Journal of Mathematical Sciences, Vol. 94, No. 2, 1999

M A N I F O L D S OF A L M O S T N O N N E G A T I V E C U R V A T U R E

Takao Y a m a g u c h i UDC 517.934

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The paper contains a systematic ezposition of results due to the author and K. Fukaya on manifolds of almost nonnegative curvature. The final part contains several conjectures and modern problems. Bibliography: 79 titles.

Introduction Basic methods in nonnegative curvature HausdonC[ convergence Alexandrov spaces and the splitting theorem Fibration theorem Fibering by the first Betti number Bounded almost nonnegative curvature Generalization of Bieberbach's theorem Solvability theorem and applications Three-dimensional case

Nilpotency theorem Generalized Margulis lemma Noncollapsing case Concluding remarks

D e d i c a t e d to P r o f e s s o r T s u n e r o Takahash i

o n his s i x t i e t h b i r t h d a y

w INTRODUCTION

In this article, we survey some results, mainly those of [77, 24], on almost nonnegatively curved manifolds. Throughout the paper, M is a closed n-dimensional Riemannian manifold unless stated otherwise. We

denote by Ricci m the Ricci curvature of M. Our concern is the topology of manifolds with almost nonnegative curvature. First recall the topological

characteristics of nonnegative curvature. At present these are the first Betti number and the fundamental group. In fact, we know the following results on these topological invariants of non_negatively Ricci-curved manifolds.

A classical result by Bochner is

T h e o r e m 0.1 [6]. Let RicciM >_ O. Then the t~rst Bet t i number bz(M) = r a n k H l ( M , Q ) is less than or equal to n, and bl(M) = n i f and only i f M is isometric to a fiat torus.

Cheeger and Gromoll [14] generalized this result as follows.

T h e o r e m 0.2 [14]. / fRicci M > O, then 7rl(M) contains a finite index free abelian subgroup of rank not greater than n.

There also exists a uniform bound on the first Betti number due to Gromov, which generalizes Bochner's result and holds even if one allows M to have negative Ricci curvature somewhere:

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 201-260. Original article submitted October 20, 1995.

1 2 7 0 1072-3374/99/9402-1270522.00 �9 Kluwer Academic/Plenum Publishers

T h e o r e m 0.3 [34]. /.t" the diameter and the curvature of M satisfy Ricci Mdiam (M) 2 > - D 2, then bl (M) < (n - 1) + C~. In particular, there exists a positive number en such that if Ricci Mdiam (M 2) > - e , , then bl(M) < n.

At this stage, this is the only known result about the topology of a manifold whose Ricci curvature is bounded from below by a negative constant. Gromov [34] proposed the following conjecture.

C o n j e c t u r e 0.4 (Gromov). There exists a positive number e, such that if Ricci Mdiam(M) 2 > - e , and bl (M) = n, then M has the topological type of a torus.

Unfortunately, this conjecture is still too difficult to attack. Our object here is the study of manifolds M which have almost nonnegative sectional curvature in the following sense:

KMdiam(M) 2 > - e

for a small positive number e. More precisely, if M satisfies the above inequality, then we say that M has e-nonnegative curvature. We say that a closed manifold M has almost nonnegative curvature if M admits a metric of e-nonnegative curvature for each positive e. For instance, the product of S 2 and a nilmanifold has almost nonnegative curvature, and any circle bundle over an almost non_negatively curved manifold also has almost nonnegative curvature. (See Theorem 2.8 for a more general example.)

Another motivation for our work is the following theorem on almost flat manifolds due to Gromov, with a modification by Run.

T h e o r e m 0.5 [30, 63]. There exists a positive number en such that if

I K M l d i a m ( M ) 2 < e,

then M is dif[eomorphic to an infranilmanifold.

Some of the main results discussed in the present paper are stated as follows.

T h e o r e m A [77]. There exists a positive number e, such that (a) i f M has e,-nonnegative curvature, then a certain f-mite cover of M fibers over a bl( M)-dimensional

torus. ( b ) / n the maxima/case bl ( M ) = n, M is diffeomorphic to a torus.

Next we describe our results on the fundamental groups of manifolds with almost nonnegative curvature, obtained in joint works with Kenji Fukaya [24, 3].

A group is called almost nilpotent (abelian, solvable) if it contains a nilpotent (abelian, solvable) subgroup of finite index. Let A be a solvable group. The length of polycycacity /:(A) of A is defined as the smallest integer s such that A admits a filtration

A = A0 D h i D . . . D As = {1}

in which each Ai/Ai+x is cyclic.

T h e o r e m B [24]. There exist positive numbers ~,, and c, such that i f M has e,-nonnegative curvature, then 7h ( M) is almost nilpotent and contains a solvable subgroup A satisfying

: A] < c . , (1)

c ( h ) ___ (2)

This extends Theorem 0.5 to the rl-level, and settles a conjecture stated in [31]. The finite-index subgroup of 7h(M) constructed in Theorem B can be generated by n elements and has

degree of nilpotency not greater than n, However, we have no uniform bounds on the index of the nilpotent subgroup in terms of the dimension n, although this seems to be possible. The theorem says that we have

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such a uniform bound for a solvable subgroup. Note that Theorem B is still new even for nonnegatively curved manifolds (see Remark 1.3).

The significance of the study of almost nonnegatively curved manifolds consists in the simple fact that a manifold collapses to a point while keeping a lower curvature bound, say KM > --1, if and only if it has almost nonnegative curvature. As we will see later, in the general situation where a manifold collapses to a lower-dimensional space under a lower curvature bound, an "almost nonnegatively curved space" arises as fiber of a certain fibration (see Theorem 4.1). Theorems A and B actually hold for such a fiber.

The proof of Theorem B allows us to generalize it to the class of manifolds with a lower curvature bound.

T h e o r e m C [25]. Given n and D > 0, there exist only t]nitely many finitely represented discrete groups F1, ' . . ,Fk such that i f an n-dimensional manifold M satisfies K M d i a m ( M ) 2 > - D 2, then for some i, 1 < i < k, there exists an exact sequence of the form

1 , A , 7h(M) ~ Fi , 1,

where A is an almost nilpotent group satisfying the conclusion of Theorem B.

In the case D = e. , the group l"i should be trivial by Theorem B. Theorem C says that the set of all isomorphism classes of fundamental groups of manifolds with fixed lower bounds on the sectional curvature and upper bounds on the diameter is finite modulo almost nilpotent subgroups. In a noncollapsing case, M. Anderson [2] proved the finiteness of the set of all isomorphism classes of fundamental groups of manifolds with fixed lower bounds on the Ricci curvature and the volume and an upper bound on the diameter (Theorem 12.1).

Our discussion using the Hausdorff convergence gives some new results even for nonnegatively curved manifolds (see See. 8), and one will recognize almost nonnegatively curved manifolds as natural objects of study.

The organization of the paper is as follows. In Sec. 1, we recall two basic methods in the study of nonnegative Ricci curvaturc the Bochner technique

and the Cheeger-Gromoll splitting theorem. Outlines of the proofs of Theorems 0.1 and 0.2 are also given. Our basic argument is the Hausdorff convergence introduced by Gromov [34]. We recall the fundamental

properties of this notion in Sec. 2. We give some examples of Hausdorff convergence and recall some basic facts on the pointed equivaxiant Hausdorff convergence, which we need later in the study of the first homology groups and the fundamental groups of almost nonnegatively curved manifolds.

In Sec. 3, we discuss Alexandrov spaces, which arise as the Hausdorff limits of manifolds with a lower sectional curvature bound. We give a proof of the splitting theorem for Alexandrov spaces with nonnegative curvature, which is one of our basic tools. As an application, we prove the Lie group property of the isometry group of an Alexandrov space proved in [25].

In Sec. 4, we discuss the fibration theorem, which is another basic tool. We give a proof of this theorem along the lines of [78], and discuss the properties of the fiber of a fibration.

The proof of Theorem A is given in Sec. 5. The fiber bundle version of Theorem A was proved in [77]. Here we mainly prove the case where M is almost nonnegatively curved.

In Sec. 6, we discuss almost n0nnegative Ricci curvature under the stronger assumption IKI _< 1. Under the additional assumption, one can apply Bochner's technique.

To prove Theorem B, we generalize the Bieberbach theorem in Sec. 7, and the solvability part of Theo- rem B is proved in Sec. 8.

In the proof of the nilpotency part of Theorem B, we need the notion of the covering space along the fiber introduced in [23]. After considering the three-dimensional case in Sec. 9 as an introduction, we give an outline of the proof of the nilpotency part in Sec. 10.

Theorem B is extended in Sec. 11 to a generalized Margulis lemma, which implies Theorem C. In Sec. 12, we consider the special case where manifolds have a lower volume bound. In this case, the

structure of manifolds is similar to that of nonnegatively curved manifolds. In the final section, Sec. 13, we state some conjectures and related arguments.

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w BASIC METHODS IN NONNEGATIVE CURVATURE

Before proceeding to the study of almost nonnegative curvature, we recall basic methods in the study of nonnegative curvature and give outlines of the proofs of Theorems 0.1 and 0.2.

There are two basic methods in the study of the topological structure of closed manifolds with nonnegative Ricci curvature. One is Bochner's method, which is analytic, and the other is Cheeger-Gromoll's method, which is more geometric.

First we recall Bochner's method. Let w l , . . . , wb~ be harmonic 1-forms on a closed Pdemannian manifold M, forming a basis of the first de Rham cohomology. Then the Albanese map A: M --* T b~ is defined by

A(x ) = w l , . . . , wb, ,

where p E M is fixed and

T b~ R~

{(f.roJ1,...,f.roJbl) "/E ~I(M)}" The method is based on the following Weitzenb6ck formula.

T h e o r e m 1.1. For every 1-form on M , we have

(m~,~) = ~/XI~I 2 + IDol 2 + mcci(~, ~w).

P r o o f of T h e o r e m 0.1 is as follows. If the Pdcci cuvature of M is nonnegative and w is harmonic, the the above relation implies

M [Dw[2 dx < O.

Hence IDwl - 0 and ~ must be parallel. Thus we have bl = b l (M;Q) < n and the Albanese map A : M --* T b~ is a Riemarmian submersion. In the maximal case bl = n, it follows that M is isometric to a fiat torus.

Next we recall the Cheeger-GromoU method, which is based on the following splitting theorem.

Theorem 1.2 [67, 14]. Let N be a complete manifold with nonnegative Ricci curvature, and suppose that it contains a line. Then N is isometric to the product No x R .

Now we can prove Theorem 0.2. Let M be a closed manifold with nonnegative Ricci curvature. By Theorem 1.2, M is isometric to the product M0 x R k, where we can assume that M0 is compact. Since the fundamental group I' of M preserves the splitting, we have a homomorphism T: r ~ Isom(Rk), and hence an exact sequence

1 , K , F ,qa(F) , 1,

where K is the kernel of ~. Note that K is finite. By the Bieberbach theorem (cf. [72]), qa(F) contains an abelian subgroup of finite index. Thus, F must be almost abelian. (See Lemma 7.2.) This completes the proof of Theorem 0.2.

R e m a r k 1.3. Theorem 0.2 grasps only the nontorsion part of the fundamental group. As the lens spaces S3 /Zp show, there is no bound on the index of the free abelian subgroup. However, by taking a solvable subgroup A instead of an abelian subgroup, we obtain a uniform bound on the index of A (Theorem B). This is shown in Sec. 8 for almost nonnegatively curved manifolds (see Theorem 8.1).

w HAUSDORFF CONVERGENCE

In our argument, we will use as a basic method the notion of Hausdorff distance introduced by Gromov

[34]. In this section, we present some basic properties related with the Hausdorff distance. We begin with

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Defin i t ion 2.1. A (not necessarily continuous) map f : X ~ Y between two metric spaces X and Y is called an �9 approximation i f

(1) Id( f (x) , f (y ) ) - d(x,y)[ < �9 t'or ali x , y e X , and (2) the �9 of f ( X ) covers Y.

The Hausdorff distance dH (X, Y ) is defined as the infimum of �9 such that there exist �9 approx- imations X ~ Y and Y ~ X.

The Hausdorff distance actually defines a distance on the set of all compact metric spaces. This metric is not useful for unbounded spaces, but the notion of pointed Hausdorff distance is effective. For pointed metric spaces (X ,p) and (Y, q), the pointed Hausdorff distance dp.H((X,p), (Y, q)) is defined as the infimum of �9 such that there exist �9 approximations

f : Bp(1 / �9 --+ Bq(1/ �9 + e, Y ) and g: Bq(1/�9 Y ) --* B , ( 1 / � 9 + � 9

between the metric balls, such that f (p) = q and g(q) = p.

Defin i t ion 2.2. The dilatation of a Lipschitz map f : X ~ Y is deigned as

d( f (x) , f (y)) di l ( f ) = sup

�9 ~ y e x d(z, y)

We say that f is an �9 if f is a bilipschitz homeomorphism and

I log(dilf)l + I log(dilf-1)[ < �9

Now the Lipschitz dintance dL(X, Y) between X and Y is defined as the infimum of �9 such that there exists an e-isometry between X and Y. By definition, dL(X, Y) = oo if there are no bilipschitz homeomorphisms between X and Y.

The pointed Lipschitz distance dp.L((X,p), (It, q)) is defined as the infimum of �9 such that there exist two mappings

f : B , ( 1 / e , X ) --* Y and g: Bq (1 / � 9 --+ X

which are �9 onto their images and f(p) = q, g(q) = p. Here are some simple examples.

E x a m p l e 2.3. (1) As �9 --+ 0, the product Sx(�9 x X tends to X with respect to the Hausdorff distance. (2) Let L C R 2 be the graph defined by

L = {(z ,y) lz or y is an integer},

and let d be the induced length metric on L. Then as �9 ---* O, (L, ed) tends to the norm space ( R 2, H" H) with respect to the Hausdorff distance (here I](x, Y)I] = Ix[ + [yD-

(3) For an arbitrary Riemannian manifold (M, g) of dimension n and for any p E M, the scaled metrics ((M, g/e), p) tend as �9 --* 0 to the fiat Euclidean space ( R n, 0) with respect to the pointed Lipschitz distance.

Why is the Hausdorff distance useful? The reason is the Gromov precompactness theorem.

T h e o r e m 2.4 [34]. Let k be an arbitrary t ea /number and let D > 0. Then (1) the set of all dosed n-dimensionM Riemannian manifolds M with Ricci M _~ k and d i am(M) <_ D is

relatively compact with respect to the Hausdorif distance; (2) the set of all pointed complete n-dimensionM Riemannian manifolds (M, p) with Ricci M _> k is

relatively compact with respect to the pointed Hausdorff distance.

Thus, any sequence Mi, for instance, with geometric bounds of Theorem 2.4 (1), has a subsequent�9 Mj converging to a compact metric space X. It is not difficult to see that the limit X is a length space. However, in general, it is not even a topological manifold.

