Weak collapsing and geometrisation of aspherical 3-manifolds
Laurent Bessières, Gérard Besson, Michel Boileau, Sylvain Maillot, Joan Porti
aa r X i v : . [ m a t h . G T ] J a n Weak collapsing and geometrisation ofaspherical 3-manifolds
L. Bessi`eres, G. Besson, M. Boileau, S. Maillot, J. PortiMay 26, 2018
Abstract
Let M be a closed, orientable, irreducible, non-simply connected3-manifold. We prove that if M admits a sequence of Riemannianmetrics whose sectional curvature is locally controlled and whose thickpart becomes asymptotically hyperbolic and has a sufficiently smallvolume, then M is Seifert fibred or contains an incompressible torus.This result gives an alternative approach for the last step in Perelman’sproof of the Geometrisation Conjecture for aspherical 3-manifolds. Introduction
Thurston’s Geometrisation Conjecture states that any closed, orien-table, irreducible 3-dimensional manifold M is hyperbolic, Seifert fi-bred, or contains an incompressible torus. This conjecture has beenproved recently by G. Perelman [31, 33, 32] (see also [27, 29, 8]) usingR. Hamilton’s Ricci flow. In this paper, we shall be concerned withthe case where π M is nontrivial. Our results apply in particular if π M is infinite, which under the above hypotheses is equivalent to M being aspherical.The last step of Perelman’s proof in this case relies on a ‘collapsingtheorem’ which is independent of the Ricci flow part. This result isstated without proof as Theorem 7.4 in [33]. A version of this theoremfor closed 3-manifolds is given in the appendix of [36] using deep resultsof Alexandrov space theory, including Perelman’s stability theorem(see [26]) and a fibration theorem for Alexandrov spaces [39].Our main result, Theorem 0.1 below, implies Theorem 7.4 of [33]for closed, irreducible 3-manifolds which are not simply connected,and is sufficient to apply Perelman’s construction of Ricci flow withsurgery to geometrise these manifolds. The proof of Theorem 0.1 ombines arguments from Riemannian geometry, algebraic topology,and 3-manifold theory. It uses Thurston’s hyperbolisation theorem forHaken manifolds, but avoids the stability and fibration theorems forAlexandrov spaces.In this text we call hyperbolic manifold a complete 3-manifold withconstant sectional curvature equal to − finite volume. The hy-perbolic metric, which is unique up to isometry by Mostow rigidity,will be denoted by g hyp .In the next two definitions, M is a 3-manifold. Definition.
Let g be a Riemannian metric on M and ε > x ∈ M is ε -thin with respect to g if there exists0 < ρ ≤ B ( x, ρ ), the sectional curvature isgreater than or equal to − ρ − and the volume of this ball is less than ε ρ . Otherwise we say that x is ε -thick with respect to g . The set of ε -thin points (resp. ε -thick points) is called the ε -thin part (resp. ε -thickpart ) of M .The following is a technical condition which guarantees the regu-larity of certain limits of riemannian manifolds. Definition.
Let g n be a sequence of Riemannian metrics on M . Wesay that g n has locally controlled curvature in the sense of Perelman if it has the following property: for all ε > r ( ε ) > , K ( ε ) , K ( ε ) >
0, such that for n large enough, if 0 < r ≤ ¯ r ( ǫ ), x ∈ ( M, g n ) satisfies vol( B ( x, r )) /r ≥ ε , and the sectional curvatureon B ( x, r ) is greater than or equal to − r − , then | Rm( x ) | < K r − and |∇ Rm( x ) | < K r − .Next we define a topological invariant V ( M ) which is essential tothis paper. Let M be a closed 3-manifold. For us, a link in M is a(possibly empty, possibly disconnected) closed 1-submanifold of M .A link is hyperbolic if its complement is a hyperbolic 3-manifold. Theinvariant V ( M ) is defined as the infimum of the volumes of all hyper-bolic links in M . This quantity is finite because any closed 3-manifoldcontains a hyperbolic link [30]. Since the set of volumes of hyperbolic3-manifolds is well-ordered, this infimum is always realised by somehyperbolic 3-manifold H ; in particular, it is positive. We note that M is hyperbolic if and only if H = M ; indeed, every hyperbolic Dehnfilling on a hyperbolic manifold strictly decreases the volume [4].We now state the main result of this article: heorem 0.1. Let M be a non-simply connected, closed, orientable,irreducible -manifold. Suppose that there exists a sequence g n of Rie-mannian metrics on M satisfying:(1) The sequence vol( g n ) is bounded.(2) Let ε > be a real number and x n ∈ M be a sequence such thatfor all n , x n is ε -thick with respect to g n . Then the sequence ofpointed manifolds ( M, g n , x n ) has a subsequence that convergesin the C topology towards a hyperbolic pointed manifold withvolume strictly less than V ( M ) .(3) The sequence g n has locally controlled curvature in the sense ofPerelman.Then M is a graph manifold or contains an incompressible torus. Recall that if M is a graph manifold, then M is Seifert fibredor contains an incompressible torus. Hence the conclusion of Theo-rem 0.1 implies that M satisfies the conclusion of the GeometrisationConjecture as stated at the beginning of this paper.Note that Hypothesis (2) of Theorem 0.1 may be vacuous; this isin particular the case if there is a sequence ε n → n , every point of M is ε n -thin with respect to g n . In this situation,we shall say that the sequence g n collapses. Thus Theorem 0.1 canbe viewed as a weak collapsing result in the sense that we allow thethick part to be nonempty, but require a control on its volume. Inthe special case where g n collapses, the proof of Theorem 0.1 does notuse Hypothesis (1), and leads to the conclusion that M is a graphmanifold. Hence we also obtain the following version of Perelman’scollapsing theorem: Corollary 0.2.
Let M be a closed, orientable, irreducible, non-simplyconnected -manifold. If M admits a sequence of Riemannian metricsthat collapses and has controlled curvature in the sense of Perelman,then M is a graph manifold. Sequences of metrics satisfying the hypotheses of Theorem 0.1 areprovided by Perelman’s construction and study of the Ricci flow withsurgery [33, 32]. From these deep results and Theorem 0.1 we deducecharacterisations of hyperbolic and graph 3-manifolds: these are The-orems 0.3 and 0.4. For this we shall use the invariant R ( M ) definedbelow, which was first suggested by M. Anderson [3] (cf. also [33, § § M be a closed 3-manifold. If g is a Riemannian metric on M , we denote by R min ( g ) the minimum of the scalar curvature R of g on M , and by ˆ R ( g ) the scale invariant quantity R min ( g ) vol( g ) / . ote that if M has a hyperbolic metric g hyp , then ˆ R ( g hyp ) is equal to − g hyp ) / .We let R ( M ) be the (possibly infinite) supremum of ˆ R ( g ), takenover all Riemannian metrics g on M . Theorem 0.3.
Let M be a closed, orientable, irreducible -manifold.(1) If R ( M ) ≤ − V ( M ) / , then M is hyperbolic and R ( M ) =ˆ R ( g hyp ) .(2) If − V ( M ) / < R ( M ) , then M contains an incompressibletorus or is Seifert fibred. Theorem 0.3 immediately implies the Geometrization Conjecture.Moreover, it shows that if M is a closed hyperbolic 3-manifold, thenthe hyperbolic metric realises the maximum in R ( M ), hence has thesmallest volume among all complete metrics with scalar curvaturebounded below by −
6. See [1] for an application.Assertion (1) of Theorem 0.3 follows from the proof of Theorem 2.9of [2]. We shall give another proof based on Thurston’s hyperbolicDehn filling theorem, following [35, Chap. 3]. When M is aspherical,Assertion (2) is proved using Theorem 0.1 and results of Perelman onthe long time behaviour of Ricci flow with surgery [33]. (In fact, it isenough to assume that M is not simply connected.) For completeness,we have included in the statement the case where π M is finite, whichfollows from [32, 12, 29].Our last result is a complement to Theorem 0.3. Let V ′ ( M ) bethe minimum of the volumes of all hyperbolic submanifolds H ⊂ M having the property that H is a link complement or ∂ ¯ H has at leastone incompressible component. Theorem 0.4.
Let M be a closed, orientable, irreducible -manifold.Then M is a graph manifold if and only if R ( M ) > − V ′ ( H ) / . This article is organized as follows: in Section 1, we describe thestrategy of the proof of Theorem 0.1. This proof is then given inSections 2–4. In Section 5, we recall the necessary results comingfrom the Ricci flow. In Section 6, we prove Theorems 0.3 and 0.4.
