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Reducible handle additions to weakly reducible Heegaard splittings
Topology and its Applications 242 (2018) 33–42
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Topology and its Applications www.elsevier.com/locate/topol
Reducible handle additions to weakly reducible Heegaard splittings Liang Liang a,1 , Fengling Li b,∗,2 , Jingyan Li c a b c
School of Mathematics, Liaoning Normal University, Dalian 116029, China School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China Department of Mathematics and Physics, Shijiazhuang Tiedao University, 050000, China
a r t i c l e
i n f o
Article history: Received 24 February 2018 Accepted 13 April 2018 Available online 24 April 2018 MSC: primary 57N10 secondary 57M50
a b s t r a c t Let V ∪S W be an irreducible Heegaard splitting of M and F a component of ∂− V . A simple closed curve J in F is called reducible if MJ = VJ S W is reducible, where MJ is obtained by attaching a 2-handle to M along J. In this paper, we give a sufficient condition for the diameter of the set of reducible curves in F to be bounded in C(F ) for the keen weakly reducible Heegaard splittings. Moreover, an upper bound of the diameter of the set of the reducible curves in F is given under some circumstance for the weakly reducible Heegaard splittings. © 2018 Elsevier B.V. All rights reserved.
Keywords: Handle addition Curve complex Subsurface projection Keen Heegaard splitting
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 MJ be 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. When F is a torus, it is just a Dehn filling and J is the slope of the filling. For a Heegaard splitting V ∪S W of M with an essential simple closed curve J lying in a component F ⊂ ∂− V , VJ ∪S W is a Heegaard splitting for MJ . If V ∪S W is irreducible and VJ ∪S W is reducible, we say that the handle addition (as well as, J) is reducible; otherwise, it is irreducible. If V ∪S W is ∂-irreducible and VJ ∪S W is ∂-reducible, we say that the handle addition (as well as, J) is ∂-reducible; otherwise, it is ∂-irreducible. * Corresponding author. 1 2
E-mail addresses: [email protected] (L. Liang), dutlfl@163.com (F. Li), [email protected] (J. Li). Supported in part by grant No. 11601209 of NSFC, Research Foundation for Doctor of Liaoning Province (No. 201601239). Supported in part by grants (No. 11671064 and No. 11471151) of NSFC.
https://doi.org/10.1016/j.topol.2018.04.008 0166-8641/© 2018 Elsevier B.V. All rights reserved.
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Przytycki [18] first gave a sufficient condition for a handle addition to a handlebody to be ∂-irreducible, Jaco [9] then generalized it to obtain the well-known handle addition theorem. Since then many generalizations have been given [2,19,11,12]. Let M be a hyperbolic 3-manifold and F a torus boundary component of M . The Geometrization Conjecture and Thurston’s hyperbolic Dehn surgery theorem show that there are only finitely many exceptional (including reducible and ∂-reducible) slopes in F such that the resulting manifolds are not hyperbolic. On the other hand, if g(F ) ≥ 2, there are many known examples such that infinitely many distinct handle additions yield non-hyperbolic manifolds. However, Scharlemann and Wu [22] showed that only finitely many handle additions yielding non-hyperbolic manifolds are basic. In 2001, Hempel [6] introduced the concept of the distance of a Heegaard splitting. He showed that there exist arbitrarily high distance Heegaard splittings for closed 3-manifolds, and a 3-manifold M admitting a distance at least 3 Heegaard splitting is hyperbolic. Since then, many properties related to the distance of Heegaard splittings, as well as 3-manifolds, have been obtained, see [4,1,21,10]. In [16], 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 [13], Masur and Schleimer [17] independently gave an estimation of the diameter of the image of the essential disks in a compact boundary reducible 3-manifold under the projection. By using the subsurface projection, Ido, Jang and Kobayashi [7] and Qiu, Zou and Guo [20], independently, showed that for any positive integers n and g ≥ 2 there exist closed 3-manifolds admitting a Heegaard splitting of genus g with exact distance n. In [8], Ido, Jang and Kobayashi introduced the concept of keen Heegaard splitting. A Heegaard splitting V ∪S W is called keen if its Hempel distance is realized by a unique pair of essential disks on the opposite sides of S. Moreover, they proved the existence of the strongly keen Heegaard splitting of genus g with distance n for each n ≥ 2 and g ≥ 3. In [3], E proved the existence of keen weakly reducible Heegaard splitting of genus g with g ≥ 3. In this paper, we give a method to obtain keen weakly reducible Heegaard splittings with disjoint separating essential disks on the opposite sides. Theorem 1.1. For any positive integers g, g1 , g2 with g > g1 + g2 , there is a keen weakly reducible Heegaard splitting V ∪S W with genus g such that ∂− V has only one component with genus g1 and ∂− W has only one component with genus g2 . Let V ∪S W be an irreducible Heegaard splitting, F a component of ∂− V , and J an essential simple closed curve in F . Let J be the set of reducible curves in F . In [14,15], we gave some sufficient conditions for the diameter of J to be bounded when V ∪S W is strongly irreducible. Recently, Zou [23] gave an upper bound on distance degenerating handle additions for the Heegaard splittings with distance at least 3. Suppose V ∪S W is an irreducible and weakly reducible Heegaard splitting. Let DF be an essential disk in V such that a component of V − DF is homeomorphic to F × I which we denote by V . Denote the other component of V −DF by V . Let S1 = ∂+ V ∩S and S2 = S −S1 . Then DF determines a projection πSi from vertices of C(S) to finite subsets of vertices of C(Si ) where i = 1, 2. Let f1 be the natural homeomorphism from S1 ∪ DF to ∂+ V and f2 the natural homeomorphism from S2 ∪ DF to F . Let fi = fi ◦ πSi . Then f1 is a projection from C(S) to C(∂+ V ) and f2 is a projection from C(S) to C(F ). A compression body W is called simple if W has either only one separating or only one non-separating essential disk up to isotopy. Let DW be the set of vertices in C(S) represented by boundaries of essential disks in W . If f1 (DW ) = ∅, then for each essential disk E in W , f1 (∂E) = ∅ which means ∂DF ∩ ∂E = ∅. If E is non-separating in W , then there must exist an essential disk E in W which is a band sum of E and a copy of E such that ∂E ∩ ∂DF = ∅ up to isotopy. So f1 (∂E ) = ∅, a contradiction. A similar argument as above implies that there is at most one separating essential disk in W up to isotopy. Since V ∪S W is weakly reducible, W is a nontrivial compression body. So W is a simple compression body
