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Counting open negatively curved manifolds up to tangential homotopy equivalence

Counting open negatively curved manifolds up to tangential homotopy equivalence
Counting open negatively curved manifolds up to tangential homotopy equivalence

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COUNTING OPEN NEGATIVELY CUR VED MANIFOLDS UP TO TANGENTIAL HOMOTOPY EQUIV ALENCE Igor Belegradek May 14,1998Abstract.Under mild assumptions on a group π,we prove that the class of com-plete Riemannian n –manifolds of uniformly bounded negative sectional curvatures and with the fundamental groups isomorphic to πbreaks into ?nitely many tan-gential homotopy types.It follows that many aspherical manifolds do not admit complete negatively curved metrics with prescribed curvature bounds.§1.Introduction This paper is an attempt to understand the topology of complete in?nite vol-ume Riemannian manifolds of negative sectional curvature.Every smooth open manifold admits a (possibly incomplete)Riemannian metric of negative sectional curvature [23].However,the universal cover of a complete negatively curved mani-fold is di?eomorphic to the Euclidean space,hence the homotopy type of a complete negatively curved manifold is determined by its fundamental group.For closed (or,more generally,?nite volume complete)negatively curved mani-folds,the fundamental group seems to encode most of the topological information.By contrast,complete negatively curved manifolds of in?nite volume with isomor-phic fundamental groups may be very di?erent topologically.For example,the total space of any vector bundle over a closed negatively curved manifold admits a complete Riemannian metric of sectional curvature pinched between two negative constants [1].Given a group π,a positive integer n ,and real numbers a ≤b <0,consider the class M a,b,π,n of n –manifolds with fundamental groups isomorphic to πthat can be given complete Riemannian metrics of sectional curvatures within [a,b ].In this paper we discuss under what assumptions on πthe class M a,b,π,n breaks

into ?nitely many tangential homotopy types.Recall that a homotopy equivalence of smooth manifolds of the same dimension f :N →L is called tangential if the vector bundles f ?T L and T N are stably isomorphic.For example,any map that is homotopic to a di?eomorphism is a tangential homotopy equivalence.Note that,unless N is a closed manifold,the bundles f ?T L and T N are isomorphic i?they are stably isomorphic.

2IGOR BELEGRADEK

If the cohomological dimension ofπis equal to n,every manifold from the class M a,b,π,n is closed.In that case,in fact,M a,b,π,n falls into?nitely many di?eomor-phism classes[5].

Here is a way to produce in?nitely many homotopy equivalent negatively curved manifolds of the same dimension such that no two of them are tangentially homo-topy equivalent.Let M be a closed negatively curved manifold with H4m(M,Q)=0 for some m>0.(Such examples abound.For instance,any closed complex hy-perbolic or quaternion hyperbolic manifold is such.Most arithmetic closed real hyperbolic orientable manifolds have nonzero Betti numbers in all dimensions[30].) Then,by an elementary K-theoretic argument,for any k≥dim(M)there exists a in?nite sequence of rank k vector bundles over M such that no two of them have tangentially homotopy equivalent total spaces.Yet,thanks to a theorem of An-derson[1],the total space of any vector bundle over M can be given a negatively curved metric.

The following is a simpli?ed form of our main result.

Theorem 1.1.Letπbe the fundamental group of a?nite aspherical complex. Suppose thatπis not virtually nilpotent and thatπdoes not split as a nontrivial amalgamated product or an HNN-extension over a virtually nilpotent group.

Then,for any positive integer n and any negative reals a≤b,the class M a,b,π,n breaks into?nitely many tangential homotopy types.

Furthermore,by an easy argument we can generalize the theorem1.1to certain amalgamated products and HNN-extensions.For example,suppose thatπis the fundamental group of a?nite graph of groups such that the edge groups are virtually nilpotent groups of cohomological dimension≤2.(Note that if M a,b,π,n=?,then any nontrivial virtually nilpotent subgroup ofπof cohomological dimension≤2is isomorphic to Z,Z×Z,or the fundamental group of the Klein bottle.)Fix n and

a≤b<0and suppose that,for each vertex groupπv,the class M a,b,π

v,n breaks

into?nitely many tangential homotopy types.Then so does the class M a,b,π,n.

Applying a powerful accessibility result of Delzant and Potyagailo[16],we deduce the following.

Corollary1.2.Letπbe the fundamental group of a?nite aspherical complex. Assume that any nilpotent subgroup ofπhas cohomological dimension≤2.

Then,for any positive integer n and any negative reals a≤b,the class M a,b,π,n breaks into?nitely many tangential homotopy types.

Any torsion free word-hyperbolic group is the fundamental group of a?nite aspherical cell complex[15,5.24].Moreover,any virtually nilpotent subgroup ofπis either trivial or in?nite cyclic.

Corollary1.3.Letπbe a word-hyperbolic group.Then,for any positive integer n and any negative reals a≤b,the class M a,b,π,n breaks into?nitely many tangential homotopy types.

It is worth mentioning that in the locally symmetric case a di?erent(and ele-mentary)argument yields the following.

Theorem1.4.Letπbe a?nitely presented torsion–free group and let X be a nonpositively curved symmetric space.

COUNTING NEGATIVELY CURVED MANIFOLDS3 Then the class of manifolds of the form X/ρ(π),whereρ∈Hom(π,Isom(X)) is a faithful discrete representation,falls into?nitely many tangential homotopy types.

Gromov posed a question(see[1])whether an(open)thickening of a?nite as-pherical cell complex with word-hyperbolic fundamental group admits a complete negatively curved metric.The following corollary shows that most thickenings do not carry complete negatively curved metric with prescribed curvature bounds. Corollary1.5.Let K be a?nite,connected,aspherical cell complex.Let a≤b<0

and n>max{4,2dim(K)}.Suppose that the class M a,b,π

1(K),n breaks into?nitely

many tangential homotopy types.

Then the set of di?eomorphism classes of open thickenings of K that belong to the class M a,b,π

1(K),n

is?nite.

The corollary follows from the fact that any two tangentially homotopy equiv-alent thickenings of su?ciently high dimension are di?eomorphic[29,pp.226–228]. In general,the set of di?eomorphism classes of open n-dimensional thickenings of K is in?nite provided n>max{4,2dim(K)}and⊕k H4k(K,Q)=0.Thus,for such n and K,most open n-dimensional thickenings of K do not belong to the class M a,b,π

1(K),n

.Combining1.3and1.5,we deduce the following.

Corollary1.6.Let M be a smooth aspherical manifold such thatπ1(M)is word-hyperbolic.Let a≤b<0and n>2dim(M).

Then the set of isomorphism classes of vector bundles over M whose total spaces belong to M a,b,π

1(K),n

is?nite.

Actually,we show in[6]that,in case dim(M)≥3,the assumption n>2dim(M) of the corollary1.6is redundant.

Synopsis of the paper.In the section2we prove main lemmas from the global Riemannian geometry.The3rd section provides some K-theoretic background.In the4th section we de?ne some invariant of actions.The5th section is devoted to our main results including the theorem1.1.In the section6we employ some group theory to deduce the corollary1.2.The section7deals with applications to thickenings.Applications to convex-cocompact groups are deduced in the section 8.The theorem1.4is proved in the9th section.

Acknowledgments.I am grateful to Werner Ballmann,Mladen Bestvina,Thomas Delzant,Martin J.Dunwoody,Lowell E.Jones,Misha Kapovich,Vitali Kapovitch, Bernhard Leeb,John https://www.doczj.com/doc/6215953281.html,lson,Igor Mineyev,James A.Schafer,Jonathan M.Rosen-berg and Shmuel Weinberger for helpful discussions and communications.

Special thanks are due to my advisor Bill Goldman for his constant interest and support.

§2.Two types of convergence.

By an action of an abstract groupπon a space X we mean a group homomor-phismρ:π→Homeo(X).An actionρis free ifρ(γ)(x)=x for all x∈X and all γ∈π\{id}.In particular,ifρis a free action,thenρis injective.

2.1.Equivariant pointed Lipschitz topology.LetΓk be a discrete subgroup of the isometry group of a complete Riemannian manifold X k and p k be a point of X k.

4IGOR BELEGRADEK

The class of all such triples{(X k,p k,Γk)}can be given the so-called equivariant pointed Lipschitz topology[20];whenΓk is trivial this reduces to the usual pointed Lipschitz topology.For convenience of the reader we give here some de?nitions borrowed from[20].

For a groupΓacting on a pointed metric space(X,p,d)the set{γ∈Γ: d(p,γ(p))

For i=1,2,let(X i,p i)be a pointed complete metric space with the distance function d i and letΓi be a discrete group of isometries of X i.In addition,assume that X i is a C∞–manifold.Take any?>0.

Then a quadruple(f1,f2,φ1,φ2)of maps f i:B1/?(p i,X i)→B1/?(p3?i,X3?i) andφi:Γi(1/3?)→Γ3?i is called an?–Lipschitz approximation between the triples (X1,p1,Γ1)and(X2,p2,Γ2)if the following seven condition hold:?f i is a di?eomorphism onto its image;

?for each x i∈B1/3?(p i,X i)and everyγi∈Γi(1/3?),f i(γi(x i))=φi(γi)(f i(x i));

?for every x i,x′i∈B1/?(p i,X i),e??