Acknowledgments
The authors wish to thank John Morgan forpointing out the proofs of Proposition 4.2 and Corollary 4.3 whichallowed to extend Theorem 0.1 from the aspherical case to the non-simply connected case.We also wish to thank the Clay Mathematics Institute for its finan-cial support, as well as the project FEDER-MEC MTM2006-04353. When M is aspherical, it turns out that R ( M ) is finite, equal to the Yamabe invariantof M (see e.g. [27, Section 93]), but we will not use this fact. Sketch of proof of Theorem 0.1
For classical 3-manifold theory, we use [25], [23] as main references, aswell as [6] for post-Thurston results. To avoid any confusion betweenmetric balls and topological balls, we shall call 3 -ball a 3-manifoldhomeomorphic to the closed unit ball in R . By contrast, our metricballs B ( x, r ) are open.Throughout the paper we work in the smooth category. Recall thata Haken manifold is a connected, compact, orientable, irreducible 3-manifold which contains an incompressible surface. Any connected,compact, orientable, irreducible 3-manifold whose boundary is notempty is Haken. It follows from deep work of W. Thurston and earlierwork of Jaco-Shalen and Johannson that every Haken manifold hasa canonical decomposition along incompressible tori into Seifert andhyperbolic pieces (see e.g. the references given in [6].) We call thisthe geometric decomposition of the Haken manifold M . Moreover,a Haken manifold is a graph manifold if and only if all pieces in itsgeometric decomposition are Seifert.Another key notion used in the proof of Theorem 0.1 is the simpli-cial volume, sometimes called Gromov’s norm, introduced by M. Gro-mov in [17]. Our proof relies on an additivity result for the simplicialvolume under gluing along tori (see [17, 28, 37]) which implies thatthe simplicial volume of a 3-manifold admitting a geometric decom-position is proportional to the sum of the volumes of the hyperbolicpieces. In particular, such a manifold has zero simplicial volume ifand only if it is a graph manifold.We also use in an essential way Gromov’s vanishing theorem [17,24]: if a n -dimensional closed manifold M can be covered by open sets U i such that the covering has dimension less than n and the image ofthe canonical homomorphism π U i → π M is amenable for all i , thenthe simplicial volume of M vanishes. (Recall that the dimension of afinite covering U i is the dimension of its nerve.)Below we outline the proof of Theorem 0.1. For simplicity, weexplain it in the special case where the sequence g n collapses. In thelast paragraph, we shall say a few words about the general case.Before discussing the proof proper, we give an example of a cov-ering argument which can be used to deduce topological informationon M (namely that M has zero simplicial volume) from the collapsinghypothesis.For n large enough, thanks to the local control on the curvature,each point has a neighbourhood in ( M, g n ) which is close to a metricball in some manifold of nonnegative sectional curvature, and whosevolume is small compared to the cube of the radius. These neigh- ourhoods will be called local models . From the classification of mani-folds with nonnegative sectional curvature, we deduce that these localmodels have virtually abelian, hence amenable fundamental groups.A technique introduced by Gromov [17] yields a covering of M , whosedimension is at most 2, by open sets contained in these neighbour-hoods. As a consequence, Gromov’s vanishing theorem implies thatthe simplicial volume of M vanishes.The previous scheme, together with the additivity of the simplicialvolume under gluing along incompressible tori, shows that a manifoldwhich admits a geometric decomposition and a sequence of collaps-ing metrics is a graph manifold. This is however insufficient to proveTheorem 0.1 since we do not assume that M admits a geometric de-composition! Hence we need a trick similar to those of [7] and [5],which we now explain.In the first step, we find a local model U such that all connectedcomponents of M \ U are Haken. This requirement is equivalent toirreducibility of each component of M \ U . Since M is irreducible, itsuffices to show that U is not contained in a 3-ball. This is in partic-ular the case if U is homotopically nontrivial , i.e. the homomorphism π ( U ) → π ( M ) has nontrivial image.The proof of the existence of a homotopically nontrivial local model U is done by contradiction: assuming that all local models are homo-topically trivial, we construct a covering of M of dimension less thanor equal to 2 by homotopically trivial open sets. By a result of J.C.G´omez-Larra˜naga and F. Gonz´alez-Acu˜na [14], a closed irreducible3-manifold admitting such a covering must have trivial fundamentalgroup. This is where we use the hypothesis that M is not simplyconnected.The second step, which is again a covering argument but donerelatively to some fixed homotopically nontrivial local model U , showsthat any manifold obtained by Dehn filling on Y := M \ U has acovering of dimension less than or equal to 2 by virtually abelianopen sets, and therefore has vanishing simplicial volume. We concludeusing Proposition 4.14, which states that if Y a Haken manifold withboundary a collection of tori and such that the simplicial volume ofevery Dehn filling on Y vanishes, then Y is a graph manifold. Thisfinishes the sketch of proof of Theorem 0.1 in the collapsing case.In this text, we shall not separate the case when g n collapses, butwe shall treat directly the general case. This implies that we first needto cover the thick part by submanifolds H in approximating compact As already mentioned in [7, 5], Proposition 4.14 is a consequence of the geometrisationof Haken manifolds, additivity of the simplicial volume mentioned above, and Thurston’shyperbolic Dehn filling theorem. ores of limiting hyperbolic manifolds (Section 2). We then cover thethin part by local models (Section 3). The bulk of the proof is in Sec-tion 4: assuming that M contains no incompressible tori, we considerthe covering of M by approximately hyperbolic submanifolds and lo-cal models of the thin part and perform two covering arguments: thefirst one shows that at least one of these open subsets is homotopicallynontrivial in M ; the second one is done relatively to this homotopicallynontrivial subset and proves that M is a graph manifold. Until Section 4, we consider a 3-manifold M and a sequence of Rie-mannian metrics g n satisfying the hypotheses of Theorem 0.1. For thesake of simplicity, in the sequel we use the notation M n := ( M, g n ).The goal of this section is to describe the thick part of the manifolds M n and to make the link between the topology of the thick part andthe topology of M . We denote by M − n ( ε ) the ε -thin part of M n , andby M + n ( ε ) its ε -thick part. Proposition 2.1.
Up to taking a subsequence of M n , there existsa finite (possibly empty) collection of pointed hyperbolic manifolds ( H , ∗ ) , . . . , ( H m , ∗ m ) and for every ≤ i ≤ m a sequence x in ∈ M n satisfying:(i) lim n →∞ ( M n , x in ) = ( H i , ∗ i ) in the C topology.(ii) For all sufficiently small ε > , there exist n ( ε ) and C ( ε ) suchthat for all n ≥ n ( ε ) one has M + n ( ε ) ⊂ S i B ( x in , C ( ε )) .Proof. By assumption, the sequence vol( M n ) is bounded above. Let µ > µ .If for all ε > M + n ( ε ) = ∅ for n large enough, thenProposition 2.1 is vacuously true. Otherwise, we use Hypothesis (2)of Theorem 0.1: up to taking a subsequence of M n , there exists ε > x n ∈ M + n ( ε ) such that ( M n , x n ) convergesto a pointed hyperbolic manifold ( H , ∗ ).If for all ε > C ( ǫ ) such that, for n large enough, M + n ( ε ) is included in B ( x n , C ( ε )), then we are done. Otherwise thereexists ε > x n ∈ M + n ( ε ) such that d ( x n , x n ) → ∞ .Again Hypothesis (2) of Theorem 0.1 ensures that, after taking a ubsequence, the sequence ( M n , x n ) converges to a pointed hyperbolicmanifold ( H , ∗ ).Note that for each i , and for n sufficiently large, M n contains asubmanifold C -close to some compact core of H i and whose volume isgreater than or equal to µ /
2. Moreover, for n fixed and large, thesesubmanifolds are pairwise disjoint. Since the volume of the manifolds M n is uniformly bounded above this construction has to stop. Con-dition (ii) of the conclusion of Proposition 2.1 is then satisfied for0 < ε < ε k . Remark.
By Proposition 2.1 one can choose sequences ε n → r n → ∞ such that the ball B ( x in , r n ) is arbitrarily close to a metric ball B ( ∗ i , r n ) ⊂ H i , for i = 1 , . . . , m , and every point of M n \ S i B ( x in , r n )is ε n -thin.Let us fix a sequence of positive real numbers ε n →
0. Let H , . . . ,H m be hyperbolic limits given by Proposition 2.1. For each i wechoose a compact core ¯ H i for H i and for each n a submanifold ¯ H in and an approximation φ in : ¯ H in → ¯ H i . Up to renumbering, one canassume that for all n we have M n \ S ¯ H in ⊂ M − n ( ε n ), and that the¯ H in ’s are disjoint.The hypothesis that the volume of each hyperbolic limit H i isless than V implies that for n sufficiently large no component ¯ H in ishomeomorphic to the exterior of a link in M .The logic of the proof is the following: each boundary componentof ¯ H in is a torus. If one of those tori is incompressible, then theconclusion of Theorem 0.1 is true. The interesting case is when all thetori that appear in the boundary of the thick part are compressible.The remainder of this section is devoted to the two following results: Proposition 2.2.
Up to taking a subsequence, one of the followingproperties is satisfied:(i) There exists an integer i ∈ { , . . . , m } such that ∂ ¯ H i n containsan incompressible torus for all n , or(ii) for all i ∈ { , . . . , m } , ¯ H in is embedded in a solid torus or in a -ball contained in M n for all n . Proposition 2.3.