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Let M be as in Theorem 2.4 (1). Then by the Bishop-Gromov volume comparison theorem [34], we have a uniform upper bound C(n, k, D)e-" for the number of disjoint e-balls in M. This is the key to the proof of Theorem 2.4. It immediately follows that

the Hansdorff dimension dims X of the limit

Xis less than or equal ton ,d imH X _< n. (2.5)

Let/3 be a set of geometric bounds on some Riemannian invariants. We say that a sequence of Riemannian n-manifolds Mi collapses to X under/3 if the following conditions are fulfilled:

(1) Mi satisfies/3,

(2) limi--oo d s ( U i , X) = 0, (3) dimH X < n.

A smooth manifold M is said to collapse under/3 if it admits a sequence of complete Riemannian metrics gi such that (M, gi) collapses to a space under/3.

Here are some basic questions.

Ques t ion 2.6. (1) What can one say about the singularities of X ? (2) What can one say about topological relations between Mi and X ? (3) Which manifolds collapse under prescribed geometric bounds/3 ?

In the case where/3 = {IKI <- 1,diam ___ D}, Fu ya [19, 20] studied questions (1) and (2). In this case, the limit X has a nice stratification in the Cl'a-metric category. Under the bound 73 = {IK[ <_ 1}, Cheeger and Gromoll [16] and Cheeger, Fulmya, and Gromov [12] studied question (3). If one assumes only the lower curvature bound K > -1 , we know that X is an Alexandrov space, whose class is discussed in the next section.

Next we exhibit an example of collapsing under a lower curvature bound. Let G be a compact connected Lie group. Clearly, G with bi-invariant metrics collapses to a point under the lower curvature bound zero. This can be generalized in the following form.

T h e o r e m 2.7 [77]. Let G act on a compact manifold M, and let g be a G-invariant metric on M. Then M collapses to the quotient space (M, g)/G under a lower sectional curvature bound.

For a generalization of Theorem 2.7, see [61]. To give a specific example related with the above theorem, consider the circle action on S 2"+2 defined

as follows: fix a great hypersphere S 2n+l C S 2"+2. Then let the circle act on each hypersphere parallel to S z"+l as the Hopf fibration. This defines a smooth circle action on the sphere S 2"+2. By Theorem 2.7, we can find a sequence of metrics on S 2"+1 converging to the quotient $2"+1/S 1, the suspension over the complex projective space CP n, which is not a topological manifold (cf. [38]).

When a sequence Mi of Riemannian manifolds converges to a manifold of smaller dimension under a lower sectional curvature bound, we can describe the topological relation between Mi and the limit (see Theorem 4.1).

In a way similar to Theorem 2.7, we have the following family of almost nonnegatively curved manifolds.

T h e o r e m 2.8 [24]. Let F ~ M -+ N be a ~ber bundle with structure group G, which is a compact Lie group, such that

(1) N has almost nonnegative curvature, (2) F has G-invariant metric os nonnegative curvature.

Then M has almost nonnegative curvature.

Since we need to understand the convergence of isometric group actions in later sections, here we present the definition and some properties of the pointed equivariant Hausdorff distance, which was introduced by

Fukaya [18].

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Def in i t ion 2.9. We say that a triple (X,F ,p ) belongs to Ad~q i f every metric ball in X is reIativeIy .compact, p E X , and F is a dosed subgroup o/ ' Isom(X), the group of isometrics of X . For R > O, put

F(R) = {7 e r ld(Tp, p) < R}.

For (X, r,p), (Y, A, q) ~ M~q, we say that a triple ( f , qo, r represents an e-pointed equivariant Hau~dorff approzimation (X, F, p) ~ (Y, A, q) if

(1) f : Bp(1 /e ,X) ~ Bq(1/e + e ,Y) is an e-Hausdorff approximation with f (p) = q; (2) ~ : F(1/e) ---, h and r h ( 1 / e ) r satisfy the following conditions: (2.1) i f 7 E F(1/e) and z, Tz E Bp(1/e ,X) , then d ( f (Tx~p(7 ) ( f x ) ) < e; (2.2)/f~ E A(1/e) and z,r e Bp(1/e,X), then d( f (r < e.

Now the pointed equivariant Hausdorff distance dp.~.H((X, F,p), (Y, A, q)) is defined as the infimum of such that there exist e-pointed equivalent Hausdorff approximations (X, F,p) ~ (Y, A, q) and (Y, A, q) (x,r,p).

When the pointed eqnivariant Hausdorff distance between (X, F, p) and (Y, A, q) is small, the definition says that the F-action on X is close to the A-action on Y via the Hausdorff approximation f . This implies

P r o p o s i t i o n 2.10 [18]. / f a sequence (Xi, Fi,Pi) converges to ( r , A, q) with respect to the pointed equi- variant Hausdor-ff distance, then the quotient space (Xi/Fi,Pi) converges to (Y/A,~) with respect to the pointed Hausdorff distance.

E x a m p l e 2.11. Let 7i be the autoisometry of 1~ 3 defined by

7~(~, y, z) = (R(1 / i ) (~ , y), z + 1/ i ~),

where R(O) denotes the rotation on the (x, y)-plane around the origin about the angle 0, and let Fi be the group generated by 3q. Then (R 3, Fi, 0) converges to (R 3, S 1 x R, 0). Note that the limit depends on the choice of the reference points. For instance, if we take pi with d(O,pi) = i as the reference points, then (R3,Fi ,pi) converges to ( R 3 , R x Z,0).

When the spaces converge, one can always construct the limit of groups as follows:

T h e o r e m 2.12 [77, 24]. Let (Xi ,F i ,p i ) E Meq and assume that (Xi,pi) converges to (Y,q) with respect to the pointed Hansdorff distance. Then there exists a dosed subgroup A C Isom (Y) such that a certa/n subsequence ( X i , r i , p i ) converges to (Y, A, q) with respect to the pointed equivariant Hausdorff distance.

Note that A may be a continuous group even if the groups r i are discrete.

Proof. For simplicity, we give the proof in the case where Y is compact. In this case, the pointed Hausdorff convergence (Xi ,pi) --~ (Y, q) coincides with the Hausdorff convergence X~ --} Y. For each positive integer j , take a finite set ~ j C Y such that

(1))"]'i C ~']~jq-x, (2) the union U ~J is dense in Y.

Let fi : Xi -+ Y and gi : Y ~ Xi be e/-Hausdorff approximations such that d(gi , f ix , x) < 2ei for all x E Xi, where ei = dH(Xi, Y ) ~ O. We consider the set Ai(j) consisting of all elements of the form ~ = fi o 7 o gi restricted to ~j , where 7 runs through Fi. Since Ej is finite, for i sufficiently large compared with j , Ai(j) is a subset of the compact metric space

f ~ ( J ) = { q v : E j ~ Y I 2 - 1 < d ( ~ x ' T y ) d(x,y) _ foral l x, y E ~ j }

equipped with the L~176 Thus for each fixed j , after passing to a subsequence, we may assume that (the closure of) AI(j) converges to a compact set A(j) C/Z(j) with respect to the (classical) Hausdorff distance in s Note that each element of A(j) is an isometric embedding of Ej into Y. By the diagonal argument, we have a sequence A(1), A(2) , . . . such that for j < k, the set A(j) is contained in the restriction A(k)l~, j . Now

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we can define the direct limit A = limj-oo A(j). Since each element in A, which is an isometric embedding of Ej into Y, extends to an isometry of Y, we can consider A as a closed set of Isom (Y). Furthermore, by the choice of fi and g~, it is easy to see that A is a group. The convergence (X~, Fi) ~ (Y, A) follows from the construction. []

To prove Theorem B, we first prove that the fundamental group of an almost nonnegatively curved manifold is almost solvable. Recall that a solvable group is the result of several extensions of abelian groups. The following result plays an essential role in finding such extensions (see Sec. 8).

T h e o r e m 2.13 [24, 25]. Let (x i , ri ,pi) converge to (Y,G,q) and let G' be a normaJ subgroup of G, and suppose that

(1) G / G' is discrete, (2) Y/G is compact, (3) Xi is simply connected and the action otfFi iS free and properly discontinuous, (4) there exists a positive number tto such that G' is generated by G' ( Ro ). Then GIG' is finitely presented and for each i there exists a normal subgroup F~ of Fi such that (5) (gi, F},pi) converges to (Y, G', q) for a subsequence, (6) r , lr: is isomo hic to v i a ' su ciently large i, (7) for every e > 0, the group F~ is generated by F~(R0 + e) for sufSciently large i.

In the case where G' is the identity component of G, the group F~ constructed above is called the collapsing part of F~. For instance, in the convergence (R 3, Fi,pi) --~ (R 3, R x Z, 0) of Example 2.11, the group F~ generated by 7i is the collapsing part. However, as the following example shows, this is not always possible if we make no assumptions in Theorem 2.13.

E x a m p l e 2.14. Consider the product R x (S 3, igo), where go is the standard metric on S 3. Let S ~ C SO(4) be a subgroup freely acting on S 3. Let 7i be the isometry of X defined by

7i(x, y) = (x + i -2, O{2y), where Ot = e 2'rv'-zY/t e S 1 �9

Let Fi be the group generated by 7i. Then we can check that (Xi,F{,pi) converges to (R4, Z x R,0). However, there are no subgroups of Fi converging to 0 x R.

The proof of Theorem 2.13 is too long to be presented here. We describe only the construction of F~. Let R be a sufficiently large number compared with max{D, R0}, where D is the diameter of Y/G.

Let (fi, ~oi, r represent an ei-pointed equivariant Hausdorff approximation ( Xi, F{, pi) ~ (Y, G, q), where lim{--.oo ei = 0. Then we have a map Ti : ri(1/e~) --, G(1/ei). Put

r~(R) = (7 C r,(R) l ~,(7) e G').

Then the required F~ is defined as the group generated by F~(R).

w ALEXANDROV SPACES AND THE SPLITTING THEOREM

As explained in w the Hausdorff limit of a sequence of Riemannian manifolds with a lower bound on the sectional curvature is an Alexandrov space. The properties of the limit space are likely to approximate those of manifolds in the sequence. Recently Burago, Gromov, and Perel 'man [7] made some important progress in the geometry of Alexandrov spaces.

Let X be a locally compact length space. Then locally there exists a minimizing segment joining every two points nearby, which is called a minimal geodesic, or simply a geodesic. A geodesic joining z and y in X is denoted by xy. For three points x, y, z E X, we denote by A(x, y, z) the geodesic triangle consisting of three geodesics joining them. For a real number k, we use the notation ~x(z, y, z) to denote a geodesic triangle A(~', ~', z~ in M2(k), the simply connected complete surface of constant curvature k, with the same side lengths as A(x, y, z) if it exists. We also denote b y / z y z the angle between y-~ and y-~.

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Def in i t ion 3.1. Under the above notation, X is caIied an Alexandrov space with curvature greater than .or equal to k if the following condition is [ult~11ed:

(1) for any point p E X , there exists an open set U containing p such that for every x, y, z E U and any w on the geodesic yz, we have d(x, z) >_ d(~, ~), where ~ is the point corresponding to w on the side ~'~;

or equivalently, (2) for every two geodesics 7 and a going from p, let xs and yt be the points on 7 and a respectively

such that d(p, xs) = s and d(p, yt) = t. Then Zxspyt is monotone nonincreasing in s and t.

By definition, the angle between 7 and a is defined as

L(7, a) = lim Zz,pyt. 0,0-.(o,o)

It is an important property of such spaces that geodesics do not branch, which immediately follows from the definition. However, note that a geodesic may not be extended anymore at some point.

From now on we consider only Alexandrov spaces with curvature bounded from below and with finite Hausdorff dimension. We call them Alexandrov spaces.

E x a m p l e 3.2. (1) A Riemannian manifold with sectional curvature greater than or equal to k is an Alexandrov space with the same lower bound on the curvature. More generally, let Mi be a sequence of complete Riemannian manifolds with KM, > ki, lim ki = k, and suppose that (Mi,Pi) converges to (X, x0) with respect to the pointed Hausdorff distance. Then X is an Alexandrov space with curvature greater than or equal to k.

(2) Let (X, do) be a length space with diameter not greater than 7r, and consider the Euclidean cone

K ( X ) = X • [0, o o ) / X x 0

over X with distance

d((~,s),(y,~)) =(s2 Jl-t 2 -28~cosdo(x,y)) 1/2, x ,y e X.

Then K ( X ) is an Alexandrov space with nonnegative curvature if and only if (X, do) is an Alexandrov space with curvature greater than or equal to one.

(3) Let (X, do) be a length space with diameter not greater than 7r. Then the spherical suspension

s ( x ) = x • [0, x {0,

with distance cos d((x, 5), (y, t)) = cos s cos t + sin s sin t cos do (x, y)

is an Alexandrov space with curvature greater than or equal 1 if and only if (X, d0) is an Alexandrov space with curvature greater than or equal to one.

A detailed argument for Examples 3.2 (2), (3) is given in [7]. A comparison theorem of Toponogov type is still valid in Alexandrov spaces.

T h e o r e m 3.3 [7]. /.f X is a complete Alexandrov space, then the triangle comparison (1) and the angle comparison (2) of Definition 3.1 hold true for arbitrary triangles A(x, y, z) and greater than or equa/ to arbitrary minimal geodesics 7 and a in X with 3'(0) = a(0).

Before proceeding to the splitting theorem, observe the following elementary principle.

Sp l i t t ing P r inc ip l e 3.4. Let X be a complete Alexandrov space with curvature greater than or equal to - k 2, and let 3' : [ - a , a] --+ X be a minimal geodesic joining p and q. Let # >> 5 > 0 be given. Then for any

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we have Apzq >_ Zpzq > r - r~,~(5),

where B~(6) is the 6-neighborhood of 7 and

~k,~,(5) ---- const(coth k# sin_h ]g~)l/2.

In particular, i f k# is su~ciently small (e.g., k = 0), then rk,~,(6) = const(6/#) 1/2 .

Proof. We give a proof for k -- 0. The general case is considered similarly. By the cosine law, we have

d(p, q)2 = d(p, x) 2 + d(q, x) 2 - 2d(p, x)d(q, z) cos 2pxq.

Since z 6 B.~(6) by the assumption, we have

d(p, q) > d(p, x) + d(q, x) - 26.

The conclusion immediately follows from the inequalities d(p, z) > it and d(q, x) >_ it. []

Coro l l a ry 3.5. /n addition to the assumption of 3.4, suppose that X is a Riemannian manifold. Then there e~sts an open set V~ cont~_mg B~(5) \ (B,(it) U Bq(it)) such that

(1) V~ is diffeomorphic to a product W6 x [0, 1], (2) V, is contained in the ~-(&)-neighborhood of By(6) \ (Bp(it) U Bq(it)), where r(&) = rk,,,~(~) with

lim~--.oo v(&) = 0.

