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On unstabilized genus three critical Heegaard surfaces
Topology and its Applications 165 (2014) 98–109
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Topology and its Applications www.elsevier.com/locate/topol
On unstabilized genus three critical Heegaard surfaces Jungsoo Kim Research Institute of Mathematics, Seoul National University, 1 Gwanak-ro, Gwanak-Gu, Seoul 151-747, Republic of Korea
a r t i c l e
i n f o
Article history: Received 28 June 2013 Received in revised form 20 January 2014 Accepted 21 January 2014 MSC: 57M50 57M25
a b s t r a c t In this article, we give a sufficient condition for a Heegaard splitting of nonsplittable, tunnel number two, three component link exterior to be critical. Moreover, we prove that if F is a genus three, unstabilized, topologically minimal Heegaard surface of a closed, irreducible 3-manifold and it is not an amalgamation of two genus 2 Heegaard surfaces along a torus, then the topological index of F is at most two. © 2014 Elsevier B.V. All rights reserved.
Keywords: 3-manifold Heegaard splitting Critical surface Topologically minimal surface Topological index
1. Introduction and result Throughout this paper, all surfaces and 3-manifolds will be taken to be compact and orientable. In [1], Bachman introduced the concept a “critical surface” and proved several theorems about incompressible surfaces, the number of Heegaard splittings with respect to each genus, and the minimal genus common stabilization. Bachman also proved Gordon’s conjecture and found some counterexamples to the Stabilization Conjecture by using the sequence of generalized Heegaard splittings and critical Heegaard splittings [2,4]. In his recent work, he generalized the idea of a critical surface to the concept a “topologically minimal surface”, which includes incompressible surfaces, strongly irreducible surfaces, critical surfaces, and so on [3]. Here, he gave a “topological index” to a topologically minimal surface F by using the disk complex of F in a given 3-manifold. Indeed, a topologically minimal surface intersects an incompressible surface so that the intersection is essential on both surfaces up to isotopy if the manifold is irreducible, where this is a generalization of the way how a strongly irreducible surface or a critical surface intersects an incompressible surface [15,1]. Moreover, he gave a conjecture so that a topologically minimal surface of index n would be isotopic to a geometrically minimal surface of index at most n in a 3-manifold in [3]. E-mail address: [email protected]. 0166-8641/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2014.01.014
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Although topologically minimal surfaces have many powerful properties as proved in Bachman’s recent works, it is not easy to determine whether a surface is topologically minimal or not. But some recent results give several examples of topologically minimal surfaces, especially in Heegaard surfaces. For example, Johnson and Bachman proved that an n-fold cover of M (1), say M (n), has a topologically minimal surface of index n with genus (n + 1), where M (1) is obtained from the exterior of a specific 2-bridge link [5]. J. Lee proved that an amalgamation of specific strongly irreducible Heegaard splittings becomes to be critical [8]. The author proved that a connected sum of two (1, 1)-knots gives a critical Heegaard splitting in its exterior [7]. F. Lei and E. Qiang proved that a self-amalgamation of a specific Heegaard splitting becomes to be critical [9]. In this article, we prove the following theorems. Theorem 1.1 (Theorem 2.18). Let L be a non-splittable, tunnel number two, three-component link in S 3 and (V, W; F ) be the induced Heegaard splitting of the exterior of L. If we can choose two weak reducing pairs (V, W ) and (V , W ) such that ∂V ∩ ∂W = ∅ and ∂V ∩ ∂W = ∅ up to isotopy, then F is critical. If we untelescope an unstabilized, genus three Heegaard splitting in a closed, orientable, irreducible 3-manifold along a weak reducing pair, then it may be possible that the thin surface of the corresponding generalized Heegaard splitting cannot be a torus. For example, the thin surface obtained by untelescoping the standard genus three Heegaard splitting of T 3 must consist of two tori (see [6]). In this case, we get the following theorem. Theorem 1.2 (Theorem 3.5). Suppose F is an unstabilized, genus three, topologically minimal Heegaard surface in a closed, orientable, irreducible 3-manifold. If F is not an amalgamation of two genus 2 Heegaard surfaces along a torus, then the topological index of F is at most two. 2. The proof of Theorem 1.1 Definition 2.1. A compression body is a 3-manifold which can be obtained by starting with some closed, orientable, connected surface F , forming the product F × I, attaching some number of 2-handles to F × {1} and capping off all resulting 2-sphere boundary components that are not contained in F × {0} with 3-balls. The boundary component F × {0} is referred to as ∂+ . The rest of the boundary is referred to as ∂− . Definition 2.2. A Heegaard splitting of a 3-manifold M is an expression of M as a union V ∪F W, denoted as (V, W; F ) (or (V, W) if necessary), where V and W are compression bodies that intersect in a transversally oriented surface F = ∂+ V = ∂+ W. If (V, W; F ) is a Heegaard splitting of M then we say F is the Heegaard surface of this splitting. If V or W is homeomorphic to a product, then we say the splitting is trivial. If there are compressing disks V ⊂ V and W ⊂ W such that V ∩ W = ∅, then we say the splitting is weakly reducible and call the pair (V, W ) a weak reducing pair. If (V, W ) is a weak reducing pair and ∂V is isotopic to ∂W in F , then we call (V, W ) a reducing pair. If the splitting is not trivial and we cannot take a weak ¯) reducing pair, then we call the splitting strongly irreducible. If there is a pair of compressing disks (V¯ , W ¯ ¯ such that V intersects W transversely in a point in F , then we call this pair a canceling pair and say the splitting is stabilized. Otherwise, we say the splitting is unstabilized. From now on, we will use letter “V ” to denote compressing disks in V and “W ” to denote those in W. Definition 2.3. Let F be a surface of genus at least two in a compact, orientable 3-manifold M . Then the disk complex D(F ) is defined as follows: 1. Vertices of D(F ) are isotopy classes of compressing disks for F . 2. A set of m+1 vertices forms an m-simplex if there are representatives for each that are pairwise disjoint.
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Definition 2.4. (Bachman, [3].) The homotopy index of a complex Γ is defined to be 0 if Γ = ∅, and the smallest n such that πn−1 (Γ ) is non-trivial, otherwise. We say a separating surface F with no torus components is topologically minimal if its disk complex D(F ) is either empty or non-contractible. When F is topologically minimal, we say its topological index is the homotopy index of D(F ). If F is topologically minimal and its topological index is two, then we call F a critical surface. Note that Bachman originally defined a critical surface in a different way in [2] and proved it is equivalent to being index two in [3]. Definition 2.5. Consider a Heegaard splitting (V, W; F ) of an orientable, irreducible 3-manifold M . Let DV (F ) and DW (F ) be the subcomplexes of D(F ) spanned by compressing disks in V and W respectively. We call these subcomplexes the disk complexes of V and W. Let DVW (F ) be the subcomplex of D(F ) spanned by the simplices which have at least one vertex in D(V), and at least one vertex in D(W). In a weak reducing pair, if a disk belongs to V, then we call it a V-disk. Otherwise, we call it a W-disk. If a simplex Δ ⊂ DVW (F ) meets both DV (F ) and DW (F ), then we say Δ is non-trivially embedded in DVW (F ). If a non-trivially embedded 2-simplex Δ is determined by two disks from V and one disk from W, then we call it a V-face. Similarly, we can define a W-face. Lemma 2.6. (McCullough, [11].) DV (F ) and DW (F ) are contractible. Definition 2.7. Let (V, W ) and (V , W ) be two weak reducing pairs for a Heegaard surface F . If there is a finite sequence of weak reducing pairs written by (V0 = V, W0 = W ),
(V1 , W1 ),
...,
Vm = V , W m = W
such that 1. each (Vi+1 , Wi ) for i = 0, . . . , m − 1 is also a weak reducing pair, 2. for 1 i m, Vi−1 is disjoint from, or equal to, Vi , and 3. for 1 i m, Wi−1 is disjoint from, or equal to, Wi , then we say that two weak reducing pairs are connected by a chain of alternately compressible loops on F . Note that the chain is equivalent to a sequence of non-trivially embedded 2-simplices in DVW (F ) where adjacent 2-simplices share a weak reducing pair. Proposition 2.8. (Bachman, Lemma 8.5 of [2].) If there are two weak reducing pairs (V, W ) and (V , W ) not connected by a chain of alternately compressible loops on the Heegaard surface, then the surface is critical. Definition 2.9. Suppose W is a compressing disk for F ⊂ M . Then there is a subset of M that can be identified with W × I so that W = W × { 12 } and F ∩ (W × I) = (∂W ) × I. We form the surface FW , obtained by compressing F along W , by removing (∂W ) × I from F and replacing it with W × (∂I). We say the two disks W × (∂I) in FW are the scars of W . From now on, we will consider only unstabilized Heegaard splittings of an irreducible 3-manifold. If a Heegaard splitting of a compact 3-manifold is reducible, then the manifold is reducible or the splitting is stabilized [12]. Hence, we can exclude the possibilities of reducing pairs among weak reducing pairs. Lemma 2.10. (Lustig and Moriah, Lemma 1.1 of [10].) Suppose that M is an irreducible 3-manifold and (V, W; F ) is an unstabilized Heegaard splitting of M . If F is obtained by compressing F along a collection of disks, then no S 2 component of F can have scars from disks in both V and W.