?f i(B1/?(p i,X i))?B(1/?)??(p3?i,X3?i)andφi(Γi(1/3?))?Γ3?i(1/3???);

?f i(B(1/?)??(p i,X i))?B1/?(p3?i,X3?i)andφi(Γi(1/3???))?Γ3?i(1/3?);

?f3?i?f i|B

(1/?)??(p i,X i)

=id andφ3?i?φi|Γ

i(1/3???)

=id;

?d3?i(f i(p i),p3?i)

We say a sequence of triples(X k,p k,Γk)converges to(X,p,Γ)in the equivariant pointed Lipschitz topology if for any?>0there is k(?)such that for all k>k(?), there exists an?–Lipschitz approximation between(X k,p k,Γk)and(X,p,Γ).

Notice that if all the groupsΓk are trivial,thenΓis trivial;in this case we say that that(X k,p k)converges to(X,p)in the pointed Lipschitz topology.Note that if X k is a complete Riemannian manifold for all k,then the space X is necessarily a C∞–manifold with a complete C1,α–Riemannian metric[22].

Remark2.2.For those with the Kleinian groups background we note that the equivariant pointed Lipschitz topology is closely related to the so-called Chabauty topology[7][14].Since the Kleinian group theory is an important source of exam-ples,we describe the precise relation.Let X be a complete Riemannian manifold (e.g.a hyperbolic space)with the isometry group G.The set of discrete subgroups of G has the Chabauty topology induced by a natural topology on the set of closed subsets of G,namely a sequence of closed subsets S i converges to S if for any compact K?G,the sets S i∩K converge to S∩K in the Hausdor?topology on K.Consider the product topology on the set of pairs(Γ,p)whereΓis a dis-crete subgroup of G and p∈X.This product topology is in fact equivalent to the equivariant pointed Lipschitz topology where(Γ,p)is thought of as a triple (X,Γ,p).

2.3.Pointwise convergence topology.Suppose that,for some p k∈X k,the sequence(X k,p k)converges to(X,p)in the pointed Lipschitz topology i.e.,for any?>0there is k(?)such that for all k>k(?)there exists an?–Lipschitz approximation(f k,g k)between(X k,p k)and(X,p).(Note that if each X k is a Hadamard manifold,then for any p k∈X k,the sequence(X k,p k)is precompact in the pointed Lipschitz topology because the injectivity radius of X k at p k is uniformly bounded away from zero[20].)We say that a sequence x k∈X k converges to x∈X if for some?

d(f k(x k),x)→0as k→∞

COUNTING NEGATIVELY CURVED MANIFOLDS5 where d(·,·)is the distance function on X and f k comes from the?–Lipschitz ap-proximation(f k,g k)between(X k,p k)and(X,p).Trivial examples:if(X k,p k) converges to(X,p)in the pointed Lipschitz topology,then p k converges to p;fur-thermore,if x∈X,the sequence g k(x)converges to x.

Given a sequence of isometriesγk∈Isom(X k)we say thatγk converges,if for any sequence x k∈X k that converges to x∈X,γk(x k)converges.The limiting transformationγthat takes x to the limit ofγk(x k)of X is necessarily an isometry. Furthermore,ifγk andγ′k converge toγandγ′respectively,thenγk·γ′k converges

toγ·γ′.In particular,γ?1

k converges toγ?1since the identity maps id k:X k→X k

converge to id:X→X.

Letρk:π→Isom(X k)be a sequence of isometric actions of a groupπon X k.We say that a sequence of actions(X k,p k,ρk)converges in the pointwise convergence topology to an action(X,p,ρ)ifρk(γ)converges toρ(γ)for everyγ∈π.The limiting mapρ:Γ→Isom(X)that takesγto the limit ofρk(γ)is necessarily a homomorphism.Ifπis generated by a?nite set S,then in order to prove that ρk converges in the pointwise convergence topology it su?ces to check thatρk(γ) converges,for everyγ∈S.

A sequence of actions(X k,p k,ρk)is called precompact in the pointwise conver-gence topology if every subsequence of(X k,p k,ρk)has a subsequence that converges in the pointwise convergence topology.

Repeating the proof in[28,4.7],it is easy to check that a sequence of isometries γk∈Isom(X k)has a converging subsequence if,for some converging sequence x k∈X k,the sequence d k(x k,γk(x k))is bounded(where d k(·,·)is the distance function on X k).

Suppose thatπis a countable group and assume that for eachγ∈πthe sequence d k(p k,ρk(γ)(p k))is bounded.Then(X k,p k,ρk)is precompact in the pointwise convergence topology.(Indeed,letγ1...γn...be the list of all elements ofπ. Take any subsequenceρk,0ofρk.Pass to subsequenceρk,1ofρk,0so thatρk,1(γ1) converges.Then pass to subsequenceρk,2ofρk,1such thatρk,2(γ2)converges,etc. Thenρk,k(γn)converges for every n.)

Note that ifπis generated by a?nite set S,then to prove thatρk is precompact it su?ces to check that d k(p k,ρk(γ)(p k))is bounded,for allγ∈S because it implies that d k(p k,ρk(γ)(p k))is bounded,for eachγ∈π.

2.4.Motivating example.Let X be a complete Riemannian manifold.Consider the isometry group Isom(X)of X and letπbe a group.The space Hom(π,Isom(X)) has a natural topology(which is usually called“algebraic topology”or“pointwise convergence topology”),namelyρk is said to converge toρif,for eachγ∈π,ρk(γ) converges toρ(γ)in the Lie group Isom(X).Note that ifπis?nitely generated, this topology on Hom(π,Isom(X))coincide with the compact-open topology.Cer-tainly,for any p∈X,the constant sequence(X,p)converges to itself in pointed Lipschitz topology.Then,obviously,the sequence(X,p,ρk)converges in the point-wise convergence topology(as de?ned in2.3)if and only ifρk∈Hom(π,Isom(X)) converges in the algebraic topology.

Lemma2.5.Letρk:π→Isom(X k)be a sequence of isometric actions of a dis-crete groupπon complete Riemannian n-manifolds X k such thatρk(π)acts freely. If the sequence(X k,p k,ρk(π))converges in the equivariant pointed Lipschitz topol-ogy to(X,Γ,p)and(X k,p k,ρk)converges to(X,p,ρ)in the pointwise convergence topology,then

6IGOR BELEGRADEK

(1)Γacts freely,and

(2)ρ(π)?Γ,and

(3)ker(ρ)?ker(ρk),for all large k.

Proof.(1)Assumeγ∈Γandγ(x)=x.Choose?∈(0,1/10)so that there is an?–approximation(f k,g k,φk,τk)of(X k,p k,ρk)and(X,p,Γ)and x∈B(p,?/10). Then g k(x)=g k(γ(x))=τk(γ)(g k(x)).Sinceρk(π)acts freely,τk(γ)=id.By the same argumentτk(id)=id.Hence id=φk(τk(id))=φk(id)=φk(τk(γ))=γas desired.

(2)We need to show thatρ(γ)∈Γ,for anyγ∈π.We can assumeρ(γ)=id. Choose?∈(0,1/10)so that the ball B(p,1/11?)containsρ(γ)(p)and consider an ?–approximation(f k,g k,φk,τk)of(X k,p k,ρk)and(X,p,Γ).

Then for all large enough k,ρk(γ)∈B(p k,1/10?).Look atτk(ρk(γ))∈Γ(1/9?). Since the setΓ(1/9?)is?nite,we can pass to subsequence to assume thatτk(ρk(γ)) is equal toγ?∈Γ(1/9?);thusρk(γ)=φk(γ?).

Take an arbitrary x∈B(p,1/9?).Then g k(x)→x and,hence,ρk(γ)(g k(x)) converges toρ(γ)(x).Notice thatρk(γ)(g k(x))=φk(γ?)(g k(x))→γ?(x).So ρ(γ)(x)=γ?(x)for any x∈B(p,1/9?).

Thus,for any small enough?,we have foundγ?∈Γthat is equal toρ(γ)on the ball B(p,1/9?).SinceΓacts freely,γ?=γ?′for all?′≤?,that is the elementγ?∈Γis independent of?.Thusρ(γ)=γ?everywhere and henceρ(γ)∈Γ.

(3)Assumeρ(γ)=id.Fix any?∈(0,1/10).Take x∈B(p,?/10)and consider an?–approximation(f k,g k,φk,τk)of(X k,p k,ρk)and(X,p,Γ).We have g k(x)→x andρk(γ)(g k(x))→ρ(γ)(x)=x.Note that d(x,φk(ρk(γ))(x))is equal to

d(f k(g k(x)),f k(ρk(γ)(g k(x))))

0.

k→∞Therefore,for all large k,φk(ρk(γ))=id,becauseΓis a discrete subgroup that acts freely.Henceρk(γ)=τk(φk(ρk(γ)))=τk(id)=id as claimed.

Lemma2.6.Letρk:π→Isom(X k)be a sequence of isometric actions of a dis-crete groupπon complete Riemannian n-manifolds X k such thatρk(π)acts freely. Suppose that the sequence(X k,p k,ρk(π))converges in the equivariant pointed Lip-schitz topology to(X,Γ,p)and(X k,p k,ρk)converges to(X,p,ρ)in the pointwise convergence topology.

