Liyuan MA()

School of Mathematical Sciences,Dalian University of Technology,Dalian 116024,China

E-mail:liyuan ma@126.com

Liang LIANG()

School of Mathematics,Liaoning Normal University,Dalian 116029,China

E-mail:liang liang@aliyun.com

Fengchun LEI()?

School of Mathematical Sciences,Dalian University of Technology,Dalian 116024,China E-mail:fclei@dlut.edu.cn

Abstract In this paper,we will give a sufficient condition for the self-amalgamation of a handlebody to be strongly irreducible.

Key words Heegaard splitting;self-amalgamation;strongly irreducble

Dedicated to Professor Banghe LI on the occasion of his 80th birthday

1 Introduction

LetMbe a compact connected orientable 3-manifold.If there is a separating surfaceFproperly embedded inMsuch thatFcutsMinto two pieces,M1andM2,withF1??M1andF2??M2as the two cutting sections ofF,thenMis called a surface sum ofM1andM2alongF1andF2.In particular,ifMis a surface sum of two compression bodiesVandWsuch that?+V=?+W=S,thenV∪SWis called a Heegaard splitting ofM.Here,Sis called the Heegaard surface and the genus ofSis called the Heegaard genus.A Heegaard splittingV∪SWforMis said to be stabilized if there are essential disksDinVandEinWsuch that|D∩E|=1.

Suppose that a compact orientable 3-manifoldMis a surface sum ofM1andM2alongF1andF2,andofMi,i=1,2.Schultens [13]found a natural Heegaard splitting forMinduced bythis is called an amalgamation ofIn general,an amalgamation of Heegaard splittings is weakly reducible,so it is natural to ask:when is the amalgamation of two Heegaard splittings unstabilized?

In the case whereSis a 2-sphere,Bachman [1]and Qiu-Scharlemann [11]independently proved that the connected sum of unstabilized Heegaard splittings is unstabilized.However,many counterexamples show that an amalgamation of two unstabilized Heegaard splittings might be stabilized wheng(F)≥1(we refer to [2],[6],or [14]).On the other hand,if the factor manifolds have high distance Heegaard splittings,see [5,16],or the gluing map of the surface sum is complicated enough,see [7–9,15],then the amalgamation of Heegaard splittings is unstabilized.

Suppose thatFis a non-separating surface properly embedded inM.LetM1=andF1,F2??M1be two cutting sections ofF.ThenMis called the self surface sum ofM1alongF1andF2.By a similar construction,each Heegaard splitting ofM1will induce a natural Heegaard splitting ofMwhich is called the self-amalgamation ofM1.It has been proven that if the Heegaard splitting is complicated enough,then the self-amalgamation is unstabilized;see [3,17].

In the present paper,we will consider the self-amalgamation of a handlebody.It is a fact that the Heegaard splittings of a handlebody are unique(see [12]),i.e.,each Heegaard splittingV∪FWof a handlebodyHnis isotopic to one which is a finite number of elementary stabilizations of the trivial splittingwhereF0is a surface inHnparallel to?Hn.The main result of the paper is that we give a sufficient condition for such a self-amalgamation to be strongly irreducible.

The rest of this article is organized as follows:in Section 2,we review some necessary preliminaries.The statement and proof of the main result are given in Section 3.

2 Preliminary

Suppose thatMis a compact orientable 3-manifold.A Heegaard splittingV∪SWforMis said to be reducible if there are essential disksDinVandEinWsuch that?D=?E;otherwise,it is called irreducible.If there are essential disksDinVandEinWsatisfying?D∩?E=?,thenV∪SWis said to be weakly reducible;otherwise,it is strongly irreducible.

SupposeFis a compact orientable surface.A simple closed curve inFis said to be essential if it is neither contractible nor?-parallel inF.A subsurfaceF′ofFis said to be essential inFif each component of?F′is essential inF.

The curve complex ofF,first introduced by Harvey [4],is defined as follows:each vertex inC(F)is the isotopy class of an essential simple closed curve inFand(k+1)vertices determine ak-simplex if they can be realized by pairwise disjoint curves.Masur and Minsky [10]extended the definition to the case in whichFis a torus or once-punctured torus.