Proof. Consider the function f ( x ) = d(x, 3/) of distance to 7, which is differentiable almost everywhere. For any x �9 B.r(5 ) \ (Bp(it) U Bq(it)), let y �9 3' be the point closest to z. It follows from 3.4 that

7F 7[ IZp=y- < l l q = y - <

Consider the following C'-approximation dp of the distance function dp(z) = d(p, x):

_ 1 [ ! d(x,y)dy. (3.6) volBp(e) Bp(e)

By (3.6), the gradients of f and d'v are almost perpendicular, and the integral curve of ~ is contained in the ~-(5)-neighborhood of B.y(5) \ (Bp(it) U Bq(it)). Let W~ be the set of all intersections of such integral

curves of d'p with the level ~-x(it). Then the required diffeomorphism W6 x [0, 1] ~ V6 can he defined by

using the integral curves of dp. []

R e m a r k 3.7. By a recent work by Perelman [59], one can obtain a topological version of Corollary 3.5 for an Alexandrov space.

In the special case where k = 0 and a = o% we have the following result, a generalization of Theorem 1.2, which will play an essential role in our discussion.

Spl i t t ing T h e o r e m 3.8 [39, 77]. Let X be a complete AIexandrov space with nonnegative curvature. If X contains a line, then X is isometric to the product Xo x It.

Proof. Let l be a line in X. We say that another line l ~ is biasymptotic to l if

supd(l(t) ,I ') < oo and supd(l '( t) , l) < oo.

Applying the splitting principle 3.4 to p = l(eo) and q = l(--ee), we obtain

at each point x �9 X, there is a unique line l~ biasymptotic to /wi th l~(0) = x. (3.9)

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Let f = ft be the Busemann function associated with the ray l][0, er

f ( x ) = lira (s -- d(x, l (s))) .

The line l~ is characterized by the equation

f(l~(t)) = t + f (x ) . (3.10)

Next we show that the level sets o f f are total ly geodesic. (3.11)

We prove that L0 = f - 1 (0) is totally geodesic. Let r be a point on a minimal geodesic joining two points p, q E L0. Pu t x~ = l(s) and y~ = l ( - s ) for large s > 0. By the definition of f , we have

[d(x~,p) - s[ < o~, I d ( x s , q ) - sl < os, (3 .12)

w h e r e lim,__,r = 0. By Theorem 3.3, we have d(r ,x , ) > d(~',}',), where ~" E ~ C ~x(p,q,x,) is the point corresponding to r. It easily follows from (3.12) that Id(}',, r-') - s I < o~, and hence d(r, zs) > s - o~. Similarly, we have d(r,y~) > s - os. On the other hand, d(r, zs) + d ( r ,y , ) - 2s = 0, since l is a line. It follows that Id(r, z~) - s I < o~, which implies r E L0. Thus, L0 is totally geodesic.

Now consider the following situation: let two lines 11 and 12 biasymptot ic to I intersect two level sets L, and L2 of f at zi and yi (i = 1, 2) where zi = l Cl Li and yi = 12 CI Li. Put

a = d ( L 1 , L 2 ) , b i=d(x i ,Y i ) , c .=d(xl ,Y2) .

To complete the proof of Theorem 3.4, it suffices to prove that

bl = b2 and c 2 = a 2 + b~. (3.13)

Let us assume that f (L1 ) < f (L2) and prove that bl = b2. Let z~ be the intersection of the segment yll l (s) with L2 for large s > 0. Then Theorem 3.3 applied to A ( z l , y , , ll(s)) implies that

b2 = lim d(zs,y2) > bl.

Similarly, bl > b2. To prove the second half, let x2 be the point corresponding to z2 on the comparison triangle

~ ( z 1, Vx, l l (s)) and put ds = d(~'2, yl )- Since lim,_-,~ d~ = (a 2 + b, 2- )1/2, Theorem 3.3 implies that d(z2, yl ) > x/2 (a 2 + bi) . The opposi te inequality d(x2, Yl) < (a 2 + b 2)1/2 follows immediately. []

�9 r n A system of pairs of points (pi, qt)i=l is called an (m, 6)-atrainer at p in an Alexandrov space X if it satisfies the following conditions:

2pipqi > 7r - 6, (1)

2piPpj > r / 2 -- 6,

ZqiPqi > 7r/2 - 6, (2)

Zpipqj > r r / 2 - 6 , (i r j).

Note that if 6 is small, condit ion (1) shows that the segments pip and pqi form an almost minimizing broken geodesic, and condition (2) shows that those m broken geodesics are independent in a certain sense. Also note that the existence of certain independent lines imposes a strong restriction on the space of nonnegative curvature (Theorem 3.8). This is also the case if there exists an (m, 6)-strainer for a small 6. In fact, the following result can be proved by essentially using the splitting principle 3.4.

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T h e o r e m 3.14 [7]. Let X be an Alexandrov space. Then (1) the t tausdor f f dimension n of X is an integer, (2) there exists a posit ive number 6 = 6,, such that the set X6 of all (n, 6)-strained points in X is open

and dense in X , (3) each point in X~ has a smaJ1 neighborhood which is T(6)-isometric to an open neighborhood in R n,

with r(6) = rn(g), }im ~ r(6) = O.

In the recent Russian version of [7], it is proved that the Hausdorff codimension of the complement of X6 is at least two. See also [56] regarding this topic and the Cl-differentiability (in a weak sense) of X0.

The following result is closely related with Theorem 3.14.

T h e o r e m 3.15 [24, 25]. Let X be an Alexandrov space with curvature bounded from below. Then for any p0 6 X and any ri -* o% there exists a sequence pi converging to po such that for a subsequence ((X, r id) ,p i ) converges to (R'*, 0), where n is the dimension of X .

Proof. We may assume that ( (X , r id) ,po) converges to (Y0, Y0). Since II0 has nonnegative curvature, it follows from splitting theorem 3.8 that Y0 is isometric to the product R k x L, where L contains no lines. Let ol denote the pointed Hansdorff distance between ( (X , r id) ,po) and (Y0, y0)- Take a sequence ri -+ 0 such that

eiri < 1/Oi and eiri ~ o0.

Put y0 = (0,z0) E 0 x L, and let Yi E 0 x L and qi E (X , rid) be such that d(O, yi) = eiri and qi is Hausdorff close to Yi. Let pi be the midpoint of a geodesic joining p0 and qi, and xi a point in R k x L which is Hausdorff close to pi. Note that

(1) dx(Po,Pi) -+ O, (2) r idx(Po ,p i ) --~ eo, (3) dH(Bpi(eiri , (X , r id)) , Bzi(eiri , R k x L)) --* O.

We change the reference point to pi. For a subsequence, the geodesic poqi converges to a line I. There also exist k independent lines in Y1 perpendicular to 1 coming from the Rk-factor of Y0. Thus, by the splitting theorem again, Y1 is isometric to the product R TM x L1. If L1 is a point, then pi give the required sequence. Otherwise, after repeating a similar argument finitely many times, we can get the required points pi- []

Finally, we discuss the isometry group of an Alexandrov space.

T h e o r e m 3.16 [25]. Let X be an Alexandrov space. Then the group I som(X) of isometrics of X is a Lie group.

Example 3.17. Let X be the union of (infinitely many) circles Si of length 1/i (i = 1 ,2 , . . . ) such that there is a unique point p at which any two of the circles Si intersect. Then I som(X) - 1-I Z2, which is totally disconnected but not discrete. Hence it is not a Lie group. Note that X has curvature - c r at p.

In the proof of Theorem 3.16, we essentially use the following result:

T h e o r e m 3.18 [28, 75]. Let G be a topological group such that there exists a neighborhood U C G of the identity containing no nontrivial subgroups. Then G is a Lie group.

Proof of Theorem 3.16. To avoid technical complexities, we give the proof only for the special case where X is a Riemannian manifold. (Of course, this case is known as the Myers-Steenrod theorem [52].) Then the essential point in the proof of the general case becomes clear.

Suppose that X is a Pdemannian manifold of dimension n and that Isom (X) is not a Lie group. For B = B p ( 1 , X ) , by Theorem 3.18, we can take a sequence Gi of closed subgroups of G such that if we put

5i = sup{ g(x, x)19 e a i , x B},

then limi--oo 5i = 0, where 59(x , X ) = d(gz, z) . Take Pi 6 Bi and gi E Gi such that 5i = 5gi(pi). Since X is Riemannian manifold,

( (X , (1 /~i )d) ,p i )converges to (R" , 0) with respect to the pointed Hausdorff distance, (3.19)

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By Theorem 2.12, we may assume that ((X,(1/~i)d) ,Gi,pi) converges to (PJ~,G,0) with respect to the pointed equivariant Hausdorff distance. Note that G is nontrivial and compact. Now we should choose h E G and y0 E R " such that

(1) 5~(Y0, R " ) > 2, (2) if Yi is a point in (X, (1/Si)d) tending to y0, then yi is contMned in Bi. Let hi be an element of Gi tending to h under the convergence ((X, (1/Si)d), Gi,pi) ~ (R" , G, 0). Then

~hi(yi,X) >/f i (2 - oi) > 5i

for a sufficiently large i, where limoi = 0. This is a contradiction. []

For an Alexandrov space X, (3.19) does not hold. Hence we should use Theorem 3.12 to take points qi near Pi such that ((X, (1]Si)d), Gi,qi) --~ (It", G,O). Note that in this case, the limit group G can be trivial. If G is nontrivial, the above proof would cause a contradiction. If G is trivial, then one can think of Gi like a "small subgroup" of Isom(X, (1/Si)d) and repeat the above argument. Because of the Hausdorif closeness between (X, (1/Si)d), qi and (R" , 0), if we changed the reference point within some ball of fixed size around qi and rescaled the metric of (X, (1/Si)d), it would converge to (R.n, 0). Thus, a modification of the above proof would work to yield the proof in the general case.

w FIBRATION THEOREM

We call a CX-map f : M ~ N an e-Riemannian submersion if

t 'df(~)l[,[ 1t<e for all tangent vectors ~ orthogonal to the fibers.

In this section, we prove the following theorem.

T h e o r e m 4.1 [77]. Given n and/~ > 01 there exists a positive number e = e,~(#) with the following property: let M and N be complete manifolds such that

(1) KM k --1, KN k --1, and (2) dim N = n, inj (N) _> #. I f the Hausdorff distance between M and N is less than e, then there exists a locally trivial fiber bundle

f : M --~ N such that (3) f is a 7"(e)-Riemanz6az~ submersion, and (4) f is a r(e)-Hausdorffapproximation, where r(e) = r,,~(e) with lim~--.0 = 0.

For simplicity, in the proof below we assume that [KN] _< 1. The general case is proved in [78] (cf. Remark 4.21).

Let a be a positive number such that e << cr << min{1,/~}. Both e and a will be determined in the final step. Let h : R ~ [0, 1] be a smooth cut off function such that

h ( t ) = l on (-cx~,(r/10], h ( t ) = 0 on [~r, ec),

on [2o/10,8 /10], (4.2) -11 <_ h'(t) <_ O, Ih"(t)l < 100a 2.

Let L2(N) be the space of all LZ-functions on N with norm normalized as follows:

Ilfll = b(~) If(z)12dz' N

where b(cr) is the volume of a ~-ball in R". Consider the smooth map

fN : N ~ L2(N), fN(p)(x) = h(d(p,x)) (x E N).

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Let ~o : N --+ M and r : M ---* N be e-Hausdorff approximations such that d(r162 x) < 2e and d(cpCx , x) < 2e, where we can assume that ~o is measurable. Put

fM(p)(x) = h(d(p, c2(x))) (= e N),

where 1 / d(p, ~o(x)) -- vol B,(qo(x)) d(p, y)dy.

B , ( ~ ( ~ ) )

Then fM : M ~ L2(N) is a Cl-map. We need this averaging since there is no lower bound for the injectivity radius of M. The derivatives of both maps are given by

dfN(~)(x) = h'(d(p,x))~(d,), ~ C UpN,

dfM(~)(z) = h'(d(q,~(z))~(~(,)) , ~ e G M , (4.3)

where dz(-) = d(-,x), d'~,(z)(') = d'(.,~0(x)), and

~(~(~)) _ 1 [ )@, vol B , (~ (= ) ) j

S,(~(x)) \C(~(=))

where C(T(z)) is the cut locus of T(z). We easily obtain

IIfN(C(P)) - IM(P)II < const-, (4.4)

for every p E M. From now on, cl, c2, . . , denote positive constants depending only on n and #. We denote by UpN the

set of all unit vectors at p E N. Since N has bounded geometry, (4.2) and (4.3) straightfowardly imply the following:

L e m m a 4.5. The map f N is a smooth embedding whose derivative is well controlled: (1) for every ~ 6 UpN we have

Cl <~ IldfN({)ll < c~;

(2) for every p and q with d(p, q) < o we have

II/N(P) -- fN(q)ll c3 < d(p, q) < c4.

Next we s tudy the tubular neighborhood of fN(N) in L2(N) and the properties of the normal projection. We start with the following lemma.

L e m m a 4.6. For any points p, q 6 N with d(p, q) <_ a and for any ~ 6 UpN, let ~ 6 UqN be obtained by parallel translation of ~ along the minimal geodesic from p to q. Then

l~(d=) - "E(d,)l < ~ d(p,q) for every x such that a/lO _< d(p, x) <_ o'.

Proof. Put v = exp~ -1 q and w = exp~ -1 x, and let t~ be obtained by parallel translation of w along the minimal geodesic from p to q. Since [KN[ < 1, a standard comparison argument (see [8]) shows that

d(expp(v + w),expq @) < ~=1.1, d(exp,(v + w),exp, w) < (1 + o~)1.1.

(1 + r which implies that the angle between exp~ -1 and t~ is less than Hence d(expp w, expq z~) < const Ivl/~. Thus,

const Iol IZ(r w ) - Z(~',expq -1 =)1 <

The result follows immediately. []

By using Lemma 4.6, we can control how rapidly the tangent spaces of fN (N) change. We put

Tp = TfN<p)(fN(N)), Np = {fN(P) + ~l~ A_ Tp}.

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L e m m a 4.7. For every p and q with d(p, q) < a we have

C6 / ( f N ( q ) -- fN(P), Tp) < - - d(p, q).

6r

Proof. Let 7: [0, l] ~ N , where l = d(p, q), be a minimal geodesic joining p and q, and let ~, = "~(t). Then for each x E Bp(2a, N ) we have

[h(d(q, x)) - h(d(p, x)) - lh'(d(p, x))~(d~)[ = l[h'(d(7(t), x))~t(d~) - h'(d(p, x))~(d~)[

('." mean value theorem)

l 2 _ ~- ('." L e m m a 4.6),

which implies fN(q) -- fN(p) [ I

-l -- dfN(~) < -

The lemma follows immediately. []

Now we define t h e not ion of the angle between two subspaces in a Hilbert space H. Let V and W be closed subspaces of H , and let 7rv : H ~ V and 7rw : H ~ W be the or thogonal project ions. T h e n the angle between V and W is defined as

f z(y, w) = ]

O b v i o u s l y , / ( V • W • = Z(V, W),

L e m m a 4.8 .

sup L(w, zrv(w)) if WM V • = {0}, w ~ O E W

sup Z(v, Trw(v)) if VM W • = {0}, v~OEV

rr/2 otherwise.

