Spanning Pre-disks in a Compression Body
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.
![pagenumber_ebook=84,pagenumber_book=178](https://qkk.lw71.com/uploads/image/2018/2/9b592bd3f3e2e3c8e0696b7245d05e00.png)
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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![pagenumber_ebook=86,pagenumber_book=180](https://qkk.lw71.com/uploads/image/2018/2/c54ff217e4a7858ab402bc19f441005c.png)
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
![pagenumber_ebook=88,pagenumber_book=182](https://qkk.lw71.com/uploads/image/2018/2/9dd3c56c292739c225e75157591ed9db.png)
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:
![pagenumber_ebook=88,pagenumber_book=182](https://qkk.lw71.com/uploads/image/2018/2/4a4d910315f06ab3ceb0bbfd0e73f7b4.png)
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,
![pagenumber_ebook=88,pagenumber_book=182](https://qkk.lw71.com/uploads/image/2018/2/f8ba9efe6406688df4bfbb2e692b5846.png)
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
![pagenumber_ebook=89,pagenumber_book=183](https://qkk.lw71.com/uploads/image/2018/2/2edccc7b3d8ca4ae533f65603bcbc1fc.png)
Thus
![pagenumber_ebook=89,pagenumber_book=183](https://qkk.lw71.com/uploads/image/2018/2/03fddca2fc4b0e695a0081f516968652.png)
This completes the proof of Theorem 1.2.
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