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We introduce the following lemma. Lemma 2.11. (J. Kim, [7].) Suppose that M is an irreducible 3-manifold and (V, W; F ) is a genus three, unstabilized Heegaard splitting of M . If there exist three mutually disjoint compressing disks V , V ⊂ V and W ⊂ W, then either V is isotopic to V , or one of ∂V and ∂V bounds a punctured torus T in F and the other is a non-separating loop in T . Moreover, we cannot choose three weak reducing pairs (V0 , W ), (V1 , W ), and (V2 , W ) such that Vi and Vj are mutually disjoint and non-isotopic in V for i = j. Definition 2.12. Let us consider a 1-dimensional graph as follows. 1. We assign a vertex to each V-face in DVW (F ). 2. If a V-face shares a weak reducing pair with another V-face, then we assign an edge between these two vertices in this graph. We call this graph the graph of V-faces of DVW (F ). If there is a maximal subset FC V of V-faces in DVW (F ) representing a connected component of the graph of V-faces and the component is not an isolated vertex, then we call FC V a V-facial cluster. Similarly, we define the graph of W-faces and a W-facial cluster. In a V-facial cluster, every weak reducing pair gives the common W-disk, and vise versa. Lemma 2.13. Assume M and F as in Lemma 2.11. If there are two adjacent V-faces f1 represented by {V0 , V1 , W } and f2 represented by {V1 , V2 , W }, then ∂V1 is non-separating, and ∂V0 , ∂V2 are separating in F . Therefore, there is a unique weak reducing pair in a V-facial cluster which can belong to two or more faces in the V-facial cluster. Proof. By Lemma 2.11, we have the following two cases. Case 1. ∂V0 and ∂V2 are non-separating, and ∂V1 is separating in F . Case 2. ∂V0 and ∂V2 are separating, and ∂V1 is non-separating in F . We will show that only Case 2 holds. Suppose that Case 1 holds. Let FV1 = FV 1 ∪ FV1 , where g(FV 1 ) = 2 and g(FV1 ) = 1. Let V¯ be the part of V that FV1 bounds in V. If we consider the V-face f1 , then ∂V0 ⊂ FV1 ¯ Hence, we can compress F along the disk V0 in V¯ and get a by Lemma 2.11. This means that V0 ⊂ V. V1 2-sphere S. Since ∂− V does not have a 2-sphere component, the inside of S in V must be a 3-ball, i.e. V¯ is a ¯ If we use Lemma 2.11 for the V-face f2 , then V2 is also a meridian solid torus and V0 is a meridian disk of V. disk of V¯ similarly. Therefore, V0 is isotopic to V2 in V¯ and we can assume that this isotopy misses V1 . This means that V0 is isotopic to V2 in V, violating the assumption. 2 Definition 2.14. By Lemma 2.13, there is a unique weak reducing pair in a V-facial cluster belonging to two or more faces in the cluster. We call it the center of a V-facial cluster. We call the other weak reducing pairs hands of a V-facial cluster. See Fig. 1. Note that if a V-face f is represented by two weak reducing pairs, then one is the center and the other is a hand. Lemma 2.13 means that the V-disk from the center of a V-facial cluster is non-separating, and the V-disks from hands are all separating. Lemma 2.15. Assume M and F as in Lemma 2.11. Every V-face belongs to some V-facial cluster. Moreover, every V-facial cluster has infinitely many hands. Proof. Suppose that there is a V-face f in DVW (F ) represented by {V0 , V1 , W }. By Lemma 2.11, we can assume that ∂V0 is separating and ∂V1 is non-separating in F . If we compress F along V1 , then FV1 bounds V in V such that V ∩ V1 = ∅. Let V1 and V1 be the scars of V1 in FV1 . Let us consider two disks V¯1 and V¯1 such that V¯1 and V¯1 are obtained by extending V1 and V1 slightly on FV1 respectively and pushing the interior of each disk into V so that it is properly embedded in V (see Fig. 2), and an arc α connecting ∂ V¯1
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Fig. 1. An example of a V-facial cluster in DVW (F ). (V0 , W ) is the center and the other weak reducing pairs are hands.
¯ . Fig. 2. An arc α connecting ∂ V¯1 and ∂ V 1
Fig. 3. There are infinitely many isotopy classes of α.
and ∂ V¯1 on FV1 which misses ∂ V¯1 , ∂ V¯1 , and ∂W in its interior. If we perform the band sum of V¯1 and V¯1 along α in V , then we get a properly embedded disk V0α in V which is ∂-parallel in V . V0α becomes an essential separating disk in V and V0α misses V1 and W . Therefore, we get the V-face f represented by {V1 , V0α , W }. This means that f and f belong to the same V-facial cluster. We claim that there are infinitely many isotopy classes for such V0α in V. If we compress FV1 again along W , then we can see that the genus of each component of FV1 W must be one, otherwise we can find an S 2 -component in FV1 W having scars from both V1 and W , violating Lemma 2.10. Let F¯V1 W be the component of FV1 W containing α. Since g(F¯V1 W ) = 1, there are infinitely many such arcs up to isotopies fixing V¯1 and V¯1 (see Fig. 3). Moreover, if such arcs are non-isotopic, then the resulting disks become non-isotopic in V. 2 The next lemma describes the situation when a V-face and a W-face share a weak reducing pair. Lemma 2.16. Assume M and F as in Lemma 2.11. If there is a V-face f1 and a W-face f2 in DVW (F ) and one shares a weak reducing pair with the other, then the four disks coming from f1 and f2 are mutually
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disjoint or the V-facial cluster and the W-facial cluster containing f1 and f2 respectively have a common center. In the latter case, we can replace a separating disk among the four disks by a new separating disk in the same compression body so that the resulting four disks are mutually disjoint. Proof. Assume that f1 and f2 are represented by {V0 , V1 , W0 } and {V1 , W0 , W1 } respectively. Since V0 , V1 , and W0 are mutually disjoint and so are V1 , W0 , and W1 , we will prove that 1. ∂W1 ∩ ∂V0 = ∅, or 2. both V- and W-facial clusters guaranteed by Lemma 2.15 have a common center. Let the separating one be Vi and the non-separating one be Vj between V0 and V1 . Let FVi = FV i ∪ FVi , where g(FV i ) = 2 and g(FVi ) = 1. We already know that ∂W0 ⊂ FV i and ∂Vj ⊂ FVi by applying Lemma 2.11 to f1 . Case 1. ∂W1 ∩ ∂Vj = ∅, i.e. j = 0 (so i = 1). In this case, the existence of the weak reducing pair (V1 , W1 ) implies that ∂W1 ∩ ∂Vi = ∅. Moreover, ∂W1 ⊂ FVi from ∂W1 ∩ ∂Vj = ∅. But if ∂W1 is non-separating in F , then ∂W1 is essential in FVi , i.e. W1 and Vi determine an S 2 component of FVi W1 , violating Lemma 2.10. If ∂W1 is separating in F , then it must be isotopic to ∂Vi , violating the fact that every weak reducing pair is not a reducing pair. Case 2. ∂W1 ∩ ∂Vj = ∅. If j = 0, then this leads to the result. Therefore, assume that j = 1 (so i = 0). If we apply Lemma 2.11 to the W-face f2 , then we can assume that ∂Wk is the separating one and ∂Wl be the non-separating one for {k, l} = {0, 1}. Here, ∂V1 (∂Wl resp.) goes into the genus two component (the genus one component resp.) of FWk again by Lemma 2.11. Case 2-1. ∂W1 ∩ ∂Vi = ∅. This case means that ∂W1 ∩ ∂V0 = ∅, i.e. this leads to the result. Case 2-2. ∂W1 ∩ ∂Vi = ∅. Suppose that l = 1 (so k = 0). Let FWk = FW ∪ FW , where g(FW ) = 2 and g(FW ) = 1. In this case, the k k k k existence of the weak reducing pair (V0 , W0 ) implies that ∂Wk ∩ ∂Vi = ∅. Here, ∂Vi must go into FW (if k ∂Vi ⊂ FWk , then it must be parallel to ∂Wk since ∂Vi is separating in F and g(FWk ) = 1). Since ∂Wl ∩∂Vi = ∅ from the assumption of Case 2-2, ∂Wl also goes into FW , violating Lemma 2.11. Therefore, we get l = 0 k (so k = 1), i.e. both ∂V1 and ∂W0 (∂V0 and ∂W1 resp.) are non-separating (separating resp.) in F . Let FC V and FC W be the V-facial cluster and the W-facial cluster containing f1 and f2 respectively guaranteed by Lemma 2.15. Since the V-disk (W-disk resp.) for the center of FC V (FC W resp.) is non-separating in V (W resp.) by Lemma 2.13, the weak reducing pair (V0 , W0 ) ((V1 , W1 ) resp.) cannot be the center of FC V (FC W resp.). Therefore, (V1 , W0 ) is the common center of both FC V and FC W . Now we will prove that there is another W-face f3 represented by {V1 , W0 , W2 } in FC W such that W2 ∩ V0 = ∅. If we compress F along V0 , then FV0 = FV 0 ∪ FV0 where g(FV 0 ) = 2 and g(FV0 ) = 1. We get ∂W0 ⊂ FV 0 and ∂V1 ⊂ FV0 by applying Lemma 2.11 to the V-face f1 . If we compress FV 0 again along W0 , then we get a torus F¯V0 W0 . Here, there are two scars W0 and W0 of W0 and one scar V0 of V0 in F¯V0 W0 . If we use the band sum argument by using an arc connecting W0 and W0 missing ∂W0 , ∂W0 , and ∂V0 in its interior as in the proof of Lemma 2.15, then we can find an essential separating disk W2 in W such that W2 ∩ V0 = W2 ∩ V1 = W2 ∩ W0 = ∅. Since we already knew that V0 , V1 , and W0 are mutually disjoint by the existence of the V-face f1 , W2 ∩ V0 = ∅ means that V0 , V1 , W0 , and W2 are mutually disjoint. 2 Note that Lemma 2.16 implies that if there is a V-face and a W-face in DVW (F ) and one shares a weak reducing pair with the other, then there is a weak reducing pair consisting of non-separating disks and a weak reducing pair consisting of separating disks. Indeed, we can imagine a situation as in Fig. 4 if there
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Fig. 4. An example of the boundaries of the resulting four disks in Lemma 2.16.