For every p and q with d(p, q) <_ a, we have

C7 Z(Tp, Tq)Z(Np,Nq) < - - d(p,q).

cr

Proof. For any ~ E UpN, let ~" E UqN be as in Lem m a 4.6. Th en L e m m a 4.6 yields

I[dfN(r - dfn(~)ll < const d(p, q) O"

[]

Let v be the normal bundle of f N ( N ) in LZ(N). For c > 0, we put

~(c) = {(x,~) ~ ~lll~ll < c}, w ( c ) = { x + ~ l ( x , ~ ) e ~(c)}.

L e m m a 4.9. There exists a positive number k = c8/a such that W ( k ) is a tubular neighborhood of fN( N), i.e., �9 + ~ # u + v for every (~, ~) # (U, v) in ~(k).

Proof. Let z = fN(P), Y = fN(q), and suppose tha t the intersection K = Np f) Nq contains x + u = y + v. First consider the case d(p, q) < a 2. Let z E Ix" and w E Nq be such tha t (i) d(z, z) = d(z, K) , (ii) d(z ,w) = d(x, Nq). T h e n A z z w < Z(Up,Nq) , and

II~ - wll = II~ - zll sin L x z w < II~ - zll sin L(Yp , Nq)

--- cA~ t1~ - zlld(p, q) ('." L e m m a 4 .8 ) (4 .10 )

___ c~ I1~ - zll I1~ - Yll ( L e m m a 4 .5) . C 3 O"

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Lemma 4.7 implies tha t IZ(= - y, N,) - 7r/21 < c6a, and by the choice of w, we have Z(x - w, Nq) = r / 2 . It follows tha t I]Y - wll < const ~11~ - yll. Combined together with (4.10), this implies that IIx - zll > c/o, and hence I1~11 > c /o , where c = const.

Next we consider the general case. It follows from the argument above tha t

the normal exponential mapping exp" : ix --+ L2(N),

e x p ' ( = , u ) = = + ~ , is n o n s i n g ~ a r o n ~(c/o). (4.11)

Let k0 = c'/a, where the constant c' will be determined later. We suppose tha t (x, y), (y, v) E v(ko) satisfy x + y = y + v (= z). Let 7 : [0, l] ~ N be the minimal geodesic joining p and q, put c(s) = fN(7(s)), and consider the mapping

a : [ 0 , 1 ] x [ 0 , 1 l ~ L 2 ( N ) , a ( s , t ) = ( 1 - t ) c ( s ) + t z .

Since [I x - YII < 2k0, Lemma 4.5 with the triangle inequality implies that

a(s,t) E W((2c4/c3 4- 1)ko).

Hence if c' is sufficiently small, a is contained in W(c/2a). Here we need the following sublemma due to Katsuda [44].

S u b l e m m a 4.12. There exists a smooth map ~ : [0, l] x [0, 1] ---+ u(c/2a) such that

e x p " ( ~ ( , , t ) ) = ~ ( ~ , t ) a n d ~ ( s , 0 ) = ( c ( ~ ) , 0 ) .

In particular, we would have ~(s, 1) -- z, which contradicts (4.11). Thus the sublemma completes the proof of Lemma 4.9.

Proof of Sublernrna 4.12. Let T be the set of those t E [0, 1] for which there exists a lift ~ : [0, l] x [0,t] v(c/2a) of a. Clearly, T 9 0. In view of (4.11), T is open. Let us show that T is closed. For ~ , t2 E T, we have

_< -lit,-t l _ IId(exp")-~ll Ih - tel

< CIh - h h

where C is a constant independent of s and t. Therefore, if ti E T converges to to', then fft~ = if(', ti) is a Canchy sequence in the space of continuous curves in v(c/2a) with L~176 Hence, to E T. []

Now let 7r: v(k) ~ fN(N) be the projection along the fibers of the normal bundle u.

L e m m a 4.13. For every x E W(k) with zr(x) = fN and every unit vector r _k Np, we have

C9 IId~=(~) - ~11 < --I1~ - ~(x)ll.

O"

Proof. Put y = x + t ~ for small t > 0 and 7r(y) = fN(q). Let No be the affine space in L2(N) of codimension n which is parallel to Np and contains y, and let z and w be the intersections of Nq and No with the n-plane 7r(x) + Tp tangent to fN(N) at 7r(x). Then similarly to (4.10) we have

I I z -wl l ~ I ! z - Yll((N0, gp)

<_ CTIlz - ull II~(x) - ~(u)ll O"

2c7 _< llx - ,~(z)l l t1~(=) - ~ (y ) l l .

O"

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It f o l l ows - f r om y - x --- to - 7r(x) and the smal lness o f Ilz - ~ ( y ) l l << t that

I I ( ~ ( u ) - ~ ( ~ ) ) - ( v - =)11 -< I1~ - ~11

_< const I1~ - ~ (~ )11 I 1 ~ ( ~ ) - ~ ( v ) l l , o"

and so I [ const

I I ~ ( y ) - ~ ( x ) l l _ 1 < ~ l l ~ - ~ ( x l l l , t o-

~- ~ < ~ o I1~ - ~-(~)11.

Letting t --~ 0, we obtain the conclusion. []

If e << a, then f M ( U ) is contained in W(k) by inequality (4.4) and the map f = fN 1 oTrofM : M --+ N is defined. Inequality (4.4) also shows that d(fx , Cx) < const e, and hence f is a Cl0e-Hausdorff approximation. To prove that f is a fibration, it suffices to prove the following lemma.

L e m m a 4.14. For every p E M and ~ E UI(p)N, let ~ ~ UpM be the velocity vector of a minimal geodesic from p to ~(exp/(p) a~). Then we have

Idfff) -E l < ~(~)+T(do) -

R e m a r k 4.15. Here the constant on the right-hand side can be expressed in the form

o ( # ~ ) + o ( v ~ ) + o ( ~ ) .

However, to avoid technical complexities, we will not perform this explicit calculation. To prove Lemma 4.14, we need the following triangle comparison lemma.

L e m m a 4.16. Let A(x , y, z) and A(~, Y, 2) be triangles in M and N respectively such that

d ( ~ ( ~ ) , ~ ) < c,o~, d (~ (~) ,y ) < c10~, d(~(~), z) < c,0~.

Suppose that

Then we have

dr

L < d(~, y) < ~ a n d / - 6 < d(x, z) < ~. 10 . . . .

[Zyzz - ZO~l < r ( a ) + r ( e / a ) .

Assume for a moment that Lemma 4.16 is proved and prove Lemma 4.14. For every ~ such that a/lO < d(f(p), ~2) <_ cr, let ~ E UI(p)N and r 1 E UpM be the velocity vectors of minimal geodesics from f(p) to 5 and from p to ~(x) respectiyely. Then Lemma 4.16 yields

IL(s q) - Lff, ~)1 < T(~) + ~(d~).

It follows that

[[dfM(~) -- dfN(~)[[ 2 = b(~. i {h'(d(p,~(x)))~('d~(=)) - h'(d(f(p), x))~(d=)}2dx,

N

where const

I h ' ( d ( p , ~ ( x ) ) ) - h'(d(/(p),x))l < G

('." (4.2)),

(4.17)

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and

Thus we have

Lemma 4.13 then implies that

Idd~( , ) ) - ~ ( d : ) l < r (a ) + ~(~/~) ( v Lemma 4.16).

I l d f M ( Q - dfN(()ll < r(a) + ~'(do)-

IId~ o dfM(~) - df(,~)ll < -,-(o-) + ~-(do-).

Now Lemma 4.14 follows from Lemma 4.5.

Proof of Lemraa 4.16. Put

= d(x, y), t = d(x, z), ~ = d(~ ,0) , ~ = d(~, ~), 0 = Zy~z,

By the assumption, KM ~ --1. Toponogov's comparison theorem implies that

d(y, z): < s 2 + ? - 2~t cos e + 0 ( ~ 4).

Since N has bounded geometry,

[d(O, ~,) _ (,~:2 + y _ 2~-cos 0)1 < O(a4).

The inequality Id(y, z ) - d ( 9 , e)l < 2c10e implies

0 > 0 - , ( o ) - . ( , / o ) .

Next take the point ~ E N such that

40, ~) = d(Y, x) + d(~, ~), d(~, ~) = a,

and put w = ~o(~). Let 0* = / z x w . Then in the same way as (4.17) we have

0* > ~ - 0 - T(~) - ~(~/~) ,

and

0 = Zyzz.

(4.18)

(4.19)

z y ~ > ~- - r(o-) - ~ - (do) .

The last inequality implies that I 0 + 0* - ~l < v(a) + r (e /a) .

The result follows from (4.18) and (4.19). []

Finally we prove that f is an almost Riemannian submersion. For any p E M, set { ( )~ a~ = ~ (4.20) Hp = ~ E Ut, M [ d p, expv]-- 6

which can be considered as the space of horizontal directions. By Lemma 4.14, dfp induces a r-Hansdonef approximation between Hp and Sn-x(1) = Uf(p)g, where v = r (a) + r(e/a). This implies that if 77 e UpM satisfies [((~,Hp) - 7r/2[ < r, then [((r/,~) - 7r/2[ < v for all ~ e Hp. In view of (4.2) and (4.3), we have [df(q)[ < r, and hence f is a r-Riemannian submersion, as required. This completes the proof of Theorem 4.1. []

R e m a r k 4.21. The embedding technique used here is originally due to Gromov [34]. Katsuda [44] overcame some gaps in [34, Chap. 8].

The fibration theorem for [KM[ _< 1 was proved by Fukaya [19]. He proved that in that case the fiber is diffeomorphic to an infranilmanifold, which generalizes Theorem 0.5. The proof presented here comes from that in [78], where an extension of Theorem 4.1 to Alexandrov spaces is discussed.

R e m a r k 4.22. It is proved in [77] that the fiber of f has v(e)-nonnegative curvature in a certain generalized sense, which can be formulated in terms of deviation of the fibers from the total geodesicity. (Note that if the fiber is totally geodesic, then because of Theorem 4.1 (3) it has ~-(e)-nonnegative curvature in the usual sense.) In what follows, we present a weaker version of this fact by making use of Splitting Theorem 3.8.

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P r o p o s i t i o n 4.23. For given m >_ n, # > O, and 0 < p < 1, there exists a positive number e = em,n,~,(p) satisfying the following condition:

let M and N satisfy the assumptions of Theorem 4.1 for# and e with d i m M = m, and let f : M --* N be the fibration constructed there. Let F be a fiber o f f with diameter 6F, and d the distance in M restricted to F. Then we have the following:

(1) i f7 is a minimal geodesic joining x, y E F with d(x, y) >_ p6y, then the angle between 7 and F is less than

(2) there exists an Alexandrov space X with nonnegative curvature such that

dH((F,d /6F) ,X) < T(e),

where r(e) = rm,.,.,a(e) with lira,--.0 r(e) = O.

Proof. This is proved by contradiction. If the proposition is not true, we have sequences of m-manifolds Mi and n-manifolds Ni with dl-l(Mi, Ni) < ei --* 0 satisfying the assumption of Theorem 4.1 such that the r(ei)-Riemasmian submersions fi : Mi ~ Ni constructed there do not satisfy the conclusion for some p < 1. Namely, there exists a point qi E N~ such that the fiber Fi = f~-a(q~) satisfies the following condition:

(1) for some xi, yi E Fi with d(xi, yi) > p6f, the angle between Fi and a minimal geodesic 7i joining xi and Yi is greater than a positive number 00 independent of i, or

(2) there exists a positive number c such that dH((Fi,d/6F~),X) > c for any Alexandrov space X with nonnegative curvature.

Put 7/= +if0) and let qh E H ~ be such that

= z ( , , H , , ) ,

where Hx, is as in (4.20). Let ~ = df(,)/ldf(,)l, and let { E U,,Mi be the velocity vector of a minimal geodesic ci joining xi and the point zi with fi(zi) = exp ai~, where ai is a positive number, ei << ai << #, as in the proof of Theorem 4.1. Then Lemma 4.14 implies that Z(~, ~?h) < 7"(ei).

Now consider the scaling of the metrics: let

gi = gM,/~F~ and hi = gN~/~Fi,

where gM,, gN, are the original metrics. We denote by di the distance function of (Mi,gi). Let wi be the point on ci with di(xi,wi) = 1. Suppose that l imdi(xi ,yi) = s. Since the angle between 7i and ci is less than r /2 - 00/2, Toponogov's theorem yields

di(Yi, wi) < (1 + s2) 1/2 -- c,

where c is a positive constant depending only on 00. For a point Pi E Fi, we may assume that (Mi,gi ,pi) and (Ni, hi,qi) converge to (X, xo) and (Y, yo)

respectively. By using the inequality inj (Ni) >_ #, one can verify that Y is isometric to R " . Noting that X is a complete Alexandrov space with nonnegative curvature, we see from Theorem 3.8 that X is isometric to the product X0 x R k. Note that the Lipschitz maps fi: (Mi,gi,pi) -* (Ni, hiqi) also converge to a Lipschitz map f : X ~ Y with Lipschitz constant equal to 1. It is not difficult to show that k = n and f : X0 • R n ---* R '~ is the projection up to a translation in R n (see [24] for details). This shows in particular that

lira di(Yi,Wi) = (1 + s2) U2, i---*oo

lim dH((Fi,di) ,Xo) = 0,

which is a contradiction. []

By using Proposition 4.23, one can prove the following result, which is useful when studying the properties of the fiber.

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T h e o r e m 4.24 [77]. Given m >_ n and #' > 0, there exists a positive number e' = e,,,,n(#') satisfying the following condition: let M and N satisfy the assumptions of Theorem 4.1 for # and e = en(#) with dim M = m, and let f : M -+ N be the fibration constructed there. Let F be a fiber of f with diameter 6p, and let d be the distance in M restricted to F. Suppose that there exists a Riemannian manifold P such that

K p _ > - l , inj (P) _> #', (1)

dH(P, (F, d/bF)) < e'. (2)

Then there exists a locally trivial fibre bundle fF : (F, d/hF) --+ P such that (1) f F is a r( e, e')-Riemannian submersion, (2) f F is a 7"( e')-Hausdon~ approximation,

where lim,,~,--.0 r(e, e') = 0, lime_0 r(e') = 0.

Outline of the proof. Let U be a small neighborhood of F in (M, d/bF). Proposition 4.23 shows that for any p E F, the set Hp of horizontal directions in U (see (4.20)) is almost parallel to the tangent space TpF. Thus, the almost Riemannian submersion f u : U ~ P constructed in Theorem 4.1 induces an almost Riemannian submersion f F : F --+ P.

