The∂-Stabilization of a Heegaard Splitting with Distance at Least 6 is Unstabilized∗

Yanqing ZOU Qilong GUO Ruifeng QIU

1 Introduction

LetMbe a compact orientable 3-manifold.If there exists a closed surfaceXwhich cutsMinto two compression bodyVandWso thatX=∂+V=∂+W,then we sayMhas a Heegaard splitting,denoted byM=V∪XW.In this case,Xis called a Heegaard surface,andg(X)is called the genus of the Heegaard splitting.The Heegaard splittingM=V∪XWis said to be stabilized if there exist essential disksBinVandDinWsuch thatBintersectsDin just one point;otherwise,it is said to be unstabilized.M=V∪XWis said to be reducible if there exists an essential simple closed curve onXwhich bounds disks in bothVandW;otherwise,it is said to be irreducible.M=V∪XWis said to be weakly reducible if there exist an essential diskDinVandBinWsuch thatD∩B=∅;otherwise,it is said to be strongly irreducible.M=V∪XWis said to be∂-reducible if there exists an essential disk ofMwhich intersectsXin an essential simple closed curve;otherwise,it is said to be∂-irreducible.The distance of two essential simple closed curvesαandβonX,denoted byd(α,β),is defined to be the smallest integern≥0 so that there exists a sequence of essential simple closed curvesα0=α,···,αn=βonXsuch thatαi−1is disjoint fromαifor 1≤i≤n.The distance of the Heegaard surfaceX,denoted byd(X),is defined to be min{d(α,β)},whereαbounds a disk inVandβbounds a disk inW(see[2,4]).

LetMbe a compact orientable 3-manifold,andFbe a component of∂M.LetM=V∪XWbe a Heegaard splitting.ThenV∪XWinduces another Heegaard splitting ofMcalled the∂-stabilization ofV∪XWas follow.

Without loss of generality,we may assume thatF⊂∂−W.Now there exists an essential diskBwhich dividesWintoF×IandW−F×(0,1).Assume thatF=F×{0}.ThenF×{1}−intBis a sub-surface ofX.Letpbe a point onF,andN(p)be a regular neighborhood ofponFsuch thatN(p)×{1}is disjoint fromB.Now letV∗=V∪N(p)×I∪F×[0,1],andW∗be the closure ofM−V∗.ThenV∗andW∗are two compression bodies such thatHenceis also a Heegaard splitting ofM,called the∂-stabilization ofV∪XW.In this case,g(X∗)=g(X)+g(F)(see[8]).

Now a natural question is the following question.

Question 1.1LetM=Wbe an unstabilized Heegaard splitting,andM=be the∂-stabilization ofV∪XW.IsM=unstabilized?

Remark 1.1IfM=is unstabilized,thenMhas two unstabilized Heegaard splittings with different Heegaard genera.Moreover,this implies a way to find Haken closed 3-manifolds which have unstabilized Heegaard splittings with different Heegaard genera:LetMbe a Haken closed 3-manifold,andFbe a closed incompressible surface which cutsMinto two 3-manifoldsM1andM2with∂Miconnected.Now letbe a Heegaard splitting,andbe the∂-stabilization ofNow if one ofandsayis unstabilized,thenMhas two natural Heegaard splittings,one of which is the amalgamation ofW1andand the other is the amalgamation ofandThus we can only consider if the two amalgamations are unstabilized.Bachman[1]announced a result on this topic.

Scharlemann and Tomova[9]proved that ifM=is a Heegaard splitting,then,for any Heegaard splittingM=eitherorM=is obtained by doing∂-stabilizations and stabilizations fromM=V∪XW.Scharlemann-Tomova theorem implies that ifM=V∪XWhas high distance while∂Mhas at least two components,then the∂-stabilization ofM=V∪XWalong a minimal genus component of∂Mis unstabilized.We know little on Question 1.1 when∂Mis connected except thatMis anI-bundle of a genusgclosed surfaceFg.In this case,the∂-stabilization of the trivial Heegaard splitting ofFg×Iis unstabilized.The main result of this paper is the following theorem.

Theorem 1.1Let M be a compact orientable3-manifold with∂M connected.Then the∂-stabilization of a Heegaard splitting of M with distance at least 6 is unstabilized.Furthermore,M admits two unstabilized Heegaard splittings with different genera.