If Conclusion (ii) of Proposition 2.2 is satisfied,then either M is a lens space or there exists for each n a submanifold W n ⊂ M n such that:(i) S ¯ H in ⊂ W n .(ii) Each connected component of W n is a solid torus, or containedin a -ball and homeomorphic to the exterior of a knot in S . iii) The boundary of each component of W n is a component of S i ∂ ¯ H in . Subsection 2.2 is devoted to general topological results concerningcompressible tori in 3-manifolds and abelian submanifolds. Proposi-tion 2.2 and 2.3 will be proved in the Subsection 2.3.
Let X be an orientable, irreducible 3-manifold and T be a compressibletorus embedded in X . The Loop Theorem shows the existence of a compression disc D for T , that is, a disc D embedded in M such that D ∩ T = ∂D and the curve ∂D is not null homotopic in T . By cuttingopen T along an open small regular neighbourhood of D and gluingtwo parallel copies of D along the boundary curves, one constructs anembedded 2-sphere S in X . We say that S is obtained by compressing T along D .Since X is assumed to be irreducible, S bounds a 3-ball B . Thereare two possible situations depending on whether B contains T ornot. The following lemma collects some standard results that we shallneed. Lemma 2.4.
Let X be an orientable, irreducible -manifold and T bea compressible torus embedded in X . Let D be a compression disc for T , S be a sphere obtained by compressing T along D , and B a ballbounded by S . Then:i) X \ T has two connected components U, V , and D is containedin the closure of one of them, say U .ii) If B does not contain T , then B is contained in ¯ U , and ¯ U is asolid torus.iii) If B contains T , then B contains V , and ¯ V is homeomorphic tothe exterior of a knot in S . In this case, there exists a homeo-morphism f from the boundary of S × D into T such that themanifold obtained by gluing S × D to ¯ U along f is homeomor-phic to X . Remark. If T is a component of ∂X and T is a compressible torus,the same argument shows that X is a solid torus. Lemma 2.5.
Let X be a closed, orientable, irreducible -manifold.Let ¯ H ⊂ X be a connected, compact, orientable, irreducible submani-fold of X whose boundary is a collection of compressible tori. If ¯ X isnot homeomorphic to the exterior of a (possibly empty) link in X , then ¯ H is included in a connected submanifold Y whose boundary is one ofthe tori of ∂ ¯ H and which satisfies one of the following properties: i) Y is a solid torus, or(ii) Y is homeomorphic to the exterior of a knot in S and containedin a ball B ⊂ X .Proof. By hypothesis the boundary of ¯ H is not empty. We denoteby T , . . . , T m the components of ∂ ¯ H . If one of them bounds a solidtorus containing ¯ H , we can choose this solid torus as Y . Henceforthwe assume that this is not the case.Each T j being compressible, it separates and thus bounds a sub-manifold V j not containing ¯ H . Up to renumbering the boundary com-ponents of ¯ H , we may assume that V , . . . , V k are solid tori, but not V k +1 , . . . , V m . At least one of the V j ’s is not a solid torus, otherwise¯ H would be homeomorphic to the exterior of a link in X .For the same reason, at least one V j , for some j > k , is not con-tained in a 3-ball. Otherwise each of the V k +1 , . . . , V m is homeomor-phic to the exterior of a knot in S , by Lemma 2.4, and one couldthen replace each V j , k + 1 ≤ j ≤ m by a solid torus without changingthe topological type of X . Hence ¯ H would be homeomorphic to theexterior of a link in X .Pick a V j , for j > k , which is not contained in a ball. Thencompressing surgery on the torus T j = ∂V j yields a sphere S boundinga ball B in X , which contains ¯ H by the choice of V j . This shows thatconclusion (ii) is satisfied with Y = X \ int V j . Proof of 2.2.
If Assertion (i) of Proposition 2.2 is not satisfied, then,up to a subsequence, one may assume that for all i ∈ , . . . , m andfor all n , each component of ∂ ¯ H in is compressible in M . We fix aninteger i ∈ , . . . , m . From the hypotheses of Theorem 0.1, we getthe inequality vol( H i ) < V ( M ), which implies that H i is not home-omorphic to the complement of a link in M . In particular since ¯ H in is homeomorphic to the compact core of H i , it is not homeomorphicto the exterior of a link in M . Lemma 2.5 allows to conclude thatAssertion (ii) of Proposition 2.2 holds true. Proof of 2.3.
For each n and each i ∈ , . . . , m , we choose a subman-ifold Y in containing ¯ H in , given by Lemma 2.5. We take W n to be theunion of the Y in . Then Assertion (i) of Proposition 2.3 is straightfor-ward.Assume that M is not a lens space. Then M n cannot be the unionof two submanifolds Y in and Y jn , otherwise M n can be covered eitherby two solid tori, or by a solid torus and a ball or by two balls. In thefirst case M n would be homeomorphic to a lens space by [16], while n the other two cases M n would be covered by three balls and thushomeomorphic to the 3-sphere S by [22], see also [15]. Thus for all i , i , the submanifolds Y i n Y i n are disjoint or one contains the other,because they have disjoint boundaries. In this case each componentof W n is homeomorphic to one of the Y in . This yields Assertions (ii)and (iii) of Proposition 2.3. In this section, it is implicit that any quantity depending on a point x ∈ M n is computed with respect to the metric g n on M n and thusdepends also on n .Let us choose a sequence ε n → x ∈ M − n ( ε n ), we choose a radius 0 < ρ ( x ) ≤
1, suchthat on the ball B ( x, ρ ( x )) the curvature is ≥ − ρ − ( x ) and the volumeof this ball is < ε n ρ ( x ).In the following proposition we use Cheeger-Gromoll’s soul theo-rem [10]. Proposition 3.1.
For all
D > there exists n ( D ) such that if n >n ( D ) , then for all x ∈ M − n ( ε n ) we have the following alternative:(a) Either M n is D -close to some closed nonnegatively curved -manifold, or(b) there exists a radius ν ( x ) ∈ (0 , ρ ( x )) and a complete noncompactRiemannian -manifold X x , with nonnegative sectional curva-ture and soul S x , such that the following properties are satisfied:(1) B ( x, ν ( x )) is D -close to a metric ball in X x .(2) There exists an approximation f x : B ( x, ν ( x )) → X x suchthat max { d ( f ( x ) , S x ) , diam S x } ≤ ν ( x ) D . (3) vol( B ( x, ν ( x ))) ≤ D ν ( x ) . Remark.
Since ν ( x ) < ρ ( x ), the sectional curvature on B ( x, ν ( x )) isgreater than or equal to − ρ ( x ) , which is in turn bounded below by − ν ( x ) . Remark.
The only closed, orientable and irreducible 3-manifold con-taining a projective plane is RP , which is a graph manifold. There-fore if the manifold M is not homeomorphic to RP , then the soul S x can be homeomorphic to a point, a circle, a 2-sphere, a 2-torus or aKlein bottle. In this case, the ball B ( x, ν ( x )) is homeomorphic to B , S × D , S × I , T × I or to the twisted I -bundle on the Klein bottle. efore starting the proof of this proposition, we prove the followinglemma and its consequence: Lemma 3.2.
There exists a universal constant
C > such that forall ε > , for all x ∈ M n , and for all r > , if the ball B ( x, r ) hasvolume ≥ ε r and curvature ≥ − r − , then for all y ∈ B ( x, r ) andall < r ′ < r , the ball B ( y, r ′ ) has volume ≥ C · ε ( r ′ ) and curvature ≥ − ( r ′ ) − . We use the function v − κ ( r ) to denote the volume of the ball ofradius r in the 3-dimensional hyperbolic space with curvature − κ .Notice that v − κ ( r ) = κ − v − ( κ r ). Proof.
The lower bound on the curvature is a consequence of themonotonicity of the function − r − with respect to r . In order to es-timate from below the normalised volume we apply Bishop-Gromov’sinequality twice. First to the ball around y , increasing the radius r ′ to r : vol( B ( y, r ′ )) ≥ vol( B ( y, r )) v − r − ( r ′ ) v − r − ( r ) . Using that v − r − ( r ′ ) = r v − ( r ′ r ) ≥ r (cid:16) r ′ r (cid:17) C for C > v − r − ( r ) = r v − ( ), and that the ball B ( y, r ) contains B ( x, r ),we have vol( B ( y, r ′ )) ≥ vol( B ( x, r )) (cid:16) r ′ r (cid:17) C . Applying again the Bishop-Gromov inequality:vol( B ( x, r )) ≥ vol( B ( x, r )) v − r − ( r ) v − r − ( r ) ≥ r ε v − ( ) v − (1) = r ε C . Hence vol( B ( y, r ′ )) ≥ ( r ′ ) ε C .We deduce an ‘improvement’ of the controlled curvature in thesense of Perelman, in which the conclusion is valid at each point ofsome metric ball, not only the centre. The only price to pay is thatthe constants can be slightly different. Corollary 3.3.