For simplicity,we do not distinguish an essential simple closed curve or its isotopy class.For any two verticesαandβinC(F),dC(F)(α,β)is the minimal number of 1-simplices among all possible simplicial paths jointingαtoβ.For any two sets of simple closed curvesAandBinC(F),the distance betweenAandB,denoted bydC(F)(A,B),isdefined to be min {dC(F)(α,β)|?α∈A,?β∈B}.

Suppose thatHis a handlebody withg(H)≥1.LetF1andF2be two disjoint homeomorphic subsurfaces in?Hwhere each component of?F1and?F2is essential in?H.LetMbe the 3-manifold obtained fromHby gluingF1andF2together along a homeomorphismf.Denote thatdi=dC(?H)(D,Ci),where D is the set of boundaries of essential disks inHand Ciis the set of simple closed curves inFithat are essential in?Hfori=1,2.Then we have the following proposition.

Proposition 2.1LetMbe the self surface sum ofHalongF1andF2.Ifdi≥2,then?Mis incompressible inM.

ProofOtherwise,suppose that?Mis compressible inMand thatDis a compressing disk.Denote the surfaceF1=F2byFinM.IsotopeDsuch that|D∩F|is minimal.

First weconsider the caseg(H)>1.IfD∩F=?,thenDisan essential disk inHsatisfyingD∩Fi=?fori=1,2.Thus,for any essential simple closed curveCi∈Ci,Ci∩?D=?wherei=1,2,sodC(?H)(D,Ci)≤dC(?H)(?D,Ci)≤1,which is a contradiction.

ThusD∩F? up to isotopy.The surfaceFis incompressible inM.Otherwise,suppose that there is a compressing diskDFforFand?DF∩?H?F1.ThenDFis a compressing disk forF1inH.This means that there is an essential simple closed curveCinF1such thatC=?D.ThusdC(?H)(D,C1)≤dC(?H)(?D,C)=0,which is a contradiction.ThereforeFis incompressible inM.

An innermost closed curveargument impliesthat there is no simple closed curvecomponent inD∩F,soD∩Fconsists of arcs.Choose an arcαinD∩Fsuch thatαis outermost inD.Thenαcuts a diskDαfromDsuch thatDα∩F=α.Dαis essential inH.Otherwise,we can isotopeDto reduce|D∩F|,which is a contradiction.Suppose thatDα∩F1=α.Ifαis inessential inF,then we isotopeDαsuch thatDα∩F1=?.A similar argument as to that above implies thatdC(?H)(D,C1)≤1,which is a contradiction.Thusαis essential inFandF1.IfF1ia an annulus,then there is a core curveCinF1satisfyingC∈C1and|C∩α|=1.It is clear thatC∩D=C∩α.Then|C∩D|=1 and?N(C∪D)bounds an essential disk inHwhich is disjoint fromC,whereN(C∪D)is the regular neighborhood ofC∪DinH.This implies thatdC(?H)(D,C1)≤1,which is a contradiction.IfF1is not an annulus,then there always exists a curveCinF1satisfyingC∩α=?andC∈C1.SinceDα∩F1=α,C∩D=?.SodC(?H)(D,C1)≤1,which is a contradiction.Thus the proposition holds in this case.

For the caseg(H)=1,a similar argument implies that the conclusion holds.

This completes the proof of the proposition. □

There is a Heegaard splitting forMcalled the self-amalgamation constructed as follows:

LetH=?H×I,whereV1is homeomorphic toHand?H=?H×{0}.Choose a pointp1inF1and a pointp2inF2such thatf(p1)=p2.Letαi=pi×Ibe a spanning arc in?H×I.Denote the surfaceF1=F2byFand the pointp1=p2bypinM.Letα=andN(α)=N(α1)∪N(α2)be the regular neighborhood ofαinMwhereN(αi)=αi×Dis the regular neighborhood ofαiin?H×Ifori=1,2.LetDi=pi×{1}×DandD=p×D.ThenN(α)∩S1=D1∪D2andN(α)∩F=D.LetV=V1∪N(α)andW=.ThenV∪SWis a Heegaard splitting ofMcalled the self-amalgamation ofH.