Then,for any?>0and for any?nite subset S?π,there is k(?,S)with the property that for each k>k(?,S)there exists an?–Lipschitz approximation (f k,g k,φk,τk)between(X k,p k,ρk(π))and(X,p,Γ)such thatφk(ρk(γ))=ρ(γ) andρk(γ)=τk(ρ(γ))for everyγ∈S.

Proof.Pickδ

Note that for everyγ∈S,the sequenceρk(γ)(g k(p))converges toρ(γ)(p).Hence the sequence f k(ρk(γ)(g k(p)))=φk(ρk(γ))(f k(g k(p)))=φk(ρk(γ))(p)converges to ρ(γ)(p)in X.

SinceΓis discrete and acts freely,φk(ρk(γ))=ρ(γ)for large enough k.Moreover, this is true for anyγ∈S,because S is?nite.Alsoρk(γ)=(τk?φk)(ρk(γ))=τk(ρ(γ)).Thus,thisδ-Lipschitz approximation has all desired properties.

COUNTING NEGATIVELY CURVED MANIFOLDS7 Proposition2.7.Let X k be a sequence of Hadamard manifolds with sectional curvatures in[a,b]for a≤b<0and letπbe a?nitely generated group that is not virtually nilpotent.Letρk:π→Isom(X k)be an arbitrary sequence of free and isometric actions such that(X k,p k,ρk)converges in the pointwise convergence topology.

Then the sequence(X k,p k,ρk)is precompact in the equivariant pointed Lipschitz topology.

Proof.Let(X,p,ρ)be the limit of(X k,p k,ρk)in the pointwise convergence topol-ogy.Choose r so that the open ball B(p,r)?X contains{ρ(γ1)(p),...ρ(γm)(p)} where{γ1,...γm}generateπ.Passing to subsequence,we assume that B(p k,r) contains{ρk(γ1)(p),...ρk(γm)(p)}.

Show that,for every k,there exists q k∈B(p k,r)such that for anyγ∈π\{id}, we haveρk(γ)(q k)/∈B(q k,μn/2)whereμn is the Margulis constant.Suppose not. Then for some k,the whole ball B(p k,r)projects into the thin part{InjRad<μn/2}under the projectionπk:X k→X k/ρk(π).Thus the ball B(p k,r)lies in a connected component W of theπk–preimage of the thin part of X k/ρk(π).Accord-ing to[2,p111]the stabilizer of W inρk(π)is virtually nilpotent and,moreover, the stabilizer contains every elementγ∈ρk(π)withγ(W)∩W=?.Therefore,the whole groupρk(π)stabilizes W.Henceρk(π)must be virtually nilpotent.Asρk is injective,πis virtually nilpotent.A contradiction.

Thus,(X k,q k,ρk(π))is Lipschitz precompact[20]and,hence passing to subse-quence,one can assume that(X k,q k,ρk(π))converges to some(X,q,Γ).

It is a general fact that follows easily from de?nitions that whenever(X k,q k,Γk) converges to(X,q,Γ)in the equivariant pointed Lipschitz topology and a sequence of points p k∈X k converges to p∈X,then(X k,p k,Γk)converges to(X,p,Γ)in the equivariant pointed Lipschitz topology.

Corollary2.8.Let X k be a sequence of Hadamard manifolds with sectional cur-vatures in[a,b]for a≤b<0and letπbe a?nitely generated group that is not virtually nilpotent.Letρk:π→Isom(X k)be an arbitrary sequence of free and isometric such that(X k,p k,ρk)converges to(X,p,ρ)in the pointwise convergence topology.

Thenρis a free action,in particularρis injective.

Proof.Pass to a subsequence so that(X k,p k,ρk)converges to(X,p,Γ)in the equi-variant pointed Lipschitz topology.By2.5(3),ρis injective.Furthermore,ρ(π) acts freely because it is a subgroup ofΓ.

Corollary2.9.Let X be a complete Riemannian manifold with sectional curva-tures in[a,b]where a≤b<0.Assumeπis a torsion free,?nitely generated group that is not virtually nilpotent.Consider a subset of faithful discrete representations S?Hom(π,Isom(X))that is precompact in the pointwise convergence topology. Then,

(1)for any p∈X,the set{(X,p,ρ(π)):ρ∈S}is precompact in the equivariant pointed Lipschitz topology.

(2)the closure of S in Hom(π,Isom(X))consists of faithful discrete representa-tions.

Proof.Proposition2.7implies(1)and Corollary2.8implies(2).

8IGOR BELEGRADEK

Proposition2.10.Assume thatπis a?nitely presented discrete group,that is not virtually nilpotent and does not have a nontrivial decomposition into an amal-gamated product or an HNN-extension over a virtually nilpotent group.

Letρk:π→Isom(X k)be an arbitrary sequence of free and isometric actions of πon Hadamard n-manifolds X k.Assume that the sectional curvatures of X k lie in [a,b]for a≤b<0.

Then,for some p k∈X k,the sequence(X k,p k,ρk)is precompact in the pointwise convergence topology.

Proof.Let S?πbe a?nite subset that generatesπand contains{id}.For x∈X k,

we denote D k(x)the diameter of the setρk(S)(x).Set D k=inf x∈X

k D k(x).

Suppose D k is unbounded.Then it follows from a work of Bestvina and Paulin [8],[32],[33](cf.[27]),that there exists an action ofπon a real tree with no proper in-variant subtree and virtually nilpotent arc stabilizers.For completeness we brie?y review this construction.The rescaled pointed Hadamard manifold1

D k ·X k,p k,ρk).Repeat-

ing an argument of Paulin[33,§4],we can pass to subsequence that converges to a triple(X∞,p∞,ρ∞).(For the de?nition of the convergence see[32],[33].Paulin calls it“convergence in the Gromov topology”.)

The limit space X∞is a length space of curvature?∞,that is a real tree. Because of the way we rescaled,the limit space has a natural isometric actionρ∞ofπwith no global?xed point[32],[33].Then it is a standard fact that there exists a uniqueπ–invariant subtree T of X∞that has no properπ–invariant subtree.In fact T is the union of all the axes of all hyperbolic elements inπ.Since the sectional curvatures are uniformly bounded away from zero and?∞,the Margulis lemma implies that the stabilizer of any non-degenerate segment is virtually nilpotent (cf.[32]).

Note that any increasing sequence of virtually nilpotent subgroups ofπis sta-tionary.Indeed,since a virtually nilpotent group is amenable,the union U of an increasing sequence U1?U2?U3?...of virtually nilpotent subgroups is also an amenable group.If the fundamental group of a complete manifold of pinched neg-ative curvature is amenable,it must be?nitely generated[13],[10].In particular, U is?nitely generated,hence U n=U for some n.Thus,theπ–action on the tree T is stable[9,Proposition3.2(2)].

We summarize that theπ–action on T is stable,has virtually nilpotent arc stabi-lizers and no properπ–invariant subtree.Therefore,the Rips machine[9,Theorem 9.5]produces a splitting ofπover a virtually solvable group.Any amenable sub-group ofπis virtually nilpotent[13],[10],henceπsplits over a virtually nilpotent group.This is a contradiction with the assumption that D k is unbounded.

Thus the sequence D k(p k)is bounded,therefore as we observed in2.3,the se-quence(X k,p k,ρk)is precompact in the pointwise convergence topology.

§3.Vector bundles and

KO-theory.

This section provides some

KO-theoretic background.All of the facts below are well-known to experts;however it is usually not easy to locate a reference.

In this paper we deal with rank n real vector bundles over open smooth n–manifolds.It is well-known that any open smooth manifold N is homotopy equiv-alent to CW–complex of dimension

COUNTING NEGATIVELY CURVED MANIFOLDS9 over N are isomorphic i?they are stably isomorphic[26,8.1.5].The set of stable isomorphism classes of real vector bundles over a space X forms an abelian group

KO(X).The correspondence X→

KO(X)is clearly a functor.This functor is isomorphic to the functor[?,BO]on the category of connected CW-complexes of uniformly bounded dimension[26,8.4.2].

We now recall the de?nition of the

KO?–theory.By the Bott periodicity,?8(BO×Z)is weakly homotopy equivalent to BO×Z[37,11.60].We can use the

KO?on spectrum?n(BO×Z)to de?ne a generalized(reduced)cohomology theory

the category of all pointed connected?nite–dimensional CW-complexes and point-

KO n(X,x)=[X,x;?n(BO×Z),?](cf.[37, preserving maps.In other words,we set

8.42]).

Let F be a forgetful map from the category of pointed connected CW–complexes to the category of connected CW–complexes.We now show that the functors

KO0(?)and

KO(F(?))are isomorphic on the category of pointed connected CW-complexes of uniformly bounded dimensions.Indeed,since we deal with connected

KO0(?)=[?;BO×Z),?]is isomorphic to the func-CW-complexes,the functor

KO(?)~=[?,BO].Therefore,it remains tor[?;BO,?].As we mentioned above

to show that the transformation of functors[?;BO,?]→[F(?);BO]is an iso-morphism.In other words,it su?ces to check that the map[X,x;BO,?]→[F(X,x);BO]=[X,BO]of based homotopy classes into free homotopy classes is bijective.Indeed,any map X→BO is homotopic to a basepoint preserving map[36,7.3.Lemma2]which is unique up to based homotopy because BO is a

KO(F(?)).Most path-connected H-space.[36,7.3.Theorem5].Thus,

KO0(?)~=

KO(?)