For all ε > there exists ¯ r ′ ( ε ) > , K ′ ( ε ) , K ′ ( ε ) such that for n large enough, if < r ≤ ¯ r ′ ( ε ) , x ∈ M n and the ball B ( x, r ) has volume ≥ ε r and sectional curvatures ≥ − r − then, forall y ∈ B ( x, r ) , | Rm( y ) | < K ′ r − and |∇ Rm( y ) | < K ′ r − .Proof. It suffices to apply Lemma 3.2, setting ¯ r ′ ( ε ) = ¯ r ( Cǫ ), K ′ ( ε ) = K ( Cǫ ) and K ′ ( ε ) = K ( Cǫ ). roof of Proposition 3.1. Let us assume that there exists D > x n ∈ M − n ( ε n ) such that neither of theconclusions of Proposition 3.1 holds with D = D .Set ε := D . We shall rescale the metrics using the followingradii: Definition.
For x ∈ M n , definerad( x ) = inf { r > | vol( B ( x, r )) /r ≤ ε } . We gather in the following lemma some properties which will beuseful for the proof:
Lemma 3.4. (i) For n large enough and x ∈ M − n ( ε n ) , one has < rad( x ) < ρ ( x ) .(ii) For n sufficiently large and x ∈ M − n ( ε n ) , one has vol( B ( x, rad( x )))rad( x ) = ε . (iii) For L > , there exists n ( L ) such that for n > n ( L ) and for x ∈ M − n ( ε n ) we have L rad( x ) ≤ ρ ( x ) . In particular lim n →∞ rad( x n ) = lim n →∞ rad( x n ) ρ ( x n ) = 0 .Proof of Lemma 3.4. Property (i) follows from continuity, by compar-ing the limit vol( B ( x, δ )) /δ → π when δ → B ( x, ρ ( x ))) /ρ ( x ) < ε n → . Assertion (ii) is also proved by continuity.We prove (iii) for
L > f x ( s ) = vol( B ( x, s rad( x )))( s rad( x )) . One has f x (1) = ε ; for all s ∈ [1 , L ], f x ( s ) ≥ ε s ≥ ε L . Furthermore,for s < f x ( s ) > ε by the definition of rad( x ). It suffices then tochoose n so that for all n ≥ n one has ε n < ε L . Remark.
For n large enough, from the preceding Lemma, we haverad( x n ) < ¯ r ′ ( ε ). orollary 3.5. There exists a constant
C > such that any sequenceof x n ∈ M n satisfies inj( x n )rad( x n ) ≥ C for n large enough.Proof. Let us first remark that, since rad( x n ) < ρ ( x n ), the sectionalcurvatures on B ( x n , rad( x n )) are ≥ − ρ ( x n ) > − x n ) . Moreover, asrad( x n ) < ¯ r ′ ( ε ), Corollary 3.3 shows that the curvature on the ball B ( x n , rad( x n )3 ) is bounded above by K ′ ( ε ) / rad( x n ) . This rescaledball 1rad( x n ) B ( x n , rad( x n ))has radius ≤
1, volume ≥ ε C (where C is a universal constant com-ing from Bishop-Gromov) and curvatures ≤ K ′ ( ε ). Using Cheeger’spropeller lemma [9, Thm. 5.8], the injectivity radius at the centre ofthe rescaled ball is bounded below by some constant C >
0. Thisproves Corollary 3.5.Having proved Lemma 3.4 and its corollary, we continue the proofof Proposition 3.1. Let us consider the rescaled manifold M n = x n ) M n . We look for a limit of the sequence ( M n , ¯ x n ), where ¯ x n isthe image of x n . The ball B (¯ x n , ρ ( x n )rad( x n ) ) ⊂ M n has sectional curva-ture bounded below by − (cid:16) rad( x n ) ρ ( x n ) (cid:17) , which goes to 0 when n → ∞ ,as follows from Assertion (iii) of Lemma 3.4.Given L >
0, the ball B (¯ x n , L ) is obtained by rescaling the ball B ( x n , L rad( x n )). Since 3 L rad( x n ) < ρ ( x n ), the sectional curvatureon B ( x n , L rad( x n )) is ≥ − ρ ( x n ) ≥ − L rad( x n )) . Moreover, we havevol( B ( x n , L rad( x n ))(3 L rad( x n )) ≥ ε (3 L ) . By applying Corollary 3.3 for n sufficiently large so that we have3 L rad( x n ) ≤ ¯ r ′ ( ε (3 L ) ), one gets that the curvature is locally controlledin the sense of Perelman at each point of the ball B ( x n , L rad( x n )).Therefore the curvature and its first derivative can be bounded aboveon any ball B (¯ x n , L ) ⊂ M n with a given radius L > x n is bounded belowalong the sequence, this upper bound on the curvature allows to useGromov’s compactness theorem [18, Chap. 8, Thm. 8.28], [34] andits versions with regularised limit [21, Thm. 2.3] or [13, Thm. 4.1 and5.10]. It follows that the pointed sequence ( M n , ¯ x n ) subconverges inthe C -topology towards a 3-dimensional smooth manifold ( X ∞ , x ∞ ), ith a complete riemannian metric of class C with nonnegative sec-tional curvature. This limit manifold cannot be closed, because thatwould contradict the assumption that the conclusion of Proposition 3.1does not hold.Hence X ∞ is not compact. Let S be its soul. Let us choose ν ( x n ) = L rad( x n ) where L ≥ S ∪ { x ∞ } ) D . for n large (to be specified later) we set X x n = rad( x n ) X ∞ , and S x n = rad( x n ) S. We then havediam( S x n ) = rad( x n ) diam( S ) < ν ( x n ) /D . Let ¯ f n : B (¯ x n , L ) → ( X ∞ , x ∞ ) be a δ n -approximation, where δ n isa sequence going to 0. After rescaling f n : B ( x n , L rad( x n )) → X x n isalso a δ n -approximation. We get: d ( f n ( x n ) , S x n ) = rad( x n ) d ( ¯ f n (¯ x n ) , S ) ≤ rad( x n ) (cid:0) d ( ¯ f n (¯ x n ) , ¯ x ∞ ) + d (¯ x ∞ , S ) (cid:1) ≤ rad( x n ) δ n + ν ( x n )2 D ≤ ν ( x n ) D . This proves assertion (2) of Proposition 3.1.Using the fact that ν ( x n ) = L rad( x n ) < ρ ( x n ), the curvature on B ( x n , ν ( x n )) is ≥ − /ν ( x n ) , L > B ( x n , ν ( x n ))) v − ν xn ) ( ν ( x n )) ≤ vol( B ( x n , rad( x n ))) v − ν xn ) (rad( x n )) = ε rad( x n ) v − ν xn ) (rad( x n )) = ε (cid:18) rad( x n ) ν ( x n ) (cid:19) v − ( rad( x n ) ν ( x n ) ) = ε L v − ( L ) . Taking now L sufficiently large, we find that:vol( B ( x n , ν ( x n ))) ≤ ε L v − (1) v − ( L ) ν ( x n ) ≤ ε ν ( x n ) = 1 D ν ( x n ) , where the last equality comes from the definition of ε .Hence we get the contradiction required to conclude the proof ofProposition 3.1. Constructions of coverings
We begin by making some reductions for the proof of Theorem 0.1.If case (a) of Proposition 3.1 occurs, then M is a closed, orientable,irreducible 3-manifold admitting a metric of nonnegative sectional cur-vature. By [19, 20], M is spherical or Euclidean, hence a graph mani-fold. Therefore we may assume that all local models are noncompact.For the same reasons, since lens spaces are graph manifolds, wecan also assume that M is not homeomorphic to a lens space, and inparticular does not contain a projective plane.If there exists an integer i ∈ , . . . , m such that, up to a subse-quence, ∂ ¯ H in contains an incompressible torus for all n , then Theo-rem 0.1 is proved. We thus assume that for all i ∈ , . . . , m and forall n , each component of ∂ ¯ H in is compressible in M n and thus Propo-sitions 2.2 and 2.3 apply and give for each n a submanifold W n .Assume that there exists a component X of W n which is not a solidtorus. From Proposition 2.3(ii), X is a knot exterior and contained ina 3-ball B ⊂ M n . By Lemma 2.4, it is possible to replace X by a solidtorus Y without changing the global topology. Let us denote by M ′ n the manifold thus obtained. We can endow M ′ n with a Riemannianmetric g ′ n , equal to g n away from Y and such that an arbitrarily largecollar neighbourhood of ∂Y in Y is isometric to a collar neighbourhood ∂X in X . When n is large, this neighbourhood is thus almost isometricto a long piece of a hyperbolic cusp, and this geometric property willbe sufficient for our covering arguments.Repeating this construction for each component of W n which isnot a solid torus, we obtain a Riemannian manifold ( M ′′ n , g ′′ n ) togetherwith a submanifold W ′′ n satisfying the following properties:(i) M ′′ n is homeomorphic to M n .(ii) M ′′ n \ W ′′ n is equal to M n \ W n and the metrics g n and g ′′ n coincideon this set.(iii) M ′′ n \ W ′′ n = M n \ W n is ε n -thin.(iv) When n goes to infinity, there exists a collar neighbourhood of ∂W ′′ n in W ′′ n of arbitrarily large diameter isometric to the corre-sponding neighbourhood in W n .(v) Each component of W ′′ n is a solid torus.For simplicity, we use the notation M n , g n , W n instead of M ′′ n , g ′′ n , W ′′ n . This amounts to assuming in the conclusion of Proposition 2.3that all components of W n are solid tori. .2 Existence of a homotopically nontrivial o-pen set We say that an arcwise connected set U ⊂ M is homotopically trivial (in M ) if the image of the homomorphism π ( U ) → π ( M ) is trivial.More generally, we say that the subset U ⊂ M is homotopically trivialif all its arcwise connected components have this property.We recall that the dimension of a finite covering { U i } i of M isthe dimension of its nerve, hence the dimension plus one equals themaximal number of U i ’s containing a given point. Proposition 4.1.
