1 Introduction

Let M be a compact and orientable 3-manifold and F a component of∂M.For an essential simple closed curve J in F,let MJbe the manifold obtained by attaching a 2-handle to M along J and filling in the resulting possible 2-sphere with a 3-ball.Such an operation is called a handle addition to M along J.If∂MJis incompressible in MJ,we say that the handle addition(as well as,J)is∂-irreducible.

Przytycki[1] first gave a sufficient condition for a handle addition to a handlebody to be∂-irreducible,Jaco[2]then generalized it to obtain the well-known handle addition theorem.Since then many generalizations have been given(see[3]–[6]).

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In[7],Masur and Minsky introduced the subsurface projection to study the structure of the curve complex.They showed the diameter of the image of a geodesic under the projection is bounded by a constant M,and M only depends on the genus of the surface.Li[8],Masur and Schleimer[9]independently gave an estimation of the diameter of the image of the essential disks in a compact boundary reducible 3-manifold under the projection.

Let V be a nontrivial compression body which is not simple.An essential simple closed curve J in ∂+V is called weakly disk-busting if∂+V −J has only one compressing disk up to isotopy.In this paper,we give an upper bound of the diameter of the image of boundaries of essential disks in V under any projection determined by a weakly disk-busting curve.Moreover,we give a sufficient condition for the handle additions to be boundary irreducible,which can be seen as an extension of Theorem 2.2 of[4].

The article is organized as follows.In Section 2,we review some necessary preliminaries.In Section 3,we give the main results and proofs.

2 Preliminaries

Let M be a 3-manifold and F a properly embedded surface which is not a 2-sphere.F is called compressible if F is a disk parallel to the boundary of M or there exists a disk D such that D∩F= ∂D and∂D is essential in F.In the second case,we call D a compressing disk for F.

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For any two vertices α,β in C(F),the distance between α and β,denoted by dC(F)(α, β),is defined to be the minimal number of 1-simplices in all possible simplicial paths connecting α to β.The simplicial path realizes the distance between α and β is called a geodesic.Let A and B be any two sets of vertices in C(F).The diameter of A,denoted by diamC(F)(A),is defined to be max{d(x,y)|x,y∈A}.The distance between A and B,denoted by dC(F)(A,B),is defined to be min{d(x,y)|x∈A,y∈B}.

Let V be a nontrivial compression body and S a subsurface of∂+V.Then S determines a projection πSfrom C(∂+V)to C(S).If πS(∂D) Ø for any essential disk D in V,then S is called a hole for V.If S is a hole for V which is compressible in V,then S is called a compressible hole for V.Masur and Schleimer[9]showed that:

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and

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For any two vertices α1,α2 ∈ C(F),if πF′(αi) Ø for i=1,2,then

Let F′be a subsurface of F such that each component of∂F′is essential in F.By the definition of projections to subsurfaces in[7],there is a natural map κF′from vertices of C(F)to finite subsets of vertices of AC(F′)defined as follows:For every vertex[γ]in C(F),take a curve γ in the isotopy class such that|γ ∩ F′|is minimal.If γ ∩ F′= Ø,then κF′([γ])= Ø.If γ ∩ F′ Ø,then κF′([γ])is the union of the isotopy classes of essential components of γ ∩ F ′.Furthermore,there is a natural map σF′ from vertices of AC(F′)to finite subsets of vertices of C(F ′):For every vertex[β]in AC(F ′),if[β]is the isotopy class of an essential simple closed curve in F ′,then σF ′([β])=[β];if[β]is the isotopy class of an essential arc,then σF′([β])is the union of the isotopy classes of essential boundary components of the regular neighborhood of β ∪ ∂F′.Then πF′= σF′° κF′is a map from vertices of C(F)to finite subsets of vertices of C(F′).

where i=1,2.

The following disk image theorem is proved by Li[8],Masur and Schleimer[9]independently.