KO0(?)and

of the time we suppress the base points and treat the functors

as isomorphic.

KO i(X)can be easily computed in terms of cohomology Rationally the group

of X as explained in the lemma below.Recall that the Bott periodicity implies

KO–groups of the0–sphere are given by that,after tensoring with rationals,the

KO i(S0)?Q=0otherwise[26,15.12.3]. KO i(S0)?Q~=Q if i≡0(mod4)and

Lemma3.1.On the category of pointed connected CW-complexes of uniformly

KO i(?)?Q and⊕n>0H n(?)?bounded dimension the contravariant functors

KO i?n(S0)?Q are isomorphic.

KO?(S0)?Q where Proof.Consider the generalized cohomology theory H?(?)?

H?(?)is the ordinary reduced cohomology.Its coe?cients are isomorphic to the

KO?(?)?Q.Since the coe?cients are rational vector coe?cients of the theory

spaces,[24,3.22(ii)]implies that the theories are naturally equivalent on the cate-gory of pointed?nite-dimensional CW–complexes.

For any connected?nite-dimensional complex X, H n(X)=0,for all n≤0. Clearly for n>0,the functors H n(?)and H n(?)are isomorphic.Hence we have an isomorphism of functors

KO i(?)?Q~=⊕n≥0 H n(?)?

KO i?n(S0)?Q.

KO i?n(S0)?Q~=⊕n>0H n(?)?

10IGOR BELEGRADEK

Remark3.2.In particular,we have isomorphisms of functors

KO?1(?)?Q~=⊕n>0H4n?1(?,Q).

KO0(?)?Q~=⊕n>0H4n(?,Q)and

Corollary3.3.Let N be an open smooth manifold such that the torsion subgroup KO(N)is?nite and let n≥dim(N).

of

Then the set of isomorphism classes of rank n vector bundles over N is in?nite if and only if H4k(N,Q)=0for some k>0.

Proof.Any open manifold N is homotopy equivalent to a CW-complex of dimension

Lemma3.4.Let Y be a connected?nite-dimensional CW-complex such thatπ1(Y) is?nitely generated and dim Q H4k(Y,Q)<∞for all k>0.Then there exists a ?nite connected subcomplex X?Y such that the inclusion i:X→Y induces a

KO(Y)?Q→

KO(X)?Q surjection on the fundamental groups and an injection

Proof.It is a standard fact[19,p117]that for any homology classα∈H i(Y,Q) there exists a?nite(not necessarily connected)CW-complex Xαand a continuous map fα:Xα→Y such thatα∈fα?(H i(Xα,Q)).For every k>0,choose a?nite basis in the Q–vector space H4k(Y,Q)and construct such a?nite CW–complex Xαfor every basis elementα.

Let X0be the the disjoint union of all these complexes over all k and all the basis elements.Since Y is?nite-dimensional with?nite4k-th Betti numbers,the CW–complex X0is?nite.Note that X0comes with a continuous map f:X0→Y that induces a surjection on rational4k-th homology,and therefore,an injection on rational4k-cohomology for all k>0.

Finally,set X to be an arbitrary connected?nite subcomplex of Y such that f(X0)?https://www.doczj.com/doc/6215953281.html,ing thatπ1(Y)is?nitely generated,we add to X?nitely many1-cells of Y to make sure the inclusion i:X?→Y induces a surjection of fundamental groups.Moreover,by construction i induces an injection on rational4k-cohomology

KO(X)?Q is injective.

KO(Y)?Q→

for all k>0.By the remark3.2,

§4.An invariant of actions.

Let K be a?nite-dimensional connected CW-complex with a reference point q. Denote?K→K the universal cover of K;choose?q∈?K that is mapped to q by the covering projection.Consider a pointed contractible manifold X and an arbitrary actionρ:π1(K,q)→Di?eo(X).

Since X is contractible,the X-bundle?K×ρX over K has a section.Any two sections are homotopic through sections.

We now de?ne a certain invariant of actions ofπ1(K,q)on X.Given such an actionρ,consider the vertical vector bundle?K×ρT X over?K×ρX and setτ(ρ) to be the pullback of the vertical bundle via an arbitrary section s:K→?K×ρX Clearly,two actions that are conjugate in Di?eo(X)have same invariants.

Any section can be lifted to aρ-equivariant continuous map?K→?K×X→X.Any twoρ-equivariant continuous maps?g,?f:?K→X,areρ-equivariantly homotopic.Indeed,?f and?g descend to sections K→?K×ρX that must be homotopic.This homotopy lifts to aρ–equivariant homotopy of?f and?g.

COUNTING NEGATIVELY CURVED MANIFOLDS11 Assume now thatρ(π1(K,q))acts freely and properly discontinuously on X, so the mapπ:X→X/ρ(π1(K))is a covering.Then the map?f descends to a continuous map f:K→X/ρ(π1(K)).Thus,to any actionρ:π1(K,q)→Homeo(X)such thatρ(π1(K,q))acts freely and properly discontinuously on a contractible manifold X,we associate a(unique up to homotopy)continuous map f.Observe that ifρis injective,then f induces an isomorphism of fundamental groups.

We now observe that,in caseρ(π1(K,q))acts freely and properly discontinuously on X,the invariantτ(ρ)is equal to f?T X/ρ(π1(K)),the pullback of the tangent bundle to X/ρ(π1(K))via f.

If we consider only orientation-preserving actions the vector bundleτ(ρ)comes with a natural orientation.

§5.Main results

Throughout this section K is a?nite-dimensional,connected CW-complex with a reference point q.Let?K be the universal cover of K,?q∈?K be a preimage of q∈https://www.doczj.com/doc/6215953281.html,ing the point?q we identifyπ1(K,q)with the group of automorphisms of the covering?K→K.Recall thatτis an invariant of actions de?ned in§4. Theorem5.1.Let the complex K is?nite.Letρk:π=π1(K,q)→Isom(X k)be a sequence of isometric actions ofπ1(K)on Hadamard n-manifolds X k such that ρk(π)is a discrete subgroup of Isom(X k)that acts freely.

Suppose that,for some p k∈X k,(X k,p k,ρk(π))converges in the equivariant pointed Lipschitz topology to(X,p,Γ)and(X k,p k,ρk)converges to(X,p,ρ)in the pointwise convergence topology.

Thenτ(ρk)=τ(ρ)for all large k.

Proof.Let?q∈F??K be a?nite subcomplex that projects onto K.Clearly,the ?nite set S={γ∈π1(K,q):γ(F)∩F=?}generatesπ1(K,q).

Note that X is contractible,indeed any spheroid in X lies in the di?eomor-phic image of a metric ball in X j.Any metric ball in a Hadamard manifold is contractible.Thusπ?(X)=1.

Using§4,we?nd aρ-equivariant continuous map?h:?K→X.Recall that by de?nitionτ(ρ)is the pullback via h:K→X/ρ(π1(K,q)of the tangent bundle to X/ρ(π1(K,q).Letˉh:K→X/Γbe the composition of h and the covering X/ρ(π1(K,q)→X/Γ.Therefore,τ(ρ)is also the pullback viaˉh of the tangent bundle to X/Γ.

Choose?>0so small that?h(F)lies in the open ball B(p,1/10?)?X.For large k,we?nd an?-Lipschitz approximation(?f k,?g k,φk,τk)between(X k,p k,ρk(π))and (X,p,Γ).By lemma2.6we can assume thatτk(ρ(γ))=ρk(γ)for allγ∈S.Hence, the map?h k=?g k??h:F→X k isρk-equivariant.Extend it by equivariance to a ρk-equivariant map?h k:?K→X k.Passing to quotients we get a map h k:K→X k/ρk(π1(K))such thatτ(ρk)is the pullback via h k of the tangent bundle to X k/ρk(π1(K)).

By construction,h k=g k?ˉh where g k is the drop of?g k.Being a di?eomorphism g k preserves tangent bundles.Therefore,the pullback via h k of the tangent bundle to X k/ρk(π1(K))is equal to the pullback viaˉh of the tangent bundle to X/ρ(π1(K,k). In other wordsτ(ρk)=τ(ρ).The proof is complete.

12IGOR BELEGRADEK

Remark5.2.In the above theorem it is possible to keep track of orientations provided all the actionsρk on X k preserve orientations(it makes sense because being a contractible manifold X k is orientable).Indeed,?x an orientation on X (which is also contractible)and choose orientations of X k so that di?eomorphisms g k preserve orientations.Then,obviously,the theorem5.1is still true where the vector bundle isomorphismτ(ρk)=τ(ρ)preserves orientations.

Theorem 5.3.Assume that dim Q

KO(K)?Q<∞.Letπ1(K)be a?nitely generated group andρk:π=π1(K)→Isom(X k)be a sequence of isometric actions ofπon Hadamard n-manifolds X k such thatρk(π)is a discrete subgroup of Isom(X k)that acts freely.

