1 Introduction

Let M be an orientable compact 3-manifold.A natural question is whether there exists a properly embedded connected incompressible surface in M with genus g and b boundary components for given g and b.Jaco[1]showed that the answer is positive when b equals to 1 or 2 for the handlebody of genus 2(therefore,for the handlebody of genus n≥2).The examples constructed by Jaco are non-separating in the handlebody.Examples of such separating surfaces in a handlebody were given independently by Eudave-Mu˜noz[2],Howards[3]and Qiu[4].Nogueira and Segerman[5]gave a generalized description of such surfaces in a handlebody with genus at least 2 or a 3-manifold with a compressible boundary component with genus at least 2.

Another question is whether the number of components in a maximal collection of pair-wise disjoint,non-parallel,incompressible surfaces in a compact 3-manifold is bounded.The Kneser-Haken Finiteness Theorem says that this is true if the surfaces are further assumed to be∂-incompressible(for a proof see[1]and[6]).The conclusion is not true if the assumption of the∂-incompressibility for the surfaces is removed.On the other hand,B.Freedman and M.H.Freedman[7]showed that for a given compact 3-manifold if the Betti numbers of surfaces are bounded,then the number of surfaces is bounded.Eudave-Mu˜noz and Shor[8]showed that there is a bound of the number of surfaces depending on the Heegaard genus of 3-manifold and the Betti numbers of surfaces.There are also other results about the embedding of a maximal collection of essential annuli in a handlebody,see[9]–[11].

The pre-disk in a 3-manifold was first introduced by Jaco[12].Let M be a 3-manifold,and J an essential simple closed curve on a boundary component F.An essential planar surface P properly embedded in M is called a pre-disk with respect to J if one boundary component C of P is not coplanar with J,and all other boundary components of P are coplanar with J in F.Jaco showed if∂M −J is incompressible,then there is no properly embedded pre-disk with respect to J in M.A handle addition theorem was given by Jaco as an application of this result.

We consider spanning pre-disks in a compression body.Let V be a nontrivial compression body with ∂−V Ø and J an essential simple closed curve in ∂−V.A properly embedded essential planar surface P(not a disk)in V is called a spanning pre-disk with respect to J,if one boundary component of P is lying in∂+V and all other boundary components of P are lying in ∂−V and coplanar with J.

Let V be a nontrivial compression body and F a component of∂−V.Then we have the following theorem:

Theorem 1.1 Let C be an essential simple closed curve in ∂−V and n a positive integer.If there exists a non-separating essential disk in V or the component of∂−V containing C has genus at least 2,then there is a spanning pre-disk P with respect to C in V such that|∂P|≥ n.

Let be the band sum of DCand D along γ.Then is a non-separating essential disk in VCand ∩D= Ø.So whether C is separating or not,there always exists a non-separating essential disk in VCdisjoint from D.Thus VCis not simple.Let DVCbe the set of boundaries of essential disks of VC.By Lemma 3.1,diamC(∂+V)(DVC)= ∞.So for any positive integer n,there exists an essential disk D′in VCsuch that

Claim 3.3 There is a separating essential disk D1in V such that|∂D1∩D|and|∂D1∩E|are minimal up to isotopty and Fi−∂D1is a collection of disks,where i=1,2.

Let F be a closed orientable surface.If the genus of F is at least 2,then the curve complex of F, first defined by Harvey[13],is the complex whose vertices are the isotopy classes of essential simple closed curves in F,and k+1 vertices determine a k-simplex if they can be represented by pairwise disjoint curves.If F is a torus,then the curve complex of F,defined by Masur and Minsky[14],is the complex whose vertices are the isotopy classes of essential simple closed curves in F,and k+1 vertices determine a k-simplex if they can be represented by a collection of curves any two of which intersect in only one point.

The article is organized as follows.In Section 2,we review some necessary preliminaries.A key lemma is given in Section 3.The proofs of the main results are given in Section 4.