exists a V-face and a W-face sharing a weak reducing pair and the corresponding four disks are mutually disjoint. The thick curves are separating and the thin curves are non-separating in F . Here, we introduce the key theorem of this section. Theorem 2.17. Assume M and F as in Lemma 2.11. Suppose that we cannot choose a weak reducing pair ¯ ) such that V¯ cuts off a solid torus from V and W ¯ also cuts off a solid torus from W. If we can choose (V¯ , W two weak reducing pairs (V, W ) and (V , W ) such that ∂V ∩ ∂W = ∅ and ∂V ∩ ∂W = ∅ up to isotopy, then F is critical. Proof. At first, we give the following claim. Claim. Suppose that no V-face shares a weak reducing pair with a W-face in DVW (F ). If we can choose two weak reducing pairs (V, W ) and (V , W ) such that ∂V ∩ ∂W = ∅ and ∂V ∩ ∂W = ∅ up to isotopy, then F is critical. Proof of Claim. Suppose that (V, W ) and (V , W ) are connected by a chain of alternately compressible loops on F . That is, there is a finite sequence of weak reducing pairs, (V0 = V, W0 = W ),
(V1 , W1 ),
...,
Vm = V , W m = W
such that 1. each (Vi+1 , Wi ) for i = 0, . . . , m − 1 is also a weak reducing pair, 2. for 1 i m, Vi−1 is disjoint from, or equal to, Vi , and 3. for 1 i m, Wi−1 is disjoint from, or equal to, Wi . By reading the above sequence from the left to the right, we get a sequence of V- and W-faces Δ0 , . . . , Δn such that 1. (V0 , W0 ) ⊂ Δ0 and (Vm , Wm ) ⊂ Δn , 2. Δi−1 shares a weak reducing pair with Δi for i = 1, . . . , n. Note that the assumption V ∩ W = ∅ and V ∩ W = ∅ implies that Δ0 = Δn . Here, Lemma 2.15 divides the proof into the following three cases. Case 1. Δ0 and Δn are contained in the same V-facial cluster. In this case, W0 = Wm . But the weak reducing pair (V0 , W0 ) means that ∂V0 ∩ ∂W0 = ∅, i.e. ∂V0 ∩ ∂Wm = ∂V ∩ ∂W = ∅, violating the assumption of Claim.
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Case 2. Δ0 belongs to some V-facial cluster and Δn belongs to another V-facial cluster. Since these V-facial clusters are different, there must be a W-face among Δ1 , . . . , Δn−1 . Therefore, we can assume that Δi is a V-face and Δi+1 is a W-face for some 0 i n − 2. But this violates the assumption of Claim. Case 3. Δ0 belongs to some V-facial cluster and Δn belongs to some W-facial cluster. Since two clusters are of different types, we can assume that Δi is a V-face and Δi+1 is a W-face for some 0 i n − 2. This gives a contradiction as in Case 2. Therefore, (V, W ) and (V , W ) are not connected by a chain of alternately compressible loops on F . Hence, Proposition 2.8 leads to the result. 2 We claim that no V-face shares a weak reducing pair with a W-face. Suppose that some V-face f1 shares ¯, a weak reducing pair with a W-face f2 . In this case, we can assume that the four disks V¯ , V˜ ⊂ V, and W ˜ ¯ W ⊂ W coming from f1 and f2 are mutually disjoint by Lemma 2.16. Indeed, we can assume that V and ¯ are separating and V˜ and W ˜ are non-separating by Lemma 2.11. W ¯ = ∅ and ∂ V¯ is not isotopic to ∂ W ¯ , ∂ V¯ ∪ ∂ W ¯ cuts F into three pieces F , F , and F , Since V¯ ∩ W ¯ resp.) in F and F is the where F (F resp.) is the once-punctured torus determined by only ∂ V¯ (∂ W ¯ ¯ ¯ ¯ twice-punctured torus determined by both ∂ V and ∂ W in F . Here, V (W resp.) cuts off a solid torus or T 2 × I from V (W resp.) where T 2 is a torus. By the assumption of this theorem, we can assume that at ¯ , say V¯ , cuts off T 2 × I from V, say V . Let T be the torus component of ∂− V which least one of V¯ and W realizes the product V . Our assumption means that V is the part of V which F ∪ V¯ bounds in V. Since ∂ V˜ is a non-separating curve of F from Lemma 2.11, V˜ is a compressing disk of F . Since V¯ is a separating disk in V and V˜ is disjoint from V¯ , we get V˜ ⊂ V , i.e. F is compressible in V ∼ = T 2 × I, this is a contradiction. Hence, Claim leads to the result. This completes the proof of Theorem 2.17. 2 Finally, we reach Theorem 2.18. Theorem 2.18. Let L be a non-splittable, tunnel number two, three-component link in S 3 and (V, W; F ) be the induced Heegaard splitting of the exterior of L. If we can choose two weak reducing pairs (V, W ) and (V , W ) such that ∂V ∩ ∂W = ∅ and ∂V ∩ ∂W = ∅ up to isotopy, then F is critical. Proof. Let T1 , T2 , and T3 be the three boundary components of the exterior of L. Assume that V is the compression body containing T1 , T2 and T3 and the two tunnels, and W is the handlebody S 3 − V. Since the tunnel number of L is two and the number of components of L is three, the induced Heegaard splitting (V, W; F ) is of genus three and unstabilized. Moreover, every separating disk V¯ ⊂ V must divide V into two compression bodies with non-empty minus boundaries. This means that we cannot choose a weak reducing ¯ ) such that V¯ cuts off a solid torus from V and W ¯ also cuts off a solid torus from W. Hence, pair (V¯ , W Theorem 2.17 leads to the result. 2 For example, the tunnel system of the three component chain in Fig. 5 induces a critical Heegaard splitting, where V and V come from the cocore disks of the tunnel a1 and a2 respectively. Note that Sedgwick proved that this splitting is not of minimal genus in [16]. 3. The proof of Theorem 1.2 In this section, we give the proof of Theorem 1.2. First, we consider the following lemmas (see [14] for the definition of an “amalgamation”).
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Fig. 5. A tunnel system of the three component chain.
Fig. 6. If we amalgamate two thick surfaces along a torus, then we can find a weak reducing pair such that the disks are separating.