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with only one separating essential disk, which we denote by DS . In this case, f2 (∂DS ) is an essential simple closed curve in F which is the only reducible curve in F . Suppose W is not simple, then f1 (DW ) = ∅. If V is not a trivial compression body, then define d = dC(∂+ V ) (f1 (DW ), DV ), where DV is the set of vertices in C(∂+ V ) represented by boundaries of essential disks in V . If V is a trivial compression body, define d = ∞. The following is another main theorem of the paper. Theorem 1.2. Let V ∪S W be an irreducible keen weakly reducible Heegaard splitting where W is not simple and F a component of ∂− V . If d ≥ 1, then for each J ∈ J , dC(F ) (J, f2 (DW )) ≤ 1. In addition, if W∂DF is not a surface product I-bundle or a simple compression body, then diamC(F ) (J ) is bounded. For a general irreducible and weakly reducible Heegaard splitting, we have Theorem 1.3. Let V ∪S W be an irreducible weakly reducible Heegaard splitting where W is not simple and F a component of ∂− V . If d ≥ 1, then for each J ∈ J , dC(F ) (J, f2 (DW )) ≤ 1. If further W∂DF is not a surface product I-bundle or a simple compression body, we have (1) If S1 and S2 are incompressible in W , then diamC(F ) (f2 (DW )) ≤ 12 and diamC(F ) (J ) ≤ 14. (2) If S1 has only one compressing disk in W up to isotopy and S2 is incompressible in W , then diamC(F ) (f2 (DW )) ≤ 20 and diamC(F ) (J ) ≤ 22. (3) If S1 is incompressible in W and S2 has only one compressing disk in W up to isotopy, then diamC(F ) (f2 (DW )) ≤ 10 and diamC(F ) (J ) ≤ 12. (4) If both S1 and S2 have only one compressing disk in W up to isotopy, then diamC(F ) (f2 (DW )) ≤ 18 and diamC(F ) (J ) ≤ 20. The article is organized as follows. In section 2, we review some necessary preliminaries. In section 3, we give the construction and some properties of keen weakly reducible Heegaard splittings. The proofs of the main results are given in section 4. 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 embedded in M such that D ∩ F = ∂D and ∂D is essential in F . In the second case, we call D a compressing disk for F . A Heegaard splitting V ∪S W is called reducible (weakly reducible) if there are essential disks D and E in V and W respectively such that ∂D = ∂E (∂D ∩ ∂E = ∅). Otherwise, the Heegaard splitting is irreducible (strongly irreducible). A reducible Heegaard splitting is clearly weakly reducible, see [2]. Suppose F is a compact orientable surface of genus at least 1. A simple closed curve c in F is called essential if c does not cut a disk or annulus component from F . If F is not a torus or once-punctured torus, the curve complex of F , first introduced by Harvey [5], 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 [16], 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 ). 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
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paths connecting α to β. Any 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}. Suppose V ∪S W is a Heegaard splitting of M . The distance of V ∪S W is defined to be dC(S) (DV , DW ) where DV and DW are sets of vertices in C(S) represented by boundaries of essential disks in V and W , respectively. 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 joining α to β over all such possible paths. 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 [16], there is a natural projection 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 = ∅, then κF ([γ]) = ∅. If γ F = ∅, then κF ([γ]) is the union of the isotopy classes of essential components of γ F . Furthermore, there is a natural projection 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 projection from vertices of C(F ) to finite subsets of vertices of C(F ). For any two vertices α1 , α2 ∈ C(F ), if dC(F ) (α1 , α2 ) ≤ 1 and πF (αi ) = ∅ for i = 1, 2, then dC(F ) (πF (α1 ), πF (α2 )) ≤ 2 and diamC(F ) (πF (αi )) ≤ 2 where i = 1, 2. The following Disk Image Theorem is proved by Li [13], Masur and Schleimer [17] independently. Lemma 2.1. Let M be a compact orientable and irreducible 3-manifold and F a component of ∂M . Suppose ∂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 V be a nontrivial compression body and S a subsurface of ∂+ V such that each component of ∂S is essential in ∂+ V . Then S determines a projection πS from C(∂+ V ) to C(S). If πS (∂D) = ∅ 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 [17] showed that: 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 D1 for S in V such that dAC(S) (πS (∂D), ∂D1 ) ≤ 3. 3. Keen weakly reducible Heegaard splittings In this section, we give a method to obtain keen weakly reducible Heegaard splittings. Proof of Theorem 1.1. Let S be a closed connected orientable surface of genus g. For any positive integer g1 with g1 < g, there exists a separating essential simple closed curve J1 in S such that a component of S − J1 is a once-punctured surface of genus g1 . Denote the other component of S − J1 by S1 . For any positive integer g2 with g2 < g − g1 , there also exists a separating essential simple closed curve J2 in S1 such that a component of S1 − J2 is a once-punctured surface of genus g2 . Then one component of S − J2 is a once-punctured surface of genus g2 , denote the other component of S − J2 by S2 .
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Let V1 be the compression body obtained from S × I by attaching a 2-handle along J1 . Since J1 is separating in S, there is only one essential disk in V1 up to isotopy which we denote by DJ1 . Let F be the component of ∂− V1 homeomorphic to S1 ∪ DJ1 . Let fS1 be the natural homeomorphism from S1 ∪ DJ1 to F . Then fS1 (J2 ) is an essential simple closed curve in F . By Theorem 2.7 in [6], there is a full simplex X in C(F ) such that dC(F ) (DHX , fS1 (J2 )) ≥ 2, where HX is handlebody obtained from F × I by attaching a collection of 2-handles along X and 3-handles to cap off the possible 2-spheres and DHX is the set of vertices in the curve complex C(F ) represented by boundaries of essential disks in HX . Let V = V1 ∪ HX . It is clear that V is a compression body and ∂− V has only one component with genus g1 . Then we have the following claim: Claim 1. There is only one compressing disk for S2 in V . Proof. Since J1 = ∂DJ1 lies in S2 , DJ1 is a compressing disk for S2 in V . Let F1 = ∂− V . Then DJ1 separates V into F1 × I and a handlebody which we denote by H. In fact, H is homeomorphic to HX . Since dC(F ) (DHX , fS1 (J2 )) ≥ 2, ∂H − J2 is incompressible in H. Suppose there is another compressing disk D for S2 in V which is not isotopic to DJ1 . Then we can isotope D and DJ1 such that |D ∩ DJ1 | is minimal. If D ∩ DJ1 = ∅, then D is an essential disk in H which is disjoint from J2 . So ∂H − J2 is compressible, a contradiction. So D ∩ DJ1 = ∅. Since |D ∩ DJ1 | is minimal, an innermost closed curve argument implies that D ∩ DJ1 consists of arcs. Choose an arc γ from D ∩ DJ1 such that γ is outermost in D. Then γ cuts a disk Dγ from D such that Dγ ∩ DJ1 = γ. Denote the components of DJ1 − γ by D1 and D2 . Let Di = Di ∪γ Dγ where i = 1, 2. Both D1 and D2 are essential disks disjoint from DJ1 in V . Otherwise, we can isotope DJ1 to reduce |D ∩ DJ1 |. If Dγ lies in F1 × I, then both D1 and D2 lie in F1 × I. So both D1 and D2 are isotopic to DJ1 . While DJ1 can be recovered by a band sum of D1 and D2 , a contradiction. So Dγ lies in H. Then both D1 and D2 are essential disks in H. Otherwise, we can isotope DJ1 to reduce |D ∩ DJ1 |. So Di is a compressing disk for ∂H − J2 in H, a contradiction. Thus Claim 1 holds and there is only one compressing disk DJ1 for S2 in V up to isotopy. 