Suppose that,for some p k∈X k,(X k,p k,ρk(π))converges in the equivariant pointed Lipschitz topology to(X,p,Γ)and(X k,p k,ρk)converges to(X,p,ρ)in the pointwise convergence topology.

KO(K)?Q are equal. Then,for all large k,the images ofτ(ρk)andτ(ρ)in

Proof.By3.4there exists a?nite connected subcomplex X?K such that the inclusion i:X→K induces aπ1–epimorphism and a

KO-monomorphism.

The sequence of isometric actionsρk?i?ofπ1(X,x)on X k satis?es the assump-tions of the theorem5.1,therefore,τ(ρk?i?)=τ(ρ?i?)for all large k.In other words,the pullback bundles i?τ(ρk)are isomorphic to the vector bundle i?τ(ρ).

KO–groups,the images ofτ(ρk)and Since i induces a monomorphism of rational

KO(K)?Q are equal for all large k.

τ(ρ)in

KO(K)is?nitely generated.Letπ1(K) Corollary5.4.Assume that the group

be a?nitely generated group and letρk:π=π1(K)→Isom(X k)be a sequence of isometric actions on Hadamard n-manifolds X k such thatρk(π)is a discrete sub-group of Isom(X k)that acts freely.Suppose that,for some p k∈X k,(X k,p k,ρk)is precompact in both pointwise convergence topology and equivariant pointed Lipschitz topology.

Then the set{τ(ρk)}falls into?nitely many stable isomorphism isomorphism classes.

Proof.Argue by contradiction.Pass to subsequence to assume that allτ(ρk)are di?erent and that(X k,p k,ρk(π1(K,q)))converges to(X,p,Γ)in the equivariant pointed Lipschitz topology and(X k,p k,ρk)converges to(X,p,ρ)in the pointwise convergence topology.

KO(K)?Q)<∞.

KO(K)is?nitely generated,then dim Q

Note that if the group

KO(K)?Q are all equal for large Then by the theorem5.3the images ofτ(ρk)in

KO(K)are all equal modulo k.In other words for large k,the images ofτ(ρk)in

KO(K)is?nitely generated,its torsion subgroup torsion.Since the abelian group,

is?nite.Therefore,passing to subsequence we can assume that the imagesτ(ρk) KO(K)are the same,a contradiction.

in

Corollary5.5.Assume that the complex K is aspherical.Suppose that the groups

KO(K)andπ1(K)are?nitely generated.Letρk:π=π1(K,q)→Isom(X k) be a sequence of free,isometric actions ofπ1(K)on Hadamard n-manifolds X k. Suppose that,for some p k∈X k,(X k,p k,ρk(π))is precompact in the equivariant pointed Lipschitz topology and(X k,p k,ρk)is precompact in the pointwise conver-gence topology.

COUNTING NEGATIVELY CURVED MANIFOLDS13 Then the set of manifolds{X k/ρk(π)}falls into?nitely many tangential equiv-alence types.

Proof.By the corollary5.4the set{τ(ρk)}falls into?nitely many stable isomor-phism isomorphism classes.Let?h k:?K→X k beρk–equivariant maps constructed in§3.Passing to quotients we get homotopy equivalences h k:K→X k/ρk(π)=N k with the property that h?k T N k~=τ(ρk).Therefore,X k/ρk and X j/ρj are tangential homotopy equivalent i?τ(ρk)~=τ(ρj).

Corollary5.6.Suppose that the complex K is aspherical and the group

KO(K) is?nitely generated.Letπ1(K)be a?nitely presented group that is not virtually nilpotent.Assume thatπ1(K)does not split as a nontrivial amalgamated product or an HNN-extension over a virtually nilpotent group.

Then,for any a≤b<0and an integer n≥2,the class M a,b,π

1(K),n breaks into

?nitely many tangential homotopy types.

Proof.Apply the corollary5.5and the proposition2.7,2.10.

Corollary5.7.Letπbe the fundamental group of a?nite aspherical CW-complex. Suppose thatπis not virtually nilpotent and thatπdoes not split as a nontrivial amalgamated product or an HNN-extension over a virtually nilpotent group.

Then,for any a≤b<0and an integer n≥2,the class M a,b,π

1(K),n breaks into

?nitely many tangential homotopy types.

Proof.Letπbe the the fundamental group of a?nite complex K.Thenπis ?nitely presented and the group

KO(K)is?nitely generated[24,p52].Hence the corollary5.6applies.

§6.Graphs of groups and Accessibility.

In this section we explain how to generalize Theorem1.1to certain amalgamated products and HNN–extensions,or more generally,to some graphs of groups.Then we deduce the corollary1.2.

Recall that a graph of groups is a graph whose vertices and edges are labeled with vertex groupsπv and edge groupsπe and such that every pair(v,e)where the edge e is incident to the vertex v is labeled with a group monomorphismπe→πv. We only consider?nite connected graphs of groups.To each graph of groups one can associate its fundamental group which is a result of repeated amalgamated products and HNN–extensions of vertex groups over the edge groups(see[3]for more details).

Proposition6.1.Letπbe the fundamental group of a?nite graph of groups with the vertex groupsπv.Assume that the homomorphism

KO(K(π,1))→⊕v

KO(K(πv,1))

induced by the inclusionsπv→πhas?nite kernel.

Fix n and a≤b<0and suppose that,for each vertex groupπv,the class

M a,b,π

v,n breaks into?nitely many tangential homotopy types.Then so does the

class M a,b,π,n.

Proof.Assume,by contradiction that there exist a sequence N k∈M a,b,π,n of manifolds that are not pairwise tangentially homotopy equivalent.It de?nes an in?nite sequence of distinct elements in

KO(K(π,1)).

14IGOR BELEGRADEK

The homomorphism KO

(K (π,1))→⊕v KO (K (πv ,1))has ?nite kernel and the set of vertices is ?nite,therefore,for some vertex v we get an in?nite sequence of elements in KO (K (πv ,1)).Each of elements of the in?nite sequence comes from the tangent bundle to a manifold ˉN

k ∈M a,b,πv ,n ,namely,ˉN k is the covering of N k induced by the inclusion πv →π.Thus,for this vertex M a,b,πv ,n falls into in?nitely many tangential homotopy types,a contradiction.

Corollary 6.2.Let πbe the fundamental group of a ?nite graph of groups such that each edge group is a virtually nilpotent group of cohomological dimension ≤2.

Fix n and a ≤b <0and suppose that,for each vertex group πv ,the class M a,b,πv ,n breaks into ?nitely many tangential homotopy types.Then so does the class M a,b,π,n .

Proof.We can assume that M a,b,π,n =?.Then M a,b,πe ,n =?for every edge group πe .Hence all the edge groups are ?nitely generated [10].Being a ?nitely generated virtually nilpotent group,each edge group πe is a fundamental group of a closed aspherical manifold [17]which has to be of dimension ≤2since cd(πe )≤2.

In particular,the group KO

?1(K (πe ,1))is a ?nitely generated abelian group [24,p52].According to 3.2,there is an isomorphism of functors KO

?1

(?)?Q ~=⊕k>0H 4k ?1(?,Q )on the category of connected CW–complexes of uniformly

bounded dimension.Hence,for each edge group πe ,the group KO

?1(K (πe ,1))is ?nite.

Following [34]we can assemble the cell complexes K (πv ,1)and K (πe ,1)×

[?1,1]into an K (π,1)cell complex by using edge–to–vertex monomorphisms.Con-sider the KO

?–Mayer-Vietoris sequence applied to the K (π,1)complex (cf.[12,VII.9]).The group ⊕e KO ?1(K (πe ,1))is ?nite,so by exactness, KO (K (π,1))→⊕v KO

(K (πv ,1))has ?nite kernel.We now apply 6.1to complete the proof. 6.3.An accessibility result.Delzant and Potyagailo have recently proved a powerful accessibility result which we state below for reader’s convenience.The de?nition 6.4and the theorem 6.5are taken from [16].

De?nition 6.4.A class E of subgroups of a group πis called elementary provided the following four conditions hold.

(i)E is closed under conjugation in π;

(ii)any in?nite group from E is contained in a unique maximal subgroup from E ;

(iii)if a group from the class E acts on a tree,it ?xes a point,an end,or a pair of ends;

(iv)each maximal subgroup from E is equal to its normalizer in π.

Note that the condition (iii)holds if any group in E is amenable [31].

Theorem 6.5[16].Let πbe a ?nitely presented group without 2-torsion and let E be an elementary class of subgroups of π.Then there exists an integer K >0and a ?nite sequence π0,π1,...,πm of subgroups of πsuch that

(1)πm =π,and

(2)for each k with 0≤k

COUNTING NEGATIVELY CURVED MANIFOLDS15

(3)for each k with K≤k≤m,the groupπk is the fundamental group of a?nite graph of groups with edge groups from E,vertex groups from{π0,π1,...,πk?1},and proper edge-to-vertex homomorphisms.

Proposition6.6.If M a,b,π,n=?,then the class of virtually nilpotent subgroup of πis elementary.

Proof.The conditions(i)is trivially satis?ed.Any virtually nilpotent group is amenable,hence(iii)holds thanks to[31].