R e m a r k 4.25. One can iterate Theorem 4.24 as follows: let E be a fiber of the fibration fF : (F, d/,hF) ---, P in Theorem 4.24, and let d/bE be the distance of E rescaled by its diameter. Then if (E, d/bE) is Hausdorff- close to a Riemalmian manifold Q, then again we have an almost Riemannian submersion fE : (E, d3E) ---* Q. This procedure is possible as long as the rescaled fiber is Hausdorff-close to a Pdemannian manifold of smaller dimension.

w FIBERING BY THE FIRST BETTI NUMBER

In this section, we give the proof of Theorem A. For a metric space X with F = ah(X), let h : F ---+ F/[F, F] be the Hurewicz homomorphism, and let ~2

be the torsion part of r / J r , r ] Then A = r / h - ' ( n ) is a free abelian group of rank bl = bl(X; Q). Suppose that X has a universal covering space X and consider the abelian covering of X:

.2 = 2 / h - ' ( a ) X.

For a point p E H, we endow A with the norm Ibll = d(Tp, p) The notation A(R) is as in Definition 2.9. The following lemma is due to Gromov [34].

L e m m a 5.1. Let X be compact. Then for every e > 0, there exists a subgroup Ar of A such that (1) A~ has rank bl, (2) 113̀ 11 -> for every nontrivial 3' E Ac, (3) there exist generators 71,.-- ,761 of Ar such that Ibill - 2(D+e) , 1 < i < ba, where D is the diameter

of X.

Proof. Since r is generated by F(2D), A is also generated by A(2D). Note that #A(e) < c~. First take the subgroup A0 generated by linearly independent elements 3`1,-.-, 7bl E A(2D). If Ao(e) is trivial, then A0 is the required subgroup. If 7 is a nontrivial element in A0(e), then we can find an integer m >_ 2 such that

-< Ib"l l -< 2(D+~). Let i E {1,. . . ,bl} be such that the 3`i-component of 3 ̀is nonzero, and define A1 to be the group generated by 3`1,.-- , 7 " , - - - ,3'bl with 7i replaced by 3`"*. Note that 3 ̀r A, and that A1 satisfies (1) and (3). After replacing A0 with A1, repeat the argument to get A2 such that ~A2(e) <_ #A(e) - 2. After repeating this procedure finitely many times, we get the required A~. []

PTo4 of Theorem A (a). We prove Theorem A (a) by contradiction. Suppose that the assertion is not true. Then there exists a sequence of closed Pdemarmian n-manifolds Mi such that

KM, > - - e i - + 0 , d i a m ( M i ) = 1, b l - - -b l (Mi )> 0,

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and no finite cover of Mi fbers over a bl-dimensional torus. Let M"'i -~ Mi be the abelian cover and let �9 pi E Mi. By Theorems 2.4 and 2.12, we may assume that (Mi, Ai,pi) converges to (X, G, x0) with respect to

the pointed equivariant Hansdorff distance. Note that X is a complete Alexandrov space with nonnegative curvature. Since X is noncompact and the action of G on X is cocompact, we can find a line in X. Then splitting theorem 3.8 implies that X is isometric to the product Y x R k, where Y contains no lines. If Y were noncompact, it would contain a line by the same reason. Hence Y must be compact.

Let 6i = max{v/B7, v ~ } , where oi = dp.H((Mi,pi), (X, xo)). We take the scaling of the original metric gMi; let gi = 6igM i �9 Note that inf Kg~ ~ 0, diama~ --~ 0, and that the pointed Hausdorff distance between

((iV//, gi), Pi) and ((X, 6id), xo) tends to zero as i ---* oo. Obviously, ((X, rid), xo) converges to ( R k, 0). Thus,

((Mi, gi), Pi) converges to ( I t k, 0). (5.2)

Again we may assume that ((~Mi,gi),Ai,pi) converges to (Rk ,H,0) . Since the group H is abelian and acts on R k transitively, it must be the vector group R k. Now consider the subgroup Ai = (Ai)I C Ai constructed in Lemma 5.1 for e = 1. We may assume that ((Mi, gi), Ai,pi) converges to ( R k, A, 0), where A C H is a free abelian group of rank bx. In particular, we obtain

bl < k. (5.3)

Next we need a pseudogroup technique. For a large positive number R >> k, put B(.R) = Bo(R, R k) and

LR = B(R) n Z k,

where Z k is the integer lattice of H, and consider LR as a pseudogroup of isometric embeddings of B(R) into

B(2R). Similarly, put Bi(R) = Bp,(R, (Mi,gi)), and for the canonical basis e l , . . . , ek of Z k, let el,i,..., ek,i be the elements in Ai tending to ca, . . . , ek respectively. Consider the pseudogroup

al ak ak Li,R = { 7 = e l , i . . , ek,i E Aile~... ek E LR, 11711 --- R}

consisting of isometric embeddings of Bi(R) into Bi(2R). We can show that

there exist an R and a positive integer IR such that for every i > In there exists

a bijective pseudohomomorphism r : Li,n ~ LR;

furthermore, r induces the equivariant Hausdorff convergence (5.4)

(Bi(R), Li,n) ---* (B(R), LR).

Note that M* = Bi(R)/Li,R covers Mi and that Mi* converges to the flat k-torus B(R) /LR ('." Theorem 2.9). Now fibration theorem 4.1 provides a fibering of M* over the k-toms, which is a contradiction. This completes the proof of Theorem A (a). []

In the above proof, we have used the pseudogroup Li,R to find the finite cover M*. The author believes

that this can be avoided. Probably, there should exist a subgroup Li C Ai such that ((Mi,gi),Li,pi) converges to ( R k, Z, 0). More generally,

C o n j e c t u r e 5,5. Let a (free) abeL/an group Ai act s on Xi, and assume that (Xi ,Ai ,Pi) converges to (Y, G, qo) and that Y /G is compact. Then for a given cocompact subgroup A C G, there exists a subgroup Ai of Ai such that (Xi ,Ai ,Pi) converges to ( r ,A, qo).

If Xi is simply connected, we can prove the above conjecture by using an argument similar to the proof of Theorem 2.13.

R e m a r k 5.6. (1) If we do not assume Ai to be abelian, then the above conjecture is false. For instance, consider the three-dimensional simply connected nilpotent group

N = 1 0

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with left-invariant metric g~ such that

g~ = e2(dx 2 + dy 2) + e4dz 2

at the unit element, and let I" be the integer lattice of N. Then ( (N ,g ( ) , I', e) converges to (R a, R a, 0). However, no subgroups of I" tend to Z a C R 3 under this convergence.

(2) If we do not assume the compactness of Y / G , then the above conjecture is also false. For instance, consider the convergence (R 3, I'~, p~) --* (its, R x Z, 0) in Example 2.11. Then no subgroups of Fi tend to Z x Z .

The proof of Theorem A (a) easily implies

T h e o r e m 5.7. Let Mi be a sequence of compact Riemannian n-manifolds with i ~ K M~ ~ O, diam(Mi) = 1, b l (Mi) = bl, and let Mi be the universal cover of Mi. Then for any ~i E Mi, there exists a sequence 6i ~ 0 such that i f gi = 6igMi, then

(1) inf Kgi --* 0,

(2) ( (M, ,gd ,~d converges to (itZ,O), where 1 >_ b~.

Proof. As before, we may assume that (Mi, P'i) and (Mi, Pi) converge to (X x R t) and (Y x R k) respectively,

where both X and Y axe compact and the covering map 7ri:.~_.Mi --* Mi takeslpp'i to Pi. In the same way as in (5.2), we can find/ii ---* 0 such that if gi = 6igM,, then ((Mi, gi), P'i) and ((Mi, gi), pi) converge to (R z, 0)

and (R k, 0) respectively. By Ascoli's theorem, 7ri : ((Mi, gi),p'i) --~ ((Mi, gi),pi) converges to a Lipschitz map rroo : (R t, 0) --, (R k, 0). Thus, l >_ k, and we already know that k > bl. []

Next we prove Theorem A (b) by contradiction. Let Mi be a sequence of closed Riemannian n-manifolds such that

KM, > --ei ---+ 0, diam(Mi) --* 0, bl(Mi) - n, (5.8)

but M~ is not diffeomorphic to a torus. Theorem A (a) implies that M~ is diffeomorphic to R" . From (5.3),

we see that (Mi,p'i) converges to (I t" , 0). Thus by fibration theorem 4.1,

(Mi,~i) converges to ( I t" , 0) with respect to the pointed Lipschitz distance. (5.9)

L e m m a 5.10. Pi is abelian for a//sufliciently laxge i.

Proof. Let pi : Fi --~ Ai be the projection, where Ai = r i / h - l ( a ) is as before. Let Ai = (Ai)~ be as in Lemma 5.1 for e = 1. Recall that

Ai has rank n I,

[[All _> 1 for every nontrivial A E Ai, (5.11)

there exist generators Ax, . . . , A, of Ai with II~jll <- 4.

Take 7j E p~--l()~j) such that II~ill = I1~11, where the norm is taken with respect to pi E M-i and piMP/. Suppose that the lemma does not hold. Then we have a sequence 61,62, . . . , E [I'i, ri] such that 2j _< 116i[[4j.

Now for a = ( a l , . . . , a , + i ) E ( Z + F +1, consider the norm Ilall = ~ a j . P u t 7,, = 7 [ 1 - . - 7~ '6~ .+~ E r i . Note that

If r >> 1, then using (5.12) we obtain

II~all ~ 41lall, II~r21~rbll ~ 1 for a # b. (5.12)

c o n s t n r n + l ~,~ # { a e Z~+ll liall ~_ r}

_< #{~ . I ltaiI _< 4r}

Vol B~,(4r + 1/2, Mi) < - Vol B~, ( 1 / 2 , Mi)

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In view of (5.9), the right-hand side of the above inequality converges to (8r+ l ) n . This is a contradiction. []

As a result of the above lemma, we see that Mi is a homotopy torus. However, it is known that there exist exotic tori with nonstandard differentiable structures and that every homotopy torus can be covered by the standard torus (see Theorem 5.13 below).

We will prove that Mi is diffeomorphic to the standard torus for large i. First we consider the case of higher dimensions, n _> 5. In this case the structure of the set S ( T ~) of all differentiable structures with underlying n-manifold homotopy-equivalent to T ~ is well understood.

T h e o r e m 5.13 [42, 69, 45]. l f n >_ 5, then S (T n) is finite and has an abelian group structure.

The identity of S ( T n) corresponds to the standard torus. It is also known that any covering T n ~ T ~ naturally acts on S(T~), whence we obtain the following

corollary.

Coro l l a ry 5.14. Let dn be a positive number which is relatively prime with the order of S(Tn). Let M ~ be a compact differentiable homotopy torus (n > 5). Then a dn-fold covering of M is diffeomorphic to the standard torus if and only i f M is diffeomorphic to the standard torus.

Proof of Theorem A (b) for n > 5. Let Mi be as in (5.8). Then Lemma 5.10 implies that (M~,Fi,~i) converges to ( R n , R n , 0 ) . We will find some subgroup of ~i which has controlled index and does not "collapse."

Since diam(Mi) ~ 0, we can find a subgroup Al,i C ~i such that

[Fi : A1] is a power ofd~,

1 < diaaxt (Mi) _< 2. (5a5)

N

Passing to a subsequence, we may assume that (Mi, AI,i,p/) converges to ( R n, G1,0) with G1 isomorphic

to Z kl ~3 R "-kl , where kl > 0 by (5.15). Put Ml,i = Mi/Ax,i . Since Mx,i converges to the flat torus T k~ = R" /G1 , we have a fibration

Fi "--, Ma,i T k' �9

In particular, we have a decomposition AI,i = Hl,i �9 rl , i , where Ha,i is isomorphic to Z k~ and Fx,i is the fundamental group of the fiber Fi, which is the collapsing part of Fi. Next we choose a subgroup A2,1 C Fx,i as follows. Let B C T k~ be a contractible ball around the reference point f i(pi) (rood Ax,~) over

which Fi is the fiber of f i . Let Ui = f/ '-l(B) and let ~ ' /be the component of rr~-I(ui) containing p"/, where

rri : Mi --* M~,i is the projection. Note that 7ri: Ui ~ Ui is a universal covering, and that Fi = 7r~-X(Fi) is also a universal cover of Fi .

S u b l e m m a 5.16. There exists a subgroup A2,i C •l,i SUCh that (1) [rl,i: A ,i] is a p o w e r d . ,

(2) 1 _< diana (-Pl,i/A2,i) <_ 2.

Proof. Under the convergence (Mi, pi) "-* (R" , 0), (-~i, Pi) converges to an ( n - k~ )-plane. Thus it converges to (R "-k~ , 0) with respect to the pointed Lipschitz distance (see Theorem 4.24). The assertion follows from the convergence diam (El) ~ O. []

Now we may assume that (M'], Hl,i~A2,i , Pi) converges to ( R n, G2,0), where G2 is isomorphic to Z kt+k2 ~3 R " - k~ - k2 and k2 > 0. Repeating the procedure finitely many times, we obtain subgroups Ha,i, H2,i,. �9 �9 , Hi,l of Fi such that

r'* = Hl,i G . . . �9 Hl,i(C Fi) is a direct sum,

[Fi: F*] is a power o ld , , (5.17) . r l * (Vi, F i ,P'i) converges to (R , A , 0) with A* isomorphic to Z n.

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Hence M* = Mi/F* converges to the torus R " / A * , and it is diffeomorphic to a torus. This completes the proof of Theorem A (b) for n >_ 5. []

Finally we give the proof of Theorem A (b) for n < 5. Here we consider a metr ic on Mi of d iameter equal to one. We know that Mi converges to a flat torus T k of positive dimension and admits a f ibrat ion

Fi ~ Mi "-'* T k,

where 7ra(Fi) ~- Z " - k . We need some informat ion on the space of diffeomorphisms of a torus.

T h e o r e m 5 .18 [17, 68]. (1) The group of diffeomorphisms of any closed orientable surface that axe homo- topic to the ident i ty is connected.

(2) The group o f PL-homeomorphisms of an irreducible, sufficiently large dosed PL- three-mani fo ld M 3 that are homotopic to the ident i ty is connected.

Note that T 3 is irreducible and sufficiently large. We begin with the lowest possible dimension.

C a s e (1): n = 2. This is trivial.

C a s e (2): n = 3. If k _> 2, then, clearly, Mi is diffeomorphic to T 3. If k = 1, then Fi ~ T 2 and Mi can be iden t i fed with the quotient T 2 x [0, 1J/g, where g : T 2 x 0 ---* T 2 x 1 is the gluing diffeomorphism. Since Fi is abelian, f should be homotopic to the identity. By Theorem 5.18 (1), it is diffeotopic to the identity, and M is diffeomorphic to T 3, as required.