Lemma 3.1. Let M be a closed 3-manifold and (V, W; F ) be a genus three Heegaard splitting of M . Then F has a non-reducing, weak reducing pair consisting of separating disks if and only if M = M1 ∪T M2 , where each Mi has a genus two Heegaard splitting for i = 1, 2, T is a torus, and F can be represented as an amalgamation of these genus two splittings along T . Proof. (⇐) Let the Heegaard splitting of M1 (M2 resp.) be (V+ , W+ ; F1 ) ((V− , W− ; F2 ) resp.), and W+ ∩ V− = T . Hence, W+ is homeomorphic to (T × I) ∪ (a 1-handle) and also V− is. Let the 1-handles for W+ and V− be hW+ and hV− respectively. If we take T to be the common 0-level of two T × I’s belonging to W+ and V− , then the attaching disks for hW+ and hV− go into the 1-levels in these products. In order to do the amalgamation, isotope the attaching disks of hW+ and hV− so that they are disjoint after the projection into T (see Fig. 6). Note that the projection function may also be changed continuously during this isotopy so that the projection images of these attaching disks do not intersect one another. Let us consider the curves CW+ and CV− in T such that CW+ (CV− resp.) bounds a disk in T and this disk contains the projection images of the attaching disks of hW+ (hV− resp.) in its interior. Make sure that the disk for CW+ is disjoint from that of CV− . Here, we can find the curve C¯W+ in ∂+ W+ (C¯V− in ∂+ V− resp.) as the preimage of the projection for CW+ (CV− resp.). Consider an essential disk W ⊂ W+ (V ⊂ V− resp.) such that ∂W = C¯W+ (∂V = C¯V− resp.) like Fig. 6. Obviously, V and W are separating in their compression bodies. After the amalgamation, V and W become disjoint separating disks and the boundary of one is not isotopic to that of the other in the resulting Heegaard surface. (⇒) Suppose that there is a non-reducing, weak reducing pair (V, W ) consisting of separating disks. Now FV W consists of three tori FV W , FV W , and FVW , where FV W (FVW resp.) bounds a solid torus V in V (W in W resp.). Find a meridian disk V (W resp.) in V (W resp.) such that V (W resp.) is
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disjoint from V (W resp.). Since V ∩ W = ∅, (V , W ) is a weak reducing pair. Let T be the surface FV W . Then F is obtained from T by attaching boundaries of two 1-handles whose cocores are V and W and removing the interiors of the attaching disks of these handles. Now we get the right one of Fig. 6. Therefore, if we weakly reduce the splitting (V, W; F ) along (V , W ), then we get the generalized Heegaard splitting {∅, T1 , t1 , T2 , ∅}, where the thick surfaces T1 = FW and T2 = FV are of genus two and the unique non-empty thin surface t1 = FV W is a torus (see [12] and [13] for the definition of a “generalized Heegaard splitting”). 2 Lemma 3.2. Suppose that M is an irreducible 3-manifold and (V, W; F ) is a genus three, unstabilized Heegaard splitting of M . If there is a V-face in DVW (F ) whose W-disk is non-separating, then DVW (F ) must have a V-face and a W-face such that one shares a weak reducing pair with the other. Proof. Suppose that there is a V-face f1 represented by {V0 , V1 , W }, ∂V0 is separating, ∂V1 is non-separating, and ∂W is non-separating in F by Lemma 2.11 and the assumption of this lemma. Let FV 0 be the genus two component of FV0 . We can check that ∂W ⊂ FV 0 and ∂V1 ∩ FV 0 = ∅ by Lemma 2.11. If we compress FV 0 along W again, then (FV 0 )W is a torus. Let W and W be the scars of W0 and V0 be the scar of V0 in (FV 0 )W . If we using the band sum arguments by using an arc connecting W and W missing ∂W , ∂W , and ∂V0 in its interior as in the proof of Lemma 2.15, we get a family {Wα } of infinitely many essential separating disks in W which miss W and V0 . If we choose a disk Wα from {Wα }, then we get a W-face f2 represented by {V0 , Wα , W }. Therefore, f1 and f2 are the wanted V- and W-faces such that one shares a weak reducing pair with the other. 2 Corollary 3.3. Assume M and F as in Lemma 3.2, and add the assumption that M is closed. If there is a V- or W-face in DVW (F ), then DVW (F ) must have a V-face and a W-face such that one shares a weak reducing pair with the other. Proof. Without loss of generality, suppose that there is a V-face f1 represented by {V0 , V1 , W } in DVW (F ), where ∂V0 is separating and ∂V1 is non-separating in F by Lemma 2.11. If ∂W is non-separating in F , then this leads to the result by Lemma 3.2. Therefore, assume that ∂W is separating in F , i.e. W is separating in W. Hence, W cuts W into a solid torus W and a genus two handlebody, where W is bounded by F˜ ∪ W in W. (F˜ is the once-punctured torus which ∂W cuts off from F .) Let Wm be a meridian disk of W . Isotope Wm in W so that ∂Wm is contained in the interior of F˜ . Let F¯ be the once-punctured genus two component of F − ∂V0 . Since ∂W ⊂ F¯ by Lemma 2.11 and ∂W cuts F¯ into a twice-punctured torus and a once-punctured torus (otherwise, ∂W is isotopic to ∂V0 in F¯ ), we get either F˜ ⊂ F¯ or F˜ is of genus two. The latter case violates the assumption that F˜ is a once-punctured torus. Hence, ∂V0 ∩ ∂Wm = ∅, i.e. we get a W-face f2 represented by {V0 , Wm , W }. Therefore, f1 and f2 are the wanted V- and W-faces such that one shares a weak reducing pair with the other. 2 Now we get the following three equivalent statements when M is closed. Corollary 3.4. Let M be a closed, orientable, irreducible 3-manifold and (V, W; F ) be an unstabilized, weakly reducible, genus three Heegaard splitting. Then the following three statements are equivalent. (1) There is no weak reducing pair consisting of separating disks. (2) F cannot be represented as an amalgamation of genus two splittings along a torus. (3) If there are different weak reducing pairs, then they do not share a disk. Proof. Since M is irreducible and (V, W; F ) is unstabilized, a weak reducing pair cannot be a reducing one, i.e. (1) and (2) are equivalent by Lemma 3.1. Hence, we will prove that (1) and (3) are equivalent.
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(1) ⇒ (3) If there are two weak reducing pairs sharing a disk, then this means that there is a V- or W-face in DVW (F ) (Lemma 8.4 of [2]). Hence, DVW (F ) must have a V-face and a W-face such that one shares a weak reducing pair with the other by Corollary 3.3. Therefore, if we use Lemma 2.16, then we get a weak reducing pair consisting of separating disks. (3) ⇒ (1) Suppose that there is a weak reducing pair (V, W ) consisting of separating disks. Let the solid torus which V cuts off from V be V and Vm be a meridian disk of V . If we isotope Vm in V so that ∂Vm misses V , then we get a V-face represented by {V, Vm , W }, so the weak reducing pairs (V, W ) and (Vm , W ) share a disk. 2 Finally, we reach Theorem 3.5. Theorem 3.5. Suppose F is an unstabilized, genus three, topologically minimal Heegaard surface in a closed, orientable, irreducible 3-manifold. If F is not an amalgamation of two genus 2 Heegaard surfaces along a torus, then the topological index of F is at most two. Proof. Let the Heegaard splitting be (V, W; F ). If F is strongly irreducible, then the topological index of F is one, leading to the result. Assume that F is weakly reducible. Suppose that the topological index of F is at least three. Then there is a non-trivial embedding ι : S n → D(F ), where n 2. Triangulate S n so that ι would be simplicial. Then we can assume that ι(S n ) is a finite union of n-cells. Since the embedding is non-trivial and both DV (F ) and DW (F ) are contractible, there must be an n-cell Δn containing at least one vertex in DV (F ) and at least one vertex in DW (F ). Since n 2, we can assume that there are two weak reducing pairs sharing a disk by using a vertex of Δn from DV (F ) and two vertices from DW (F ), or vise versa. Hence, Corollary 3.4 means that F can be represented as an amalgamation of genus two splittings along a torus, violating the assumption. 2 Note that Johnson proved that a weak reducing pair of the standard genus three Heegaard splitting of T 3 must consist of only non-separating disks in Lemma 6 of [6]. Moreover, he proved that the thin surface obtained by untelescoping the splitting along a weak reducing pair must consist of two tori, not a torus in Lemma 5 of [6]. This is a good example of the Heegaard splitting satisfying the assumption of Theorem 3.5. Indeed, this splitting is critical by Theorem 2.17. Acknowledgements The author is grateful to the referee for a number of helpful suggestions for improvement in the paper. The author is grateful to Prof. Jung-Hoon Lee for reading this article and giving me helpful advices. The author is grateful to E. Qiang for giving me an advice for polishing the proof of Lemma 2.16. The author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (NRF-2010-0019516). References [1] [2] [3] [4]
D. Bachman, Critical Heegaard surfaces, Trans. Amer. Math. Soc. 354 (2002) 4015–4042. D. Bachman, Connected sums of unstabilized Heegaard splittings are unstabilized, Geom. Topol. 12 (2008) 2327–2378. D. Bachman, Topological index theory for surfaces in 3-manifolds, Geom. Topol. 14 (2010) 585–609. D. Bachman, Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds, Math. Ann. 355 (2013) 697–728. [5] D. Bachman, J. Johnson, On the existence of high index topologically minimal surfaces, Math. Res. Lett. 17 (2010) 389–394. [6] J. Johnson, Automorphisms of the three-torus preserving a genus-three Heegaard splitting, Pacific J. Math. 253 (2011) 75–94. [7] J. Kim, On critical Heegaard splittings of tunnel number two composite knot exteriors, J. Knot Theory Ramifications 22 (2013) 1350065 (11 pp.).