Let X be a Hadamard manifold such that X/π∈M a,b,π,n.Being the funda-mental group of an aspherical manifold,the groupπis torsion free.According to [11],a nontrivial torsion-free virtually nilpotent discrete isometry groupΓof X is characterized by its?xed-point-set at in?nity.In fact,either the?xed-point-set is a point or it consists of two points.Conversely,any discrete torsion-free subgroup of Isom(X)that?xes a point at in?nity is virtually nilpotent.Thus,a virtually nilpotent subgroup N ofπis maximal if and only if N contains every element ofπthat?xes the?xed-point-set of N.This characterization proves(ii).

Finally,we deduce(iv).Let g be an element of the normalizer of a maximal virtually nilpotent subgroup N.Then g preserves the?xed-point-set of N setwise. If the?xed-point-set is a point,we conclude g∈N.If the?xed-point-set consists of two points p and q and g/∈N,then g(p)=q.Hence g must?x a point on the geodesic that joins p and q,so the element g is elliptic.This is a contradiction because no discrete torsion-free group contains an elliptic element.

Theorem6.7.Letπbe a?nitely presented group such that any nilpotent subgroup ofπhas cohomological dimension≤2.Assume that the group

KO(K(π,1))is ?nitely generated.

Then,for any positive integer n and any negative reals a≤b,the class M a,b,π,n breaks into?nitely many tangential homotopy types.

Proof.Applying the theorem6.5,we get a sequenceπ0,π1,...,πm of subgroups ofπ.In particular,for every k with0≤k

KO(K(π,1))is?nitely generated.Therefore,by the theorem5.6,the class

M a,b,π

k ,n

breaks into?nitely many tangential homotopy types.

Repeatedly applying the corollary6.2,we deduce that M a,b,π,n breaks into ?nitely many tangential homotopy types.

Corollary6.8.Letπbe a word-hyperbolic group.Then for any n and a≤b<0, the class M a,b,π,n breaks into?nitely many tangential homotopy types.

Proof.We can assume that M a,b,K(π,1),n=?,henceπis torsion–free.Any torsion–free word–hyperbolic group is the fundamental group of a?nite CW-complex K[15, 5.24],in particularπis?nitely presented and has?nitely generated

KO(K)[24, p52].Any nilpotent subgroup of a torsion free word hyperbolic group is either trivial or in?nite cyclic(see e.g.[4]).The result now follows from the previous theorem.

Proposition6.9.Letπbe the fundamental group of a?nite graph of groups with virtually nilpotent edge groups.Assume M a,b,π,n=?.Then

(1)πis?nitely presented i?all the vertex groups are?nitely presented,and

16IGOR BELEGRADEK

(2)dim Q⊕k>0H4k(π,Q)<∞i?dim Q⊕k>0H4k(πv,Q)<∞for every vertex groupπv.

KO(π)is?nitely generated i?for every vertex groupπv the group (3)the group

KO(π)is?nitely generated.

Proof.All the edge groups are?nitely generated[10].Being a?nitely generated virtually nilpotent group,each edge groupπe is a fundamental group of a closed

KO?(πe)as well as H?(πe)is a?nitely aspherical manifold[17].In particular,

generated abelian group.Hence,the parts(2)and(3)follow from the Mayer-Vietoris sequence(cf.[12,VII.9]).

We now prove(1).It trivially follows from de?nitions that if all vertex groups are?nitely presented,then so isπ.

Assume now thatπis?nitely presented.Then every vertex groupπv?nitely generated[3,Lemma13,p.158].The fundamental groupπof a graph of groups has a presentation that can be described as follows.The set of generators S for for this presentationπis the union of the(?nite)sets of generators of all the vertex groups;so S is?nite.The set of relations R is a union of the sets of relations of all the vertex groups and(?nitely many)relations coming from amalgamations and HNN-extensions over the edge groups.

Sinceπis?nitely presented,the groupπhas a presentation S|R′ on the same set of generators S where R′is a?nite subset of R[3,Theorem12,p52].By adding ?nitely many(redundant)relations,we can assume that all the relations coming from the edge groups still belong to the new set of relations R′.This de?nes a new graph of groups decomposition ofπwith the same underlying graph,same edge groups and with?nitely presented vertex groupsπ′v.By construction,the subgroupsπ′v andπv ofπhave the same set of generators,henceπ′v=πv.Thus, we have found a?nite presentation for every vertex group.

§7.Finiteness for thickenings

Throughout this section K is a?nite,connected CW-complex.

By an n–thickening of K we mean a compact smooth manifold L of dimension ≥dim(K)+3such that L is simply homotopy equivalent to K,and the inclusion ?L→L induces an isomorphism of fundamental groups(see[38]).To avoid low-dimensional complications,we always assume that n>4.

A thickening L of K is always di?eomorphic to the regular neighborhood of a ?nite simplicial subcomplex K′?L with dim(K′)≤dim(K)[25,p219][35].

Ifπ1(K)is isomorphic to the fundamental group of a complete manifold of sec-tional curvature pinched between two negative constants,any homotopy equivalence K≈L is necessarily simple because Wh(π1(K))=0[18].

By an open n–thickening of a?nite connected CW–complex K we mean an open smooth manifold that is di?eomorphic to the interior of a thickening of K. Example7.1(Totally geodesic).Let N be a complete manifold of nonpositive curvature of dimension>4that is homotopy equivalent to a?nite cell complex K of dimension≤dim(N)?3.Assume that N contains a(possibly noncompact) totally geodesic embedded submanifold M of dimension≤dim(N)?3such that the inclusion M→N is a homotopy equivalence.

Then N is an open thickening of K.(Indeed,the exponential map identi?es N with the total space of the normal bundle of M in N.Then,according to[35],the

COUNTING NEGATIVELY CURVED MANIFOLDS17 manifold N has exactly one end that isπ1-stable and the natural homomorphism of the fundamental group at in?nity intoπ1(N)is an isomorphism.By[25]N is simple homotopy equivalent to a simplicial subcomplex K′of dimension≤dim(K)≤dim(N)?3.Hence,N is di?eomorphic to the regular neighborhood of K′in N [35].Thus,N is an open thickening of K.)

7.3.Existence of thickenings.For each vector bundleξover K of rank n> max{5,dim(K)}there exists an n-thickening L of K and a simple homotopy equiv-alence f:K→L such that f?T L~=ξ[38,5.1].

7.4.Uniqueness of thickenings.Any tangential homotopy equivalence of n-thickenings is homotopic to a di?eomorphism provided n>5.[38,5.1].

7.5.Uniqueness of open thickenings.Any tangential homotopy equivalence of open n-thickenings is homotopic to a di?eomorphism provided n>4.[29,pp.226–228]

7.6.Theorem.Let K be a?nite,connected,aspherical CW-complex and let n be

an integer with n>max{4,2dim(K)}.Suppose that the class M a,b,π

1(K),n breaks

into?nitely many tangential homotopy types.

Then,for any a≤b<0,the set of di?eomorphism classes of open thickenings of K that belong to M a,b,π

1(K),n

is?nite.

Proof.Apply7.5.

7.6.Theorem.Let K be a?nite,connected,aspherical CW-complex and let n be

an integer with n>max{5,2dim(K)}.Suppose that the class M a,b,π

1(K),n breaks

into?nitely many tangential homotopy types.

Then,for any a≤b<0,the set of di?eomorphism classes of thickenings of K whose interiors belong to M a,b,π

1(K),n

is?nite.

Proof.Apply7.4.

7.7.Corollary.Let K be a?nite,connected,aspherical CW-complex and let n be an integer with n>max{4,2dim(K)}.Suppose that either

?any nilpotent subgroup ofπ1(K)has cohomological dimension≤2,or

?πis not virtually nilpotent andπdoes not split as a nontrivial amalgamated product or an HNN-extension over a virtually nilpotent group.

Then,for any a≤b<0,the set of di?eomorphism classes of open thickenings of K that belong to M a,b,π

1(K),n

is?nite.

https://www.doczj.com/doc/6215953281.html,bine5.7,6.7,and7.6.

7.8.Theorem.Let K be a?nite,connected,aspherical CW-complex and let n be an integer with n>max{4,2dim(K)}.Supposeρk is a sequence of free isometric actions ofπ1(K)on Hadamard n-manifolds X k such that,for some p k∈X k, (X k,p k,ρk)is precompact in both pointwise convergence topology and equivariant pointed Lipschitz topology.Assume that for each k,the manifold X k/ρk(π1(K))is an open thickening of K.

Then the set{X k/ρk(π1(K))}breaks into?nitely many di?eomorphism types. Proof.Since K is a?nite complex,the group

KO(K)is?nitely generated.Hence 5.5implies that the set of manifolds{X k/ρk(π1(K))}falls into?nitely many tan-gential homotopy types.According to7.6,{X k/ρk(π1(K))}breaks into?nitely many di?eomorphism types.

18IGOR BELEGRADEK

§8.Finiteness for convex-cocompact groups First,we recall some basic facts on convex-cocompact groups that are well known in the constant negative curvature case(see[11]for more details).