Lemma 2.1 Let M be a compact orientable and irreducible 3-manifold and F a component of∂M.Suppose that∂M −F is incompressible in M.Let D be the disk complex for F.Let S be a compact connected subsurface of F and suppose every component of∂S is disk-busting.Then either M is an I-bundle over a compact surface,S is a component of the horizontal boundary of this I-bundle,and the vertical boundary of this I-bundle is a single annulus,or diamC(S)(πS(D))≤ 12.

Let F be a compact orientable surface of genus at least 1 with nonempty boundary.Denote the arc and curve complex of F by AC(F).Vertices of AC(F)are isotopy classes of essential arcs or curves in F and(k+1)vertices determine a k-simplex if they can be represented by pairwise disjoint arcs or curves.The distance between two vertices α, β,denoted by dAC(F)(α,β),is defined to be the minimal number of 1-simplices in a simplicial path jointing α to β over all such possible paths.

Lemma 2.2 Let V be a nontrivial compression body and S a compressing hole for V.Then for any essential disk D in V there is a compressing disk D1for S in V such that

3 Main Results

and the conclusion holds in this case.

Theorem 3.1 Suppose that V is not simple and J is a weakly disk-busting curve in∂+C.If VJis not simple,then diamC(Si)(πi(D))is at most 20.

Proof. If J is not separating in∂+V,then

So we denote∂+V −J by S.By Lemma 2.2,

Let V be a non-trivial compression body with g(∂+V) ≥ 2 and J a weakly disk-busting curve in∂+V.If V has either only one separating or only one nonseparating essential disk up to isotopy,then V is called simple.Denote the components of∂+V −J by S1and S2(maybe S1=S2).Let πibe the projection from C(∂+V)to C(Si)determined by J and D the set of vertices in C(∂+V)represented by boundaries of essential disks in V.Then we have the following theorem:

Suppose that J is separating in ∂+V and J cuts∂+V into two components S1and S2.Since J is weakly disk-busting,one of S1and S2,say S1,has only one separating compressing disk in V.Denote the separating compressing disk of S1by D.

Hence

Since∂D is separating in S1and J is separating in ∂+V,D is separating in V.Denote the component of V −D which J lies in by V′.Then J cuts∂+V′into two components one of which is S2.Since V is not simple,V′is a nontrivial compression body.Since J is weakly disk-busting in V,each component of∂+V ′− J is incompressible in V ′.

Let D′be the set of vertices in C(∂+V ′)represented by boundaries of essential disks in V′.Since VJis not simple,which is obtained from V′by attaching a 2-handle along J is not a trivial I-bundle of a closed orientable surface.So,by Lemma 2.1,

Since D ∩S2= Ø,π2(∂D)= Ø.Let D0be an essential disk in V which is not isotopic to D.If D0∩D= Ø,then D0is an essential disk in V ′.Otherwise,D0is a compressing disk for S1in V,a contradiction.So

Suppose D0∩D Ø,isotope D0such that|D0∩D|is minimal and|∂D0∩J|is minimal.An innermost closed curve argument implies that D0 ∩ D consists of arcs.Choose an arc γ from D0 ∩ D such that γ is outermost in D0.Then γ cuts an disk Dγ from D0such that D∩Dγ=γ.Suppose γ cuts D intoand .Let Di=∪γDγ,where i=1,2.

Since S1has only one compressing disk D in V,both D1and D2must be essential disks in V ′.Otherwise,we can isotope D0to reduce|D0∩D|,a contradiction.Since ∂+V ′−J is incompressible in V ′,∂Di∩ J Ø up to isotopy.