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[8] J. Lee, Critical Heegaard surfaces obtained by amalgamation, Topology Appl. 160 (2013) 111–116. [9] F. Lei, E. Qiang, Critical Heegaard surfaces obtained by self-amalgamation, J. Knot Theory Ramifications 22 (2013) 1350015 (7 pp.). [10] M. Lustig, Y. Moriah, Closed incompressible surfaces in complements of wide knots and links, Topology Appl. 92 (1999) 1–13. [11] D. McCullough, Virtually geometrically finite mapping class groups of 3-manifolds, J. Differential Geom. 33 (1991) 1–65. [12] T. Saito, M. Scharlemann, J. Schultens, Lecture notes on generalized Heegaard splittings, arXiv:math/0504167. [13] M. Scharlemann, A. Thompson, Thin position for 3-manifolds, AMS Contemp. Math. 164 (1994) 231–238. [14] J. Schultens, The classification of Heegaard splittings for (compact orientable surface) × S 1 , Proc. London Math. Soc. 67 (1993) 425–448. [15] J. Schultens, Additivity of tunnel number for small knots, Comment. Math. Helv. 75 (2000) 353–367. [16] E. Sedgwick, Genus two 3-manifolds are built from handle number one pieces, Algebr. Geom. Topol. 1 (2001) 763–790.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
On unstabilized genus three critical Heegaard surfaces Jungsoo Kim Research Institute of Mathematics, Seoul National University, 1 Gwanak-ro, Gwanak-Gu, Seoul 151-747, Republic of Korea
a r t i c l e
i n f o
Article history: Received 28 June 2013 Received in revised form 20 January 2014 Accepted 21 January 2014 MSC: 57M50 57M25
a b s t r a c t In this article, we give a sufficient condition for a Heegaard splitting of nonsplittable, tunnel number two, three component link exterior to be critical. Moreover, we prove that if F is a genus three, unstabilized, topologically minimal Heegaard surface of a closed, irreducible 3-manifold and it is not an amalgamation of two genus 2 Heegaard surfaces along a torus, then the topological index of F is at most two. © 2014 Elsevier B.V. All rights reserved.
Keywords: 3-manifold Heegaard splitting Critical surface Topologically minimal surface Topological index
1. Introduction and result Throughout this paper, all surfaces and 3-manifolds will be taken to be compact and orientable. In [1], Bachman introduced the concept a “critical surface” and proved several theorems about incompressible surfaces, the number of Heegaard splittings with respect to each genus, and the minimal genus common stabilization. Bachman also proved Gordon’s conjecture and found some counterexamples to the Stabilization Conjecture by using the sequence of generalized Heegaard splittings and critical Heegaard splittings [2,4]. In his recent work, he generalized the idea of a critical surface to the concept a “topologically minimal surface”, which includes incompressible surfaces, strongly irreducible surfaces, critical surfaces, and so on [3]. Here, he gave a “topological index” to a topologically minimal surface F by using the disk complex of F in a given 3-manifold. Indeed, a topologically minimal surface intersects an incompressible surface so that the intersection is essential on both surfaces up to isotopy if the manifold is irreducible, where this is a generalization of the way how a strongly irreducible surface or a critical surface intersects an incompressible surface [15,1]. Moreover, he gave a conjecture so that a topologically minimal surface of index n would be isotopic to a geometrically minimal surface of index at most n in a 3-manifold in [3]. E-mail address: [email protected]. 0166-8641/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2014.01.014
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Although topologically minimal surfaces have many powerful properties as proved in Bachman’s recent works, it is not easy to determine whether a surface is topologically minimal or not. But some recent results give several examples of topologically minimal surfaces, especially in Heegaard surfaces. For example, Johnson and Bachman proved that an n-fold cover of M (1), say M (n), has a topologically minimal surface of index n with genus (n + 1), where M (1) is obtained from the exterior of a specific 2-bridge link [5]. J. Lee proved that an amalgamation of specific strongly irreducible Heegaard splittings becomes to be critical [8]. The author proved that a connected sum of two (1, 1)-knots gives a critical Heegaard splitting in its exterior [7]. F. Lei and E. Qiang proved that a self-amalgamation of a specific Heegaard splitting becomes to be critical [9]. In this article, we prove the following theorems. Theorem 1.1 (Theorem 2.18). Let L be a non-splittable, tunnel number two, three-component link in S 3 and (V, W; F ) be the induced Heegaard splitting of the exterior of L. If we can choose two weak reducing pairs (V, W ) and (V , W ) such that ∂V ∩ ∂W = ∅ and ∂V ∩ ∂W = ∅ up to isotopy, then F is critical. If we untelescope an unstabilized, genus three Heegaard splitting in a closed, orientable, irreducible 3-manifold along a weak reducing pair, then it may be possible that the thin surface of the corresponding generalized Heegaard splitting cannot be a torus. For example, the thin surface obtained by untelescoping the standard genus three Heegaard splitting of T 3 must consist of two tori (see [6]). In this case, we get the following theorem. Theorem 1.2 (Theorem 3.5). Suppose F is an unstabilized, genus three, topologically minimal Heegaard surface in a closed, orientable, irreducible 3-manifold. If F is not an amalgamation of two genus 2 Heegaard surfaces along a torus, then the topological index of F is at most two. 2. The proof of Theorem 1.1 Definition 2.1. A compression body is a 3-manifold which can be obtained by starting with some closed, orientable, connected surface F , forming the product F × I, attaching some number of 2-handles to F × {1} and capping off all resulting 2-sphere boundary components that are not contained in F × {0} with 3-balls. The boundary component F × {0} is referred to as ∂+ . The rest of the boundary is referred to as ∂− . Definition 2.2. A Heegaard splitting of a 3-manifold M is an expression of M as a union V ∪F W, denoted as (V, W; F ) (or (V, W) if necessary), where V and W are compression bodies that intersect in a transversally oriented surface F = ∂+ V = ∂+ W. If (V, W; F ) is a Heegaard splitting of M then we say F is the Heegaard surface of this splitting. If V or W is homeomorphic to a product, then we say the splitting is trivial. If there are compressing disks V ⊂ V and W ⊂ W such that V ∩ W = ∅, then we say the splitting is weakly reducible and call the pair (V, W ) a weak reducing pair. If (V, W ) is a weak reducing pair and ∂V is isotopic to ∂W in F , then we call (V, W ) a reducing pair. If the splitting is not trivial and we cannot take a weak ¯) reducing pair, then we call the splitting strongly irreducible. If there is a pair of compressing disks (V¯ , W ¯ ¯ such that V intersects W transversely in a point in F , then we call this pair a canceling pair and say the splitting is stabilized. Otherwise, we say the splitting is unstabilized. From now on, we will use letter “V ” to denote compressing disks in V and “W ” to denote those in W. Definition 2.3. Let F be a surface of genus at least two in a compact, orientable 3-manifold M . Then the disk complex D(F ) is defined as follows: 1. Vertices of D(F ) are isotopy classes of compressing disks for F . 2. A set of m+1 vertices forms an m-simplex if there are representatives for each that are pairwise disjoint.
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Definition 2.4. (Bachman, [3].) The homotopy index of a complex Γ is defined to be 0 if Γ = ∅, and the smallest n such that πn−1 (Γ ) is non-trivial, otherwise. We say a separating surface F with no torus components is topologically minimal if its disk complex D(F ) is either empty or non-contractible. When F is topologically minimal, we say its topological index is the homotopy index of D(F ). If F is topologically minimal and its topological index is two, then we call F a critical surface. Note that Bachman originally defined a critical surface in a different way in [2] and proved it is equivalent to being index two in [3]. Definition 2.5. Consider a Heegaard splitting (V, W; F ) of an orientable, irreducible 3-manifold M . Let DV (F ) and DW (F ) be the subcomplexes of D(F ) spanned by compressing disks in V and W respectively. We call these subcomplexes the disk complexes of V and W. Let DVW (F ) be the subcomplex of D(F ) spanned by the simplices which have at least one vertex in D(V), and at least one vertex in D(W). In a weak reducing pair, if a disk belongs to V, then we call it a V-disk. Otherwise, we call it a W-disk. If a simplex Δ ⊂ DVW (F ) meets both DV (F ) and DW (F ), then we say Δ is non-trivially embedded in DVW (F ). If a non-trivially embedded 2-simplex Δ is determined by two disks from V and one disk from W, then we call it a V-face. Similarly, we can define a W-face. Lemma 2.6. (McCullough, [11].) DV (F ) and DW (F ) are contractible. Definition 2.7. Let (V, W ) and (V , W ) be two weak reducing pairs for a Heegaard surface F . If there is a finite sequence of weak reducing pairs written by (V0 = V, W0 = W ),
(V1 , W1 ),
...,
Vm = V , W m = W
such that 1. each (Vi+1 , Wi ) for i = 0, . . . , m − 1 is also a weak reducing pair, 2. for 1 i m, Vi−1 is disjoint from, or equal to, Vi , and 3. for 1 i m, Wi−1 is disjoint from, or equal to, Wi , then we say that two weak reducing pairs are connected by a chain of alternately compressible loops on F . Note that the chain is equivalent to a sequence of non-trivially embedded 2-simplices in DVW (F ) where adjacent 2-simplices share a weak reducing pair. Proposition 2.8. (Bachman, Lemma 8.5 of [2].) If there are two weak reducing pairs (V, W ) and (V , W ) not connected by a chain of alternately compressible loops on the Heegaard surface, then the surface is critical. Definition 2.9. Suppose W is a compressing disk for F ⊂ M . Then there is a subset of M that can be identified with W × I so that W = W × { 12 } and F ∩ (W × I) = (∂W ) × I. We form the surface FW , obtained by compressing F along W , by removing (∂W ) × I from F and replacing it with W × (∂I). We say the two disks W × (∂I) in FW are the scars of W . From now on, we will consider only unstabilized Heegaard splittings of an irreducible 3-manifold. If a Heegaard splitting of a compact 3-manifold is reducible, then the manifold is reducible or the splitting is stabilized [12]. Hence, we can exclude the possibilities of reducing pairs among weak reducing pairs. Lemma 2.10. (Lustig and Moriah, Lemma 1.1 of [10].) Suppose that M is an irreducible 3-manifold and (V, W; F ) is an unstabilized Heegaard splitting of M . If F is obtained by compressing F along a collection of disks, then no S 2 component of F can have scars from disks in both V and W.