2 Preliminaries

Let V be a nontrivial compression body.A set D of disjoint essential disks in V is called to be a minimal complete collection if V−D is homeomorphic to∂−V×I.Assume that∂−V Ø and C is an essential simple closed curve in ∂−V.Let D be a disk and A a regular neighborhood of C in∂−V.Then there exists a homeomorphism from A to∂D×I.Denote the manifold V∪fD×I by VC.We say that VCis obtained by attaching a 2-handle to V along C.

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Denote the curve complex of F by C(F).For any two vertices x and y in C(F),define the distance dC(F)(x,y)to be the minimal number of 1-simplices in a simplicial path jointing x to y over all such possible paths.Let A and B be two sets of vertices of C(F).The diameter of A,which is denoted by diamC(F)(A),is defined to be max{d(x,y)|x,y∈A}.The distance between A and B,which is denoted by dC(F)(A,B),is defined to be min{d(x,y)|x∈A,y∈B}.

Let F be a compact surface of genus at least 1 with non-empty boundary.Define the arc and curve complex AC(F)as follows:Each vertex of AC(F)is the isotopy class of an essential simple closed curve or an essential properly embedded arc in F,and a set of vertices forms a simplex of AC(F)if these vertices are represented by pairwise disjoint arcs or curves in F.For any two disjoint vertices,define the distance dAC(F)(α,β)to be the minimal number of 1-simplices in a simplicial path jointing x to y over all such possible paths.

A subsurface F′of F is called to be essential if∂F′consists of essential curves in F.Fix a compact essential subsurface F′⊂ F.By the definition of projections to subsurfaces in [14],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′),σF′(β)is the union of the isotopy classes of essential boundary components of the regular neighborhood of β ∪ ∂F ′.It is obvious that if β is the isotopy class of a simple closed curve in F′,then σF′(β)= β.Then we have a map πF′= σF′°κF′from vertices of C(F)to finite subsets of vertices of C(F′).

Lemma 2.1[14](Bounded Geodesic Image Theorem) Let F′be an essential subsurface of F,and γ a geodesic segment in C(F),such that πF′(v) Ø for every vertex v of γ.Then there is a constant M depending only on F so that diamC(F′)(πF′(γ)) ≤ M.

Suppose that N is a compressible boundary component of a compact irreducible orientable 3-manifold M and(S,∂S)⊂ (M,∂M)is an orientable properly embedded essential surface in M in which some essential component is incident to N and no component is a disk.Let D and S denote respectively the sets of vertices in the curve complex for N represented by boundaries of compression disks and by boundary component of S.

Lemma 2.2[15]Suppose that Q is essential in M.Then d(D,S)≤1−χ(S).

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3 In finite Diameter

Let V be a compression body and D denote the set of vertices in the curve complex C(∂+V)represented by boundaries of essential disks.A compression body V is called simple if V has either only one separating or only one non-separating essential disk up to isotopy.We require the following lemma in order to prove the main results:

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Lemma 3.1 Let V be a nontrivial compression body with g(∂+V)≥ 2.If V is not simple,then diamC(∂+V)(D)= ∞.

Proof. If g(∂+V)=2,since V is not simple,then V is a handlebody with genus 2.By Theorem 2.6 in[16],diamC(∂+V)(D)= ∞.

So assume g(∂+V)≥ 3.There are two cases.

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Case 1. There exists at least one non-separating essential disk in V.

Assume that D is a non-separating essential disk in V.Then we have the following claim:

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Claim 3.1 There must exists an essential disk D1in V such that D∩D1Ø up to isotopy.

Next suppose that each essential disk in V is separating.If V is simple,then V has only one separating essential disk.Since g(F)≥2,VCis not simple.So the conclusion holds.

Isotope D1such that|D1∩D|is minimal.Let F= ∂+V−D.Then F∩∂D1is a collection of essential arcs in F.Then we have the following claim:

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Claim 3.2 There is an essential disk D2in V such that dC(∂+V)(∂D1,∂D2) ≥ 3.