There exists D > such that for all D > D , forevery n greater than or equal to the number n ( D ) given by Proposi-tion 3.1, one of the following assertions is true:(a) some connected component of W n is not homotopically trivial, or(b) there exists x ∈ M n \ int( W n ) such that the image of π ( B ( x, ν ( x ))) → π ( M n ) is not homotopically trivial. In [14] J.C. G´omez-Larra˜naga and F. Gonz´alez-Acu˜na have com-puted the 1-dimensional Lusternik-Schnirelmann category of a closed3-manifold. One step of their proof gives the following proposition(cf. [14, Proof of Prop. 2.1]:)
Proposition 4.2.
Let X be a closed, connected 3-manifold. If X hasa covering of dimension 2 by open subsets which are homotopicallytrivial in X , then there is a connected -dimensional complex K anda continuous map f : X → K such that the induced homomorphism f ⋆ : π ( X ) → π ( K ) is an isomorphism. Standard homological arguments show the following, cf. [14, § Corollary 4.3.
Let X be a closed, connected, orientable, irreducible3-manifold. If X has a covering of dimension 2 by open subsets whichare homotopically trivial in X , then X is simply connected.Proof. By Proposition 4.2, let f : X → K be a continuous map from X to a connected 2-dimensional complex K , such that the inducedhomomorphism f ⋆ : π ( X ) → π ( K ) is an isomorphism. Let Z be a K ( π ( X ) ,
1) space. Let φ : X → Z be a map from X to Z realizingthe identity homomorphism on π ( X ) and let ψ : K → Z be themap from K to Z realizing the isomorphism f − ⋆ : π ( K ) → π ( X ).Then φ is homotopic to ψ ◦ f and the induced homomorphism φ ∗ : H ( X ; Z ) → H ( Z ; Z ) factors through ψ ∗ : H ( K ; Z ) → H ( Z ; Z ).Since H ( K ; Z ) = { } , the homomorphism φ ∗ must be trivial. X φ −→ Zf ↓ ր ψ K f π ( X ) is infinite, then X is aspherical and φ ∗ is an isomorphism.Therefore π ( X ) is finite.If π ( X ) is finite of order d >
1, then let e X be the universal cov-ering of X . The covering map p : e X → X induces an isomorphismbetween the homotopy groups π k ( e X ) and π k ( X ) for k ≥
2. Since π ( X ) = { } , π ( e X ) = { } , and by the Hurewicz theorem, the canon-ical homomorphism π ( e X ) → H ( e X ; Z ) = Z is an isomorphism. Itfollows that the canonical map π ( X ) = Z → H ( X ; Z ) = Z is themultiplication by the degree d > p : e X → X . It iswell known that one can construct a K ( π ( X ) ,
1) space Z by addinga 4-cell to kill the generator of π ( X ) = Z , and adding further cellsof dimension ≥ φ : X → Z induces the identity on π ( X ) and a surjection φ ∗ : H ( X ; Z ) = Z → H ( Z ; Z ) = Z /d Z . Therefore X must be simplyconnected.In the proof of Proposition 4.1 we argue by contradiction usingCorollary 4.3 and the fact that π ( M ) is not trivial.With the notation of Proposition 3.1, we may assume that forarbitrarily large D there exists n ≥ n ( D ) such that the image of π ( B ( x, ν ( x ))) → π ( M n ) is trivial for all x ∈ M n \ int( W n ) as well asfor each component of W n .Then for all x ∈ M n \ int( W n ) we set:triv( x ) = sup r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π ( B ( x, r )) → π ( M n ) is trivial and B ( x, r ) is contained in B ( x ′ , r ′ ) withcurvature ≥ − r ′ ) and vol( B ( x ′ ,r ′ ))( r ′ ) ≤ /D By hypothesis, we have triv( x ) ≥ ν ( x ). The proof of Proposi-tion 4.1 follows by contradiction with the following assertion. Assertion 4.4.
There exists a covering of M n by open sets U , . . . ,U p such that: • Each U i is either contained in some B ( x i , triv( x i )) or in a subsetthat deformation retracts to a component of int( W n ) . In partic-ular, U i is homotopically trivial in M . • The dimension of this covering is at most . Since M is irreducible and non-simply connected, this contradictsCorollary 4.3.To prove Assertion 4.4, we define r ( x ) = min {
111 triv( x ) , } . emma 4.5. Let x, y ∈ M n \ int( W n ) . If B ( x, r ( x )) ∩ B ( y, r ( y )) = ∅ ,then(a) / ≤ r ( x ) /r ( y ) ≤ / ;(b) B ( x, r ( x )) ⊂ B ( y, r ( y )) .Proof. We may assume that r ( x ) ≤ r ( y ) and that r ( x ) = triv( x ) <
1. From the triangle inequality, we get:triv( x ) ≥ triv( y ) − r ( x ) − r ( y ) , hence 11 r ( x ) = triv( x ) ≥ r ( y ) − r ( x ) − r ( y ) ≥ r ( y ) . Consequently, we have 1 ≥ r ( x ) /r ( y ) ≥ / ≥ /
4, which shows (a).Now (b) follows because 2 r ( x ) + r ( y ) < r ( y ).If n is sufficiently large, we can choose points x , . . . , x q ∈ ∂W n in such a way that a tubular neighbourhood of each component ofthe boundary of W n contains precisely one of the x j ’s and that theballs B ( x j ,
1) are disjoint, have volume ≤ D and sectional curvatureclose to −
1. Furthermore, we may assume that B ( x j ,
1) is includedin a submanifold W ′ n which contains W n and can be retracted bydeformation onto it. In particular W ′ n and B ( x j ,
1) are homotopicallytrivial, since we have assumed that W n is. This implies that triv( x j )is close to 1.Moreover, for n large enough, we may assume that B ( x j , r ( x j ))contains an almost horospherical torus corresponding to a boundarycomponent of W n . We can even arrange for both components of B ( x j , r ( x j )) \ B ( x j , r ( x j )) to also contain a parallel almost horo-spherical torus, which allows to retract W n on the complement of B ( x j , r ( x j )).We complete x , x . . . , x q to a sequence x , x , . . . in M n \ int( W n )such that the balls B ( x , r ( x )), B ( x , r ( x )) , . . . are pairwise dis-joint.Such a sequence is necessarily finite, since M n \ int( W n ) is compact,and Lemma 4.5 implies a positive local lower bound for the function x r ( x ). Let us choose a maximal finite sequence x , . . . , x p withthis property. Lemma 4.6.
The balls B ( x , r ( x )) , . . . , B ( x p , r ( x p )) cover M n \ int( W n ) .Proof. Let x ∈ M n \ int( W n ) be an arbitrary point. By maximality,there exists a point x j such that B ( x, r ( x )) ∩ B ( x j , r ( x j )) = ∅ . FromLemma 4.5, we have r ( x ) ≤ r ( x j ) and d ( x, x j ) ≤ ( r ( x ) + r ( x j )) ≤ r ( x j ), hence x ∈ B ( x j , r ( x j )). et us define r i := r ( x i ). If W n, , . . . , W n,q are the components of W n , so that the almost horospherical torus ∂W n,i ⊂ B ( x i , r i ), we set: • V i := B ( x i , r i ) ∪ W n,i , for i = 1 , . . . , q . • V i := B ( x i , r i ), for i > q .Furthermore, each component of W n,i can be retracted in ordernot to intersect V j when j = i .The construction of the open sets V i and Lemma 4.6 imply thefollowing: Lemma 4.7.
The open sets V , . . . , V p cover M n . Let K be the nerve of the covering { V i } . We will use this cov-ering and the complex K to build the required map from M to a2-dimensional complex. The idea is first to map M to K and then toimprove this mapping by pushing it into the 2-skeleton K (2) of K .The following Lemma shows that the dimension of K is boundedabove by a uniform constant. Lemma 4.8.