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We introduce the following lemma. Lemma 2.11. (J. Kim, [7].) Suppose that M is an irreducible 3-manifold and (V, W; F ) is a genus three, unstabilized Heegaard splitting of M . If there exist three mutually disjoint compressing disks V , V ⊂ V and W ⊂ W, then either V is isotopic to V , or one of ∂V and ∂V bounds a punctured torus T in F and the other is a non-separating loop in T . Moreover, we cannot choose three weak reducing pairs (V0 , W ), (V1 , W ), and (V2 , W ) such that Vi and Vj are mutually disjoint and non-isotopic in V for i = j. Definition 2.12. Let us consider a 1-dimensional graph as follows. 1. We assign a vertex to each V-face in DVW (F ). 2. If a V-face shares a weak reducing pair with another V-face, then we assign an edge between these two vertices in this graph. We call this graph the graph of V-faces of DVW (F ). If there is a maximal subset FC V of V-faces in DVW (F ) representing a connected component of the graph of V-faces and the component is not an isolated vertex, then we call FC V a V-facial cluster. Similarly, we define the graph of W-faces and a W-facial cluster. In a V-facial cluster, every weak reducing pair gives the common W-disk, and vise versa. Lemma 2.13. Assume M and F as in Lemma 2.11. If there are two adjacent V-faces f1 represented by {V0 , V1 , W } and f2 represented by {V1 , V2 , W }, then ∂V1 is non-separating, and ∂V0 , ∂V2 are separating in F . Therefore, there is a unique weak reducing pair in a V-facial cluster which can belong to two or more faces in the V-facial cluster. Proof. By Lemma 2.11, we have the following two cases. Case 1. ∂V0 and ∂V2 are non-separating, and ∂V1 is separating in F . Case 2. ∂V0 and ∂V2 are separating, and ∂V1 is non-separating in F . We will show that only Case 2 holds. Suppose that Case 1 holds. Let FV1 = FV 1 ∪ FV1 , where g(FV 1 ) = 2 and g(FV1 ) = 1. Let V¯ be the part of V that FV1 bounds in V. If we consider the V-face f1 , then ∂V0 ⊂ FV1 ¯ Hence, we can compress F along the disk V0 in V¯ and get a by Lemma 2.11. This means that V0 ⊂ V. V1 2-sphere S. Since ∂− V does not have a 2-sphere component, the inside of S in V must be a 3-ball, i.e. V¯ is a ¯ If we use Lemma 2.11 for the V-face f2 , then V2 is also a meridian solid torus and V0 is a meridian disk of V. disk of V¯ similarly. Therefore, V0 is isotopic to V2 in V¯ and we can assume that this isotopy misses V1 . This means that V0 is isotopic to V2 in V, violating the assumption. 2 Definition 2.14. By Lemma 2.13, there is a unique weak reducing pair in a V-facial cluster belonging to two or more faces in the cluster. We call it the center of a V-facial cluster. We call the other weak reducing pairs hands of a V-facial cluster. See Fig. 1. Note that if a V-face f is represented by two weak reducing pairs, then one is the center and the other is a hand. Lemma 2.13 means that the V-disk from the center of a V-facial cluster is non-separating, and the V-disks from hands are all separating. Lemma 2.15. Assume M and F as in Lemma 2.11. Every V-face belongs to some V-facial cluster. Moreover, every V-facial cluster has infinitely many hands. Proof. Suppose that there is a V-face f in DVW (F ) represented by {V0 , V1 , W }. By Lemma 2.11, we can assume that ∂V0 is separating and ∂V1 is non-separating in F . If we compress F along V1 , then FV1 bounds V in V such that V ∩ V1 = ∅. Let V1 and V1 be the scars of V1 in FV1 . Let us consider two disks V¯1 and V¯1 such that V¯1 and V¯1 are obtained by extending V1 and V1 slightly on FV1 respectively and pushing the interior of each disk into V so that it is properly embedded in V (see Fig. 2), and an arc α connecting ∂ V¯1
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Fig. 1. An example of a V-facial cluster in DVW (F ). (V0 , W ) is the center and the other weak reducing pairs are hands.
¯ . Fig. 2. An arc α connecting ∂ V¯1 and ∂ V 1
Fig. 3. There are infinitely many isotopy classes of α.
and ∂ V¯1 on FV1 which misses ∂ V¯1 , ∂ V¯1 , and ∂W in its interior. If we perform the band sum of V¯1 and V¯1 along α in V , then we get a properly embedded disk V0α in V which is ∂-parallel in V . V0α becomes an essential separating disk in V and V0α misses V1 and W . Therefore, we get the V-face f represented by {V1 , V0α , W }. This means that f and f belong to the same V-facial cluster. We claim that there are infinitely many isotopy classes for such V0α in V. If we compress FV1 again along W , then we can see that the genus of each component of FV1 W must be one, otherwise we can find an S 2 -component in FV1 W having scars from both V1 and W , violating Lemma 2.10. Let F¯V1 W be the component of FV1 W containing α. Since g(F¯V1 W ) = 1, there are infinitely many such arcs up to isotopies fixing V¯1 and V¯1 (see Fig. 3). Moreover, if such arcs are non-isotopic, then the resulting disks become non-isotopic in V. 2 The next lemma describes the situation when a V-face and a W-face share a weak reducing pair. Lemma 2.16. Assume M and F as in Lemma 2.11. If there is a V-face f1 and a W-face f2 in DVW (F ) and one shares a weak reducing pair with the other, then the four disks coming from f1 and f2 are mutually
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disjoint or the V-facial cluster and the W-facial cluster containing f1 and f2 respectively have a common center. In the latter case, we can replace a separating disk among the four disks by a new separating disk in the same compression body so that the resulting four disks are mutually disjoint. Proof. Assume that f1 and f2 are represented by {V0 , V1 , W0 } and {V1 , W0 , W1 } respectively. Since V0 , V1 , and W0 are mutually disjoint and so are V1 , W0 , and W1 , we will prove that 1. ∂W1 ∩ ∂V0 = ∅, or 2. both V- and W-facial clusters guaranteed by Lemma 2.15 have a common center. Let the separating one be Vi and the non-separating one be Vj between V0 and V1 . Let FVi = FV i ∪ FVi , where g(FV i ) = 2 and g(FVi ) = 1. We already know that ∂W0 ⊂ FV i and ∂Vj ⊂ FVi by applying Lemma 2.11 to f1 . Case 1. ∂W1 ∩ ∂Vj = ∅, i.e. j = 0 (so i = 1). In this case, the existence of the weak reducing pair (V1 , W1 ) implies that ∂W1 ∩ ∂Vi = ∅. Moreover, ∂W1 ⊂ FVi from ∂W1 ∩ ∂Vj = ∅. But if ∂W1 is non-separating in F , then ∂W1 is essential in FVi , i.e. W1 and Vi determine an S 2 component of FVi W1 , violating Lemma 2.10. If ∂W1 is separating in F , then it must be isotopic to ∂Vi , violating the fact that every weak reducing pair is not a reducing pair. Case 2. ∂W1 ∩ ∂Vj = ∅. If j = 0, then this leads to the result. Therefore, assume that j = 1 (so i = 0). If we apply Lemma 2.11 to the W-face f2 , then we can assume that ∂Wk is the separating one and ∂Wl be the non-separating one for {k, l} = {0, 1}. Here, ∂V1 (∂Wl resp.) goes into the genus two component (the genus one component resp.) of FWk again by Lemma 2.11. Case 2-1. ∂W1 ∩ ∂Vi = ∅. This case means that ∂W1 ∩ ∂V0 = ∅, i.e. this leads to the result. Case 2-2. ∂W1 ∩ ∂Vi = ∅. Suppose that l = 1 (so k = 0). Let FWk = FW ∪ FW , where g(FW ) = 2 and g(FW ) = 1. In this case, the k k k k existence of the weak reducing pair (V0 , W0 ) implies that ∂Wk ∩ ∂Vi = ∅. Here, ∂Vi must go into FW (if k ∂Vi ⊂ FWk , then it must be parallel to ∂Wk since ∂Vi is separating in F and g(FWk ) = 1). Since ∂Wl ∩∂Vi = ∅ from the assumption of Case 2-2, ∂Wl also goes into FW , violating