2 Let W1 be the compression body obtained from S × I by attaching a 2-handle along J2 . It is obvious that there is only one separating essential disk in W1 which we denote by DJ2 . Since J1 ∩ J2 = ∅, DJ1 is disjoint from DJ2 . The proof of Claim 1 implies that W1 ∪S V is a keen weakly reducible Heegaard splitting. By the proof of Claim 1, any essential disk in V has nonempty intersection with S2 . So S2 is a compressible hole for V . Let πS2 be the projection from C(S) to C(S2 ) determined by S2 . For any essential disk D in V , since there is only one compressing disk DJ1 for S2 in V , by Lemma 2.2 dC(S2 ) (πS2 (∂D), ∂DJ1 ) ≤ 3. So for α ∈ πS2 (∂D), dC(S2 ) (α, ∂DJ1 ) ≤ dC(S2 ) (πS2 (∂D), ∂DJ1 ) + diamC(S2 ) (πS2 (∂D)) ≤ 5. Hence diamC(S2 ) (πS2 (DV )) ≤ 10, where DV is the set of vertices in C(S) represented by boundaries of essential disks in V . Let F2 be the component of ∂− W1 homeomorphic to S2 ∪ DJ2 and fS2 the natural homeomorphism from S2 ∪ DJ2 to F2 . Let f = fS2 ◦ πS2 . Then f is a projection from C(S) to C(F2 ) and diamC(F2 ) (f (DV )) = diamC(S2 ) (πS2 (DV )) ≤ 10. By Theorem 2.7 in [6], there is a full simplex X in C(F2 ) such that dC(F2 ) (DHX , f (DV )) ≥ 3,
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where HX is handlebody obtained from F2 ×I by attaching a collection of 2-handles along X and 3-handles to cap off the possible 2-spheres and DHX is the set of vertices in the curve complex C(F2 ) represented by boundaries of essential disks in HX . Let W = W1 ∪ HX . It is clear that W is a compression body and ∂− W has only one component with genus g2 . Then we have the following claim Claim 2. V ∪S W is a keen weakly reducible Heegaard splitting. Proof. It is clear that DJ1 and DJ2 is a pair of disjoint essential disks in V and W respectively. Suppose there is another pair of disks D in V and E in W such that D ∩ E = ∅. Since S2 is a compressible hole and D is an essential disk in V , ∂D ∩ S2 = ∅. If ∂E ∩ S2 = ∅, then E is isotopic to DJ2 . So D is a compressing disk for S2 in V which is not isotopic to DJ1 , a contradiction. So ∂E ∩ S2 = ∅. Isotope D and E such that |∂D ∩ S2 | and |∂E ∩ S2 | are minimal. If E ∩ DJ2 = ∅, then ∂E lies in S2 and f (∂E) = fS2 (∂E) which bounds an essential disk in HX . If E ∩ DJ2 = ∅, an innermost closed curve argument implies that E ∩ DJ2 consists of arcs. Choose an arc α from E ∩ DJ2 such that α is outermost in E. Then α cuts a disk E0 from E such that E0 ∩ DJ2 = α. Denote the components of DJ2 − α by E1 and E2 . Let Ei = Ei ∪α E0 where i = 1, 2. Both E1 and E2 are essential disks disjoint from DJ2 in W . Otherwise, we can isotope DJ2 to reduce |∂E ∩ S2 |. A similar argument as the proof of Claim 1 implies that ∂Ei lies in S2 which is not isotopic to J2 . Hence f (∂Ei ) bounds an essential disk in HX and f (∂Ei ) ∈ f (∂E) where i = 1, 2. So f (∂Ei ) ∈ DHX . Since D ∩ E = ∅, D ∩ S2 = ∅ and E ∩ S2 = ∅, dC(F2 ) (f (∂D), f (∂Ei )) ≤ 2. Since D is an essential disk in V and f (∂Ei ) ∈ DHX , dC(F2 ) (DHX , f (DV )) ≤ dC(F2 ) (f (∂Ei ), f (∂D)) ≤ 2, a contradiction. 2 So V ∪S W is a keen weakly reducible Heegaard splitting and the theorem holds. This completes the proof of the theorem. 2 Remark 3.1. In fact, following the lines of the proof of Theorem 1.1, if we take a suitable subcomplex of the full complex to attach 2-handles, we can obtain keen weakly reducible Heegaard splittings with arbitrarily many boundary components. Proposition 3.2. Suppose V1 ∪S V2 is an irreducible keen weakly reducible Heegaard splitting and Di is an essential disk in Vi such that ∂D1 ∩ ∂D2 = ∅, where i = 1, 2. (1) If Di is non-separating in Vi , then Dj is non-separating in Vj and ∂D1 ∪ ∂D2 is separating in S, where {i, j} = {1, 2}. (2) If Di is separating in Vi , then Dj is separating in Vj for {i, j} = {1, 2} and there is a component Fi of ∂− Vi such that a component of Vi − Di is homeomorphic to Fi × I, where i = 1, 2. Proof. Since V1 ∪S V2 is irreducible and weakly reducible, g(S) ≥ 3. Suppose one of D1 and D2 , say D1 , is non-separating. Let S1 = S − ∂D1 . Then g(S1 ) ≥ 2. If D2 is separating in V2 , then the components of ∂S1 lie in the same component of S1 − ∂D2 . So there exists an arc α properly embedded in S1 such that α ∩ ∂D2 = ∅ and the endpoints of α lie in different components of ∂S1 . Let D0 the band sum of D1 and a copy of D1 along α. Then D0 is an essential disk in V1 and ∂D0 ∩ ∂D2 = ∅, which contradicts to the assumption V1 ∪S V2 is a keen Heegaard splitting. So D2 is non-separating in V2 .
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If ∂D2 is non-separating in S1 , with a similar argument as above we can prove V1 ∪S V2 is not keen. So ∂D2 is separating in S1 . Thus ∂D1 ∪ ∂D2 is separating in S. Suppose D1 is separating in V1 . By the argument above, D2 is also separating in V2 . Denote the components of V1 − D1 by V1 and V1 . Since D1 is separating, then ∂D2 lies in one component of V1 − D1 , say V1 . If V2 is nontrivial, then each essential disk in V2 is disjoint from D2 , a contradiction. So V2 is a trivial compression body and the conclusion holds. 2 4. Reducible handle additions In this section, let F, DF , V , V , S1 , S2 , f2 , f1 , f2 be the symbols defined in the paragraph above Theorem 1.2. Suppose V ∪S W is an irreducible keen weakly reducible Heegaard splitting and W is not simple. Let D and E be essential disks in V and W such that D ∩ E = ∅. We are now in a position to prove the Theorem 1.2. Proof of Theorem 1.2. First we assume g(F ) ≥ 2. Suppose V is not simple. For J ∈ J , VJ ∪S W is reducible. So there are essential disks DJ in VJ and B in W such that ∂DJ = ∂B. Isotope DJ such that |DJ ∩ DF | is minimal. Suppose DJ ∩ DF = ∅. If ∂DJ lies in S1 , then DJ is an essential disk in V and V ∪S W is reducible, a contradiction. So ∂DJ lies in S2 . Let VJ be the compression body obtained from V by attaching a 2-handle along J and DJ the essential disk in VJ determined by J such that f2 (∂DJ ) = J. Then VJ is a simple compression body. By [14, Lemma 3.1], dC(∂+ VJ ) (∂DJ , ∂DJ ) ≤ 1. Since f2 (∂DJ ) = f2 (∂DJ ) and ∂+ VJ = S2 ∪ DF , dC(F ) (J, f2 (DW )) ≤ dC(F ) (J, f2 (∂B)) = dC(F ) (J, f2 (∂DJ )) = dC(F ) (f2 (∂DJ ), f2 (∂DJ )) = dC(∂+ VJ ) (∂DJ , ∂DJ ) ≤1 Next suppose DJ ∩ DF = ∅. An innermost closed curve argument implies that DJ ∩ DF consists of arcs. Let α be an arc of DJ ∩ DF which is outermost in DJ . Then α cuts a disk Dα from DJ such that Dα ∩ DF = α. Suppose α cuts DF into D1 and D2 . Let Di = Dα ∪α Di where i = 1, 2. Then both D1 and D2 are essential in VJ . Otherwise we can isotope DJ to reduce |DJ ∩ DF |. If Dα lies in V , then Di is an essential disk in V and ∂Di ∈ f1 (∂DJ ). So dC(∂+ V ) (f1 (∂DJ ), DV ) = 0. Since ∂DJ bounds an essential disk in W , f1 (∂DJ ) ⊂ f1 (DW ). So d = dC(∂+ V ) (f1 (DW ), DV ) = 0, a contradiction. So Dα lies in VJ . Thus Di is an essential disk in VJ and f2 (∂Di ) = f2 (∂Di ) ∈ f2 (∂DJ ) where i = 1, 2. With a similar argument as above, dC(F ) (J, f2 (DW )) ≤ 1. So for any J1 , J2 ∈ J , there is an essential disk DJi in VJi such that ∂DJi also bounds an essential disk in W and dC(F ) (Ji , f2 (DW )) ≤ dC(F ) (Ji , f2 (∂DJi )) ≤ 1 for i = 1, 2. Thus dC(F ) (J1 , J2 ) ≤ dC(F ) (J1 , f2 (∂DJ1 )) + diamC(F ) (f2 (DW )) + dC(F ) (J2 , f2 (∂DJ2 )) ≤ diamC(F ) (f2 (DW )) + 2