C a s e (3): n = 4. If k >__ 3, then, clearly, Mi is diffeomorphic to a torus. Let k = 1. Th en it is easy to show that the fiber Fi (with metr ic scaled so that diam (Fi) = 1) converges to a flat t o r u s T I. By Th eo rem 4.24 we have a f ibrat ion Ei ~ Fi --* T I. The argument in Case (2) shows tha t Fi ~ T a. Now M is identified with the quotient space T a x [0, 1]/g, where g : T 3 x 0 -~ T a x 1 is the gluing diffeomorphism. Since Fi is abelian, g is homotopic to the identity. It follows from Theorem 5.18 (2) tha t M is PL-homeomorph ic to T 4, and hence by [50] it is diffeomorphic to a toms. If k = 2, use the project ion T 2 --* S 1 x 0 to the

first factor to get the f ibrat ion Ni ~ Mi/ -~ S 1. We also have the f ibration T 2 r Ni f/IN, 0 • S 1 . By the argument used in Case (2), we see that Ni is diffeomorphic to T 3. Thus this case is reduced to the case

k = l : T 3 ~ Mi ~ S 1 �9

The proof of T h e o r e m A (b) is now complete. By using T h e o r e m 4.24, we can prove Theorem A for the fiber of a f ibrat ion as in Th eo rem 4.1.

T h e o r e m 5 .19 [77]. Given m >_ n and # > O, there exists a positive number e = em,n(#) satisfying the following: let M "n and N n be complete manifolds such that

KM >_ --1, KN >_ --1, i n j ( N ) > #,

dH(M, N ) < e.

Let F be a f iber of a fibration as in Theorem 4.1. Then (a) a finite cover of F fibers over a b l (F)-dimensionM torus, (b) i f bl ( M ) = m - n, then F is diffeomorphic to T m-n .

w BOUNDED ALMOST NONNEGATIVE CURVATURE

In this section, we consider almost nonnegative Ricci curvature under the s t ronger assumption I/(I _< 1. By using the Bochner technique, one can generalize the argument of Sec. 1 as follows.

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T h e o r e m 6.1 [76]. Given n and D > O, there exists a positive number e = e~(D) such that i f M satisfies

IKM[ < l , d i am(M) < D, RicciM > - e ,

then every harmonic 1-form on M does not vanish. In particular, the Albanese map A: M ~ T bl , as in Sec. 1, is a fiber bundle.

Proof. Let w be a harmonic 1-form on M. By the Weitzenbgck formula (1.1), we have

fMlDWl ~ rUcci(~w,~w) = 0. +

Consider the following norm on 1-forms:

1 fM Iwl dx. I1 11 - vol M

Assume tha t I1 11 = 1. Then it fol!ows from Gallot 's and Li's inequalities [26, 46] tha t

I lc0 -< coas t I1 11 = cons t .

Now RicciM > --e implies

1 fM tDw[2 < coast e. vol M

By using the assumption [KMI < 1, we will prove

(6.2)

ID lco < (6.3) where r(e) = "rn,D(e) with lira,--.0 v(e) = 0.

For any p E M, put B = B0(1, TpM) and B ' = B0(1/2, TpM). Let g be the metric on B pulled back by the exponential map expp : B ~ M. Since the injectivity radius of (B' ,g) is greater t han 1/2, it follows from Jost and Karcher 's results [43] tha t there exists a harmonic coordinate ball of fixed size centered at any prescribed point in B ' such tha t the eigenvalues of the metric tensor with respect to these coordinates have some positive uniform bounds both from below and from above on B' . Consider the harmonic form

= exp; w on B and take a function f on B such that df = ~. Since A f = 8~ = 0, it follows from the Schauder est imate (see [27]) tha t

[flB,,c~.o < const If Is,c0.

Since we may assume tha t IflB,C0 -< coast I lB,co, we see that

< e o n s t P l B , c 0

_< eonst I~lc0 (6.4) = const.

Thus, we have proved tha t [WlM,Ct,~ <_ coast. Now (6.3) follows from (6.2), (6.4), and the inequality d i m ( M ) < D.

Now we see tha t w is almost parallel if e is small. Thus if w l , . . . , wk are harmonic I-forms of M forming a basis of the first de Rham cohomology group, they axe pointwise linearly independent . Hence the Albanese map is a fibration. []

Note tha t the mapping A : M ~ T b~ is harmonic, and by (6.3) it is a r (e ) -Riemannian submersion.

P r o b l e m 6.5. F ind a geometric or a topological property of the fiber of A.

Lacouturier and Robelt have recently obtained a generalization of Theorem 6.1. They replaced the uniform bound IKMI < 1 by a bound on some integral norm on KM.

If we assume no bounds on the sectional curvature, then Theorem 6.1 does not hold anymore because of the following result due to Anderson.

1294

T h e o r e m 6.6 [3]. Let n >_ 4 and 1 <_ k <_ n - 1. Then by doing surgery on T" we can construct an n-dimensional dosed manifold M with bl( M) = k such that

(1) no finite cover of M fibers over a bl-dimensional torus, (2) for every e > 0, there exists a metric ge on M such that

I Riccig, I < e, d iam(g , ) -- 1,

(3) (M,g , ) converges to a flat torus T k with respect to the Hausdorfl:distance.

Note that the Gromov conjecture 0.4 is still open. Next we present a result for aspherical manifolds, i.e., for manifolds with lri -- 0 for i > 2, with bounded

sectional and almost nonnegative Pdcci curvature. Note that every asphericat manifold with nonnegative Ricci curvature is fiat [14], which directly follows from splitting theorem 1.2. This can be generalized as follows.

T h e o r e m 6.7 [23]. Given n and D > O, there exists a positive number e = e . ( D ) such that i f M is an aspherical Riemannian manifold with

[KM[ ~ 1, diatoM _< D, PdcciM > --e,

then it is diffeomorphic to an infranihnanifold.

Outline of the proof. The proof proceeds by contradiction. Suppose that the assertion is not true. Then we have a sequence Mi of n-dimensional aspherical manifolds with given bounds on the sectional curvature and the diameter and with PdcciM~ > -e i ~ 0, such that Mi is not diffeomorphic to an infrardlmanifold. By

the topological assumption, we see from [21] that the universal cover Mi does not collapse. Namely, in view

of the bound IKt _< 1, (Mi,pi) converges to a pointed space (N, q) of the same dimension with respect to the Cl,~-topology on compact subsets. More generally, this convergence holds in the L2'P-topology, where p > n (see, for example, [53]). Thus the metric tensor of N lies in L 2,p, and hence has second derivatives almost everywhere. Therefore, we can get the splitting theorem for N (cf. [9]). Since N is contractible (see [21]), it must be isometric to R n. Thus, Mi is L2'p-almost flat in a certain sense. If it were C%almost flat, then Theorem 0.5 would work. Since this is not the case, we use the technique of covering space along the fiber introduced in [23]. By applying this argument, we obtain a (singular) fibration Fi ~ Mi ~ R m / A over a fiat orbifold R '* /A, where the fiber Fi is an infranilmanifold and the structure group can be reduced to a certain special form (similar to that in Proposition 10.1). Using that information on the structure group, we can construct smooth almost flat metrics on Mi for sufficiently large i (see [23] for details). []

w G E N E R A L I Z A T I O N OF BIEBERBACII'S T H E O R E M

In this section, we give a generalization of Bieberbach's theorem which is needed in the proof of Theorem B. As indicated in Sec. 1, the proof of Theorem 0.2 depends on splitting theorem 1.2 and the Bieberbach theorem. In our case, we co.nsider the equivariant Hausdortf convergence of the isometric action of funda- mental groups on universal covering spaces. The limit group is not necessarily discrete. Thus we need the following result:

T h e o r e m 7.1 [24]. Let G be a dosed subgroup o f l som(R n) . Then 7to(G) = G/Go is almost abe//an. More precisely, ~ro(G) contains a free subgroup A of [inite index such that rank(A) < d im(R~/G) .

The special case of Theorem 7.1 where G is discrete is Bieberbach's theorem. We need the following elementary lemma from group theory. The proof is omitted (see [24]).

L e m m a 7.2. Let a group G enter the exact sequence

1 --+ f~ -+ G -+ Zk --+ 1, "

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where f~ is t~nite of order I. Then G contains a free abe/ian subgroup A of rank k such that [G : A] < C(k, l).

Proof of Theorem 7.1. The proof proceeds by induction on the dimension of G. The case dim G = 0 is the Bieberbach theorem. Let dim G >_ 1 and let R be the radical of G. Put N = [R, R]. We divide the proof into the following three cases:

(1) d i m N > 0, (2) dim R > 0 and R is abelian, (3) R = {1} . Here we consider only case (1). The proofs in the other cases axe similar. Let G be the center of N. For

g 6 G, put

min(g) = {x 6 Rnld(gx, x) <_ d(dy, y)for ally E It"},

L---- N min(g)- gEC

Note that L is a nonempty G-invariant convex set without boundary. Hence L ~ R l C R. '~, and we obtain a homomorphism ~: G --* Isom ( R t) such that ~(G) is closed. Now putting K = ker(~), we obtain exact sequences

1 --, K -~ G -* ~(G) ~ 1,

~'o(K) ~ ~ro(a) ~ ~'o(%o(G)) ~ 1,

where 7r0(K) is finite because of the compactness of K. Hence in view of Lemma 7.2, it suffices to prove the theorem for (p(G) C Isom (Rt).

Case (a): ~ ( C ) = {1}. Since

d im~(G) = d i m G / g <_ d imG/C < dimG,

the inductive hypothesis works. Case (b): ~(C) # {1}. Since v (C) is normal in ~ (a ) , ~(C) acts on Rt/~(C) ~- I~t/I~ t - " = R " , where

m < I. Let r ~(G) ---* I s o m ( R m) be the induced homomorphism. Putting g ' = ker(r we obtain exact sequences

1 ~ g ' ---* v(G) --~ r ---+ 1,

~r0(K') ---* ~r0(T(G)) ~ ~'0(r --~ 1.

Since ~(C) C g ' C I som(Rt-m) , we have K' /~(C) C O(l - m). It follows that zr0(g') is finite. It is easy to show that r is closed. Thus it suffices to prove the theorem for r C Isom(Rm). Since

d imr _< dim~(G)/T(C) = d imG/C < dimG,

the inductive hypothesis can be applied. []

C o r o l l a r y 7.3 [24]. /n addition to the assumptions of Theorem 7.1, assume that R " / G is compact. Then G contains a normM subgroup G' such that

(1) [G: G'] < c. , and (2) R"/G' is isometric to a fiat torus.

Proof. This is also done by induction on dim(G). The case dim G = 0 is the Bieberbach theorem. Let dim G > 0. We use the same notation as in the proof of Theorem 7.1, and assume that case (1) takes place. Since L is G-invariant and convex and Ftn/G is compact, we see that L -- I t" . Hence ~p = identity. Note that C is normal in G. Since

R " I G = (R"IC)I (GIC) = R m l ( a f c )

and dim (G/C) < dim G, the inductive hypothesis can be applied. []

In Theorem 7.1, the index [Tr0(G) : A] admits no uniform bounds depending only on n even if I:U'/G is compact.

E x a m p l e 7.4. Let G be the subgroup of Isom (R 2) generated by the translations of p,.2 and the rotation about the angle 27r/p around the origin. Then R2/G is a point and ~r0(G) = Zp.

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w SOLVABILITY THEOREM AND APPLICATIONS

In this section, we give the proof of the solvability part of Theorem B.

T h e o r e m 8.1 [24]. There exist positive numbers e, and c, such that i f M has e,-nonnegative curature, then 7rl (M) contains a solvable subgroup A such that

[Tri(M) : A] < c,.,, (1)

/:(A) ~ n. (2)

A solvable group is obtained by several extensions of abelian groups. In the proof below we will see that Theorems 3.8 and 7.1 provide each building block, and tha t Theorems 2.12 and 4.1 provide each extension.

We start with an algebraic lemma.

L e m m a 8.2. Consider the exact sequence

1 , A ,F , Z k ;1 ,

where A contains a solvable subgroup A' such that [A : A'] = s and s = m. Then F contains a solvable subgroup r' such that [r: r'] < e f t ) and z (r ' ) < k + m.

Proof. Take a subgroup A* of A' which is characteristic in A (see [24]). Then A* is normal in F and we have an exact sequence

A F Zk A* A*

Now the conclusion follows from Lemma 7.2. []

Proof of Theorem 8.1. This is done by contradiction and induction on the dimension of M. We will prove only the almost solvability. Suppose that the assertion is not true. Then we have a sequence of closed n-manifolds Mi such tha t KM, > --ei, ei ---* 0 as i ---* r and diam(Mi) = 1, but Fi = ~rl(Mi) is not almost solvable. Passing to a subsequence, we may assume tha t Mi converges to a compact Alexandrov space X. Since X might not be a manifold, we must consider the action of Fi on the universal cover Mi. For pi 6 Mi, we may assume by Theorems 2.4 and 2.12 that (Mi, Fi,pi) converges to a triple (Y, G,q) with respect to the pointed equivariant Hausdorff distance. Then splitting theorem 3.8 shows tha t Y is isometric to the product R k x Y0, where Y0 is compact.

A s s e r t i o n 8.3. G/Go contains a free abe/ian subgroup of 6nite index and rank not greater than k.

Proof. Since G preserves the splitting R k x Y0, we obtain a homomorphism ~0: G --* Isom(Rk). If K = kerfs0), then G / K is a closed subgroup of I som(R k) and we see by Theorem 7.1 tha t

(a /K) / (a /K)o =~ G/aoK

contains a free abelian subgroup of finite index and rank not greater than k. In view of Lemma 7.2, the assertion follows from the exact sequence

GoK G G I , - - , - - , ~ ~1,

Go Go GoK

where GoK/Go is finite. []

Note that G/Go is discrete by Theorem 3.16 and that ( R k x Y) /G = X is compact. Hence we can apply Theorem 2.12 to get a normal subgroup F~ of Fi such that

(M~, r~,pi) converges to (Y, G0, q);

r,/q is isomorphic to G/Go for large i; (8.4.)

for any e > 0, the group F' i is generated by F'i(e ) for large i _> f(e).

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A s s e r t i o n 8.5. s is almost solvable/"or every sufllciently large i.

"Proof. Put x0 -- q mod G E X. For given rj ~ oo, by Theorem 3.15 we can choose xj E X such that ((X, r jd) ,x i ) converges to (Rm,0). Note that m _> 1 since diam(X) = 1. Set dH(Mi ,X ) = oi and dp.H(((X, r jd) ,xj) , (R'n,O)) = o1, and let pij E Mi be a point which is Hausdorff close to xj. By the triangle inequality, we have

dp.H(((Mi, rjgM,),Pij), (R rn, 0)) < rjoi -~ Oj.

Hence we can make the above Hausdorff distance as small as we like if we take a large j = j0 and any large i _> i0. Thus for some choice of such j0 and i0, we have an almost Pdemannian submersion

fi :Bp,(1,(Mi,rjogM,))--~Bo(2, Rm)

over its image such that the fiber Fi = f i "-1 (0 ) has almost nonnegative curvature in the generalized sense. In view of Proposition 4.23 and Theorem 4.24, we can apply the inductive hypothesis to the fiber. Therefore, 7rl(Fi) is almost solvable for all sufficiently large i. Let e = 1/(lOrjo ). Since Bp,(e, Mi) is contained in the neighborhood f/-X(B0(1, Rm)) -~ B0(1, R m) x Fi for large i, it follows from (8.4) that F~ is contained in the image of the inclusion homomorphism,

r~cIm[~1(Fi)-~ri].