Let X be a Hadamard manifold with sectional curvatures pinched between two negative constants and letΓbe a discrete subgroup of the isometry group of X. ThenΓacts by homeomorphisms on the ideal boundary?∞X of X.The set of points?(Γ)??∞X whereΓacts properly discontinuously is called the domain of discontinuity ofΓ.Its complementΛ(Γ)=?∞X\?(Γ)is called the limit set ofΓ.

Fix any?>0.Let C?(Γ)be the closed?-neighborhood of the convex hull of Λ(Γ)in X.

There is aΓ-equivariant homeomorphism of X∪?(Γ)and C?de?ned as follows. The?bers of orthogonal projection p:X→C?(Γ)are geodesic rays orthogonal to ?C?(Γ).Any point of?(Γ)is the endpoint of such a ray and no two of these rays have the same endpoint.Thus,the map p extends to an equivariant continuous map ˉp:X∪?∞X→C?(Γ)∪Λ(Γ)which is the identity on the limit set.Contracting along the rays de?nes an equivariant homeomorphism of X∪?∞X and C2?(Γ)∪Λ(Γ). which descends to a homeomorphism of X∪?(Γ))/Γand C2?(Γ)/Γ.

We say thatΓis convex-cocompact if the quotient X∪?(Γ)/Γis compact.

Two convex–cocompact groupsΓ1andΓ2are called topologically equivalent if there exists a homeomorphism h:X1∪?∞X1→X2∪?∞X2that is equivariant with respect to a certain isomorphism ofΓ1andΓ2.

Given negative reals a≤b,a torsion-free groupπ,and an integer n,de?ne a

class of convex-cocompact groups CC a,b,π,n,π

1(?)=1as follows.

A convex-cocompact groupΓof isometries of a Hadamard manifold X is said to

belong to CC a,b,π,n,π

1(?)=1provided the following three conditions hold

?Γis isomorphic toπ;

?dim(X)=n and the sectional curvature of X is within[a,b];

?the domain of discontinuity?(Γ)ofΓis simply-connected.

Proposition.8.1.LetΓbe a torsion–free convex–cocompact subgroup of the isom-etry group of a Hadamard manifold X of dimension>5such that?(Γ)is simply-connected.Assume thatΓis the fundamental group of a?nite aspherical CW–complex K of dimension≤dim(X)?3.

Then(X∪?(Γ))/Γis a thickening of K.

Proof.SinceΓbe a torsion–free convex–cocompact group,the quotient(X∪?(Γ))/Γis a compact manifold with boundary that is homotopy equivalent to K.

The homotopy equivalence is necessarily simple because Wh(Γ)=0.(Farrell and Jones[18]proved the vanishing of the Whitehead group of the fundamental group of any complete manifold of pinched negative curvature.)

Finally,since?(Γ)is simply-connected,the inclusion?(Γ)/Γ→(X∪?(Γ))/Γinduces aπ1–isomorphism.

Proposition.8.2.Let K be a?nite,connected,aspherical CW-complex and let n be an integer with n>2dim(K)and n>5.Given i∈{1,2},letΓi be a torsion–free subgroup of the isometry group of a Hadamard manifold X i such that

Γi∈CC a,b,π

1(K),n,π1(?)=1.

If X1/Γ1and X2/Γ2are tangentially homotopy equivalent,thenΓ1andΓ2are topologically equivalent.

COUNTING NEGATIVELY CURVED MANIFOLDS19 Proof.Since C?(Γi)/Γi is homeomorphic to X∪?(Γi),the proposition8.1implies that C?(Γi)/Γi is a thickening of K.

Moreover,C?(Γi)/Γi is a codimension zero submanifold of X/Γi,hence the homo-topy equivalence C?(Γi)/Γi?→X/Γi is tangential.Thus,C?(Γ1)/Γ1and C?(Γ2)/Γ2 are tangentially homotopy equivalent thickenings,and hence they are di?eomorphic.

The di?eomorphism of compact manifolds C?(Γ1)/Γ1→C?(Γ2)/Γ2is necessarily bilipschitz.Hence,it lifts to an equivariant bilipschitz di?eomorphism d:C?(Γ1)→C?(Γ2).

Note that C?(Γi)is a Gromov hyperbolic space with ideal boundaryΛ(Γi). Therefore,d extends to an equivariant homeomorphism C?(Γ1)∪Λ(Γ1)→C?(Γ2)∪Λ(Γ2)[15,p35].Finally,using equivariant homeomorphisms of X∪?∞X and C?(Γi)∪Λ(Γi),we produce a topological equivalence ofΓ1andΓ2.

8.3.Theorem.Let K be a?nite,connected,aspherical CW-complex and let n be an integer with n>2dim(K)and n>5.

Then the class CC a,b,π

1(K),n,π1(?)=1falls into?nitely many topological equiva-

lence classes.

Proof.Any convex–cocompact groupΓis word-hyperbolic because it acts isometri-cally and cocompactly on a negatively curved space C?(Γ).Hence the result follows from6.8and8.2.

§9.Locally symmetric nonpositively curved

manifolds up to tangential homotopy equivalence.

For completeness we present a proof of the following result.

Theorem9.1.Letπbe a?nitely presented torsion–free group and let X be a nonpositively curved symmetric space.

Then the class of manifolds of the form X/ρ(π),whereρ∈Hom(π,Isom(X)) is a faithful discrete representation,falls into?nitely many tangential homotopy types.

Proof.We can assume that Hom(π,Isom(X))contains a faithful discrete represen-tation.Let K be the corresponding quotient manifold.

First note that,for any two faithful discrete representationsρ1andρ2that lie in the same connected component of the analytic variety Hom(π1(K),Isom(X)), we haveτ(ρ1)~=τ(ρ2).Indeed,by the covering homotopy theorem the X-bundles

?K×

ρ1X and?K×ρ

2

X over K are isomorphic.In particular,the pullbacks to K of

the vertical bundles?K×ρ

1T X and?K×ρ

2

T X are isomorphic as desired.

Thus,if the analytic variety Hom(π,Isom(X))has?nitely many connected com-ponents,the set of the manifolds of the form X/ρ(π)whereρis discrete and faithful falls into?nitely many tangential equivalence classes.This is the case ifπis?nitely presented and X is a nonpositively curved symmetric space.Indeed,represent X as a Riemannian product Y×R k where Y is a nonpositively curved symmetric space without Euclidean factors.By the de Rham’s theorem this decomposition is unique,so Isom(X)~=Isom(Y)×Isom(R k).The group Isom(Y)is semisimple with trivial center,hence the analytic variety Hom(π,Isom(Y))has?nitely many connected components[21,p.567].The same is true for Hom(π,Isom(R k))because Isom(R k)is real algebraic[21,p.567].Hence the analytic variety

Hom(π,Isom(X))~=Hom(π,Isom(Y))×Hom(π,Isom(R k))

20IGOR BELEGRADEK

has?nitely many connected components.

9.2.Theorem.Let K be a?nite,connected,aspherical CW-complex and let n be

an integer with n>max{4,2dim(K)}.

Then the set of di?eomorphism classes of open n-thickenings of K that admit complete locally symmetric metrics of nonpositive sectional curvature is?nite.

Proof.Note that there exist only?nitely many symmetric Hadamard manifolds of

a given dimension.Hence,the result follows from7.5and9.1.

9.3.Theorem.Let K be a?nite,connected,aspherical CW-complex and let X be

a symmetric negatively curved Hadamard n-manifold with n>max{5,2dim(K)}.

Letρ1andρ2be injective representations ofπ1(K)into the isometry group of X

that lie in the same connected component of the representation variety Hom(π1(K),Isom(X)). Suppose that,for i∈{1,2},ρi(π1(K))is a convex-cocompact group with simply-connected domain of discontinuity.

Thenρ1andρ2are conjugate be a homeomorphism of X∪?∞X

Proof.Recall that the are four kinds of symmetric spaces of negative sectional curvature,namely,they are hyperbolic spaces over the reals,complex numbers, quaternions and Cayley numbers.The spaces have sectional curvatures pinched between?4and?1.Applying8.1we conclude that X∪?(ρi(π1(K)))/ρi(π1(K))

is a thickening of K for i=1,2.

It follows from the proof of9.1that the homotopy equivalence induced by

ρ1?(ρ2)?1is tangential becauseρ1andρ2lie in the same connected component.

Thus,by7.4,the homotopy equivalence is homotopic to a di?eomorphism that lifts

to a smooth conjugacy ofρ1andρ2on X∪?(ρ1(π1(K)))and X∪?(ρ2(π1(K))). Repeating the argument of8.2,we deduce that the conjugacy extends to an equi-

variant self-homeomorphism of X∪?∞X.

References

1.M.T.Anderson,Metrics of negative curvature on vector bundles,Proc.Amer.Math.Soc.

99(1987),no.2,357–363.

2.W.Ballmann,M.Gromov,and V.Schroeder,Manifolds of nonpositive curvature,Birkh¨a user,

Progress in mathematics,vol.61,1985.

3.G.Baumslag,Topics in combinatorial group theory,Lectures in Mathematics ETH Z¨u rich.

Birkh¨a user,Basel,1993.

4.I.Belegradek,Intersections in hyperbolic manifolds,preprint(1997).

5.

,Negatively curved vector bundles,pinching,and accessibility,preprint(1998).