There exists a universal upper bound N on the numberof open sets V i which intersect a given V k .Proof. If V i ∩ V k = ∅ , then B ( x i , r i ) ∩ B ( x k , r k ) = ∅ and B ( x i , r i ) ⊂ B ( x k , r i + r k ) ⊆ B ( x k , r k ). On the other hand, for all i = i suchthat V i and V i intersect V k one has d ( x i , x i ) ≥ ( r i + r i ) ≥ r k .The number of V i intersecting V k is thus bounded above by:vol( B ( x k , r k ))vol( B ( x i , r k )) ≤ vol( B ( x i , r k ))vol( B ( x i , r k )) ≤ vol( B ( x i , r i ))vol( B ( x i , r i )) . As B ( x i , r i ) is included in a ball B ( x ′ , r ′ ) with curvature ≥ − r ′ ) ,by Bishop-Gromov inequality this is bounded above by: v − r ′ )2 (11 r i ) v − r ′ )2 ( r i ) = v − ( r i r ′ ) v − ( r i r ′ ) ≤ N. Let ∆ p − ⊂ R p denote the standard unit simplex of dimension p −
1. With a partition of unity ( φ i ) adapted to the ( V i ) and withcertain metric properties, we construct a map: f = 1 P i φ i ( φ , . . . , φ p ) : M n → ∆ p − ⊂ R p , We view K as a subcomplex of ∆ p − , so that the range of f iscontained in K , whose dimension is at most N . Moreover, f maps he components of W n onto distinct vertices of the 0-skeleton of K .We first estimate the Lipschitz constant of the map f : M n → K , bychoosing the φ i ’s. Lemma 4.9.
There exists L N > such that the partition of unity canbe chosen so that the restriction f | V k is L N r k -Lipschitz.Proof. Let τ : [0 , → [0 ,
1] be an auxiliary function with Lipschitzconstant bounded by 4, which vanishes in a neighbourhood of 0 andverifies τ | [ , ≡
1. Let us define φ k := τ ( r k d ( ∂V k , · )) on V k and let usextend it trivially on M n . Then φ k is r k -Lipschitz.Let x ∈ V k . The functions φ i have Lipschitz constant ≤ · r k , andall φ i vanish at x except at most N + 1 of them. Since the functions( y , . . . , y N ) y k P Ni =0 y i are Lipschitz on { y ∈ R N +1 | y ≥ , . . . , y N and P Ni =0 y i ≥ } ,and each x ∈ M n belongs to some V k with d ( x, ∂V k ) ≥ r k , the conclu-sion follows.We shall now inductively deform f by homotopy into the 3-skeleton K (3) , while keeping the local Lipschitz constant under control. Lemma 4.10.
For all d ≥ and L > there exists L ′ = L ′ ( d, L ) > such that the following assertion holds true:Let g : M n → K ( d ) be a Lr k -Lipschitz map defined on V k and suchthat the pull-back of the open star of the vertex v V k ∈ K (0) is containedin V k . Then g is homotopic rel K ( d − to a map ˜ g : M n → K ( d − with the same properties as g , L being replaced by L ′ .Proof. It suffices to find a constant θ = θ ( d, L ) > d -simplex σ ⊂ K contains a point z whose distance to ∂σ and to theimage of g is ≥ θ . In order to push g into the ( d − σ with the radial projection from z . This increasesthe Lipschitz constant by a multiplicative factor bounded above by afunction of θ ( d, L ), and decrease the inverse image of the open starsof the vertices.If θ does not satisfy the required property for some d -simplex σ ,then image ( g ) ∩ int ( σ ) contains a set of cardinality at least C ( d ) · θ d of points whose pairwise distances are ≥ θ . Let A ⊂ M n be a setcontaining exactly one point of the inverse image of each of thesepoints. Since g maps W n into the 0-skeleton, A ⊂ B ( x k , r k ). As g is Lr k -Lipschitz on V k , the distance between any two distinct points in A s bounded below by L r k θ . Hence the cardinal of A is bounded aboveby vol( B ( x k , r k )vol( B ( y, r k θ L )) ≤ vol( B ( y, r k )vol( B ( y, r k θ L )) ≤ v − r ′ )2 (2 r k ) v − r ′ )2 ( r k θ L ) = v − (2 r k r ′ ) v − ( θ L r k r ′ ) ≤ C (cid:18) Lθ (cid:19) , where y is any point in A . In order to apply Bishop-Gromov, weused the fact that B ( x k , r k ) is included in a ball of radius r ′ withcurvature ≥ − / ( r ′ ) . The inequality C ( d ) · θ d ≤ C · ( Lθ ) gives apositive lower bound θ ( d, L ) for θ . Consequently, any θ < θ has thedesired property. Lemma 4.11.
There exists a constant C such that if D is largeenough, then vol( B ( x i , r i )) ≤ C D r i for all i. Proof.
We know that vol( B ( x i , ν x i )) ≤ D ν x i . Furthermore, r i ≥ ν xi and B ( x i , r i ) is included in a ball B ( x ′ , r ′ ) with curvature ≥ − r ′ . As r ′ ≥ r i , the curvature on B ( x i , r i ) is ≥ − r i . The Bishop-Gromovinequality gives: vol( B ( x i , ν xi )) v − r i ( ν xi ) ≥ vol( B ( x i , r i )) v − r i ( r i ) . Equivalently,vol( B ( x i , ν x i
11 )) ≥ vol( B ( x i , r i )) v − (1) v − ( ν x i r i ) ≥ vol( B ( x i , r i )) 1 C (cid:18) ν x i r i (cid:19) , for some uniform C >
0. Hencevol( B ( x i , r i )) ≤ C (cid:18) r i ν x i (cid:19) vol( B ( x i , ν x i
11 )) ≤ Cr i D .
Finally we push f into the 2-skeleton. Lemma 4.12.
For a suitable choice of
D > , there exists a map f (2) : M n → K (2) such that:i. f (2) is homotopic to f rel K (2) . i. The inverse image of the open star of each vertex v V k ∈ K (0) iscontained in V k .Proof. The inverse image by f of the open star of the vertex v V k ∈ K (0) is contained in V k . Using Lemma 4.10 several times, we find a map f (3) : M n → K (3) homotopic to f and a universal constant ˆ L suchthat ( f (3) ) − ( star ( v V k )) ⊂ V k and f (3) | V k is ˆ Lr k -Lipschitz.It now suffices to show that no 3-simplex σ ⊂ K can lie entirely inthe image of f (3) . Indeed, once we know this, we can push f (3) intothe 2-skeleton of K using a central projection in each simplex, withcentre in the complement of this image. Note that here no metricestimate is required in the conclusion.Let us thus assume that there exists a 3-simplex σ contained in theimage of f (3) . The inverse image of int ( σ ) by f (3) is a subset of theintersection of the V j ’s such that v V j is a vertex of σ . Let V k be oneof them. As vol( f (3) ( V k )) ≤ vol( f (3) ( B ( x k , r k ))), Lemma 4.11 yields:vol( image ( f (3) ) ∩ σ ) ≤ vol( f (3) ( V k )) ≤ ( ˆ Lr k ) vol( B ( x k , r k )) ≤ C ˆ L D with uniform constants C and ˆ L . Hence, if D is sufficiently large, thenone has vol( image ( f (3) ) ∩ σ ) < vol( σ ).The retraction used to push f into the 2-skeleton does not involvethe 0-skeleton of K . As a consequence, the inverse images of the openstars of the vertices v k still satisfy ( f (2) ) − ( star ( v V k )) ⊂ V k , and thus( f (2) ) − ( star ( v V k )) is homotopically trivial in M . This proves theassertion and ends the proof by contradiction of Proposition 4.1. Proposition 4.1 implies:
Corollary 4.13.
There exists D > such that if D > D and n ≥ n ( D ) , then there exists a compact submanifold V ⊂ M n with thefollowing properties:(i) V is either a connected component of W n or a tubular neigh-bourhood of the soul of the local model of some point x ∈ M n \ int( W n ) .(ii) V is a solid torus, a thickened torus or the twisted I -bundle onthe Klein bottle.(iii) V is homotopically non-trivial in M n . roof. We recall that each component of W n is a solid torus. If oneof them is homotopically non-trivial, then we choose it. Otherwise,by Proposition 4.1, there exists a point x ∈ M n \ int( W n ) such that B ( x , ν x ) is homotopically non-trivial; one of the remarks followingProposition 3.1 shows that B ( x , ν x ) is necessarily a solid torus, athickened torus or a twisted I -bundle over the Klein bottle. Indeed, S can be neither a point nor a 2-sphere, otherwise B ( x , ν x ) wouldbe homeomorphic to B or S × I , which have trivial fundamentalgroup.As V is not contained in any 3-ball, each component Y of its com-plement is irreducible, hence a Haken manifold whose boundary isa union of tori. In particular, Y admits a geometric decomposition.Here is an important consequence of Thurston’s hyperbolic Dehn fill-ing theorem (cf. [5, Prop. 10.17], [6, Prop. 9.36]): Proposition 4.14.
Let Y be a Haken 3-manifold whose boundary isa union of tori. Assume that any manifold obtained from Y by Dehnfilling has vanishing simplicial volume. Then Y is a graph manifold. In order to prove that M n is a graph manifold, it is sufficient toshow that each component of M n \V is a graph manifold. To concludethe proof of Theorem 0.1, it suffices to show the following proposition: Proposition 4.15.