This shows that F~ also is almost solvable. []

Finally, we have the exact sequence

1 , F~ , r i , G I G 0 , 1, (8 .6)

where F~ is almost solvable and G/Go is almost abelian. Therefore, the preceding lemma implies that Fi is almost solvable, which is a contradiction.

This argument also gives a uniform bound on the index of a solvable subgroup of F/ (by contradiction). The inequality for the length of polycyclicity follows from the relations

s + rank(G/Go); (Trl(Fi) ) - rn ('." induct ive hypothes i s ) ; ,

rank(a/ao) < d i m R k A o ( G ) _ d i m X = rn ('." T h e o r e m 7.1).

(8.7)

This completes the proof of Theorem 8.1, []

R e m a r k 8.8. Note that in the s~ep of induction in the above proof, we pursue the fiber properties at most n times by using Proposition 4.23 and Theorem 4.24. Although this is possible, we avoided this argument in [24], where the theorem was proved for the fiber of an almost Riemarmian submersion in fibration theorem 4.1 by using a reverse induction.

Now we give some corollaries of Theorem 8.1. Below we will see the significance of the uniform bound (2) in Theorem 8.1.

Co ro l l a ry 8.9 [24]. There exists a positive number Cn such that the following assertion is true: let M have en-nonnegative curvature, and suppose that for any solvable subgroup A of 7ra(M) with [zrl(M), A] <_ C, we have s = n. Then

(1) A is a poly-Z group,

1298

(2) the universM cover of M is diffeomorphic to R n.

Proof. We use the notat ion from the proof of Theorem 8.1. By (8.7), we have

rank(G/Go) = dim (R k/~(G)) = dim X = m,

c ( r ' i ) = - m .

By Corollary 7.3, there exists a normal subgroup Go C G C G such that R k / ~ ( G ) is isometric to a flat

torus T " . Put )~ = R t' x Y / G and consider the commutat ive diagram

R k x Y ql ~ R k

m I ~p2

q2

where Pi and qi are the natural projections, which do not increase distance. Le t R k --= R m x R k-m be the orthogonal decomposit ion for which P2 : R m ---* T m is a Pdemannian covering. Since p2 o ql is a local isometry on each Rm-factor, so is q2. It follows that )~ is isometric to a flat torus Tm. By Theorem 2.13, we can take a normal subgroup f" tending to G. Hence M = M / P tends to T 'n, and we have a f ibration

Fr J~I__. T '~

and the exact sequence I ' ~'I (F) , f" , Z m , 1.

Applying the inductive hypothesis to the fiber F , we obtain the conslusion. []

The uniform bound on the index of a solvable subgroup in Theorem 8.1 is useful. By using this theorem, we obtain the following corollaries.

C o r o l l a r y 8 .10 [24]. There exists a positive integer pn such that i f M has e,~-nonnegative curvature, then bx(M, Zp) _< n for every prime number p >_ p~,. Is the equality bx(M, Zp) = n occurs for some p, then M is diffeomorphic to a toms.

The inequality immediately follows from Theorem 8.1. The equality case follows from an argument similar to Theorem A (b).

C o r o l l a r y 8 .11. There exists a positive number Cn such that i f M has en-nonnegative curvature and ~q(M) is finite, then

where M is the universal cover of M.

diam(M)

d iam(M) < Cn,

Proof. Assume the contrary: suppose that the conclusion does not hold. Then we have a sequence of closed n-manifolds Mi with ei-nonnegative curvature, ei ~ 0, and finite fundamental groups Fi such tha t the ratio of the diameters for Mi goes to infinity as i ~ oo. By Theorem 8.1, we may assume tha t Fi is solvable with length of polycyclicity not greater than n. Let

[`i = r l ~ 3 I '11)~ "'" ~ I'l " ) = {1}

be the derived series, ['I s) [['l~-l),rls-1)l, and put M} ~) - - (~) = = M i / F i . Then we have a tower of abelian coverings,

M~i ~ M} n-l) , . . - ~ M} l) ~ Mi.

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Here we scale the metrics so that diazn(Mi) --* 0 and the lower bound of the sectional curvature of Mi

�9 still goes to zero. Then we find some s, 0 < s < n - 1, such that for a subsequence, M~ s) and M} s+l) respectively converge to a point and a space Y which is not a point. By taking the scaling carefully, we may assume that Y is noncompact and is actually isometric to Euclidean space R k, k > 1 (see Theorem 5.7). Then applying the pseudogroup technique used in Sec. 5 (see the argument after (5.3)), we can find

a covering M* of M[ s) which converges to a fiat torus T k. By fibration theorem 4.1, we have a fibration of M/* over T k, which contradicts the finiteness of Fi. []

Note that we have no explicit estimates for our constant Cn in the above result. On the other hand, for any lens space S n / F with constant curvature, we know that

diam(Sn) - 2.

di (s-/r)

It would certainly be interesting to find a (realistic) explicit constant Cn. The next corollary immediately follows from the proof of Corollary 8.11.

Coro l l a ry 8.12; Let M have en-normegative curvature and an infinite fundamenta l group. Then a finite cover of M fibers over S 1 . In particular, the ELder characteristic of M vanishes.

C o n j e c t u r e 8.13. The Pontryagin numbers of an e,~-nonnegatively curved mani fo ld with infinite funda- mental group also vanish.

w THREE-DIMENSIONAL CASE

We will give an outline of the proof of Theorem B in the next section. The key point in the proof is to develop the method of covering space along fibers introduced in the study of almost nonpositively curved manifolds (see [23]). In this section, we clarify the basic idea by considering the three-dimensional case, where we can determine the manifold structure up to a finite cover and up to the Poincard conjecture.

T h e o r e m 9.1 [24]. There exists a posit ive number e such that i f a dosed three-manifold M has e- nonnegative curvature, then a lqnite covering of M is homotopy-equivMent to S 3 or diffeomorphic to S 1 x S 2,

a nBmanifold, or a torus.

Proof. We can assume that the fundamental group of M is infinite. Then the proof of Corollary 8.11 shows that a finite cover M* of M tends to a flat torus T k. If k >__ 2, then M* is diffeomorphic to an infranilmanifold or a torus (see the argument below). Let k = 1 and suppose that the conclusion does not hold in this case. Then we have a sequence of closed three-manifolds Mi such that KM~ > --el ~ 0 and a finite cover Mi* converges to S 1 but Mi does not satisfy the conclusion of the theorem. By Theorem 4.1, there exists a fibration

Pi -* M* ~ S 1,

where Fi is diffeomorphic to S 2, a projective plane p 2 a torus T 2, or a Klein bottle K 2 by Theorem 5.19. If Fi ~- S 2, then M/* is diffeomorphic to S 2 x S 1. If Fi ~- K 2, then we can take a finite cover of Mi with fiber S 2 or T 2 respectively. Thus we may assume that Fi ~ T 2. Let

1 ~ Z 2 -- '+Fi ~ Z ~ 1

be the associated exact sequence, where l-'i = 7rl(M/*). Now take a 7i E I'i which projects to 1 E Z, and define AT~ E SL(2 , Z) by

--1 A'rl (g) -- 7ig')' i

Then the following fact is easily verified.

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L e m m a 9.2. Fi is nilpotent i f and only i f AT~ is conjugate to an element of the form

(lo ;) The fiber Fi collapses to a point, and it is inconvenient to analyze the properties of A.r~. Therefore we

take a finite cover of Fi in order to "look at" the fiber. This is the basic idea of the covering space along the fiber.

For m E Z+, let FIr") be the subgroup of Fi generated by (mZ)2(C Z 2) and 7i, and let M[ r") be the

finite cover of Mi with fundamental group FI m). Let F (m) be the m2-fold covering of Fi corresponding to

M[ m). Then we can find mi so that

1 <_ diam(F[ " ' )) _< 2. (0.3)

Passing to a subsequence, we may assume that (M-i,Flm'),pi) converges to (Rk, G, 0). Note that d im(Rk /G) _> 2 because of (9.3). By Coronary 7.3, R k / G admits a finite (ramified) covering by T 2

or T 3. If M~ is the finite cover of M[ m') corresponding to it, then M~ converges to T t with l = 2 or 3. If l = 3, then M' -~ T 3. If I = 2, then we have a fibration

S 1 ~ M~ ~ T 2.

Hence the subgroup Z = 7[1(S 1) C r~l mi) is normal and A-,-invariant. Therefore, it is also normal in Fi, and choosing a basis of Z 2 = 7ra(Fi) we see that A- n is conjugate to an element of the form

,) + 1 "

Lemma 9.2 implies that Fi is almost nilpotent, and hence M/* is diffeomorphic to an infranilmanifold. []

w NILPOTENCY THEOREM

In this section, we give only an outline of the proof of Theorem B. See [24] for details. First consider the exact sequence

1 ~A , F ~A ,1 ,

where A is almost nilpotent and A is almost abelian. We need a criterion for F to be almost nilpotent.

P r o p o s i t i o n 10.1 [24]. The group F is almost nilpotent i f and only i f there exists a subset F generating F such that for any 7 G F there exists a filtration

1 =Ao c A 1 C:-'" c A k = A

and an N E Z+ such that

(1) Ai is normal and 7N-invariant, (2) i f [~Ai/Ai-I = o% it is abe/ian, (3) the automorphism AN E Aut (Ai /Ai -1) defined by

has finite order.

Note that if F is almost nilpotent, the upper central series of a normal nilpotent subgroup of F can be taken as the filtration.

1301

We prove Theorem B by contradiction and induction on dim M. If the theorem does not hold, we have a sequence of closed n-manifolds Mi such that KM~ > --ei ~ O, diarn(Mi) = 1, but Fi = ~'l(Mi) is not almost nilpotent. As in Sec. 8, we may assume that (Mi,r.pi) converges to ( R k x Y,G,q) . By (8.6), we have the exact sequence

1 --'-+ F~ ' Fi ' G/Go ~ I,

where G/Go is almost abeliaa, and we may assume that F~ is almost nilpotent by the inductive hypothesis (cf. Remark 8.8). Now for our purposes it is better to kill the compact factor Y in the splitting l:t k x Y. As in (5.2), we can choose positive numbers 8i --* 0 so that for the metric gl = 8igM~

(Mi, gi,pi) converges to ( R k, 0),

while keeping inf Kg~ ---+ 0. Note that diam(Mi, gi) --* O. From now on we use the notation

(10.2)

Xi = ( i i , g i ) , Ai = r~, A = G/Go,

and verify the condition of Proposition 10.1. Note that Fi is generated by Fi(1), and suppose that it does not satisfy the condition of Proposition

10.1. Then there exists a 7~ 6 Fi(1) not satisfying the condition. Note that [7~] E Fi/Ai must have infinite order. Now we can prove the following.

L e m m a 10.3. There exists a 7i E Fi(1) of the form "[i -~ (Ti)rnl2i, l~i E Ai, such that i f Ai(7i) denotes the group generated by Ai and 7i, then Ai(7i)(1/3) = Ai(1/3).

We may assume that Ai is solvable by Theorem 8.1. We may a/so assume that (Xi , Ai(Ti),pi) converges to (Rk ,H,0 ) , so that Ai and 7i converge to H C -~ and 700 E 2q, respectively. Let H(700 ) be the group generated by H and 700. We do not know if H is connected.

By Lemma 10.3, - ~ / H is discrete and hence H contains the identity component of H. Summing up, we obtain

P r o p o s i t i o n 10.4. (1) l imi- .00(Xi,Ai ,pi) = (Rk,H, 0), where H C ~ro. (2) hi is normal in hi(7i) ,

1 ~ Ai J' Ai(7i) , Z , 1.

(3) 7i --* 700 C /~', 1 , H , H(7r162 ) , Z , 1.

(4) Ai(Ti) is solvable with length of polycyclicity not greater than n. (5) Ai(7i)(1/3) = Ai(1/3), Ai = Ai(1).

Our purpose is to show that Ai has a filtration satisfying the condition for 7i in Proposition 10.1. Then the form of 7i would imply that Ai has a filtration satisfying the condition for 7~ in the proposition, which is a contradiction.

We know that Ai is generated by Ai(gi), where 6i --~ 0. However, we do not know if this is the case for [Ai, Ai]. For this reason we need a stronger notion.

For (X, F, p) 6 .Meq, put

r ( e ; D ) = {7 E r l d ( 7 , z ) < e for all �9 e B p ( D , X ) } .

Defin i t ion 10.5. We say that a sequence (X i ,K i ,p i ) is locally generated i f for any D > 0 there exists a sequence el(D) --* 0 such that Ki is generated by Ki(ei(D); D).

Now we can prove

L e m m a 10.6. If (Xi , Ki ,p i ) is locally generated, then (Xi, [Ki, Ki],pi) is also locMly generated.

In the first step, we replace {Ai} by a locally generated sequence. To do this, we need a technical lemma similar to Theorem 2.13.

1302

L e m m a 10.7. Let L i c A i be normM in Ai(Ti) and such that . l im(Xi ,L i ,p i ) = (Rk, L,0), and let a | ~ O O

sequence 6i "* 0 and D > 0 be given. Then for a subsequence there exists a subgroup L] C Li satisfying the conditions

(1) L~ is normal in Ai(7i), (2) L~ is locally generated, (3) L~ ~ Li(6i; D) i'or each i, (4) l imi_oo(Xi,L~,pi) = (R*,L ' ,0 ) , where L' D L0. Assume, in addition, that there exists an Ro > 0 such that Li is generated by Li(Ro). Let L(7oo )

and Li(Ti) be the groups generated by L U 700 and Li U 7i respectively. Then there exists a surjective homomorpMsm L(7oo)/L' ~ Li(Ti)/L~ which takes [700] to [7i].

In the lamina above, we do not assume the compactness of R i l L , which is the point that is essentially different from Theorem 2.13.

Applying Lamina 10.7 to Li - Ai, L = H, we obtain the following lemma, which is the first step in construction of the filtration of Ai.

L e m m a 10.8. There exists a A~ C Ai such that

(1) A~ is normal in Ai(7i), (2) (Xi, Ai,pi ) is JocaIJy generated, (3) A-r, E Aut(Ai/A~) is of finite order.

Proof. By Lemma 10.7, there exists a A~ C Ai satisfying (1) and (2) and such that there exists a surjective homomorpkism//(700)///0 --' Ai(Ti)/Ab Theorem 7.1 shows that //(700)///0 is almost Abelima, which implies (3). []

The following lamina is the main step in the proof.

L e m m a 10.9. Let [Ei, Ei] C Fi C Ei C Ai be normal in Ai(Ti), and assume that

(1) Ei and Fi are locally generated, (2) # ( E i / F , ) = ~ , (3) limi--.oo(Xi, Ei, pi) = (R k, E, 0).

Then there exist E~ such that for a subsequence

(4) Fi C E~ C El, E~ is normal in Ai(Ti), (5) E~ is locally generated, (6) A-r, E Aut(Ei /E~) is os fJnite order, (7) limi-.oo(Zi, E~,pi) = (R k, E ' ,0) , where d i m E ' < d imE.