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Let β = ∂D0 ∩ ∂Dγ.Then β ∩ ∂Di= Ø for i=1,2.Since|∂D0 ∩ J|is minimal,|β ∩ J|is minimal.If β∩J= Ø,which means Dγ∩J= Ø,then Di∩J= Ø,a contradiction.So

Isotope D1such that|∂D1 ∩ S2|is minimal and ∂D1∩ β = Ø.Let β′be a component of β∩S2.Then

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Thus

and

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where N(β′∪ ∂S2)is the regular neighborhood of β′∪ ∂S2in S2.So

Since D1is an essential disk in V ′,π2(∂D1)⊂ π2(D′).So

For any α1,α2 ∈ π2(D),there are essential disks Dαiin V such that DαiD and αi ∈π2(∂Dαi)for i=1,2.Since dC(S2)(π2(∂Dαi),π2(D′))≤ 2,

Suppose that F is a compact orientable surface of genus at least 1.The curve complex of F, first introduced by Harvey[10],is defined as follows:each vertex is the isotopy class of an essential simple closed curve in F and(k+1)vertices determine a k-simplex if they can be realized by pairwise disjoint curves.When F is a torus or once-punctured torus,the curve complex of F,defined by Masur and Minsky[7],is the complex whose vertices are isotopy classes of essential simple closed curves in F,and(k+1)vertices determine a k-simplex if they can be realized by curves which mutually intersect in only one point.Denote the curve complex of F by C(F).

Since S2is incompressible in V,each essential disk in V has nonempty intersection with S1.So S1is a compressible hole with only one separating compressing disk D.Thus,by Lemma 2.2,

Thus the conclusion holds.

Suppose that M is a compact orientable 3-manifold and J2is an essential simple closed curve in∂M.An essential simple closed curve J1in∂M −J2is called coplanar with J2if J1cuts a planar surface from∂M−J2.Let P be a planar surface properly embedded in M such that∂P⊂∂M−J2.P is called a pre-disk with respect to J2if one component C0of∂P is not coplanar with J2and each component of∂P −C0is coplanar with J2.Then we have the following theorem:

Theorem 3.2 Let V be a nontrivial and non-simple compression body and J1,J2two disjoint essential simple closed curves in ∂+V such that∂+V −(J1∪J2)is incompressible in V.If J2is weakly disk-busting and J1is not coplanar with J2,then∂VJ1,J2is incompressible in VJ1,J2.

Proof. First,we consider the case J2is separating in∂+V.Since J2is weakly disk-busting,there is a unique compressing disk for∂+V −J2in V which we denote by D.Denote the component of V − D which J2lies in by V ′.Then we have the following claim:

Claim 3.1 There is no incompressible pre-disk with respect to J2in V.

Otherwise,we can choose an incompressible pre-disk P with respect to J2such that|P ∩D|is minimal up to isotopy.Denote the component of∂P which is not coplanar with J2by J.

Suppose P ∩D=Ø.Since J is not coplanar with∂D,P is a pre-disk with respect to J2 in V ′.Since J2is weakly disk-busting,∂+V ′− J2is incompressible in V ′.So by Lemma 1 of[2],there is no pre-disk with respect to J2in V′,a contradiction.Thus

Since|P∩D|is minimal,an innermost closed curve argument implies that P∩D consists of arcs.Choose an arc γ from P ∩ D such that γ is outermost in D.Then ∂γ lies in J and γ cuts a disk Dγfrom D such that Dγ∩P= γ.Denote two components of P −γ byLet Pi=where i=1,2.Denote the component of∂Piwhich does not belong to∂P by Ci,where i=1,2.Then at least one of C1and C2is not coplanar with J2.Otherwise,suppose that both C1and C2are coplanar with J2.Since J can be obtained by a band sum of C1and C2,J is coplanar with J2or J is inessential in∂+V,a contradiction.So at least one of P1and P2is a pre-disk with respect to J2.Since|Pi∩D|＜|P∩D|,this contradicts to the assumption|P∩D|is minimal.So the claim holds.

Since J2is separating in ∂+V,∂D is separating in ∂+V −J2.By the claim above,there is only one separating essential disk D in VJ2.Since∂+V −(J1∪J2)is incompressible and J2∩∂D= Ø,J1∩∂D Ø.So ∂VJ2−J1is incompressible in VJ2.Thus,by Theorem 2 of[2],∂VJ1,J2is incompressible in VJ1,J2.Therefore the conclusion holds in this case.

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If J2is non-separating in∂+V,a similar argument implies the conclusion holds.

References

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