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There are two cases: Case 1: DF = D In this case, by Proposition 3.2, both D and E are separating and the proof of Theorem 1.1 implies that S1 has only one compressing disk E in W . We have the following claim: Claim 3. diamC(F ) (f2 (DW )) ≤ 20. Proof. Let C be the component of W − E which ∂DF lies in. Since W is not simple, by Proposition 3.2, C is a nontrivial compression body and the other component of W − E is a trivial compression body. Since V ∪S W is keen, by Theorem 1.1, ∂+ C − ∂DF is incompressible in C. C ∪ (DF × I) is not a surface I-bundle. Otherwise W ∪ (DF × I) is simple, a contradiction. So by Lemma 2.1, diamC(F ) (f2 (DC )) ≤ 12, where DC is set of vertices in C(S) represented by the boundaries of essential disks in C. Since E ∩ S2 = ∅, f2 (∂E) = ∅. Let D0 be an essential disk in W such that D0 = E. Since E ∩ DF = ∅, we can isotope D0 such that |D0 ∩ DF | and |D0 ∩ E| are minimal. Since V ∪S W is keen, D0 ∩ DF = ∅. So f2 (∂D0 ) = ∅. If D0 ∩ E = ∅, then D0 is an essential disk in C and f2 (∂D0 ) ⊂ f2 (DC ). If D0 ∩ E = ∅, choose an arc γ from D0 ∩ E such that γ is outermost in D0 . Then γ cuts an disk Dγ from D0 such that Dγ ∩ E = γ. Suppose γ cuts E into E1 and E2 . Let Ei = Dγ ∪γ Ei where i = 1, 2. Then both E1 and E2 are essential disks in C. Otherwise we can isotope D0 to reduce |D0 ∩ E|, a contradiction. Let β = ∂D0 ∩ ∂Dγ . Then β ∩ ∂Ei = ∅ for i = 1, 2. Since |∂D0 ∩ S2 | is minimal, |β ∩ S2 | is minimal. If β ∩ S2 = ∅, which means Dγ ∩ DF = ∅, then Ei is a compressing disk for ∂+ C − ∂DF in C, a contradiction. So β ∩ S2 = ∅. Isotope E1 such that |∂E1 ∩ S2 | is minimal and ∂E1 ∩ β = ∅. Let β be a component of β ∩ S2 . Then β ∩ (∂E1 ∩ S2 ) = ∅. Then dC(S2 ) (∂N (β ∪ ∂S2 ), πS2 (∂E1 )) ≤ 2 and ∂N (β ∪ ∂S2 ) ⊂ πS2 (∂D0 ), where N (β ∪ ∂S2 ) is the regular neighborhood of β ∪ ∂S2 in S2 . So dC(F ) (f2 (∂D0 ), f2 (∂E1 )) ≤ dC(S2 ) (∂N (β ∪ ∂S2 ), πS2 (∂E1 )) ≤ 2. Since E1 is an essential disk in C, f2 (∂E1 ) ⊂ f2 (DC ). So dC(F ) (f2 (∂D0 ), f2 (DC )) ≤ dC(F ) (f2 (∂D0 ), f2 (∂E1 )) ≤ 2. For any α1 , α2 ∈ f2 (DW ), there are essential disks Dαi in W such that Dαi = E and αi ∈ f2 (∂Dαi ) for i = 1, 2. Since dC(F ) (f2 (∂Dαi ), f2 (DC )) ≤ 2, dC(F ) (αi , f2 (DC )) ≤ dC(F ) (f2 (∂Dαi ), f2 (DC )) + diamC(F ) (f2 (∂Dαi )) ≤ 4. Hence dC(F2 ) (α1 , α2 ) ≤ dC(F2 ) (α1 , f2 (DC )) + diamC(F2 ) (f2 (DC )) + dC(F2 ) (f2 (DC ), α2 ) ≤ 20 Thus the claim holds. 2 Then dC(F ) (J1 , J2 ) ≤ diamC(F ) (f2 (DW )) + 2 ≤ 22 Thus the diameter of J is at most 22 and the conclusion holds in this case.
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Case 2: DF = D Since V ∪S W is keen, ∂+ W − ∂DF is incompressible in W . Since W ∪ (DF × I) is not a surface I-bundle, by Lemma 2.1, diamC(F ) f2 (DW ) ≤ 12. Then dC(F ) (J1 , J2 ) ≤ diamC(F ) (f2 (DW )) + 2 ≤ 14 Thus the diameter of J is at most 14 and the conclusion holds in this case. If V is a simple compression body, a similar argument as above implies theorem holds. So the theorem holds in the case g(F ) ≥ 2. If g(F ) = 1, a similar argument implies the theorem holds. This completes the proof of the theorem. 2 Remark 4.1. In case 1, by Theorem 1.1, S1 is a compressible hole for W and diamC(∂+ V ) (f1 (DW )) ≤ 10 and there exist infinitely many keen weakly reducible Heegaard splittings satisfying the assumptions of the theorem. In case 2, a similar method implies there also exist keen weakly reducible Heegaard splittings satisfying the assumptions in theorem. Suppose V ∪S W is an irreducible and weakly reducible Heegaard splitting. Now we give the proof of Theorem 1.3. Proof of Theorem 1.3. A similar argument as above implies that if d ≥ 1, dC(F ) (J, f2 (DW )) ≤ 1 and diamC(F ) (J ) ≤ diamC(F ) (f2 (DW )) + 2. In case (1), since S1 and S2 are incompressible and W ∪(DF ×I) is not a surface I-bundle, by Lemma 2.1, diamC(F ) (f2 (DW )) ≤ 12. So diamC(F ) (J ) ≤ 14. In case (2), suppose D1 is a compressing disk for S1 in W . Since S1 has only one compressing disk in W , D1 must be separating in S1 . Otherwise, there must exist another compressing disk for S1 constructed by a band sum of D1 and a copy of D1 , which is not isotopic to D1 , a contradiction. Since S2 is not compressible in W , a similar argument as the proof of Claim 3 implies the conclusion holds. In case (3), since S1 is incompressible and S2 is compressible, each essential disk in W has nonempty intersection with S2 . So S2 is a compressible hole with only one compressing disk in W . Then by the proof of Theorem 1.1, diamC(F ) (f2 (DW )) ≤ 10 and diamC(F ) (J ) ≤ 12. In case (4), let D1 be the compressing disk for S1 in W . The argument in case (2) implies D1 is separating in W . Denote the component of W − D1 which ∂DF lies in by C. Then C is a nontrivial compression body and ∂+ C − S2 is incompressible in C. Otherwise, each compressing disk determines a compressing disk for S1 in W which is not isotopic to D1 , a contradiction. So S2 is a compressing hole for C with only one compressing disk. By the proof of Theorem 1.1, diamC(F ) (f2 (DC )) ≤ 10, where DC is the set of vertices in C(S) represented by boundaries of essential disks in C. A similar argument as Claim 3 implies that diamC(F ) (f2 (DW )) ≤ 8 + diamC(F ) (f2 (DC )) ≤ 18. So the conclusion holds. This completes the proof of the theorem. 2 References [1] [2] [3] [4] [5]
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J. Hempel, 3-Manifolds as viewed from curve complex, Topology 40 (2001) 631–657. A. Ido, Y. Jang, T. Kobayashi, Heegaard splittings of distance exactly n, Algebraic Geom. Topol. 14 (3) (2014) 1395–1411. A. Ido, Y. Jang, T. Kobayashi, On keen Heegaard splittings, preprint, available at arXiv:1605.04513. W. Jaco, Adding a 2-handle to a 3-manifold: an application to property R, Proc. Am. Math. Soc. 92 (1984) 288–292. T. Kobayashi, R. Qiu, The amalgamation of high distance Heegaard splittings is always efficient, Math. Ann. 341 (3) (2008) 707–715. F. Lei, A proof of Przytycki’s conjecture on n-relator 3-manifolds, Topology 34 (2) (1995) 473–476. F. Lei, A general handle addition theorem, Math. Z. 221 (2) (1996) 211–216. T. Li, Images of the disk complex, Geom. Dedic. 158 (2012) 121–136. L. Liang, F. Lei, F. Li, Distance degenerating handle addition, Proc. Am. Math. Soc. 144 (2016) 423–434. L. Liang, F. Li, F. Lei, A sufficient condition for the set of distance degenerating handle additions to be bounded, Preprint. H. Masur, Y. Minsky, Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902–974. H. Masur, S. Schleimer, The geometry of the disk complex, J. Am. Math. Soc. 26 (1) (2013) 1–62. J. Przytycki, Incompressibility of surfaces after Dehn surgery, Mich. Math. J. 30 (1983) 289–308. J. Przytycki, n-Relator 3-manifolds with incompressible boundary, in: Low-Dimensional Topology and Kleinian Groups, Coventry/Durham, 1984, in: London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 273–285. R. Qiu, Y. Zou, Q. Guo, The Heegaard distances cover all nonnegative integers, Pac. J. Math. 275 (1) (2015) 231–255. M. Scharlemann, M. Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006) 593–617. M. Scharlemann, Y. Wu, Hyperbolic manifolds and degenerating handle additions, J. Aust. Math. Soc. A 55 (1) (1993) 72–89. Y. Zou, An upper bound on distance degenerate handle additions, preprint, arXiv:1707.05053, 2017.