2 Some Known Results on Arc and Curve Complexes

In this section,we assume thatSis a compact orientable surface of genusgwith at least one boundary component.A simple closed curve inSis said to be essential if it does not bound a disk inSand not parallel to∂S.A properly embedded arc inSis said to be essential if it is not parallel to∂S.

Suppose thatg≥2.Harvey[3]defined the curve complexC(S)as follows:The vertices ofC(S)are the isotopy classes of essential simple closed curves onS,andk+1 distinct verticesx0,x1,···,xkdetermine ak-simplex ofC(S)if and only if they are represented by pairwise disjoint simple closed curves.For two verticesxandyofC(S),the distance ofxandy,denoted by(x,y),is defined to be the smallest integern≥0 so there exists a sequence of verticesx0=x,···,xn=ysuch thatandxiare represented by two disjoint simple closed curves onSfor each 1≤i≤n.For two sets of vertices inC(S),d(X,Y)is defined to be min(x,y)|x∈X,y∈Y.For a Heegaard splittingV∪XWwith genus at least 2,if we denote byAthe isotopy class of essential simple closed curves onXwhich bounds a disk inV,andBthe isotopy class of essential simple closed curves onXwhich bounds a disk inW,thend(X)=d(A,B).Now letSbe a once-punctured torus or a torus.In this case,Masur and Minsky[5–6]defineC(S)as follows:The vertices ofC(S)are the isotopy classes of essential simple closed curves or essential arcs onS,andk+1 distinct verticesdetermine ak-simplex ofC(S)if and only ifandxiare represented by two simple closed curvesandcionSsuch thatintersectsciin just one point for each 1≤i≤k.

Masur and Minsky define the arc and curve complexAC(S)as follows:The vertices ofC(S)are the isotopy classes of essential simple closed curves and essential arcs onS.ThenAC(S)and(x,y)can be defined in the same way withC(S).

In the following argument,we assume thatXis a closed surface of genus at least two andSis a once-punctured subsurface ofXwithg(S)≥1.Sis said to be essential and proper if∂Sdoes not bound a disk onX.Define the mapsκS:C(X)→andσS:AC(S)→C(S)as follows:

Letα∈C(X),andαcbe a simple closed curve in the isotopy classα.αcis tight to∂Sif the geometry intersection number ofαcand∂Sis minimal among all the simple closed curves inα.Now forα∈C(X),andαc∈αwhich is tight toS,letκS(α)=α ∩S.For anyα∈C(X),α′∈σS(α)if and only ifα′is a boundary component of a regular neighborhood ofα∪∂Sand essential.Specially,letNow letWe sayα∈C(X)cutsSifIfα,β∈C(X)both cutS,we write(α,β)=

Lemma 2.1Let S be an essential subsurface of X.Suppose α,β∈C(X)are disjoint in X and both cut S.Then(β))≤2.

ProofThe lemma is immediately from Lemma 2.2 in[6].

SupposeVis a genus at least 2 handlebody with∂V=X.Define disk complexD(V)to be the collection of essential diskD⊂V,up to isotopy.Place an edge between any two verticesD1,D2∈D(V)ifD1andD2can be isotopic to being disjoint inV.LetSbe a once-punctured essential subsurface ofX.Sis called a hole forD(V)if,for anyD∈D(V),∂DcutsS.

A role tool of this paper is the following.

Lemma 2.2Suppose S is a hole for D(V),S⊂∂V.Then for any essential disk D cuts S,there exists an essential disk D′with the following properties:

(1)∂S and∂D′are tight.

(2)If S is incompressible,then D′is not boundary compressible into S and≤3.

(3)If S is compressible,then∂D′⊂S and≤3.

ProofSee the proof of Lemma 11.7 in[7].

3 The Proof of Theorem 1.1

Theorem 3.1Let M be a compact orientable3-manifold with∂M connected.Then the∂-stabilization of a Heegaard splitting of M with distance at least6is unstabilized.Furthermore,M admits two unstabilized Heegaard splittings with different genera.

ProofLetM=V∪XWbe a Heegaard splitting with distance at least 6.Recalling the definition of the∂-stabilization ofV∪XW:

In this case,we may assume thatF=∂M=∂−W.As defined in Section 1,V∗andW∗are two compression bodies such thatAndis also a Heegaard splitting ofM,called the∂-stabilization ofV∪XW.Since∂M=Fis connected,W∗is a handlebody of genusg(X∗)=g(X)+g(F).See Figure 1.

