Annulus and Disk Complex Is Contractible and Quasi-convex

GUO QI-LONG,QIU RUI-FENG,ZOU YAN-QINGAND ZHANG FA-ZE

(1.School of Mathematics Sciences,Dalian University of Technology,Dalian,Liaoning,116023) (2.Department of Mathematics,East China Normal University,Shanghai,200062)

Communicated by Lei Feng-chun

Annulus and Disk Complex Is Contractible and Quasi-convex

GUO QI-LONG1,QIU RUI-FENG2,ZOU YAN-QING1AND ZHANG FA-ZE1

(1.School of Mathematics Sciences,Dalian University of Technology,Dalian,Liaoning,116023) (2.Department of Mathematics,East China Normal University,Shanghai,200062)

Communicated by Lei Feng-chun

The annulus and disk complex is defined and researched.Especially,we prove that this complex is contractible and quasi-convex in the curve complex.

annulus and disk complex,contractible,quasi-convex

1 Introduction

Let S be a closed orientable surface with genus at least 2.Harvey[1]defined the curve complex of S as follows.The curve complex of S is the complex whose vertices are the isotopy classes of essential simple closed curves on S,and k+1 vertices in the curve complex span a k-simplex if they are represented by pairwise disjoint curves.We denote the curve complex of S by C(S).Harer[2]proved that C(S)is homotopy equivalent to a bouquet of spheres of dimension-χ(S).

If S is a boundary component of an irreducible 3-manifold M,then we can define the disk complex Δ(M,S)as in[3].A vertex of Δ(M,S)is an isotopy class of an essential curve in S which bounds a disk in M.As in the Definition of C(S),k+1 vertices in Δ(M,S) span a k-simplex if they are represented by pairwise disjoint curves.It is easy to see that Δ(M,S)is a subcomplex of C(S).McCullough[3]researched this complex and proved that it is contractible.

In Section 2,we define a new complex associated to a compression body as a generalization of both curve complex and disk complex of a handlebody.For a compression body C, we denote this new complex by AD(C)and call it annulus and disk complex.By using the techniques in[3],we prove the following theorem:

Theorem 1.1The annulus and disk complex AD(C)is contractible.

A metric space(X,d)is geodesic,if for any pair of points there is a path connecting them which is a geodesic;and a subset Y of(X,d)is K-quasi-convex if for any pair of points in Y,any geodesic in X connecting them lies in a K-neighborhood of Y.A result in[4] implies that Δ(M,S)is quasi-convex in C(S).By the aid of their results,we prove

Theorem 1.2AD(C)is K-quasi-convex in C(S),where K depends only on the genus of S.

2 Preliminaries

Definition 2.1A compression body C is a 3-manifold obtained from an orientable connected closed surface Σ by attaching 2-handles to Σ×{1}?Σ×[0,1]and 3-balls to 2-sphere boundaries thereby created.We write

When C=Σ×[0,1],we say that C is a trivial compression body.When?-C=?,we say that C is a handlebody.

Remark 2.1If F is an essential annulus properly embeded in a compression body C,then this annulus must have one boundary component in?+C as the other boundary component in?-C.Furthermore,if F1and F2are two essential annuli such that F1∩?+C is isotopic to F2∩?+C in?+C,then F1is isotopic to F2in C.

Essential annuli play an important role in the following definition.

Definition 2.2For a compression body C,the annulus and disk complex AD(C)is defined as follows:A vertex of AD(C)is an isotopy class of an essential curve on?+C which bounds an essential disk in C or cobounds an essential annulus in C with another curve in

Remark 2.2If C is a trivial compression body,then AD(C)is nothing but the curve complex C(?+C).If C is a handlebody,then AD(C)is the disk complex Δ(C,?+C).

Then we define another complex associated to a compression body C without concerning?+C.

Definition 2.3For a compression body C,the complexis defined as follows:A vertex ofis an isotopy class of an essential disk in C or an essential annulus in C. k+1 vertices[F0],···,[Fk]determine an k-simplex if and only if we can isotopy F0,···,Fkso that they are mutually disjoint.

Lemma 2.1The map]can be extended to be an isomorphism from

As in[5],we give the Definition of δ-hyperbolic spaces.

Definition 2.4A geodesic space X is said to be δ-hyperbolic with constant δ≥0,if for every geodesic triangle xyz in X,one sideis contained in the closed δ-neighborhood of the union of the other two sides

Lemma 2.2[6]C(S)is δ-hyperbolic for some constant δ＞0,where δ depends only on the genus of S.

Lemma 2.3[4]Suppose that M is a compact,orientable 3-manifold M with boundary component S.Then Δ(M,S)is a K-quasi-convex subset of C(S),where K depends only on the genus of S. and

3 Proof of Theorem 1.1

The proof of Theorem 1.1 is similar to the proof of Theorem 5.3 in[3].

It is easy to see that we only need to prove that any map from any sphere to X is null-homotopic.Suppose that the map f:Sq→X is given.

Define Ω:={(K,g):K is a triangulation of Sq,g:Sq→X is homotopic to f,and g:K→X is simplicial}.

Next,we define a complexity function P on Ω as follows.

Let E be an essential disk of C.Then a=[E]is a vertex of X.

For any pair(K,g)∈Ω,define Pi(K,g)to be the number of vertices v of K such that g(v)·a=i.The complexity P(K,g)is defined by(···,P3(K,g),P2(K,g),P1(K,g)),and the complexities are ordered lexicographically.

Suppose that(K,g)has the minimal complexity among all elements in Ω.