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Topology and its Applications www.elsevier.com/locate/topol
Reducible handle additions to weakly reducible Heegaard splittings Liang Liang a,1 , Fengling Li b,∗,2 , Jingyan Li c a b c
School of Mathematics, Liaoning Normal University, Dalian 116029, China School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China Department of Mathematics and Physics, Shijiazhuang Tiedao University, 050000, China
a r t i c l e
i n f o
Article history: Received 24 February 2018 Accepted 13 April 2018 Available online 24 April 2018 MSC: primary 57N10 secondary 57M50
a b s t r a c t Let V ∪S W be an irreducible Heegaard splitting of M and F a component of ∂− V . A simple closed curve J in F is called reducible if MJ = VJ S W is reducible, where MJ is obtained by attaching a 2-handle to M along J. In this paper, we give a sufficient condition for the diameter of the set of reducible curves in F to be bounded in C(F ) for the keen weakly reducible Heegaard splittings. Moreover, an upper bound of the diameter of the set of the reducible curves in F is given under some circumstance for the weakly reducible Heegaard splittings. © 2018 Elsevier B.V. All rights reserved.
Keywords: Handle addition Curve complex Subsurface projection Keen Heegaard splitting
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 MJ be 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. When F is a torus, it is just a Dehn filling and J is the slope of the filling. For a Heegaard splitting V ∪S W of M with an essential simple closed curve J lying in a component F ⊂ ∂− V , VJ ∪S W is a Heegaard splitting for MJ . If V ∪S W is irreducible and VJ ∪S W is reducible, we say that the handle addition (as well as, J) is reducible; otherwise, it is irreducible. If V ∪S W is ∂-irreducible and VJ ∪S W is ∂-reducible, we say that the handle addition (as well as, J) is ∂-reducible; otherwise, it is ∂-irreducible. * Corresponding author. 1 2
E-mail addresses: [email protected] (L. Liang), dutlfl@163.com (F. Li), [email protected] (J. Li). Supported in part by grant No. 11601209 of NSFC, Research Foundation for Doctor of Liaoning Province (No. 201601239). Supported in part by grants (No. 11671064 and No. 11471151) of NSFC.
https://doi.org/10.1016/j.topol.2018.04.008 0166-8641/© 2018 Elsevier B.V. All rights reserved.
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Przytycki [18] first gave a sufficient condition for a handle addition to a handlebody to be ∂-irreducible, Jaco [9] then generalized it to obtain the well-known handle addition theorem. Since then many generalizations have been given [2,19,11,12]. Let M be a hyperbolic 3-manifold and F a torus boundary component of M . The Geometrization Conjecture and Thurston’s hyperbolic Dehn surgery theorem show that there are only finitely many exceptional (including reducible and ∂-reducible) slopes in F such that the resulting manifolds are not hyperbolic. On the other hand, if g(F ) ≥ 2, there are many known examples such that infinitely many distinct handle additions yield non-hyperbolic manifolds. However, Scharlemann and Wu [22] showed that only finitely many handle additions yielding non-hyperbolic manifolds are basic. In 2001, Hempel [6] introduced the concept of the distance of a Heegaard splitting. He showed that there exist arbitrarily high distance Heegaard splittings for closed 3-manifolds, and a 3-manifold M admitting a distance at least 3 Heegaard splitting is hyperbolic. Since then, many properties related to the distance of Heegaard splittings, as well as 3-manifolds, have been obtained, see [4,1,21,10]. In [16], 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 [13], Masur and Schleimer [17] independently gave an estimation of the diameter of the image of the essential disks in a compact boundary reducible 3-manifold under the projection. By using the subsurface projection, Ido, Jang and Kobayashi [7] and Qiu, Zou and Guo [20], independently, showed that for any positive integers n and g ≥ 2 there exist closed 3-manifolds admitting a Heegaard splitting of genus g with exact distance n. In [8], Ido, Jang and Kobayashi introduced the concept of keen Heegaard splitting. A Heegaard splitting V ∪S W is called keen if its Hempel distance is realized by a unique pair of essential disks on the opposite sides of S. Moreover, they proved the existence of the strongly keen Heegaard splitting of genus g with distance n for each n ≥ 2 and g ≥ 3. In [3], E proved the existence of keen weakly reducible Heegaard splitting of genus g with g ≥ 3. In this paper, we give a method to obtain keen weakly reducible Heegaard splittings with disjoint separating essential disks on the opposite sides. Theorem 1.1. For any positive integers g, g1 , g2 with g > g1 + g2 , there is a keen weakly reducible Heegaard splitting V ∪S W with genus g such that ∂− V has only one component with genus g1 and ∂− W has only one component with genus g2 . Let V ∪S W be an irreducible Heegaard splitting, F a component of ∂− V , and J an essential simple closed curve in F . Let J be the set of reducible curves in F . In [14,15], we gave some sufficient conditions for the diameter of J to be bounded when V ∪S W is strongly irreducible. Recently, Zou [23] gave an upper bound on distance degenerating handle additions for the Heegaard splittings with distance at least 3. Suppose V ∪S W is an irreducible and weakly reducible Heegaard splitting. Let DF be an essential disk in V such that a component of V − DF is homeomorphic to F × I which we denote by V . Denote the other component of V −DF by V . Let S1 = ∂+ V ∩S and S2 = S −S1 . Then DF determines a projection πSi from vertices of C(S) to finite subsets of vertices of C(Si ) where i = 1, 2. Let f1 be the natural homeomorphism from S1 ∪ DF to ∂+ V and f2 the natural homeomorphism from S2 ∪ DF to F . Let fi = fi ◦ πSi . Then f1 is a projection from C(S) to C(∂+ V ) and f2 is a projection from C(S) to C(F ). A compression body W is called simple if W has either only one separating or only one non-separating essential disk up to isotopy. Let DW be the set of vertices in C(S) represented by boundaries of essential disks in W . If f1 (DW ) = ∅, then for each essential disk E in W , f1 (∂E) = ∅ which means ∂DF ∩ ∂E = ∅. If E is non-separating in W , then there must exist an essential disk E in W which is a band sum of E and a copy of E such that ∂E ∩ ∂DF = ∅ up to isotopy. So f1 (∂E ) = ∅, a contradiction. A similar argument as above implies that there is at most one separating essential disk in W up to isotopy. Since V ∪S W is weakly reducible, W is a nontrivial compression body. So W is a simple compression body