For n large enough, one can find a submanifold V as above such that every Dehn filling on each component Y of M n \ int( V ) has vanishing simplicial volume. We choose the set V as follows: • If some component of W n is homotopically non-trivial, then wechoose it as V . • If all components of W n are homotopically trivial, then there ex-ists a point x ∈ M n \ int( W n ) such that B ( x, ν x ) is homotopicallynon-trivial. We choose x ∈ M n \ int( W n ) such that ν = ν x ≥
12 sup { ν ( x ) | π ( B ( x, ν ( x ))) → π ( M n ) is non-trivial } . Let S be the soul of the local model B ( x , ν ). We choose V to be the metric open δ -neighbourhood with 0 < δ < ν D . Afterpossibly shrinking W n , one has V ∩ W n = ∅ , as ν ≤ U ⊂ M n is virtually abelian relatively to V ifthe image in π ( M n \ V ) of the fundamental group of each connectedcomponent of U ∩ ( M n \ V ) is virtually abelian. e set:ab( x ) = sup r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B ( x, r ) is virtually abelian relatively to V and B ( x, r ) is contained in a ball B ( x ′ , r ′ ) withcurvature ≥ − r ′ ) and vol( B ( x ′ ,r ′ ))( r ′ ) ≤ /D and r ( x ) = min {
111 ab( x ) , } . We are now led to prove the following assertion:
Assertion 4.16.
With this choice of V , for n large enough, M n canbe covered by a finite collection of open sets U i such that: • Each U i is either contained in a component of W n or in a ball B ( x i , r ( x i )) for some x i ∈ M − n ( ε n ) . In particular, U i is virtuallyabelian relatively to V . • The dimension of this covering is not greater than , and it iszero on V . Let us first show why this assertion implies Proposition 4.15.
Proof.
The covering described in the assertion induces naturally acovering on every closed and orientable manifold ˆ Y , obtained by gluingsolid tori to ∂Y . It is a 2-dimensional covering by open sets whichare virtually abelian and thus amenable in ˆ Y . Gromov’s vanishingtheorem [17, § Y vanishes, which proves Proposition 4.15.We now prove Assertion 4.16. The argument for the constructionof a 2-dimensional covering by abelian open sets is similar to the oneused in the proof of Proposition 4.1, replacing everywhere the trivialityradius triv by the abelianity radius ab. There are, however, a fewdifferences, which we now point out.If V is a connected component of W n , then for n large enough wechoose points x , x , . . . , x q ∈ ∂W n , with x ∈ ∂ V in such a way that • Every boundary component of W n contains exactly one of the x j ’s. • The balls B ( x j ,
1) are pairwise disjoint. • Every B ( x j ,
1) has normalised volume ≤ D and sectional curva-ture close to − • Every B ( x j ,
1) is contained in a thickened torus (which impliesthat this ball is abelian). urthermore, going sufficiently far in the cusp and taking n largeenough, one can assume that B ( x j , r ( x j )) contains an almost horo-spherical torus corresponding to a boundary component of W n . In thiscase the proof previously done applies without any change, since thedimension of the original covering and all those obtained by shrinkingis zero on W n (or on a set obtained by shrinking W n ).From now on we shall assume that all connected components of W n are homotopically trivial . We then choose x ∈ V ⊂ M n \ int( W n ) asabove, and points x , . . . , x q ∈ ∂W n as before.We complete the sequence x , x , . . . , x q to a maximal finite se-quence x , x , x , . . . , x p such that the balls B ( x i , r ( x i )) are disjoint.We set r i = r ( x i ), and, if W n, , . . . W n,q are the connected compo-nents of W n , then we set • V := B ( x , r ). • V i := B ( x i , r i ) ∪ W n,i , for i = 1 , . . . , q . • V i := B ( x i , r i ) \ V for i = q + 1 , . . . , p .After possibly shrinking W n , we have V ∩ B ( x i , r i ) = ∅ for i = 1 , . . . , q ,since r i ≤ V ∩ W n = ∅ . It follows that V ∩ V i = ∅ for i = 0.Under the hypothesis that the W n are homotopically trivial, thefollowing two lemmas deal with the difference with the previous proofs. Lemma 4.17.
Each x ∈ M n belongs to some V k such that d ( x, ∂V k ) ≥ r k . This lemma is used in the control of the Lipschitz constant of thecharacteristic map, Lemma 4.9. In order to prove it, we begin withthe following remark:
Remark. If n is large enough, we have V ⊂ B ( x , r ).The bounds on the diameter of S , the distance to the base point,and the radius of the neighbourhood, give V ⊂ B ( x , ν x D ). Then theremark follows. Proof of Lemma 4.17. If x ∈ W n , then there is nothing to prove. Wethus assume that x ∈ M n \ W n . If x ∈ B ( x , r ) we may choose k = 0. Let us then assume that x B ( x , r ). There exists k suchthat x ∈ B ( x k , r k ). If V k and V are disjoint, then we are done.Hence we assume that V k ∩ V = ∅ . By the previous remark, one has: d ( x, V ) ≥ d ( x, x ) − r ≥ r − r ≥ · r k > r k . This implies that d ( x, ∂V k ) ≥ r k . he second difference is that the triviality radius satisfies triv( x i ) ≥ ν x i by construction. Here we shall prove that ab( x i ) ≥ c ν x i for a uni-form c >
0. This is used in the proof of Lemma 4.11, where theinequality vol( B i ) ≤ C D r i will still be true, but with a different con-stant C . Lemma 4.18.
There exists c > such that r i ≥ c ν x i for all i .Proof. One has r ≥ ν x by construction. For all i >
0, if B ( x i , ν xi ) ∩V = ∅ , then r i ≥ ν x i . Hence we assume B ( x i , ν xi ) ∩ V = ∅ ,and we claim that d ( x i , V ) > c ′ ν x i for a uniform c ′ >
0. Since V ⊂ B ( x , r ): d ( x i , V ) ≥ d ( x i , x ) − r ≥ r − r > r ≥ ν x . (1)We distinguish two cases, according to whether V is contained in B ( x i , ν x i ) or not.If V ⊂ B ( x i , ν x i ), the image of π ( B ( x i , ν x i )) → π ( M n ) cannotbe trivial, since the image of π ( V ) → π ( M n ) is not. In addition, ν x i ≤ ν x , (2)by the choice of ν x . Equations (1) and (2) give d ( x i , V ) > ν x i / V B ( x i , ν x i ), then since V ∩ B ( x i , ν xi ) = ∅ , we havediam( V ) + ν x i ≥ ν x i . Consequently, diam( V ) /ν x i ≥ /
11. As diam( V ) ≤ ν x /D , wededuce that ν x ν x i ≥ D. Combining with (1), we get d ( x i , V ) ≥ D ν x i .For D >
30, this contradicts V ∩ B ( x i , ν xi ) = ∅ . Ricci flow with surgery
In order to apply Theorem 0.1, we shall need the following straight-forward consequence of Perelman’s work [31, 33, 32]:
Theorem 5.1.
Let M be a closed, orientable, irreducible -manifold.(1) If π M is finite, then M is spherical.(2) If π M is infinite, then for every riemannian metric g on M ,there exists an infinite sequence of riemannian metrics g , . . . , g n , . . . with the following properties:(i) The sequence ( ˆ R ( g n )) n ≥ is nondecreasing. In particular, ithas a limit, which is greater than or equal to ˆ R ( g ) .(ii) The sequence (vol( g n )) n ≥ is bounded.(iii) Let ε > be a real number and x n ∈ M be a sequence suchthat for all n , x n is ε -thick with respect to g n . Then thesequence ( M, g n , x n ) subconverges in the C topology towardssome hyperbolic pointed manifold.(iv) The sequence g n has controlled curvature in the sense ofPerelman. In this section we explain how to deduce Theorem 5.1 from Perel-man’s results on the Ricci flow. We refer to [27] and [29] for thedetails.Let M be a closed, orientable 3-manifold. R. Hamilton introducedin [19] the following equation: dgdt = − g ) , where the unknown g = g ( t ) is a family of riemannian metrics on M depending on a time parameter t ∈ R . A Ricci flow is a solution tothis equation.In [33], Perelman constructs an object he calls
Ricci flow with δ -cutoff , also known as Ricci flow with surgery . It can be viewed as a1-parameter family of (possibly disconnected) riemannian manifolds( M ( t ) , g ( t )) which satisfies Hamilton’s equation in a weak sense. Thetopology of the manifold M ( t ) is allowed to change at a discrete setof times, the change being a connected sum decomposition into primefactors, as well as RP or S × S factors, and removing componentsthat are spherical or diffeomorphic to S × S .Perelman [33, § § g on M , there exists a Ricciflow with surgery satisfying the initial condition ( M (0) , g (0)) = ( M, g ).It may happen that M ( t ) becomes empty for some finite time t ; in his case, M is a connected sum of spherical manifolds and copies of S × S . Such a Ricci flow with surgery is said to become extinct .From now on we assume that M is irreducible. Thus, if some Ricciflow with surgery with initial manifold M (0) = M becomes extinct,then M is spherical. If M is not diffeomorphic to S and ( M ( t ) , g ( t ))is a Ricci flow with surgery such that M (0) = M and which does notbecome extinct, then for each time t , the manifold M ( t ) has exactlyone component diffeomorphic to M , the others being copies of S .Thus we get a 1-parameter family of metrics on M , which we stilldenote by g ( t ), defined for all t ≥
0. (There is some freedom for thechoice of diffeomorphisms between M and the various M ( t )’s, but thefollowing discussion does not depend on this choice.)If the metric g has positive scalar curvature, then a maximal prin-ciple argument shows that any Ricci flow with surgery with initialmetric g becomes extinct in finite time. Thus M is spherical. Thesame conclusion holds if π M is finite (see [32], [29, Chapter 18], [12].)If π M is infinite, then Ricci flow with surgery cannot become ex-tinct, and M cannot admit any metric of positive scalar curvature.Let g be any riemannian metric on M . By scaling, we get a nor-malised metric ˆ g (i.e. the absolute value of its sectional curvature isbounded above by 1, and each ball of radius 1 has volume greater thanor equal to half of the volume of the Euclidean unit ball.) Startingwith the initial metric ˆ g , we get a family of metrics { g ( t ) } t ≥ . Forall integers n ≥
1, set g n := (4 n ) − g ( n ). Then it is proved in [33, § § { g n } n ≥ satisfiesProperties (ii)–(iv) of the conclusion of Theorem 5.1. Moreover, thefunction t ˆ R ( g ( t )) is nondecreasing [33, § R is scale in-variant, we have ˆ R ( g ) = ˆ R (ˆ g ), and ˆ R ( g n ) = ˆ R ( g ( n )) for all n ≥ R ( g n )) n ≥ is nondecreasing. This completes theproof of Theorem 5.1. The following proposition is Assertion (1) of Theorem 0.3:
Proposition 6.1.
