p, A(p)) are contractible (see [P]). (3) Now that there is a bound for the contractibility radius for X it is easy to get a lower bound for the filling radius for X (in fact fillrad X __> rad X/(2n + 4)) (see [G]). Gromov has informed us that the important inequality (fillrad X)" __< C, volX is true for the spaces under consideration here. Thus we get a lower volume bound for X. (A different proof of this fact was given independently in [Pe]). (4) This is proved by contradiction. Suppose that we have a sequence Xi where radX~--* n. By passing to a subsequence if necessary we can suppose that Xi converges to a space X with curvature ->_ 1 and radius = 7z. This implies that X is isometric to a sphere (see e.g. [GP2]). The fact that all of the spaces Xi have a uniform lower volume bound implies that the limit X has dimension n = dim X~. So the sequence converges to the standard sphere of the same dimension. Then the results in [BGP] (cf. also [SOY]) show that the Xi's must eventually be bi- Lipschitz equivalent to S ".

We now proceed to prove the corollary of the introduction. Let X be as in the corollary and denote the universal cover by -~. The fundamental group rh (X) = G acts by decktransformations on )~. We know that X is again an Alexandrov space of curvature > 1 and must therefore be compact. In particular G is finite. Let now g ~ G. We will show that g can only have order 1, 2, or 3. View g as an isometry on X. Since sysl (X) > n/2 we have that d(x, gx) > n/2 for all x ~)~. This immediately implies that tad X > n/2, hence X is a sphere. Now choose x ~)~ such that the displacement d(x, gx) is minimal and let a be a segment from x to gx. By minimality of the displacement it follows that the two segments a and g(a) form an angle n at gx. If the order of g is k, then the k segments g~(a), i = 1,..., k - 1 form a closed geodesic. Suppose now k > 4 and consider the four different points x, gx, g2x, and A radius sphere theorem 58l gk-1 X. Since sys 1(X) > n/2 the distance between all pairs of these points is > rt/2. Now form the two triangles T(x, gx, g2x) and T(x, g2x, gk-tx) where the sides from x to gx and gk-IX to X are chosen to be a and gk-1 a resp. Toponogov's Triangle Comparison Theorem now implies that the angles /__(gx, x, g2x) and /__ (g2x, x, gk-lx) at x are both > n/2, since all sides in the triangles are > rt/2. This, however, contradicts the fact that a and gk- ~tr form an angle n at x. Thus we have shown that X is a sphere and that G is a finite group acting freely on )~ with the property that all its elements have order 1, 2, or 3. It is proved in [MTW] that a group acting freely on a sphere has the p2 and 2p-properties for all primes p. This means that all subgroups of order pq are cyclic. Since all non-trivial elements of G have order 2 or 3 the Sylow theorems imply that the order of G is 2k3k If k or I are bigger than 2, then the Sylow theorems tell us that G has a subgroup of order p2, where p is 2 or 3. This subgroup must be cyclic by the p2-property, p = 2 or 3, but then G has elements of order > 3, which is a contradiction. The 2-3-property again tells us that G can't have order 6 = 2.3, so G must have order 1, 2, or 3. Which is what we wanted to prove.

2 Equality discussion

The Diameter-Suspension Theorem of [P] can be phrased as follows: An n-dimensional Alexandrov space X admits an Alexandrov structure with curv X > 1 and diam X > ~/2 if and only if X is the suspension of an (n - 1)- dimensional Alexandrov space E with curv E > 1. Together with the main result of this paper, this gives a complete description of Alexandrov spaces with curvature > 1 and diameter > ~t/2 or radius > ~/2. Since both results are optimal it is natural to investigate exactly how they fail, when the strict inequality is replaced by equality. This is already quite difficult in the case of Riemannian manifolds (of. [GG1, 2]) and the strategy used there seems to work only for the first few steps. Specifically, if X is an Alexandrov space with curv X > 1 and diameter = n/2 one finds two convex subsets A, B = X at maximal distance, and X is homeomorphic to the union of e-neighborhoods D~A, D~B glued together along their common boundaries OD~A ~ OD~B. This fact uses Perelman's extension of critical point theory to Alexandrov spaces. Although the Rigidity Comparison Theorem of [GM] replaces the use of parallel transport in the Riemannian setting, it does not have the same impact. There are several reasons for this, one of the more fundamental ones being that the space of directions in a Riemannian manifold is the unit sphere, whereas it can be any positively curved Alexandrov space in general. In this connection, it is interesting to point out that if any space of directions of an Alexandrov space X with curvX > 1 and radX > n/2, is topologically a sphere, then an easy induction argument would yield our result as well. We have been informed, by Perelman, that this has indeed recently been done by Petrunin. The proof is based on a very natural, yet difficult and involved, notion of (pre)quasigeodesics (el. [Pe]). With this in mind one might hope that X is a manifold if rad X = ~/2. This, however, is not correct as we shall see below. Except for the Cayley plane CaP 2 all known examples of Alexandrov spaces Z with curv Z > 1 and diam Z = n/2 (or rad Z = r~/2) are based on the topological join construction. As noted in [GM] the join Z = X * Y of Alexandrov spaces 582 K. Grove and P. Petersen