Note that condition (7) is similar to the argument in Sec. 9 (see (9.3)).

Now we put Ai,l ~= Al l , Ai,t+l ~ [Aid,hi,t].

By Proposition 10.4 (4), there exists an N such that A~,N = {1}. Applying Lemma 10.9 inductively, we obtain

A s s e r t i o n 10.10. There exist A~, k such that A~, k is normal in Ai(Ti) and

(1) i i , k ~- i~ , k ~ i 2 k ~ ' ' ' ~ h i ,k+1,

(2) A.r, E Aut(A{,/,/A{,+k 1) is o/" fhaite order,

(3) l imi- ,~(Xi , j , , i,k J = oo. Ai,t,,pi) = (Rk,Hj, k,0), where d i m H / k > d imHj+ i k if[h~,k : A/+ll

By condition (3) there exists an L independent of i such that [AiLk : Ai,k+l] is finite. Finally we get the filtration

1 Ai,K C A L 2 i,K-1 C " Ai,K_ 1 C Ai,h'-i C A L = "" i,K-2 C "" "C Ai,h'-3 C "'" C Aid = Ai C Ai

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of Ai, which satisfies the condition of Proposition 10.1. The proof of Theorem B is complete. []

�9 R e m a r k 10.11. In our argument here, we are not able to control the torsion parts of successive quotients in the filtration. This is the main reason why we cannot get a uniform bound on the index of a nilpotent subgroup.

w GENERALIZED MARGULIS LEMMA

Let M be a complete Riemannian n-manifold with [/(MI < 1. In the proof of Theorem 0.5, Gromov [30] essentially uses Margulis' lemma, which states that at any point p E M, the small loops of length not greater than e, generate an almost nilpotent subgroup of rrl(M). By using Theorem B, we can drop the upper bound on the curvature.

T h e o r e m 11.1 [24]. There exists a positive number en such that i f M is a complete Riemannian n-manifold with KM >_ -1 , then for any p E M the image

Im[~h B,(en, M) --* 7rlBp(1, M)]

under the inclusion homomorphism is almost nilpotent and satisfies the conclusion of Theorem B.

As the first step, we show that the conclusion holds for some p E M.

L e m m a 11.2. There exists a 6n > 0 such that i f M and p are as in Theorem 11.1, there exists a point q E Bp(1/2, M) such that

Im[Trl Bq( 6n, M) --* rl B,(1, M)]

is almost nilpotent and satisfies the conclusion of Theorem B.

This follows from Theorem B for the fiber of an almost Riemannian submersion and the argument used in the proof of Theorem 8.1

Outline of the proof of Theorem 11.1. Suppose that the assertion of the theorem does not hold. Then we have a sequence (Mi,pi, el), where Mi is a complete n-manifold with KMI > --1, Pi 6 Mi is a point, and ei -'-* 0, such that

r i = Im[ IB ,(ei,M ) --*

is not almost nilpotent. By Lamina 11.2, we have 6 = 6n > 0 and q~ E M~ with d(p~, qi) < 1/2 such that

r~ = :m[~: B,, (~, M,) ~ ~, B,, (1, M)]

is almost nilpotent. In the argument below, we show that [Pi : Pl n P~] is finite, which is a contradiction. Let 7rl : Xi ~ Bp,(1,Mi) be the universal covering, and let xi, yi E Xi be such that ~-i(xi) = pi,

ri(yi) = qi, and d(xi, yi) = d(pi, qi). Put Gi = 7raBp,(1, Mi). Passing to a subsequence, we may assume that (Xi, Gi,xi) converges to (Z, G, Xoo). By [30] (see also [8]),

there exists an N > 0 such that Fi can be generated by N

elements ~/i,1,... ,Ti,N with 7i,j E Fi(2ei). (11.3)

Let (fi, ~i , r with

fi : Bz , (Di ,X i ) --* B , ~ ( D i + 1 /Di ,X) , ~oi: Gi(Di) --* G, %hi : G(Di) --', Gi

be a pointed equivariant Hausdorff approximation as in Definition 2.9, where Di ---* oo. By Ascoli-Arzela's theorem, we may assume that

qOi(Ti,j) converges toTooj E G for each j. (11.4)

By (11.3), ~ , ~ j ( z ~ ) = zoo.

We may also assume that F} converges to F ~ C G~.

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S u b l e m m a 11.5. The group G/F~ is discrete.

Proof. Let yi "* yoo E X . By the definition of F~, the open set {7 E G [ d(Tyoo , yoo) < 5/2} C G is contained l

in Foo. []

We put z = xoo for simplicity. Let G~ be the isotropy subgroup of x. Lemma 11.5 shows that L = [G~ : G~ M F ' ] is finite. Since %o,j E G~, using (11.3) and (11.4) we can show that [Fi : Fi MF~] < C(N,L) for sufficiently large i. Therefore Fi is almost nilpotent. []

Proof of Theorem C. Suppose that the theorem does not hold. Then there exists a sequence of closed n-manifolds Mi with KM, > --1, diam(Mi) < D, and rl(Mi) = Fi such that

Fi ~ Fj (i # j ) modulo almost nilpotent groups.

Passing to a subsequence, we may assume that (M/,I'.pi) converges to (II, G, q). By Theorem 2.13, there exists a normal subgroup F] of Fi such that

(1) F~ converges to Go, (2) r,lr G/Go for large i, (3) F~ is generated by F~(ei), where lim~-o~ e~ = 0.

It follows from Theorem 11.1 that F~ is almost nilpotent and satisfies the conclusion of Theorem B. Theorem 2.13 also shows that G/Go is finitely represented. This is a contradiction. []

w NONCOLLAPSING CASE

In this section, we consider almost nonnegatively curved manifolds with a lower volume bound. structure of such a manifold should be similar to that of a nonnegatively curved manifold.

We start with the following finiteness theorem for fundamental groups, which is due to Anderson.

Theorem 12.1 [2]. with

is finite.

The

The set of all isomorphism classes of fundamentM groups of dosed n-masdfolds M

RicciM > --(n -- 1)k 2, d iam(M) < D, vol(M) > v0

N

Proof~ Let M satisfy the above geometric bounds, let M be the universal cover of M with reference point p E M, and let P = 7rl(M) be the deck transformation group. The point of the proof is to show

Asse r t i on 12.2. There exist positive numbers 5 and No depending only on the given constants such that

# r ( 6 ) = #{7 e Fld(Tp, p) < 6} <_ No.

Proof. Let F be the fundamental domain of F,

F = {x E M t d(p, x) < d(Tp, x) for all 7 E r}.

For any 6 > 0, put

N = # F ( 6 ) , F ( 6 ) = { T a , . . . , T N } ;

It follows from the inequality d(p, gip) < i6 that

l

U 9i(F) C Bp(18 + D) i = 1

and hence by Bishop's volume comparison theorem we have

Iv0 </vol (M) < bk(Z6 + D),

gi = 7i �9 �9 �9 71

(1 < I < N),

(1 < i < N).

(12.3)

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where bk(r) denotes the volume of an r-ball in the n-dimensional complete, simply connected space of constant curvature - k 2. Hence if, for instance, we put No =- bk(2D)/vo and 6 =_ D/No , then we obtain the required estimate. []

Now by a result of Gromov [34], F can be generated by those 7i for which d(p, TiP) <_ 2D and the relations have the form 7i717k 1 = 1. Hence to prove the theorem, it suffices to estimate # F ( 2 D ) . Let {x/} be a maximal subset of {7P[7 E F(2D)} such that d(x j , zk ) > 6 for j ~ k. Let K be the number of {xj}. By the Bishop-Gromov volume comparison theorem [34], we have

g < ;r + 6/2) < bk(2D + 6/2)

volB (6/2) - b (6/2)

It follows that #r(2D) < K#F(5) < No bk(2D + 6/2)

- - bk(612) (12.4)

[]

For almost nonnegative Ricci curvature, Wei proved an estimate for the growth of fundamental groups by using the idea of Theorem 12.1.

T h e o r e m 12.5 [70]. Given n, D, and v0 > 0, there exists a positive number e = en(D, vo) such that ira closed n-manifold M satisfies

mcciu > - , , diana(M) < D, vol (M) _> v0,

then 7q ( M) has polynomial growth of order not greater than n.

Proof. Let M satisfy the above bounds for e > 0. By (12.4), we can take generators 7 1 , - . . , 7L of F = 7rl (M) such that d(Tio, p ) <_ 2D, where L is bounded by a uniform constant. Let g(s) be the number of words in F of length not greater than s with respect to 3'1,.-. , 7L. Similarly to (12.3), we have

g(s)Vo <_ b,(2sO + D). (12.6)

If g(s) is not of polynomial growth of order not greater than n for any sufficiently small e, there exists a sequence si ~ (x~ such that g(si) > isi. Since there axe only finitely many possibilities for the isomorphism class of F (Theorem 12.1), we can take si independent of M. On the other hand, it follows from (12.6) that for any large s we can find a small e > 0 such that g(s) < constn,D,v0 s n. This is a contradiction. []

For almost nonnegative sectional curvature, Theorem 8.1 implies

T h e o r e m 12.7 [24]. Given n, D, and Vo > O, there exists a positive number e = en(D, vo) such that f f a closed n-manifold M satisfies

KM > --e, diana(M) < D, vo l (M) > v0,

then r l ( M ) contains a free abe/ian subgroup A of rank not greater than n such that [~q(M) : A] < c , .

Proof. Suppose that the theorem does not hold. Then we have a sequence of closed n-manifolds Mi with K M , > --ei --+ 0, volM, > v0, diam(Mi) < D, and such that Fi = r l (Mi ) does not satisfy the conclusion. For some points pi in the universal cover Mi of Mi, we may assume as in the proof of Theorem 8.1 that (Mi, Fi, Pi) converges to ( R k x IF, G, q) with respect to the pointed equivariant Hausdorff distance, where Y is compact. By a result of Cheeger [10], there exists a positive number 6 = 6,(D, vo) such that Fi(6) = {1}. It follows that Go is trivial, and hence G is discrete. By Assertion 8.3, G/Go contains a free abelian subgroup of finite index and of rank not greater than k, and by Theorem 2.13, Fi is isomorphic to G for sufficiently large i, a contradiction. []

R e m a r k 12.8. In Corollary 12.5, it follows from the polynomial growth theorem [33] that 7rl (M) is almost nilpotent. Note that for any nilmanifold N ~ which is not a torus, 7q(N) has polynomial growth of order greater than n (see [48, 73]). Probably, the conclusion of Corollary 12.7 should hold under the assumptions of Corollary 12.5.

For the (topological) splitting property of a finite cover of an almost nonnegatively curved manifold with a lower volume bound, see [65, 74, 9].

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w CONCLUDING REMARKS

First note that the main methods used in our argument were splitting theorem 3.8 and fibration theorem 4.1. We have a generalization of Theorem 4.1 to the Alexandrov spaces ([7S], see also [71]). The resulting map in this case is an almost Lipschitz ~ubmersion, which is not known to be a fiber bundle yet. However it is sufficient for generalizing Theorem A (b) and the results on fundamental groups, Theorems B and C, to the Alexandrov spaces (see [78]).

The main problem still open is to extend the result to manifolds with almost nonnegative Ricci curvature. Thus we are led to

C o n j e c t u r e 13.1 [24]. Let (X ,p ) be the pointed Hausdorif lirait of a sequence (Mi,pi) o[ complete n- manifolds with RicciM~ > --el --* O. Then the splitting theorem holds t'or X.

As for the fibration theorem, Anderson's theorem 6.4 shows that it does not hold if dim N < dim M. However, the equality case dim N = dim M is open:

C o n j e c t u r e 13.2 [24]. There exists a positive number e = en(#0) such that i f the Hansdorff distance between two complete n-manifolds M and N with

RicciM > --(n-- 1), IKNI _< 1, inj(N) >_ #o

is less than e, then M and N has the same topological type.

By Perelman's recent result [59], the above conjecture is true up to homotopy equivalence if one can prove the following volume convergence:

C o n j e c t u r e 13.3. /n the situation described in Conjecture 13.2, for any p E M we have

vo lBp(r ,M) 1 < r(e), vol Bq(r, N)

where q E N is a point Hausdorff close to p and lim~--.0 v(e) = 0.

Let a sequence of n-dimensional complete Riemannian manifolds Mi converge to a aiemannian n- manifold N. Then, as the following example shows, the lower semicontinuity of volume

nm inf vol(Mi) > vol(N) i--*c~

does not hold except for n = 1, unless no curvature assumptions are made.

E x a m p l e 13.4. Let L C R 2 be as in Example 2,3 (2). For each positive integer k, divide each of the four edges of S = L Cl [0, 1] 2 into small intervals of length 6 = 2 -k. Let Sk C R 2 be the tree constructed by joining all pairs of such partition points by segments. Then we can consider Sk as a tree on the flat torus T 2 = R2/Z 2. For e > 0 much smaller than 6, let Mk(e) denote the boundary of the e-neighborhood of Sk in T 3 = T 2 x S 1. After carrying out a smoothing procedure for Mk(e), we obtain a smooth hypersurface in T 3, also denoted by Mk(e), such that it converges to T 2 with respect to the Hausdorff distance as both 6 and e << 2 -k tend to zero. Obviously, the area of Mk(e) tends to zero.

The following conjecture would be proved if one could settle Conjectures 13.1 and 13.3.

Conjecture 13.5 (Gromov). / f Pdcci diam 2 > - e , , then the fundamental group is almost nilpotent.

For some refinements of the above conjecture, see [24]. The following is closely related with the Chern conjecture stating that every abelian subgroup of the

fundamental group of a closed manifold with positive sectional curvature is cyclic.

C o n j e c t u r e 13.6. There exists a positive number c, such that if a dosed n-manifold M has positive sectional curvature, then Tel(M) contains a cydic subgroup S such that [Trl(M) : S l < cn. Thus, the fundamental groups of positively curved manifolds are essentially cyclic.

The following is a sort of a gap theorem conjecture.

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Conjec tu re 13.7. There exists a positive number e, such that i[ M" has e,-nonnegative curvature, then it has almost nonnega~ive curvature.

The gap theorems for almost flat manifolds and almost nonpositively curved manifolds were proved in [30] and [23] respectively.

So far, only a few results other than those on ~ra or bl are known for the topology of closed manifolds of nonnegative curvature except for Gromov's Betti number theorem [32], several sphere theorems, and related results.

Ques t ion 13.8. What can one say about 7ri (i > 1) [or nonnegatively curved manifolds?

See [36] for related topics. In this article, we consider almost nonnegative sectional or Ricci curvature. For positive scalar curvature,

some topological obstructions are known ([47, 35, 64]). A question related with our work is as follows.

Ques t ion 13.9. What can one say about the convergence o[ metrics o[ almost nonnegative scalar curvature on some dosed mani[old M? (For instance, for M = T".)

This would be more realistic if one assumes some geometric bounds, for instance [K] _< 1, diam _< D, V O | ~___ V 0 .

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