Figure 1 ∂-stabilization

By the definition,∂BcutsX∗into a subsurface ofX,sayS1,and a subsurface ofF×{1},sayS2.See Figure 1.

Claim 3.1S2is incompressible inW∗.

ProofSinced(X)≥6,by definitions in Section 1,V∪XWis strongly irreducible and∂-irreducible.HenceMis irreducible and∂-irreducible(see[2]).This means thatFis incompressible inM.IfS2is compressible inW∗,then∂Bbounds a disk inW∗,sayB′;otherwise,Fis compressible inM.NowB∪B′is a sphere inWsuch thatXandFlie in the two sides ofB∪B′.This means that the compression bodyWis reducible,a contradiction.

Supposeis stabilized.Sincegis a reducible Heegaard splitting.Hence there exists a spherePwhich intersectsX∗in an essential simple closed curve,sayC.ThusCcutsPinto an essential diskD1inV∗and an essential diskE1inW∗.We may assume that|C∩∂B|is minimal among all reducing sphere ofBy Claim 3.1,ifC∩∂B=∅,thenC⊂S1.In this case,by the proof of Claim 3.1,Cis not parallel to∂B.This means thatCis essential onX.This means thatV∪XWis reducible,a contradiction.Hence we have|C∩ ∂B|＞0.

Claim 3.2(1)S1is compressible inW∗.

(2)S1is a hole forD(W∗).

Proof(1)By the definition ofN(p)×Iis disjoint fromW−F×(0,1).HenceS1is compressible inW∗.

(2)LetDbe an essential disk inW∗.By Claim 3.1,S2is incompressible inW∗.Hence either∂D⊂SorDcan be isotoped so that each component of∂D∩S1and∂D∩S2is essential inS1orS2.By the definition,S1is a hole forD(W∗).

Note thatBcutsV∗intoVandF×I.Now consider the two essential disksD1inV∗andD2inW∗.By the minimality ofC∩∂B,each component of∂D∩S1and∂D∩S2is essential inS1orS2.We may assume that each component ofD1∩Bis an arc on bothD1andB.Letabe an outermost component ofD1∩Brelative toD1.This means thata,together with an arc on∂D1,bounds a diskD2such that intD2is disjoint fromB,andD2⊂VsinceFis incompressible inM.Thusa,together with an arc on∂B,does also bound a diskD3inV.Furthermore,∂D3is essential inX.SinceE1is an essential disk ofW∗,by Lemma 2.2,there exists an essential diskE2inW∗such that≤3,and∂E2⊂S1.

By Lemma 2.1,and since∂E2are contained in≤5.Note that∂D1=∂E1=C,≤5.Then≤5.Since both∂D3and∂E2are essential curves inS1,andS1is obtained by removing a diskBfromX,we have that any vertex in the path ofC(S1)connecting∂D3and∂E2is essential inX.≤5.This means thatd(X)≤5,a contradiction.Noware two unstabilized Heegaard splittings with generag(X)andg(X)+g(∂M).

Remark 3.1In fact,Lemma 2.2 is also true whenVis a compression body.By the proof of Theorem 1.1,it is also true when∂Mis not connected.We omit the argument.

Now an interesting question is to determine the sharp lower bound ofd(X),sayb,so that the∂-stabilization ofV∪XWis unstabilized.LetMbe a compact orientable 3-manifold with∂Mconnected,andV∪XWbe a Heegaard splitting ofM.We may assume that∂M=∂−W.V∪XWis said to be primitive if there exist an essential diskDinVand a spine annulus A inWsuch thatDintersectsAin just one point.IfV∪XWis primitive,thend(X)≤2,and the∂-stabilization ofV∪XWis stabilized.Furthermore,there exists primitive Heegaard splittings with distance 2.For example,Morimoto[8]constructed a non-trivial knot whose complement admits a genus two primitive Heegaard splittingV∪XW.Henced(X)=2.In this case,b≥3.So we have the following conjecture.

Conjecture 3.1LetMbe a compact orientable 3-manifold with∂Mconnected.Then the∂-stabilization of a Heegaard splitting ofMwith distance at least 3 is unstabilized.

AcknowledgementThe authors thank Tao Li and Jiming Ma for helpful discussions on this topic.

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