X and Y with curvature > 1 is again an Alexandrov space with curvature > 1. Here we give an explicit description of this so-called spherical join: IfX is an Alexandrov space with curvature > 1 then the linear cone L(X) is an Alexandrov space with curvature >__ 0, such that the space of directions at the vertex is X. If we have two Alexandrov spaces X, Y with curvature > 1 then we get an Alexandrov space L(X) x L(Y), The space of directions at the point which is (vertex, vertex) is an Alexandrov space with curvature > 1 which can be naturally identified with the join X * Y. Note that X, Y are isometrically imbedded in X * Y in such a way that all points in X have distance n/2 to all points in Y. Moreover if A c X and B c Yare Alexandrov spaces isometrically embedded in X and Y, then A * B c X Yisometrically. This can be used to compute distances between points in joins in the following way. Any point in X * Y - (X w Y) has a unique coordi- nate representation as (x, t, y) where x ~ X, y a Y, t ~ (0, n/2), so if we have two points in the join let A be a segment joining the x-coordinates and B a segment joining the y-coordinates, the distance between the original points can now be computed in A *B c $3(1).= $1(1),$1(1), because A and B can be regarded as isometrically embedded in S ~ since they have length < ~. When Yis a point we get the spherical cone over X, and when Y is the two point set we get the spherical suspension over X. In all other cases Yis assumed to be connected and have dimension > 1. With this description of the join it is easy to see that:

(i) diam X * Y = max ~, diam X, diam

It is also possible to compute rad X * K however, here we only point out the following facts:

(ii) radX, Y> ~ ~ min{radX, rad Y} >

:g 7"C (iii) radX, Y< ~ ,~ max{tad X, rad Y} < ~.

Note in particular that the classes of spaces with curvature > 1 and dia- meter = ~/2 and/or radius = 1t/2 is closed under the join operation. Another important feature of the join operation is that if a compact group G acts by isometries on both X and Ythen G also acts by isometrics on X Y. Thus we can form the quotient space (X, Y)/G which generically is different from (X/G)*(Y/G). This is interesting because: (iv) diam(X Y)/G ~ (v) rad(X * Y)/G > rad(X/G) (Y/G),

Here (iv) is trivial, whereas (v) follows from the Rigidity Distance Comparison Theorem of [GM]. The above constructions yield numerous examples of spaces with curvature > 1 and diameter = zr/2 or even radius = 7r/2. For the latter, let us point out the following simple but interesting explicit example: Let M 7 be one of the Wallach examples M = SU(3)/S t (see [AW]). Then M admits an S 1 action with M/S 1 = SU(3)/T 2 = N 6 the flag manifold. From the above inequalities we derive that rad X = ~/2 for X = (S 2"+~ M7)/S 1. Note that here the convex sets A = flIP" and B = N 6 are manifolds but X is not. Similarly rad Y = n/2, where Y = ((S 2" + 1, M 7), ($2,+ t, M 7))/S 1. Here the convex sets are both the space X from above. A radius sphere theorem 583

These examples illustrate some of the difficulties we discussed above in applying the methods of [GG1, 2] to Alexandrov spaces. There is another complication with Alexandrov spaces which is worth mention- ing. Let X be an Alexandrov space and p ~ X. If Zp is a sphere then we know that p is a manifold point. The converse, however, is not true.

Example 1 The Double Suspension Theorem of Edwards says that the double suspension of any homology sphere is a manifold and hence a sphere (see [D]). Let therefore M be any homology sphere which is also a space form of constant curvature 1 (e.g. M = Poincare's 3-dimensional homology sphere). The spherical suspension SM is an Alexandrov space of curvature > 1, but it is not a manifold. The double spherical suspension SZM = S 1 M is a manifold and an Alexandrov space of curvature > 1, but some points have SM as a space of directions.

Acknowledgements. The authors would like to thank M. Bestvina, R. Edwards, and S. Ferry for helpful conversations concerning the topological aspects of this paper Their advise has greatly simplified the topological aspects of our proof. We would also like to thank F. Wilhelm for bringing the 1-systole problem to our attention.

References

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