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If F−∂D1is a collection of disks,then let D2=D.

Otherwise,since g(∂+V)≥ 3,by Lemma 2.6 in[17]there exists an essential arc γ′in F such that ∂γ′lies in different boundary components of F and dAC(F)(γ′,∂D1∩ F)≥ 3.So F − (∂D1 ∪ γ′)is a collection of disks.Isotope γ′such that|γ′∩ ∂D1|is minimal.Let D2 be the band sum of D and a copy of D along γ′.Then D2is a separating essential disk in V and|∂D1 ∩ ∂D2|is minimal.Otherwise,we can isotope γ′to reduce|γ′∩ ∂D1|,a contradiction.

Assume that D2separates ∂+V into two components T and F′,where T is a regular neighborhood of∂D ∪ γ′in ∂+V.Then

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Since F′−∂D1is a collection of subsurfaces of F −(∂D1∪γ′),F′− ∂D1is a collection of disks.On the other hand,∂D ∩ ∂D1 Ø and γ′∩ ∂D1 Ø,so T − ∂D1is a collection of disks.So ∂+V − (∂D1 ∪ ∂D2)is a collection of disks and dC(∂+V)(∂D1,∂D2)≥ 3.

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Since|∂D1∩∂D2|is minimal and ∂+V −(∂D1∪∂D2)is a collection of disks,∂D1∪∂D2 fills ∂+V.So by the proof of Theorem 2.6 in[16],diamC(∂+V)(D)= ∞.

Case 2. There is no non-separating essential disk in V.

In this case,since V is not simple,V has at least two disjoint separating essential disks which we denote by D and E.Denote the component of∂+V −∂D disjoint from E by F1 and the component of∂+V −∂E disjoint from D by F2.Then we have the following claim:

Theorem 1.2 Let V be a nontrivial compression body with∂−VØ and C an essential simple closed curve in ∂−V.If the collection C is maximal,then

Denote the component of∂+V −(∂D∪∂E)which has two boundary components by S0.Then we can choose an arc γ0in S0connecting ∂D and ∂E.Let D0be the band sum of D and E along γ0.Obviously,D0is a separating essential disk in V.Denote the component of∂+V − (∂D0 ∪ ∂E)which ∂D lies in by S1.By Lemma 2.6 in[17],there is an arc γ1in S1where ∂γ1lie in different boundary components of S1,such that|γ1∩∂D|is minimal up to isotopy and S1−(∂D∪γ1)is a collection of disks.Let D′0be the band sum of D0and E along γ1.It is clear that|∂D′0∩∂D|is minimal and F1−∂D′0is a collection of subsurfaces of S1−(∂D∪γ1).So F1−∂D′0is a collection of disks.

Denote the component of∂+V−(∂D0∪∂D′0)which∂E lies in by SE.Then D∩SEis a collection of parallel arcs with endpoints lying in∂D′0.By Lemma 2.6 in[17],there is an arc γEin SE,where ∂γ1lie in different boundary components of SE,such that|γE ∩ ∂D|and|γE ∩ ∂E|are minimal up to isotopy and SE − (∂E ∪ γE)is a collection of disks.Let D1be the band sum of D′0and D0along γE.Then D1is a separating essential disk in V and|∂D1∩∂D|and|∂D1∩∂E|are minimal.Thus∂D′0∩F1⊂∂D1∩F1.It is clear that F1−∂D1is a collection of subsurfaces of F1−∂D′0and F2−∂D1is a collection of subsurfaces of SE −(∂E ∪γE).So Fi−∂D1is a collection of disks,where i=1,2.

