Qilong GUORuifeng QIUYanqing ZOU

1 Introduction

All manifolds in this paper are assumed to be compact and orientable,unless otherwise stated.

LetMbe a connected irreducible 3-manifold,and suppose that∂Mconsists of two homeomorphic surfacesF1andF2.LetMfbe the closed 3-manifold obtained by identifyingF1andF2using a homeomorphismBy the construction of self-amalgamation of Heegaard splitting,we haveSo a natural and interesting question is:

Question 1.1When does

In[2],Du and Qiu proved that whenMis sufficiently complicated,the equality holds.In[3],Guo and Zou proved that whenMis irreducible and∂-irreducible,and satisfies some conditions,and if the gluing map is sufficiently complicated,the equality holds.In this paper we consider the case thatMhas two compressible boundary components.Here is our result.

Theorem 1.1Let M be a connected orientable compact irreducible3-manifold,and∂M consist of two homeomorphic surfaces F1and F2which are both compressible.Suppose thatwhere g(M,F1)is the Heegaard genus of M relative to F1.Let Mf be the closed orientable3-manifold obtained by identifying F1and F2through a homeomorphism.Then there is a function defined for all such Mf,such that if d(Mf)>2g(M),then

Remark 1.1g(M,F1)is defined as in Definition 2.2,andd(Mf)is defined as in Definition 2.5.

Note that the irreducibility ofMand compressibility ofF1andF2imply thatg(F1)=g(F2)≥2.

The proof of our main result is a bit similar to that in[6],but the case that we deal with is a bit harder.

2 Preliminaries

LetMbe a 3-manifold.First let us review some notions about Heegaard splittings.

De fi nition 2.3Suppose that F is a properly embedded surface in a3-manifold M,and F splits M into two submanifolds M1and M2.We say that F is strongly irreducible if F has compressing disks on both sides,and each compressing disk in M1meets each compressing disk in M2.We say that F is∂-strongly irreducible if

(1)every compressing and∂-compressing disk in M1meets every compressing and∂-compressing disk in M2,and

(2)there is at least one compressing or∂-compressing disk on each side of F.

Definition 2.4Let F be a connected closed surface with g(F)>1.The curve complex of F is the complex whose vertices are the isotopy classes of essential simple closed curves on F,and k+1vertices in this complex determine a k-simplex if they can be represented by pairwise disjoint curves.We denote the curve complex of F by C(F)and denote its 0-skeleton by C(0)(F).

For α,β∈C(0)(F),we de fi ne

dC(F)(α,β)=min{n;there exists a sequence of essential simple closed curves c0,···,cn,

For two subsetswe de fi ne

In this paper,we do not distinguish a vertex inC(F)from a simple closed curve inFrepresenting this vertex,unless otherwise stated.

Definition 2.5Let F be a properly embedded closed bicompressible surface in M.De fi ne

Suppose thatMis connected and∂Mconsists of two boundary componentsF1andF2withLetMfbe the closed orientable 3-manifold obtained by identifyingF1andF2via a homeomorphismand letFbe the surface inMfwhich is the image ofF1andF2after gluing.It is often helpful to viewMas a sub-manifold ofMf,i.e.,M=Mf−int(N(F)),whereN(F)=F×[1,2]is a closed small regular neighborhood ofFinMf,andFican be viewed asF×{i}(i=1,2).

Definition 2.6Let M,F1,F2,f and Mfbe as above.Suppose that F1and F2are both compressible,and let Ui={[α]∈C(Fi);α bounds a disk in M}.By projection F×[1,2]to F,we may view U1and U2as subsets of C(F).Then we define

Definition 2.7Let P and Q be two closed bicompressible separating surfaces in an irreducible and∂-irreducible3-manifold M.Maximally compressing P on both sides and deleting all resulting2-sphere components,we get

where is a compression body withandis a collection(may be empty)ofclose surfaces of genus more than zero for i=1,2.In this case,P is a Heegaard surface of themanifoldSimilarly we defineP and Q are said to be well separated inM if we can have isotopyso that it is disjoint from

3 Some Lemmas

Lemma 3.1(see[4,7])If S is a Heegaard surface of a3-manifold M,andis an essential connected surface,then

Lemma 3.2(see[9])If P and Q are both strongly irreducible connected closed separating surfaces in a3-manifold M,then one of the following holds:

(1)P and Q are well separated,or

(2)P and Q are isotopic,or

(3)

(1)S may be isotopied to be transverse to Q,with every component of S−η(Q)incompressible in the respective sub-manifold of M− η(Q);

(2)S may be isotopied to be transverse to Q,with every component of S−η(Q)incompressible in the respective sub-manifold of M−η(Q),except for exactly one strongly irreducible component;

(3)S may be isotopied to be almost transverse to Q(i.e.,S is transverse to Q except for one saddle point),with every component of S−η(Q)incompressible in the respective sub-manifold of M− η(Q).

In each case,η(Q)is a suitable(open)regular neighborhood of F in M.

ProofWe only prove the final result and the first result is a special case of the final result.

By the construction of amalgamation,we can see that

ThenVis obtained by attaching 1-handles towhereis a closed neighborhood ofif,orN(∂−V)is a 3-ball if∂−V= ∅.LetNbe the number of 1-handles attached toand then

Hence

As an application of the above lemma,we prove the following result which will be used in the proof of Theorem 1.1.

ProofDenote the component ofNkwhich containsFbyand writeWithout loss of generality,we assume thatN(F)is contained inThenM?=Writefori=1,2.ThenandSincea compression body,For the same reason,Hence

whereforandSo

ative toF1,denoted byS?.By Lemma 3.5,

Note thatforandWe getSo

4 Proof of Theorem 1.1

As in[8],the untelescoping of the Heegaard splitting gives a decomposition

Letbe the surface obtained by maximally compressingFiinMand removing all resulting 2-sphere components.Thenbounds a sub-manifoldMFinM,andFis a Heegaard surface ofMF.WritewherecontainsFias in Figure 1.SinceF1andF2are compressible,we have that bothMF1andMF2are nontrivial compression bodies.

Figure 1 MF1and MF2

Recall thatαbounds a disk inand writeαbounds a disk inso then

Claim 4.1EachHican be isotopied to be disjoint withF.

ProofIf not,thenHican not be made disjoint withMFfor somei.Without loss of generality,we assume that each component ofis an essential surface ofMF.Choosing some componentH?ofwe havesoaccording to Lemma 3.1.But it is a contradiction since

By Claim 4.1,F⊂int(Nk)for somek.Denote the component ofcontainingFbyand denoteFori=1,2,writeSinceis incompressible inM,any compressing disk forFcan be isotopied intoNk.So after isotopying,we may assumeThen there are two possibilities forandF:

Case 1can not be isotopied to be disjoint withF.

By Lemma 3.3,we can assume each component ofis incompressible inMFexcept for at most one strongly irreducible component.Furthermore,we assume that each component is not∂-parallel.Then by maximally∂-compressingwe will get at least one connected surface which is either an essential surface or a strongly irreducible and∂-strongly irreducible surface,and we choose such a component,denoted byQ.Then by Lemma 3.1 and Lemma 3.4,we havewhich is a contradiction since

Case 2We can have isotopy

Without loss of generality,we assume thatand thenFmust separate,soFseparatesSinceandFare obviously not well separated andby Lemma 3.2,andFare isotopic.Without loss of generality,we assumeThen by Lemma 3.6,we getwhich is also a contradiction.

From the above,we show thatis impossible,so

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