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with only one separating essential disk, which we denote by DS . In this case, f2 (∂DS ) is an essential simple closed curve in F which is the only reducible curve in F . Suppose W is not simple, then f1 (DW ) = ∅. If V is not a trivial compression body, then define d = dC(∂+ V ) (f1 (DW ), DV ), where DV is the set of vertices in C(∂+ V ) represented by boundaries of essential disks in V . If V is a trivial compression body, define d = ∞. The following is another main theorem of the paper. Theorem 1.2. Let V ∪S W be an irreducible keen weakly reducible Heegaard splitting where W is not simple and F a component of ∂− V . If d ≥ 1, then for each J ∈ J , dC(F ) (J, f2 (DW )) ≤ 1. In addition, if W∂DF is not a surface product I-bundle or a simple compression body, then diamC(F ) (J ) is bounded. For a general irreducible and weakly reducible Heegaard splitting, we have Theorem 1.3. Let V ∪S W be an irreducible weakly reducible Heegaard splitting where W is not simple and F a component of ∂− V . If d ≥ 1, then for each J ∈ J , dC(F ) (J, f2 (DW )) ≤ 1. If further W∂DF is not a surface product I-bundle or a simple compression body, we have (1) If S1 and S2 are incompressible in W , then diamC(F ) (f2 (DW )) ≤ 12 and diamC(F ) (J ) ≤ 14. (2) If S1 has only one compressing disk in W up to isotopy and S2 is incompressible in W , then diamC(F ) (f2 (DW )) ≤ 20 and diamC(F ) (J ) ≤ 22. (3) If S1 is incompressible in W and S2 has only one compressing disk in W up to isotopy, then diamC(F ) (f2 (DW )) ≤ 10 and diamC(F ) (J ) ≤ 12. (4) If both S1 and S2 have only one compressing disk in W up to isotopy, then diamC(F ) (f2 (DW )) ≤ 18 and diamC(F ) (J ) ≤ 20. The article is organized as follows. In section 2, we review some necessary preliminaries. In section 3, we give the construction and some properties of keen weakly reducible Heegaard splittings. The proofs of the main results are given in section 4. 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 embedded in M such that D ∩ F = ∂D and ∂D is essential in F . In the second case, we call D a compressing disk for F . A Heegaard splitting V ∪S W is called reducible (weakly reducible) if there are essential disks D and E in V and W respectively such that ∂D = ∂E (∂D ∩ ∂E = ∅). Otherwise, the Heegaard splitting is irreducible (strongly irreducible). A reducible Heegaard splitting is clearly weakly reducible, see [2]. Suppose F is a compact orientable surface of genus at least 1. A simple closed curve c in F is called essential if c does not cut a disk or annulus component from F . If F is not a torus or once-punctured torus, the curve complex of F , first introduced by Harvey [5], 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 [16], 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 ). 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
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paths connecting α to β. Any 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}. Suppose V ∪S W is a Heegaard splitting of M . The distance of V ∪S W is defined to be dC(S) (DV , DW ) where DV and DW are sets of vertices in C(S) represented by boundaries of essential disks in V and W , respectively. 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 joining α to β over all such possible paths. 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 [16], there is a natural projection 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 = ∅, then κF ([γ]) = ∅. If γ F = ∅, then κF ([γ]) is the union of the isotopy classes of essential components of γ F . Furthermore, there is a natural projection 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 projection from vertices of C(F ) to finite subsets of vertices of C(F ). For any two vertices α1 , α2 ∈ C(F ), if dC(F ) (α1 , α2 ) ≤ 1 and πF (αi ) = ∅ for i = 1, 2, then dC(F ) (πF (α1 ), πF (α2 )) ≤ 2 and diamC(F ) (πF (αi )) ≤ 2 where i = 1, 2. The following Disk Image Theorem is proved by Li [13], Masur and Schleimer [17] independently. Lemma 2.1. Let M be a compact orientable and irreducible 3-manifold and F a component of ∂M . Suppose ∂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 V be a nontrivial compression body and S a subsurface of ∂+ V such that each component of ∂S is essential in ∂+ V . Then S determines a projection πS from C(∂+ V ) to C(S). If πS (∂D) = ∅ 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 [17] showed that: 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 D1 for S in V such that dAC(S) (πS (∂D), ∂D1 ) ≤ 3. 3. Keen weakly reducible Heegaard splittings In this section, we give a method to obtain keen weakly reducible Heegaard splittings. Proof of Theorem 1.1. Let S be a closed connected orientable surface of genus g. For any positive integer g1 with g1 < g, there exists a separating essential simple closed curve J1 in S such that a component of S − J1 is a once-punctured surface of genus g1 . Denote the other component of S − J1 by S1 . For any positive integer g2 with g2 < g − g1 , there also exists a separating essential simple closed curve J2 in S1 such that a component of S1 − J2 is a once-punctured surface of genus g2 . Then one component of S − J2 is a once-punctured surface of genus g2 , denote the other component of S − J2 by S2 .
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Let V1 be the compression body obtained from S × I by attaching a 2-handle along J1 . Since J1 is separating in S, there is only one essential disk in V1 up to isotopy which we denote by DJ1 . Let F be the component of ∂− V1 homeomorphic to S1 ∪ DJ1 . Let fS1 be the natural homeomorphism from S1 ∪ DJ1 to F . Then fS1 (J2 ) is an essential simple closed curve in F . By Theorem 2.7 in [6], there is a full simplex X in C(F ) such that dC(F ) (DHX , fS1 (J2 )) ≥ 2, where HX is handlebody obtained from F × I by attaching a collection of 2-handles along X and 3-handles to cap off the possible 2-spheres and DHX is the set of vertices in the curve complex C(F ) represented by boundaries of essential disks in HX . Let V = V1 ∪ HX . It is clear that V is a compression body and ∂− V has only one component with genus g1 . Then we have the following claim: Claim 1. There is only one compressing disk for S2 in V . Proof. Since J1 = ∂DJ1 lies in S2 , DJ1 is a compressing disk for S2 in V . Let F1 = ∂− V . Then DJ1 separates V into F1 × I and a handlebody which we denote by H. In fact, H is homeomorphic to HX . Since dC(F ) (DHX , fS1 (J2 )) ≥ 2, ∂H − J2 is incompressible in H. Suppose there is another compressing disk D for S2 in V which is not isotopic to DJ1 . Then we can isotope D and DJ1 such that |D ∩ DJ1 | is minimal. If D ∩ DJ1 = ∅, then D is an essential disk in H which is disjoint from J2 . So ∂H − J2 is compressible, a contradiction. So D ∩ DJ1 = ∅. Since |D ∩ DJ1 | is minimal, an innermost closed curve argument implies that D ∩ DJ1 consists of arcs. Choose an arc γ from D ∩ DJ1 such that γ is outermost in D. Then γ cuts a disk Dγ from D such that Dγ ∩ DJ1 = γ. Denote the components of DJ1 − γ by D1 and D2 . Let Di = Di ∪γ Dγ where i = 1, 2. Both D1 and D2 are essential disks disjoint from DJ1 in V . Otherwise, we can isotope DJ1 to reduce |D ∩ DJ1 |. If Dγ lies in F1 × I, then both D1 and D2 lie in F1 × I. So both D1 and D2 are isotopic to DJ1 . While DJ1 can be recovered by a band sum of D1 and D2 , a contradiction. So Dγ lies in H. Then both D1 and D2 are essential disks in H. Otherwise, we can isotope DJ1 to reduce |D ∩ DJ1 |. So Di is a compressing disk for ∂H − J2 in H, a contradiction. Thus Claim 1 holds and there is only one compressing disk DJ1 for S2 in V up to isotopy. 