With the similar argument as above,we can choose an arc γ2in S0the endpoints of which lie in different boundary components of S0,such that S0− (∂D1∪ γ2)is a collection of disks and|γ2 ∩ ∂D1|is minimal.Let D2be the band sum of D and E along γ2.Then|∂D1∩∂D2|is minimal and each component of∂+V −(∂D1∪∂D2)is a subsurface of either Fi−∂D1or S0−(∂D1∪γ2).So ∂+V −(∂D1∪∂D2)is a collection of disks.With the same argument as case 1,we can prove that diamC(∂+V)(D)= ∞.

This completes the proof of the lemma.

4 Proofs of the Theorems

We are now equipped to prove the theorems.

Proof of Theorem 1.1 Let DVbe a minimal complete collection of essential disks in V and A a spanning annulus in V such that A∩DV= Ø and A∩∂−V=C.Let VCbe the compression body obtained by attaching a 2-handle to V along C.Then A determines an essential disk in VCwhich we denote by DC.

Assume there exists at least one non-separating essential disk in V.If V is simple,then V has only one non-separating essential disk.By Lemma 3.1 in[18],diamC(∂+V)(D) ≤ 2.If C is non-separating in F,then DCis non-separating in VCand disjoint from D.If C is separating in F,then DCis separating in VCand disjoint from D.So we can choose an arc γ in ∂+V connecting∂DCand ∂D such that

Let C be a collection of mutually disjoint spanning pre-disks with respect to C in V.C is called to be maximal if whenever P is a spanning pre-disk with respect to C with P∩C= Ø,then P is parallel to a component of C in V.Then we have the following theorem:

Isotope D′such that|D′∩V|is minimal.Let P=D′∩V.It is clear that P is a spanning pre-disk with respect to C in V.By Lemma 2.2,

So the conclusion holds in this case.

If V has at least two non-separating essential disks,then DVconsists of non-separating essential disks in V.We can choose one from DVdenoted by D.Let V′=V−(DV−D).Then V′is a compression body with only one non-separating essential disk D.By the argument above,there exists a spanning pre-disk P with respect to C in V′such that|∂P|≥ n.It is clear that P is also a spanning pre-disk with respect to C in V.So the conclusion holds in this case.

Now assume that g(F)≥2.If there exists a non-separating essential disk in V,then the argument above implies the conclusion holds.

Since V is not simple,there must exists another non-separating essential disk which we denote by E such that E is not isotopic to D.If E∩DØ,then let D1=E.Otherwise,we can choose an arc γ in ∂+V −∂E such that∂γ lie in different boundary components of∂+V −∂E and γ∩ D Ø up to isotopy.Let D1be the band sum of E and a copy of E along γ,obviously,D1 ∩ D Ø.

If V is not simple,by the similar argument as above,we can prove the conclusion holds.

This completes the proof of Theorem 1.1.

We are now in a position to prove Theorem 1.2.

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Proof of Theorem 1.2 Let VCbe the compression body obtained by attaching a 2-handle to V along C and γ the co-core of the 2-handle.Then γ is a properly embedded simple arc in VC.Let Pi∈C and Dibe the essential disk determined by Piin VC,where i=1,2.

Claim 4.1 If P1∩ ∂+V is parallel to P2∩∂+V,then P1is isotopic to P2.

Since P1∩∂+V is parallel to P2∩∂+V and VCis a compression body,∂D1= ∂D2and D1 is parallel to D2in VC.So a component A of VC−(D1∪D2)is homeomorphic to D×[1,2]with D ×{i}=Di,where i=1,2.Since γ is simple in VC,γ∩A is a collection of simple arcs.Let β be a component of γ ∩ A.Then one endpoint of β lies in D1and the other one in D2.Otherwise,assume ∂β ∈ D1.Then we can choose a simple closed curve α in D1which cuts a disk Dαfrom D1such that Dα∩γ= ∂β.Since γ∩A is a collection of simple arcs,we can isotope Dαalong β such that Dα∩γ=Ø.So Dαis a compressing disk for P1in V,a contradiction.So P1is isotopic to P2.

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Thus for each P∈C,P is determined by P∩∂+V.So

Thus

This completes the proof of Theorem 1.2.

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