Let M be a closed, orientable and irreducible -manifold. Suppose that the inequality R ( M ) ≤ − V ( M ) / holds.Then M is hyperbolic and the hyperbolic metric realises R ( M ) . An immaterial difference between our statement and Perelman’s is that we use therescaling factor (4 t ) − instead of t − in order to get limits of sectional curvature − − / roof. Let H be a hyperbolic manifold homeomorphic to the comple-ment of a link L in M and whose volume realises V ( M ). To proveProposition 6.1, it is sufficient to show that L is empty. Let us as-sume that it is not true and prove that M carries a metric g ε such thatvol( g ε ) < V ( M ) and R min ( g ε ) ≥ −
6. This can be done by a directconstruction as in [2]. We give here a different argument relying onThurston’s hyperbolic Dehn filling theorem.If L = ∅ , then we consider the orbifold O with underlying space M , singular locus L local group Z /n Z with n > M corresponding to the orbifold structure, in atubular neighbourhood of L : Lemma 6.2 (Salgueiro [35]) . For each ε > there exists a Rieman-nian metric g ε on M with sectional curvature bounded below by − and such that vol( M, g ε ) < (1 + ε ) vol( O ) . For completeness we give the proof of this lemma, following [35,Chap. 3].
Proof.
Let g be the hyperbolic cone metric on M induced by thehyperbolic orbifold O . Let N ⊂ O be a tubular neighbourhood ofradius r > L . In N the local expressionof the singular metric g in Fermi (cylindrical) coordinates is: ds = dr + (cid:18) n sinh( r ) (cid:19) dθ + cosh ( r ) dh , where r ∈ (0 , r ) is the distance to L , h is the length parameter along L , and θ ∈ (0 , π ) is the rescaled angle parameter.The deformation depends only on the parameter r and consists inreplacing the metric g by a smooth metric g ′ which coincides with g outside of N , and has in N the form ds = dr + φ ( r ) dθ + ψ ( r ) dh , where for some δ = δ ( ε ) > φ, ψ : [0 , r − δ ] → [0 , + ∞ )are smooth and satisfy the following properties:(1) In a neighbourhood of 0, φ ( r ) = r and ψ ( r ) is constant.(2) In a neighbourhood of r − δ , φ ( r ) = n sinh( r + δ ) and ψ ( r ) =cosh( r + δ ). ∀ r ∈ (0 , r − δ ) , φ ′′ ( r ) φ ( r ) ≤ ε, ψ ′′ ( r ) ψ ( r ) ≤ ε and φ ′ ( r ) ψ ′ ( r ) φ ( r ) ψ ( r ) ≤ ε. The new metric is non-singular by (1), it matches the previous oneaway from N by (2) and has sectional curvature ≥ − − ε by (3).First we deal with the construction of φ . Let r = r ( δ ) > r = n sinh( r + δ ). Notice that r ≈ n − δ . The function φ is a smooth modification in a neighbourhood of r of the piecewisesmooth function r (cid:26) r on [0 , r ] n sinh( r + δ ) on [ r , r − δ ]so that: • On [0 , r ], φ ≥ r ε , φ ′ ≤ φ ′′ ≤ • On [ r , r − δ ], φ ≥ n sinh( r + δ ) ε , φ ′ ≤ n cosh( r + δ ), φ ′′ ≤ n sinh( r + δ ).We choose ψ satisfying: • On [0 , r ], ψ is constant. • On [ r , r − δ ], ψ ≥ cosh( r + δ ), ψ ′ ≤ sinh( r + δ ), ψ ′′ ≤ cosh( r + δ )(1 + ε ).Notice that ψ ′ ( r ) = 0 and ψ ′ ( r − δ ) = sinh( r ), so for a given ε > δ sufficiently small to achieve the required boundon ψ ′′ .As δ →
0, vol(
M, g ′ ) → vol( O ), since φ → n sinh( r ) and ψ → cosh ( r ). So given ε >
0, for a choice of δ sufficiently small, oneobtains a smooth Riemannian metric g ′ on M with sectional curvature ≥ − − ε and volume vol( M, g ′ ) ≤ (1 + ε ) vol( O ). Then the rescaledmetric g ε = √ ε g ′ on M has sectional curvature ≥ − M, g ε ) ≤ (1 + ε ) vol( O ).As vol( O ) < vol( H ), for ε > M such that vol( M, g ε ) < vol( H ) and R min ( g ε ) ≥−
6. In particularˆ R ( g ε ) > − H ) / = − V ( M ) / which contradicts the hypothesis. The link L is thus empty and wehave M = H . .2 Proof of Theorem 0.3 Thanks to previous section, there just remains to show Assertion (2).If π M is finite, then by Theorem 5.1(1), M is spherical, hence a graphmanifold. From now on we assume that π M is infinite. In particular, M is not simply connected.By assumption, there exists a riemannian metric g on M such that − V ( M ) / < ˆ R ( g ). Applying Theorem 5.1(2), we get a sequenceof metrics { g n } satisfying properties (i)–(iv) of this Theorem. Notethat properties (ii) and (iv) are respectively Hypotheses (1) and (3)of Theorem 0.1. Next we check Hypothesis (2), which is the contentof the following lemma: Lemma 6.3. If H is any hyperbolic -manifold which appears as apointed C -limit of the some subsequence of ( M, g n ) , then vol( H )
0. We choose a compact core ¯ H ( ξ ) of H such thatvol( ¯ H ( ξ )) ≥ (1 − ξ ) vol( H ). Since the convergence is C , there ex-ists, for n sufficiently large, a submanifold ¯ H n ( ξ ) ⊂ M n with volumeat least (1 − ξ ) vol( H ) and whose scalar curvature is less than orequal to − − ξ ). Thus, for n large, R min ( g n ) ≤ − − ξ ) andvol( M n ) ≥ vol( ¯ H n ( ξ )) ≥ (1 − ξ ) vol( H ). Letting ξ → R ( g n ) ≤ − H ) / ≤ − V ( M ) / , which gives the desired contradiction.Hence we can apply Theorem 0.1. It follows that M contains anincompressible torus or is a graph manifold. In the latter case, M contains an incompressible torus or is Seifert fibred. Let us recall that V ′ ( M ) denotes the infimum of the volumes of hyper-bolic manifolds which can be embedded in M with and incompressiblecusp in M or as complement of a (possibly empty) link in M . Sincehyperbolic volumes form a well-ordered set, this infimum is in fact aminimum, hence V ′ ( M ) > M be a graph manifold. After Cheeger-Gromov [11], one canconstruct Riemannian metrics on M with sectional curvature pinched etween − R ( M ) ≥ V ′ ( M ) >
0, this proves the ‘only if’ part of the equivalence.The ‘if’ part follows from Theorem 5.1 and the following variantof Theorem 0.1:
Theorem 6.4.
Let M be a closed, orientable, non-simply connected,irreducible -manifold. Let g n be a sequence of Riemannian metricssatisfying:(1) The sequence vol( g n ) is bounded.(2) For all ε > , if x n ∈ M is a sequence such that for all n , x n is in the ε -thick part of ( M, g n ) , then ( M, g n , x n ) subconvergesin the C topology to a pointed hyperbolic manifold with volumestrictly less than V ′ ( M ) .(3) The sequence ( M, g n ) has locally controlled curvature in the senseof Perelman.Then M is a graph manifold.Proof. Let H , . . . , H m be hyperbolic limits given by Proposition 2.1and let ε n → i wefix a compact core ¯ H i of H i and for each n a submanifold ¯ H in andan approximation φ in : ¯ H in → ¯ H i . The fact that the volume of eachhyperbolic manifold H i is les than V ′ ( M ) implies the following result: Lemma 6.5.
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