2 Let W1 be the compression body obtained from S × I by attaching a 2-handle along J2 . It is obvious that there is only one separating essential disk in W1 which we denote by DJ2 . Since J1 ∩ J2 = ∅, DJ1 is disjoint from DJ2 . The proof of Claim 1 implies that W1 ∪S V is a keen weakly reducible Heegaard splitting. By the proof of Claim 1, any essential disk in V has nonempty intersection with S2 . So S2 is a compressible hole for V . Let πS2 be the projection from C(S) to C(S2 ) determined by S2 . For any essential disk D in V , since there is only one compressing disk DJ1 for S2 in V , by Lemma 2.2 dC(S2 ) (πS2 (∂D), ∂DJ1 ) ≤ 3. So for α ∈ πS2 (∂D), dC(S2 ) (α, ∂DJ1 ) ≤ dC(S2 ) (πS2 (∂D), ∂DJ1 ) + diamC(S2 ) (πS2 (∂D)) ≤ 5. Hence diamC(S2 ) (πS2 (DV )) ≤ 10, where DV is the set of vertices in C(S) represented by boundaries of essential disks in V . Let F2 be the component of ∂− W1 homeomorphic to S2 ∪ DJ2 and fS2 the natural homeomorphism from S2 ∪ DJ2 to F2 . Let f = fS2 ◦ πS2 . Then f is a projection from C(S) to C(F2 ) and diamC(F2 ) (f (DV )) = diamC(S2 ) (πS2 (DV )) ≤ 10. By Theorem 2.7 in [6], there is a full simplex X in C(F2 ) such that dC(F2 ) (DHX , f (DV )) ≥ 3,
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where HX is handlebody obtained from F2 ×I by attaching a collection of 2-handles along X and 3-handles to cap off the possible 2-spheres and DHX is the set of vertices in the curve complex C(F2 ) represented by boundaries of essential disks in HX . Let W = W1 ∪ HX . It is clear that W is a compression body and ∂− W has only one component with genus g2 . Then we have the following claim Claim 2. V ∪S W is a keen weakly reducible Heegaard splitting. Proof. It is clear that DJ1 and DJ2 is a pair of disjoint essential disks in V and W respectively. Suppose there is another pair of disks D in V and E in W such that D ∩ E = ∅. Since S2 is a compressible hole and D is an essential disk in V , ∂D ∩ S2 = ∅. If ∂E ∩ S2 = ∅, then E is isotopic to DJ2 . So D is a compressing disk for S2 in V which is not isotopic to DJ1 , a contradiction. So ∂E ∩ S2 = ∅. Isotope D and E such that |∂D ∩ S2 | and |∂E ∩ S2 | are minimal. If E ∩ DJ2 = ∅, then ∂E lies in S2 and f (∂E) = fS2 (∂E) which bounds an essential disk in HX . If E ∩ DJ2 = ∅, an innermost closed curve argument implies that E ∩ DJ2 consists of arcs. Choose an arc α from E ∩ DJ2 such that α is outermost in E. Then α cuts a disk E0 from E such that E0 ∩ DJ2 = α. Denote the components of DJ2 − α by E1 and E2 . Let Ei = Ei ∪α E0 where i = 1, 2. Both E1 and E2 are essential disks disjoint from DJ2 in W . Otherwise, we can isotope DJ2 to reduce |∂E ∩ S2 |. A similar argument as the proof of Claim 1 implies that ∂Ei lies in S2 which is not isotopic to J2 . Hence f (∂Ei ) bounds an essential disk in HX and f (∂Ei ) ∈ f (∂E) where i = 1, 2. So f (∂Ei ) ∈ DHX . Since D ∩ E = ∅, D ∩ S2 = ∅ and E ∩ S2 = ∅, dC(F2 ) (f (∂D), f (∂Ei )) ≤ 2. Since D is an essential disk in V and f (∂Ei ) ∈ DHX , dC(F2 ) (DHX , f (DV )) ≤ dC(F2 ) (f (∂Ei ), f (∂D)) ≤ 2, a contradiction. 2 So V ∪S W is a keen weakly reducible Heegaard splitting and the theorem holds. This completes the proof of the theorem. 2 Remark 3.1. In fact, following the lines of the proof of Theorem 1.1, if we take a suitable subcomplex of the full complex to attach 2-handles, we can obtain keen weakly reducible Heegaard splittings with arbitrarily many boundary components. Proposition 3.2. Suppose V1 ∪S V2 is an irreducible keen weakly reducible Heegaard splitting and Di is an essential disk in Vi such that ∂D1 ∩ ∂D2 = ∅, where i = 1, 2. (1) If Di is non-separating in Vi , then Dj is non-separating in Vj and ∂D1 ∪ ∂D2 is separating in S, where {i, j} = {1, 2}. (2) If Di is separating in Vi , then Dj is separating in Vj for {i, j} = {1, 2} and there is a component Fi of ∂− Vi such that a component of Vi − Di is homeomorphic to Fi × I, where i = 1, 2. Proof. Since V1 ∪S V2 is irreducible and weakly reducible, g(S) ≥ 3. Suppose one of D1 and D2 , say D1 , is non-separating. Let S1 = S − ∂D1 . Then g(S1 ) ≥ 2. If D2 is separating in V2 , then the components of ∂S1 lie in the same component of S1 − ∂D2 . So there exists an arc α properly embedded in S1 such that α ∩ ∂D2 = ∅ and the endpoints of α lie in different components of ∂S1 . Let D0 the band sum of D1 and a copy of D1 along α. Then D0 is an essential disk in V1 and ∂D0 ∩ ∂D2 = ∅, which contradicts to the assumption V1 ∪S V2 is a keen Heegaard splitting. So D2 is non-separating in V2 .
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If ∂D2 is non-separating in S1 , with a similar argument as above we can prove V1 ∪S V2 is not keen. So ∂D2 is separating in S1 . Thus ∂D1 ∪ ∂D2 is separating in S. Suppose D1 is separating in V1 . By the argument above, D2 is also separating in V2 . Denote the components of V1 − D1 by V1 and V1 . Since D1 is separating, then ∂D2 lies in one component of V1 − D1 , say V1 . If V2 is nontrivial, then each essential disk in V2 is disjoint from D2 , a contradiction. So V2 is a trivial compression body and the conclusion holds. 