If Pi(K,g)=0 for each i,then each vertex of K is carried intoby g.Since g is simplicial and,by Lemmas 2.2 and 2.3,g is null-homotopic.Hence f is also null-homotopic.

Therefore,suppose that Pi(K,g)=0 for all i＞n and Pn(K,g)＞0 for some n＞0. Choose a vertex v0∈K such that g(v0)·a=n.Let v1,v2,···,vkbe the vertices in Kadjacent to v0,and we can choose representatives Fifor each g(vi)so that Fiintersects Fjminimally∩for eachand each Fiintersects E minimally(i.e.,=? for each i/=0.

中等厚度靶(中厚靶)通常是指靶板厚度與撞擊形成的坑深大致相等。如上所述，當(dāng)厚靶的厚度遠(yuǎn)大于坑的深度時，靶板的后表面，即自由面對成坑幾何尺寸沒有影響。但是實際上，靶都是有限厚的，無限厚靶屬于一種極限情況。彈丸超高速侵徹中厚靶的瞬態(tài)階段及主要侵徹階段與彈丸超高速侵徹?zé)o限厚靶的情況完全相同，只有當(dāng)靶板后表面反射的稀疏波到達(dá)侵徹坑底部之后，才會表現(xiàn)出與無限厚靶板侵徹過程的差異，即向前的應(yīng)力波和靶板后表面反射的稀疏波相遇產(chǎn)生拉伸應(yīng)力，當(dāng)拉伸應(yīng)力大于靶板的拉伸斷裂強(qiáng)度時，靶板后表面發(fā)生層裂。圖5為球形彈丸超高速侵徹中等厚度靶導(dǎo)致靶板后表面產(chǎn)生層裂的現(xiàn)象。

Since n＞0,F0is not disjoint from E.Consider an arc α of their intersection which is outermost in E.There is a disc B?E such that?B?α∪?E and int(B)∩F0=?.

There are two possibilities.

Since F0intersects E minimally,by surgery on F0along B we get two essential surfaces, one of which is an essential disk(denoted by)and the another is a spanning annulus or an essential disk,and we can see that∩.Since(F0∪B)∩Fi=? for each i/=0, and∩we get that if[Fi1],···,[Fim]and[F0]are contained in a common simplex of X for some i1,···,im,then Fi1,···,Fim,F0and F′0are mutually disjoint.Hence[Fi1], ···,[Fim],[F0]and[F′0]are also contained in a common simplex of X.

Claim 3.1is homotopic to g.

Proof.Since the difference part of the two maps is only on the carrier simplex of{v0,···,vk} in K,we canfind a homotopy gtsuch that

and gt(v0)slides on the 1-simplex〉when t increases from 0 to 1.Actually,if Δ is a simplex of K which does not contain v0,then

and if Δ is a simplex of K containing v0,then gt|Δis defined by linear expansion,and this can be done since g|t(v0),···,g|t(vk)are contained in a common simplex of X.

Let β be an arc of intersection of some Fiand B which is outermost in B and contains no arcs of any Fjwhich is disjoint from Fi(note that β may still intersect an arc of some B∩Fjfor some j/=i).Then by surgery Fialong B,we get an essential diskas above, and can see that.Hence ifand[F0](or[Fi]) are contained in a common simplex of X for some i1,···,im,then[Fi1],···,[Fim],[F0](or [Fi])andare also contained in a common simplex of X.

Claim 3.2g′is homotopic to g.

Proof.Since F0,Fi,are mutually disjoint,and[F0],[Fi],are contained in a simplex of X,which also contains,there is a segment connectingand.For each t∈[0,1],we define gt(x)=x if x/=viandto be a point in the segmentsuch that

Then we can extend each gtto the whole Sqas follows.If Δ is a simplex which does not contains,then define gt|Δ=g0|Δ.If Δ contains,then gt|Δis defined by linear expansion in Δ.It is easy to see that

and gtis a homotopy between g and g′.

Now P(K1,g1)＞P(K,g),because we have added the new vertex v′mapping toBut·[E]＜[Fk]·[E]＜n since n is maximal,and hence

Repeating finitely many times,we obtain a subdivision K2of K and a simplicial map g2:K2→X homotopic to g such that

and int(B)is disjoint from the representative surfaces for the vertices{g2(v1),···,g2(vk)}. Now,by surgery F0along B as in Case 1,we can find a simplicial map g3:K2→X which is homotopic to g2and satisfies that

and

Hence P(K2,g3)＜P(K,g),which contradicts the choice of(K,g).

4 Proof of Theorem 1.2

For any pair of two vertices[F1],[F2]in AD(C),let γ be any geodesic joining them.Let K be the constant in Lemma 2.5 in the case that M=C and S=?+C.

There are three cases about F1and F2.

Case 1.Both are disks.

In this case,Lemma 2.5 implies Theorem 1.2.

Case 2.F1is a disk,while F2is an annulus.

Choose an essential disk D of C such that D∩F2=?.Let β be the 1-simplex joining [D]and[F2],and α be a geodesic joining[D]and[F1].By the hyperbolicity of C(S),α is in the δ-neighborhood of β∪γ.By Lemma 2.6,γ is in the K-neighborhood of Δ(M,S).Since Δ(M,S)?AD(C)and β is in the 1-neighborhood of AD(C),α is in the(K+δ+1)-neighborhood of AD(C).

Case 3.Both F1and F2are annuli.

The proof of this case is similar to that in Case 2,and we can choose the constant to be K+2β+1.

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1674-5647(2013)04-0377-06

Received date:April 6,2012.

The NSF(10901029)of China.

E-mail address:guoqilong1984@hotmail.com(Guo Q L).