3 Main Result and Proof

Assume that the Heegaard splittingV∪SWis weakly reducible.Choose an essential diskBinVandEinWsuch thatB∩E=?and|E∩F|is minimal among all such pairs of disks.Furthermore,we can isotopeBsuch that|B∩D|is minimal andB∩E=?.

Assume thatγis an outermost arc inE∩F0and thatγcutsEγfromEjust the same as above.Thenγis essential inF0andEγis isotopic toγ×I.Choose a component of?D?γ,which we denote byδ.ThenEγ∪δ×Iis a spanning annulus in?H×I.It is clear thatγ∪δis an essential simple closed curve inF1.LetC=(Eγ∪δ×I)∩S1.ThenCis essential inS1andC=f1(γ∪δ)∈f1(C1).We have the following claim:

Claim 4There is an essential disk inV1which is disjoint fromC.

IfB∩D=?,thenBis an essential disk inV1,and it is clear thatB∩C=?.The claim holds in this case.

IfB∩D?,then an innermost closed curve argument implies thatB∩Dconsists of arcs.Choose an arcξinB∩Dsuchξis outermost inB.Thenξcuts a diskB0fromBsuch thatB0∩D=ξ.Since|B∩D|is minimal,B0is an essential disk inV1.IfB0∩F1=?,thenB0∩C=?,and the claim holds.

Now assume thatB0∩F1?.If? ξlie inδorthen after isotopyB0∩C=?,and the claim holds.If one component of? ξlies inδand the other one lies inthen after isotopy,|B0∩C|=1.In this case,?N(B0∪C)bounds an essential disk which is disjoint fromC,so the claim holds.

For the caseg(H)=1,a similar argument as to that above implies that the conclusion holds.

This completes the proof of the theorem. □

Remark 3.3In fact,d1andd2can be viewed as a description of the complexity for a selfamalgamation.LetF1andF2be disjoint essential homeomorphic subsurfaces in?H.If there is an essential diskDinHsuch thatD∩Fi=?,thendi≤1,wherei=1,2.By the construction ofV∪SW,we can choose an essential arcγ1?F1andγ2?F2satisfyingf(γ1)=γ2.LetE=It is clear thatEis an essential disk inWthat andDis an essential disk inVsatisfying that?D∩?E=?.Therefore in this case the self-amalgamationV∪SWis weakly reducible.

Moreover,suppose thatg(H)=2.There exist two disjoint two essential simple closed curvesC1andC2in?Hsatisfying the following conditions:

(1)?H?C2is incompressible;

(2)there exists an essential diskDinHsuch that|C1∩D|=1 and|C2∩D|=2;see Figure 2.

Figure 1 Self-amalgamation of a handlebody

Figure 2

Those two examples show that the self-amalgamation of a handlebody might be stabilized ifdi≤1 fori=1 or 2.Hence the conditions in Theorem 3.2 are optimal in sense of the complexity of the self-amalgamation.

IfFiis an annulus,then there is only one essential simple closed curve inFiup to isotopy.Thus we have the following corollary.

Corollary 3.4LetHbe a handlebody withg(H)≥1 and letF1andF2be two disjoint homeomorphic essential annuli in?H.If?H?Fiis incompressible fori=1,2,then the self-amalgamationV∪SWis strongly irreducible.

By the definition of curve complex,we have the following corollary.

Corollary 3.5LetHbe a handlebody withg(H)≥1 andF1,F2two disjoint homeomorphic essential subsurfacesin?H.IfdC(?H)(D,?Fi)≥3 fori=1,2,then the self-amalgamationV∪SWis strongly irreducible.

With a similar argument,we can extend the results to compression bodies.

Corollary 3.6Suppose thatCis a compression body withg(?+C)≥1 and thatF1andF2aretwo disjoint homeomorphic essential subsurfacesin?+C.Let DCbetheset of boundaries of essential disks inC.IfdC(?+C)(DC,Ci)≥2 fori=1,2,then the self-amalgamation ofCalongF1andF2is strongly irreducible.