Lemma 2.11. Therefore, we get l = 0 k (so k = 1), i.e. both ∂V1 and ∂W0 (∂V0 and ∂W1 resp.) are non-separating (separating resp.) in F . Let FC V and FC W be the V-facial cluster and the W-facial cluster containing f1 and f2 respectively guaranteed by Lemma 2.15. Since the V-disk (W-disk resp.) for the center of FC V (FC W resp.) is non-separating in V (W resp.) by Lemma 2.13, the weak reducing pair (V0 , W0 ) ((V1 , W1 ) resp.) cannot be the center of FC V (FC W resp.). Therefore, (V1 , W0 ) is the common center of both FC V and FC W . Now we will prove that there is another W-face f3 represented by {V1 , W0 , W2 } in FC W such that W2 ∩ V0 = ∅. If we compress F along V0 , then FV0 = FV 0 ∪ FV0 where g(FV 0 ) = 2 and g(FV0 ) = 1. We get ∂W0 ⊂ FV 0 and ∂V1 ⊂ FV0 by applying Lemma 2.11 to the V-face f1 . If we compress FV 0 again along W0 , then we get a torus F¯V0 W0 . Here, there are two scars W0 and W0 of W0 and one scar V0 of V0 in F¯V0 W0 . If we use the band sum argument by using an arc connecting W0 and W0 missing ∂W0 , ∂W0 , and ∂V0 in its interior as in the proof of Lemma 2.15, then we can find an essential separating disk W2 in W such that W2 ∩ V0 = W2 ∩ V1 = W2 ∩ W0 = ∅. Since we already knew that V0 , V1 , and W0 are mutually disjoint by the existence of the V-face f1 , W2 ∩ V0 = ∅ means that V0 , V1 , W0 , and W2 are mutually disjoint. 2 Note that Lemma 2.16 implies that if there is a V-face and a W-face in DVW (F ) and one shares a weak reducing pair with the other, then there is a weak reducing pair consisting of non-separating disks and a weak reducing pair consisting of separating disks. Indeed, we can imagine a situation as in Fig. 4 if there
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Fig. 4. An example of the boundaries of the resulting four disks in Lemma 2.16.
exists a V-face and a W-face sharing a weak reducing pair and the corresponding four disks are mutually disjoint. The thick curves are separating and the thin curves are non-separating in F . Here, we introduce the key theorem of this section. Theorem 2.17. Assume M and F as in Lemma 2.11. Suppose that we cannot choose a weak reducing pair ¯ ) such that V¯ cuts off a solid torus from V and W ¯ also cuts off a solid torus from W. If we can choose (V¯ , W two weak reducing pairs (V, W ) and (V , W ) such that ∂V ∩ ∂W = ∅ and ∂V ∩ ∂W = ∅ up to isotopy, then F is critical. Proof. At first, we give the following claim. Claim. Suppose that no V-face shares a weak reducing pair with a W-face in DVW (F ). If we can choose two weak reducing pairs (V, W ) and (V , W ) such that ∂V ∩ ∂W = ∅ and ∂V ∩ ∂W = ∅ up to isotopy, then F is critical. Proof of Claim. Suppose that (V, W ) and (V , W ) are connected by a chain of alternately compressible loops on F . That is, there is a finite sequence of weak reducing pairs, (V0 = V, W0 = W ),
(V1 , W1 ),
...,
Vm = V , W m = W
such that 1. each (Vi+1 , Wi ) for i = 0, . . . , m − 1 is also a weak reducing pair, 2. for 1 i m, Vi−1 is disjoint from, or equal to, Vi , and 3. for 1 i m, Wi−1 is disjoint from, or equal to, Wi . By reading the above sequence from the left to the right, we get a sequence of V- and W-faces Δ0 , . . . , Δn such that 1. (V0 , W0 ) ⊂ Δ0 and (Vm , Wm ) ⊂ Δn , 2. Δi−1 shares a weak reducing pair with Δi for i = 1, . . . , n. Note that the assumption V ∩ W = ∅ and V ∩ W = ∅ implies that Δ0 = Δn . Here, Lemma 2.15 divides the proof into the following three cases. Case 1. Δ0 and Δn are contained in the same V-facial cluster. In this case, W0 = Wm . But the weak reducing pair (V0 , W0 ) means that ∂V0 ∩ ∂W0 = ∅, i.e. ∂V0 ∩ ∂Wm = ∂V ∩ ∂W = ∅, violating the assumption of Claim.
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Case 2. Δ0 belongs to some V-facial cluster and Δn belongs to another V-facial cluster. Since these V-facial clusters are different, there must be a W-face among Δ1 , . . . , Δn−1 . Therefore, we can assume that Δi is a V-face and Δi+1 is a W-face for some 0 i n − 2. But this violates the assumption of Claim. Case 3. Δ0 belongs to some V-facial cluster and Δn belongs to some W-facial cluster. Since two clusters are of different types, we can assume that Δi is a V-face and Δi+1 is a W-face for some 0 i n − 2. This gives a contradiction as in Case 2. Therefore, (V, W ) and (V , W ) are not connected by a chain of alternately compressible loops on F . Hence, Proposition 2.8 leads to the result. 2 We claim that no V-face shares a weak reducing pair with a W-face. Suppose that some V-face f1 shares ¯, a weak reducing pair with a W-face f2 . In this case, we can assume that the four disks V¯ , V˜ ⊂ V, and W ˜ ¯ W ⊂ W coming from f1 and f2 are mutually disjoint by Lemma 2.16. Indeed, we can assume that V and ¯ are separating and V˜ and W ˜ are non-separating by Lemma 2.11. W ¯ = ∅ and ∂ V¯ is not isotopic to ∂ W ¯ , ∂ V¯ ∪ ∂ W ¯ cuts F into three pieces F , F , and F , Since V¯ ∩ W ¯ resp.) in F and F is the where F (F resp.) is the once-punctured torus determined by only ∂ V¯ (∂ W ¯ ¯ ¯ ¯ twice-punctured torus determined by both ∂ V and ∂ W in F . Here, V (W resp.) cuts off a solid torus or T 2 × I from V (W resp.) where T 2 is a torus. By the assumption of this theorem, we can assume that at ¯ , say V¯ , cuts off T 2 × I from V, say V . Let T be the torus component of ∂− V which least one of V¯ and W realizes the product V . Our assumption means that V is the part of V which F ∪ V¯ bounds in V. Since ∂ V˜ is a non-separating curve of F from Lemma 2.11, V˜ is a compressing disk of F . Since V¯ is a separating disk in V and V˜ is disjoint from V¯ , we get V˜ ⊂ V , i.e. F is compressible in V ∼ = T 2 × I, this is a contradiction. Hence, Claim leads to the result. This completes the proof of Theorem 2.17. 2 Finally, we reach Theorem 2.18. Theorem 2.18. Let L be a non-splittable, tunnel number two, three-component link in S 3 and (V, W; F ) be the induced Heegaard splitting of the exterior of L. If we can choose two weak reducing pairs (V, W ) and (V , W ) such that ∂V ∩ ∂W = ∅ and ∂V ∩ ∂W = ∅ up to isotopy, then F is critical. Proof. Let T1 , T2 , and T3 be the three boundary components of the exterior of L. Assume that V is the compression body containing T1 , T2 and T3 and the two tunnels, and W is the handlebody S 3 − V. Since the tunnel number of L is two and the number of components of L is three, the induced Heegaard splitting (V, W; F ) is of genus three and unstabilized. Moreover, every separating disk V¯ ⊂ V must divide V into two compression bodies with non-empty minus boundaries. This means that we cannot choose a weak reducing ¯ ) such that V¯ cuts off a solid torus from V and W ¯ also cuts off a solid torus from W. Hence, pair (V¯ , W Theorem 2.17 leads to the result. 2 For example, the tunnel system of the three component chain in Fig. 5 induces a critical Heegaard splitting, where V and V come from the cocore disks of the tunnel a1 and a2 respectively. Note that Sedgwick proved that this splitting is not of minimal genus in [16]. 3. The proof of Theorem 1.2 In this section, we give the proof of Theorem 1.2. First, we consider the following lemmas (see [14] for the definition of an “amalgamation”).
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Fig. 5. A tunnel system of the three component chain.
Fig. 6. If we amalgamate two thick surfaces along a torus, then we can find a weak reducing pair such that the disks are separating.