7.R.Benedetti and C.Petronio,Lectures on hyperbolic geometry,Universitext,Springer-Verlag,

1992.

8.M.Bestvina,Degenerations of the hyperbolic space,Duke Math.J56(1988),143–161.

9.M.Bestvina and M.Feighn,Stable actions of groups on real trees,Invent.Math.121(1995),

no.2,287–321.

10.B.H.Bowditch,Discrete parabolic groups,J.Di?erential Geom.38(1993),no.3,559–583.

11.

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2.7.4标准云听测试总结报告 测试人员:***

目录 1引言 (3) 1.1编写目的 (3) 1.2背景 (3) 1.3用户群 (3) 1.4定义 (3) 1.5 测试对象 (4) 1.6 测试阶段 (4) 1.7 测试工具 (4) 1.8 参考资料 (4) 2测试概要 (4) 2.1进度回顾 (5) 2.2测试执行 (5) 2.3 测试用例 (5) 2.3.1 功能性 (5) 2.3.2 易用性 (5) 3测试环境 (6) 4 测试结果 (6) 4.1 Bug 趋势图 (6) 4.2 Bug 严重程度 (7) 4.3 BUG分类统计占比 (8) 5测试结论 (9) 5.1功能性 (9) 5.2易用性 (9) 5.3可靠性 (10) 5.4兼容性 (10) 5.5安全性 (10) 6 分析摘要 (10) 6.1 建议 (10) 7度量 (11) 7.1 资源消耗 (11) 8典型缺陷引入原因分析 (11)

1引言 1.1编写目的 编写标准云听测试报告主要目的罗列如下: 1.通过对测试结果的分析,得到对软件质量的评估 2.分析测试的过程,产品,资源,信息,为以后制定测试计划提供参考3.评估测试执行和测试计划是否符合 4.分析系统存在的缺陷,为修复和预防bug 提供建议 1.2背景 客户需求 1.3用户群 主要使用者: (1) 电台主播(主持人) (2) 频道负责人 (3) 媒体负责人 (4) 电台听众 1.4定义 1.出现以下缺陷,定义为致命bug (1级) : (1) 系统出现闪退、崩溃; (2) 系统无响应,处于死机状态,需要其他人工修复系统才可复原;’ (3) 操作某个功能出现报错或者返回异常错误; (4) 进行某个操作(增加、修改、删除等)后,出现报错或者返回异常错误; (5) 实现功能和需求不符等; 2.出现以下缺陷,定义为严重(功能)bug (2级) : (1) 当对必填字段进行校验时,未输入必输字段,出现报错或者返回异常错误 (2) 系统定义不能重复的字段输入重复数据后,出现报错或者返回异常错误 (3) 系统刷新加载不正常,不能正确显示; (4) 显示信息与配置信息不一致等; 3.出现以下缺陷,定义为一般bug(3级): (1) 显示问题; (2) 提示问题;

常用通讯测试工具使用

常用通讯测试工具 鉴于很多MCGS用户和技术人员对通讯测试工具并不很熟悉,本文档将针对实际的测试情况,对串口、以太网通讯调试过程中所涉及到的常用的测试软件进行相关的讲解。 1. 串口测试工具: 串口调试工具:用来模拟上下位机收发数据的串口工具,占用串口资源。如:串口调试助手,串口精灵,Comm等。 串口监听工具:用来监听上下位机串口相关操作,并截获收发数据的串口工具。不占用串口资源。如:PortMon,ComSky等。 串口模拟工具:用来模拟物理串口的操作,其模拟生成的串口为成对出现,并可被大多数串口调试和监听软件正常识别,是串口测试的绝好工具。如:Visual Serial Port等。 下面将分别介绍串口调试助手、Comm、PortMon和Visual Serial Port的使用。

1.1. 串口调试助手: 为最常用的串口收发测试工具,其各区域说明及操作过程如下: 串口状态 打开/关闭串口 十六进制/ASCII 切换 串口数据 接收区 串口参数 设置区 串口数据 发送区 串口收发计数区 发送数据功能区 保存数据功能区 操作流程如下: ? 设置串口参数(之前先关闭串口)。 ? 设置接收字符类型(十六进制/ASCII 码) ? 设置保存数据的目录路径。 ? 打开串口。 ? 输入发送数据(类型应与接收相同)。 ? 手动或自动发送数据。 ? 点击“保存显示数据”保存接收数据区数据到文件RecXX.txt。 ? 关闭串口。 注:如果没有相应串口或串口被占用时,软件会弹出“没有发现此串口”的提示。

1.2. PortMon 串口监听工具: 用来监听上下位机串口相关操作,并截获收发数据的串口工具。不占用串口资源, 但在进行监听前,要保证相应串口不被占用,否则无法正常监听数据。 连接状态 菜单栏 工具栏 截获数据显示区 PortMon 设置及使用: 1). 确保要监听的串口未被占用。 如果串口被占用,请关闭相应串口的应用程序。比如:要监视MCGS 软件与串口1设备通讯,应该先关闭MCGS 软件。 说明:PortMon 虽不占用串口资源,但在使用前必须确保要监听的串口未被占用,否则无法进行监视。 2). 运行PortMon,并进行相应设置。 ? 连接设置: 在菜单栏选择“计算机(M)”->“连接本地(L)”。如果连接成功,则连接状态显示为“PortMon 于\\计算机名(本地)”。如下图:

自动化概述

一、概述 1.1 什么是自动化测试 自动化测试是把以人为驱动的测试行为转化为机器执行的一种过程。通常,在设计了测试用例并通过评审之后,由测试人员根据测试用例中描述的规程一步步执行测试,得到实际结果与期望结果的比较。在此过程中,为了节省人力、时间或 硬件资源,提高测试效率,便引入了自动化测试]的概念。 提高测试效率,保证产品质量 1.自动化测试完全取代手工测试 2.自动化测试一定比手工测试厉害,更加高大上 3.自动化可以发掘更多的bug 二、自动化层次模型 2.1 单元自动化测试 1.主要是针对于类、方法的测试。

2.此阶段测试效益最大。 3.常见测试框架:Junit 、TestNG、Unittest。 1、节省了测试成本 根据数据模型推算,底层的一个程序BUG可能引发上层的8个左右BUG,而且 底层的BUG更容易引起全网的死机;接口测试能够提供系统复杂度上升情况下的低成本高效率的解决方案。 2、接口测试不同于单元测试 接口测试是站在用户的角度对系统接口进行全面高效持续的检测。 3、效益更高 将接口测试实现为自动化和持续集成,当系统复杂度和体积越大,接口测试的成本就越低,相对应的,效益产出就越高。 4.常见工具 httpUnit (接口框架)、postman(接口调试工具)。 1、界面元素测试 2、面向用户,测试工作占比大 3、robot framework ,selenium,appium

三、自动化测试框架模型 3.1 线性测试## 独立功能测试,流水线执行 模块复用(如登录模块) 参数化 关键字封装(QTP、selenium) 1.需求变动不频繁 2.项目周期足够长 3.项目需要重复回归测试

软件测试个人总结及小结

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l 完成“测试经验交流与知识分享”简报,包括简报材料的 制作。该简报内容包括:项目测试经验介绍、测试度量、性能测 试知识介绍、loadrunner使用经验交流。 l 对现有测试规范提供改进反馈意见; l 根据以往经验,在cmbp项目中提供帮助。 3.完成所需知识的积累 这部分工作,主要是为了更好的完成工作,学习所需的知识、工具及技能。我主要是根据《新员工入职指引表》的要求进行的。主要工作内容有: l 学习金融行业业务知识 l 学习公司研发规范 l 学习研发部产品知识(保理项目、intelliworkflow、农行crm系统、工作流知识) l 参加公司或业务部门组织的培训(新员工入职培训、基于 uml的面向对象分析和设计、金融衍生工具介绍) l 学习缺陷管理工具ttp 4.工具学习及研究 根据《新员工入职指引表》的要求,我了解rational 测试解决方案和工具,并进行rational performance tester的研究。完成对rational performance tester的研究后,我提交了研究成果,包括:《rational performance tester 6 介绍.doc》、使用rational performance tester进行性能测试的例子及学习参考资

[示例文档1]软件测试计划书

[示例文档1]软件测试计划 书 标准化文件发布号:(9312-EUATWW-MWUB-WUNN-INNUL-DQQTY-

软件测试计划

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测试工具 说明本次测试使用的测试工具,包括自编测试程序,并进行确认。 测试开始时间 指明本项目测试工作的开始时间。 测试结束时间 确认测试工作预计的完成时间。 3 实施计划 测试设计工作任务分解和人员安排 测试设计工作应包括对系统功能及专业知识的学习, 编写测试大纲、设计测试用例等工作。 时间安排 测试设计开始时间:测试设计工作预计开始时间。 测试设计结束时间:测试设计工作预计结束时间。 人员安排 列出预计参加本次测试设计工作的全部测试人员。 输出要求 测试设计工作的输出应包括《测试用例》、《测试记录表》、《测试报告》。 对系统功能及专业知识学习如有必要也要形成书面材料。 由测试小组负责规定组织相关的测试人员进行评审计划。

接口自动化测试框架设计

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