2 4. Reducible handle additions In this section, let F, DF , V , V , S1 , S2 , f2 , f1 , f2 be the symbols defined in the paragraph above Theorem 1.2. Suppose V ∪S W is an irreducible keen weakly reducible Heegaard splitting and W is not simple. Let D and E be essential disks in V and W such that D ∩ E = ∅. We are now in a position to prove the Theorem 1.2. Proof of Theorem 1.2. First we assume g(F ) ≥ 2. Suppose V is not simple. For J ∈ J , VJ ∪S W is reducible. So there are essential disks DJ in VJ and B in W such that ∂DJ = ∂B. Isotope DJ such that |DJ ∩ DF | is minimal. Suppose DJ ∩ DF = ∅. If ∂DJ lies in S1 , then DJ is an essential disk in V and V ∪S W is reducible, a contradiction. So ∂DJ lies in S2 . Let VJ be the compression body obtained from V by attaching a 2-handle along J and DJ the essential disk in VJ determined by J such that f2 (∂DJ ) = J. Then VJ is a simple compression body. By [14, Lemma 3.1], dC(∂+ VJ ) (∂DJ , ∂DJ ) ≤ 1. Since f2 (∂DJ ) = f2 (∂DJ ) and ∂+ VJ = S2 ∪ DF , dC(F ) (J, f2 (DW )) ≤ dC(F ) (J, f2 (∂B)) = dC(F ) (J, f2 (∂DJ )) = dC(F ) (f2 (∂DJ ), f2 (∂DJ )) = dC(∂+ VJ ) (∂DJ , ∂DJ ) ≤1 Next suppose DJ ∩ DF = ∅. An innermost closed curve argument implies that DJ ∩ DF consists of arcs. Let α be an arc of DJ ∩ DF which is outermost in DJ . Then α cuts a disk Dα from DJ such that Dα ∩ DF = α. Suppose α cuts DF into D1 and D2 . Let Di = Dα ∪α Di where i = 1, 2. Then both D1 and D2 are essential in VJ . Otherwise we can isotope DJ to reduce |DJ ∩ DF |. If Dα lies in V , then Di is an essential disk in V and ∂Di ∈ f1 (∂DJ ). So dC(∂+ V ) (f1 (∂DJ ), DV ) = 0. Since ∂DJ bounds an essential disk in W , f1 (∂DJ ) ⊂ f1 (DW ). So d = dC(∂+ V ) (f1 (DW ), DV ) = 0, a contradiction. So Dα lies in VJ . Thus Di is an essential disk in VJ and f2 (∂Di ) = f2 (∂Di ) ∈ f2 (∂DJ ) where i = 1, 2. With a similar argument as above, dC(F ) (J, f2 (DW )) ≤ 1. So for any J1 , J2 ∈ J , there is an essential disk DJi in VJi such that ∂DJi also bounds an essential disk in W and dC(F ) (Ji , f2 (DW )) ≤ dC(F ) (Ji , f2 (∂DJi )) ≤ 1 for i = 1, 2. Thus dC(F ) (J1 , J2 ) ≤ dC(F ) (J1 , f2 (∂DJ1 )) + diamC(F ) (f2 (DW )) + dC(F ) (J2 , f2 (∂DJ2 )) ≤ diamC(F ) (f2 (DW )) + 2
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There are two cases: Case 1: DF = D In this case, by Proposition 3.2, both D and E are separating and the proof of Theorem 1.1 implies that S1 has only one compressing disk E in W . We have the following claim: Claim 3. diamC(F ) (f2 (DW )) ≤ 20. Proof. Let C be the component of W − E which ∂DF lies in. Since W is not simple, by Proposition 3.2, C is a nontrivial compression body and the other component of W − E is a trivial compression body. Since V ∪S W is keen, by Theorem 1.1, ∂+ C − ∂DF is incompressible in C. C ∪ (DF × I) is not a surface I-bundle. Otherwise W ∪ (DF × I) is simple, a contradiction. So by Lemma 2.1, diamC(F ) (f2 (DC )) ≤ 12, where DC is set of vertices in C(S) represented by the boundaries of essential disks in C. Since E ∩ S2 = ∅, f2 (∂E) = ∅. Let D0 be an essential disk in W such that D0 = E. Since E ∩ DF = ∅, we can isotope D0 such that |D0 ∩ DF | and |D0 ∩ E| are minimal. Since V ∪S W is keen, D0 ∩ DF = ∅. So f2 (∂D0 ) = ∅. If D0 ∩ E = ∅, then D0 is an essential disk in C and f2 (∂D0 ) ⊂ f2 (DC ). If D0 ∩ E = ∅, choose an arc γ from D0 ∩ E such that γ is outermost in D0 . Then γ cuts an disk Dγ from D0 such that Dγ ∩ E = γ. Suppose γ cuts E into E1 and E2 . Let Ei = Dγ ∪γ Ei where i = 1, 2. Then both E1 and E2 are essential disks in C. Otherwise we can isotope D0 to reduce |D0 ∩ E|, a contradiction. Let β = ∂D0 ∩ ∂Dγ . Then β ∩ ∂Ei = ∅ for i = 1, 2. Since |∂D0 ∩ S2 | is minimal, |β ∩ S2 | is minimal. If β ∩ S2 = ∅, which means Dγ ∩ DF = ∅, then Ei is a compressing disk for ∂+ C − ∂DF in C, a contradiction. So β ∩ S2 = ∅. Isotope E1 such that |∂E1 ∩ S2 | is minimal and ∂E1 ∩ β = ∅. Let β be a component of β ∩ S2 . Then β ∩ (∂E1 ∩ S2 ) = ∅. Then dC(S2 ) (∂N (β ∪ ∂S2 ), πS2 (∂E1 )) ≤ 2 and ∂N (β ∪ ∂S2 ) ⊂ πS2 (∂D0 ), where N (β ∪ ∂S2 ) is the regular neighborhood of β ∪ ∂S2 in S2 . So dC(F ) (f2 (∂D0 ), f2 (∂E1 )) ≤ dC(S2 ) (∂N (β ∪ ∂S2 ), πS2 (∂E1 )) ≤ 2. Since E1 is an essential disk in C, f2 (∂E1 ) ⊂ f2 (DC ). So dC(F ) (f2 (∂D0 ), f2 (DC )) ≤ dC(F ) (f2 (∂D0 ), f2 (∂E1 )) ≤ 2. For any α1 , α2 ∈ f2 (DW ), there are essential disks Dαi in W such that Dαi = E and αi ∈ f2 (∂Dαi ) for i = 1, 2. Since dC(F ) (f2 (∂Dαi ), f2 (DC )) ≤ 2, dC(F ) (αi , f2 (DC )) ≤ dC(F ) (f2 (∂Dαi ), f2 (DC )) + diamC(F ) (f2 (∂Dαi )) ≤ 4. Hence dC(F2 ) (α1 , α2 ) ≤ dC(F2 ) (α1 , f2 (DC )) + diamC(F2 ) (f2 (DC )) + dC(F2 ) (f2 (DC ), α2 ) ≤ 20 Thus the claim holds. 2 Then dC(F ) (J1 , J2 ) ≤ diamC(F ) (f2 (DW )) + 2 ≤ 22 Thus the diameter of J is at most 22 and the conclusion holds in this case.
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Case 2: DF = D Since V ∪S W is keen, ∂+ W − ∂DF is incompressible in W . Since W ∪ (DF × I) is not a surface I-bundle, by Lemma 2.1, diamC(F ) f2 (DW ) ≤ 12. Then dC(F ) (J1 , J2 ) ≤ diamC(F ) (f2 (DW )) + 2 ≤ 14 Thus the diameter of J is at most 14 and the conclusion holds in this case. If V is a simple compression body, a similar argument as above implies theorem holds. So the theorem holds in the case g(F ) ≥ 2. If g(F ) = 1, a similar argument implies the theorem holds. This completes the proof of the theorem. 2 Remark 4.1. In case 1, by Theorem 1.1, S1 is a compressible hole for W and diamC(∂+ V ) (f1 (DW )) ≤ 10 and there exist infinitely many keen weakly reducible Heegaard splittings satisfying the assumptions of the theorem. In case 2, a similar method implies there also exist keen weakly reducible Heegaard splittings satisfying the assumptions in theorem. Suppose V ∪S W is an irreducible and weakly reducible Heegaard splitting. Now we give the proof of Theorem 1.3. Proof of Theorem 1.3. A similar argument as above implies that if d ≥ 1, dC(F ) (J, f2 (DW )) ≤ 1 and diamC(F ) (J ) ≤ diamC(F ) (f2 (DW )) + 2. In case (1), since S1 and S2 are incompressible and W ∪(DF ×I) is not a surface I-bundle, by Lemma 2.1, diamC(F ) (f2 (DW )) ≤ 12. So diamC(F ) (J ) ≤ 14. In case (2), suppose D1 is a compressing disk for S1 in W . Since S1 has only one compressing disk in W , D1 must be separating in S1 . Otherwise, there must exist another compressing disk for S1 constructed by a band sum of D1 and a copy of D1 , which is not isotopic to D1 , a contradiction. Since S2 is not compressible in W , a similar argument as the proof of Claim 3 implies the conclusion holds. In case (3), since S1 is incompressible and S2 is compressible, each essential disk in W has nonempty intersection with S2 . So S2 is a compressible hole with only one compressing disk in W . Then by the proof of Theorem 1.1, diamC(F ) (f2 (DW )) ≤ 10 and diamC(F ) (J ) ≤ 12. In case (4), let D1 be the compressing disk for S1 in W . The argument in case (2) implies D1 is separating in W . Denote the component of W − D1 which ∂DF lies in by C. Then C is a nontrivial compression body and ∂+ C − S2 is incompressible in C. Otherwise, each compressing disk determines a compressing disk for S1 in W which is not isotopic to D1 , a contradiction. So S2 is a compressing hole for C with only one compressing disk. By the proof of Theorem 1.1, diamC(F ) (f2 (DC )) ≤ 10, where DC is the set of vertices in C(S) represented by boundaries of essential disks in C. A similar argument as Claim 3 implies that diamC(F ) (f2 (DW )) ≤ 8 + diamC(F ) (f2 (DC )) ≤ 18. So the conclusion holds. This completes the proof of the theorem. 2 References [1] [2] [3] [4] [5]
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