Lemma 3.1. Let M be a closed 3-manifold and (V, W; F ) be a genus three Heegaard splitting of M . Then F has a non-reducing, weak reducing pair consisting of separating disks if and only if M = M1 ∪T M2 , where each Mi has a genus two Heegaard splitting for i = 1, 2, T is a torus, and F can be represented as an amalgamation of these genus two splittings along T . Proof. (⇐) Let the Heegaard splitting of M1 (M2 resp.) be (V+ , W+ ; F1 ) ((V− , W− ; F2 ) resp.), and W+ ∩ V− = T . Hence, W+ is homeomorphic to (T × I) ∪ (a 1-handle) and also V− is. Let the 1-handles for W+ and V− be hW+ and hV− respectively. If we take T to be the common 0-level of two T × I’s belonging to W+ and V− , then the attaching disks for hW+ and hV− go into the 1-levels in these products. In order to do the amalgamation, isotope the attaching disks of hW+ and hV− so that they are disjoint after the projection into T (see Fig. 6). Note that the projection function may also be changed continuously during this isotopy so that the projection images of these attaching disks do not intersect one another. Let us consider the curves CW+ and CV− in T such that CW+ (CV− resp.) bounds a disk in T and this disk contains the projection images of the attaching disks of hW+ (hV− resp.) in its interior. Make sure that the disk for CW+ is disjoint from that of CV− . Here, we can find the curve C¯W+ in ∂+ W+ (C¯V− in ∂+ V− resp.) as the preimage of the projection for CW+ (CV− resp.). Consider an essential disk W ⊂ W+ (V ⊂ V− resp.) such that ∂W = C¯W+ (∂V = C¯V− resp.) like Fig. 6. Obviously, V and W are separating in their compression bodies. After the amalgamation, V and W become disjoint separating disks and the boundary of one is not isotopic to that of the other in the resulting Heegaard surface. (⇒) Suppose that there is a non-reducing, weak reducing pair (V, W ) consisting of separating disks. Now FV W consists of three tori FV W , FV W , and FVW , where FV W (FVW resp.) bounds a solid torus V in V (W in W resp.). Find a meridian disk V (W resp.) in V (W resp.) such that V (W resp.) is
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disjoint from V (W resp.). Since V ∩ W = ∅, (V , W ) is a weak reducing pair. Let T be the surface FV W . Then F is obtained from T by attaching boundaries of two 1-handles whose cocores are V and W and removing the interiors of the attaching disks of these handles. Now we get the right one of Fig. 6. Therefore, if we weakly reduce the splitting (V, W; F ) along (V , W ), then we get the generalized Heegaard splitting {∅, T1 , t1 , T2 , ∅}, where the thick surfaces T1 = FW and T2 = FV are of genus two and the unique non-empty thin surface t1 = FV W is a torus (see [12] and [13] for the definition of a “generalized Heegaard splitting”). 2 Lemma 3.2. Suppose that M is an irreducible 3-manifold and (V, W; F ) is a genus three, unstabilized Heegaard splitting of M . If there is a V-face in DVW (F ) whose W-disk is non-separating, then DVW (F ) must have a V-face and a W-face such that one shares a weak reducing pair with the other. Proof. Suppose that there is a V-face f1 represented by {V0 , V1 , W }, ∂V0 is separating, ∂V1 is non-separating, and ∂W is non-separating in F by Lemma 2.11 and the assumption of this lemma. Let FV 0 be the genus two component of FV0 . We can check that ∂W ⊂ FV 0 and ∂V1 ∩ FV 0 = ∅ by Lemma 2.11. If we compress FV 0 along W again, then (FV 0 )W is a torus. Let W and W be the scars of W0 and V0 be the scar of V0 in (FV 0 )W . If we using the band sum arguments by using an arc connecting W and W missing ∂W , ∂W , and ∂V0 in its interior as in the proof of Lemma 2.15, we get a family {Wα } of infinitely many essential separating disks in W which miss W and V0 . If we choose a disk Wα from {Wα }, then we get a W-face f2 represented by {V0 , Wα , W }. Therefore, f1 and f2 are the wanted V- and W-faces such that one shares a weak reducing pair with the other. 2 Corollary 3.3. Assume M and F as in Lemma 3.2, and add the assumption that M is closed. If there is a V- or W-face in DVW (F ), then DVW (F ) must have a V-face and a W-face such that one shares a weak reducing pair with the other. Proof. Without loss of generality, suppose that there is a V-face f1 represented by {V0 , V1 , W } in DVW (F ), where ∂V0 is separating and ∂V1 is non-separating in F by Lemma 2.11. If ∂W is non-separating in F , then this leads to the result by Lemma 3.2. Therefore, assume that ∂W is separating in F , i.e. W is separating in W. Hence, W cuts W into a solid torus W and a genus two handlebody, where W is bounded by F˜ ∪ W in W. (F˜ is the once-punctured torus which ∂W cuts off from F .) Let Wm be a meridian disk of W . Isotope Wm in W so that ∂Wm is contained in the interior of F˜ . Let F¯ be the once-punctured genus two component of F − ∂V0 . Since ∂W ⊂ F¯ by Lemma 2.11 and ∂W cuts F¯ into a twice-punctured torus and a once-punctured torus (otherwise, ∂W is isotopic to ∂V0 in F¯ ), we get either F˜ ⊂ F¯ or F˜ is of genus two. The latter case violates the assumption that F˜ is a once-punctured torus. Hence, ∂V0 ∩ ∂Wm = ∅, i.e. we get a W-face f2 represented by {V0 , Wm , W }. Therefore, f1 and f2 are the wanted V- and W-faces such that one shares a weak reducing pair with the other. 2 Now we get the following three equivalent statements when M is closed. Corollary 3.4. Let M be a closed, orientable, irreducible 3-manifold and (V, W; F ) be an unstabilized, weakly reducible, genus three Heegaard splitting. Then the following three statements are equivalent. (1) There is no weak reducing pair consisting of separating disks. (2) F cannot be represented as an amalgamation of genus two splittings along a torus. (3) If there are different weak reducing pairs, then they do not share a disk. Proof. Since M is irreducible and (V, W; F ) is unstabilized, a weak reducing pair cannot be a reducing one, i.e. (1) and (2) are equivalent by Lemma 3.1. Hence, we will prove that (1) and (3) are equivalent.
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(1) ⇒ (3) If there are two weak reducing pairs sharing a disk, then this means that there is a V- or W-face in DVW (F ) (Lemma 8.4 of [2]). Hence, DVW (F ) must have a V-face and a W-face such that one shares a weak reducing pair with the other by Corollary 3.3. Therefore, if we use Lemma 2.16, then we get a weak reducing pair consisting of separating disks. (3) ⇒ (1) Suppose that there is a weak reducing pair (V, W ) consisting of separating disks. Let the solid torus which V cuts off from V be V and Vm be a meridian disk of V . If we isotope Vm in V so that ∂Vm misses V , then we get a V-face represented by {V, Vm , W }, so the weak reducing pairs (V, W ) and (Vm , W ) share a disk. 2 Finally, we reach Theorem 3.5. Theorem 3.5. Suppose F is an unstabilized, genus three, topologically minimal Heegaard surface in a closed, orientable, irreducible 3-manifold. If F is not an amalgamation of two genus 2 Heegaard surfaces along a torus, then the topological index of F is at most two. Proof. Let the Heegaard splitting be (V, W; F ). If F is strongly irreducible, then the topological index of F is one, leading to the result. Assume that F is weakly reducible. Suppose that the topological index of F is at least three. Then there is a non-trivial embedding ι : S n → D(F ), where n 2. Triangulate S n so that ι would be simplicial. Then we can assume that ι(S n ) is a finite union of n-cells. Since the embedding is non-trivial and both DV (F ) and DW (F ) are contractible, there must be an n-cell Δn containing at least one vertex in DV (F ) and at least one vertex in DW (F ). Since n 2, we can assume that there are two weak reducing pairs sharing a disk by using a vertex of Δn from DV (F ) and two vertices from DW (F ), or vise versa. Hence, Corollary 3.4 means that F can be represented as an amalgamation of genus two splittings along a torus, violating the assumption. 2 Note that Johnson proved that a weak reducing pair of the standard genus three Heegaard splitting of T 3 must consist of only non-separating disks in Lemma 6 of [6]. Moreover, he proved that the thin surface obtained by untelescoping the splitting along a weak reducing pair must consist of two tori, not a torus in Lemma 5 of [6]. This is a good example of the Heegaard splitting satisfying the assumption of Theorem 3.5. Indeed, this splitting is critical by Theorem 2.17. Acknowledgements The author is grateful to the referee for a number of helpful suggestions for improvement in the paper. The author is grateful to Prof. Jung-Hoon Lee for reading this article and giving me helpful advices. The author is grateful to E. Qiang for giving me an advice for polishing the proof of Lemma 2.16. The author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (NRF-2010-0019516). References [1] [2] [3] [4]
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