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Unstabilized self-amalgamation of a Heegaard splitting
Topology and its Applications 160 (2013) 406–411
Contents lists available at SciVerse ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Unstabilized self-amalgamation of a Heegaard splitting ✩ Yanqing Zou a,∗ , Kun Du b , Qilong Guo a , Ruifeng Qiu c a b c
School of Mathematical Sciences, Dalian University of Technology, Dalian 116023, China School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China Department of Mathematics, East China Normal University, Shanghai 200241, China
a r t i c l e
i n f o
Article history: Received 22 April 2012 Accepted 28 November 2012 Keywords: Heegaard splitting Distance Stabilization
a b s t r a c t Let M be a compact orientable 3-manifold, M = V ∪ S W be a Heegaard splitting of M, and F 1 , F 2 be two homeomorphic components of ∂ M lying in the minus boundary of W . Let M ∗ be the manifold obtained from M by gluing F 1 and F 2 together. Then M ∗ has a natural Heegaard splitting called the self-amalgamation of V ∪ S W . In this paper, we prove that the self-amalgamation of a distance at least 3 Heegaard splitting is unstabilized. There are some examples to show that the lower bound 3 is the best. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Let M be a compact orientable 3-manifold. If there is a closed surface S which cuts M into two compression bodies V and W so that S = ∂+ V = ∂+ W , then we say that M has a Heegaard splitting, denoted by M = V ∪ S W . In this case, S is called a Heegaard surface, and g ( S ) is called the genus of the Heegaard splitting. A Heegaard splitting M = V ∪ S W is said to be stabilized if there are essential disks D in V and E in W such that D intersects E in just one point; otherwise, it is said to be unstabilized. M = V ∪ S W is said to be reducible if there is an essential simple closed curve on S which bounds disks in both V and W ; otherwise, it is said to be irreducible. M = V ∪ S W is said to be weakly reducible if there is an essential disk D in V and E in W such that D ∩ E = ∅; otherwise, it is said to be strongly irreducible. For any two essential simple closed curves α and β on S, d(α , β) is defined to be the smallest integer n 0 so that there is a sequence of essential simple closed curves α0 = α , . . . , αn = β on X such that αi −1 is disjoint from αi for 1 i n. The distance of the Heegaard surface S, denoted by d( S ), is defined to be min{d(α , β)}, where α bounds an essential disk in V and β bounds an essential disk in W . See [2,5,6]. Let F be a properly embedded surface in a compact orientable 3-manifold M. F is said to be compressible if either F is a 2-sphere which bounds a 3-ball or there is an essential simple closed curve on F which bounds a disk in F ; otherwise, F is said to be incompressible. An incompressible surface not parallel to ∂ M is said to be essential. Let M be a compact orientable 3-manifold which contains an essential closed surface F . We first assume that F is a 2-sphere. Then M is said to be reducible. By Kneser–Milnor’s theorem, M is the connected sum of n manifolds M 1 , . . . , M n , where M i is either irreducible or S 2 × S 1 . By Haken’s lemma, any Heegaard splitting of M is reducible. See [5]. Now if M i admits a Heegaard splitting V i ∪ S i W i for each 1 i n, then M admits a natural Heegaard splitting V ∪ S W called a connect sum of n Heegaard splittings V i ∪ S i W i . It’s well known that the connect sum of unstabilized V i ∪ S i W i is unstabilized. See [1] and [12].
✩ This work is supported by NSFC (11171108). The first author and the third author are also supported by a grant (No. 11271058/a010402) of NSFC. The second author is also supported by the Fundamental Research Funds for the Central Universities (No. lzujbky-2011-124). Corresponding author. E-mail addresses: [email protected] (Y. Zou), [email protected] (K. Du), [email protected] (Q. Guo), [email protected] (R. Qiu).
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0166-8641/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2012.11.020
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Fig. 1. The way of gluing.
We assume now that F is a separating surface with g ( F ) 1. Then F cuts M into two manifolds M 1 and M 2 . In other words, M is obtained by gluing M 1 and M 2 along a homeomorphism f from F 1 to F 2 . Now if M i admits a Heegaard splitting V i ∪ S i W i for each 1 i 2, M has a natural Heegaard splitting called the amalgamation of V 1 ∪ S 1 W 1 and V 2 ∪ S 2 W 2 . A question on amalgamation is: Question 1. Is the amalgamation of two unstabilized Heegaard splittings unstabilized? It is possible that the amalgamation of two unstabilized Heegaard splittings can be destabilized many times simultaneously, see [7] and [14]. On the other side, (1) the amalgamation of two high distance Heegaard splittings is unstabilized, see [8,15,3]; and (2) if the gluing map f is complicated enough, then the amalgamation of two minimal Heegaard splittings of M 1 and M 2 is unstabilized. See [9–11,13]. Finally we assume that F is a non-separating surface M with g ( F ) 1. Now let F × I be the regular neighborhood of F in M and M 1 = M − F × (0, 1). Then M is obtained from M 1 by gluing F × {0} and F × {1} along a homeomorphism f from F × {0} to F × {1}. Let M 1 = V 1 ∪ S 1 W 1 be a Heegaard splitting. Then M has a natural Heegaard splitting called the self-amalgamation of V 1 ∪ S 1 W 1 , see Section 2. A question on self-amalgamation is the following: Question 2. Is the self-amalgamation of an unstabilized Heegaard splitting unstabilized? It is possible that the self-amalgamation of an unstabilized Heegaard splitting is stabilized. On the other side, the selfamalgamation of a high distance Heegaard splitting is unstabilized, see [4]. However, all the known examples on stabilized amalgamations and self-amalgamations of Heegaard splittings have one of its summands with Heegaard distance at most 2. All the known results on unstabilized amalgamations and selfamalgamations of Heegaard splittings have the complexity (high distance or complicated gluing map) strongly depending on manifolds. Hence a natural question arises: Question 3. Is there a constant k such that: (1) the amalgamation of two distance at least k Heegaard splittings is unstabilized? (2) the self-amalgamation of a distance at least k Heegaard splitting is unstabilized? In this paper, we give a positive answer to Question 3(2). The main result is the following: Theorem 1. The self-amalgamation of a distance at least 3 Heegaard splitting is unstabilized. It is possible that the self-amalgamation of a strongly irreducible Heegaard splitting is stabilized, see [4]. Hence the lower bound 3 in Theorem 1 is the best. 2. The proof of Theorem 1 Let M be a compact orientable 3-manifold with homeomorphic boundary components F 1 and F 2 , and M = V ∪ S W be a Heegaard splitting such that F 1 , F 2 ⊂ ∂− W . Let M ∗ be the manifold obtained from M by gluing F 1 and F 2 along a homeomorphism f from F 1 to F 2 . Then M ∗ has a Heegaard splitting called the self-amalgamation of V ∗ ∪ S ∗ W ∗ as follows: Let p i be a point on F i such that f ( p 1 ) = p 2 . Note that W is obtained by attaching 1-handles h1 , . . . , hm to ∂− W × I . Let αi = p i × I , αi × D be the regular neighborhood of αi for i = 1, 2. We may assume that αi × D is disjoint from the 2 1-handles h1 , . . . , hm , and f ( p 1 × D ) = p 2 × D. Let W be the closure of W − i =1 αi × D. Now, in the closure of M ∗ − V , the arc α = α1 ∪ α2 has a regular neighborhood α × D which intersects ∂+ V = S in two disks D 1 and D 2 . We denote by p the point p i , D the disk p × D ⊂ α × D, and F the surface F i in M ∗ . See Fig. 1 and Fig. 2. Let V ∗ = V ∪ α × D and W ∗ be the closure of M ∗ − V . Then V ∗ and W ∗ are compression bodies. Hence M ∗ = V ∗ ∪ S ∗ W ∗ is a Heegaard splitting called the self-amalgamation of V ∪ S W . Let S 1 be the surface S − int D 1 ∪ int D 2 . Then S 1 is a sub-surface of S with two boundary components ∂ D 1 and ∂ D 2 .
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Fig. 2. Construction of V ∗ from V .
Lemma 2.1. F − int D is incompressible in W ∗ . from W by gluing F 1 − int D and F 2 − int D along the homeomorphism f , Proof. By the construction, W ∗ is obtained 2 where W is the closure of W − i =1 αi × D. In this case, F − int D = F 2 − int D = f ( F 1 − int D ). Hence we only need to prove that both F 1 − int D and F 2 − int D are incompressible in W . Suppose that one of F 1 − int D and F 2 − int D, say F 1 − int D, is compressible in W . Then there is an essential simple closed curve C on F 1 − int D which bounds a disk in W . Since F 1 is incompressible in W , C is isotopic to ∂ D on F 1 − int D. Hence there is a 2-sphere in W such that S = ∂+ W and F 1 lie in distinct sides of the sphere. This means that W is reducible, a contradiction. 2 Let α be a simple closed curve or a properly embedded arc in S 1 , and N (α ∪ ∂ S 1 ) be the regular neighborhood of α ∪ ∂ S 1 in S 1 . Then α is said to be strongly essential if one component of ∂ N (α ∪ ∂ S 1 ) is essential in S. Since S = S 1 ∪ D 1 ∪ D 2 , we have the following observations: Lemma 2.2. A simple closed curve α on S 1 is strongly essential if and only if α is essential on S. Lemma 2.3. A properly embedded arc α on S 1 is strongly essential if and only if (1) the two end points of α lie in one component of ∂ S 1 , say ∂ D j , and (2) α is an essential arc on S 1 ∪ D k , where { j , k} = {1, 2}. In order to obtain a contradiction, in the following argument, we assume that M ∗ = V ∗ ∪ S ∗ W ∗ is stabilized. This means that there are essential disks B ⊂ V ∗ and E ⊂ W ∗ such that B intersects E in one point x. Furthermore, we may assume that | B ∩ D | + | E ∩ ( F − int D )| is minimal among all stabilizing pairs of disks. By Lemma 2.1 and standard arguments, we have the following observations: Lemma 2.4. (1) Each component of B ∩ D is a properly embedded arc in both B and D, and each component of E ∩ ( F − int D ) is a properly embedded arc in both E and F − int D. (2) Each component of ∂ B ∩ S 1 is essential on S 1 , and each component of ∂ E ∩ S 1 is essential on S 1 . (3) Each component of B ∩ α × ∂ D is an arc with its two end points lying in distinct boundary components of α × ∂ D, and |∂ B ∩ ∂ D | = |∂ B ∩ ∂ D 1 | = |∂ B ∩ ∂ D 2 |. (4) Each component of E ∩ α × ∂ D is an arc with its two end points lying in distinct boundary components of α × ∂ D, and |∂ E ∩ ∂ D | = |∂ E ∩ ∂ D 1 | = |∂ B ∩ ∂ E 2 |. Suppose first that B ∩ ( D 1 ∪ D 2 ) = ∅. Let β be an outermost component of B ∩ ( D 1 ∪ D 2 ) on B. This means that β , together with an arc β ∗ on ∂ B, bounds a disk in B, say B β , such that B β intersects D 1 ∪ D 2 in β . In this case, D β is called an outermost disk related to β , and β ∗ is called an outermost arc related to β . Lemma 2.5. (1) β ∗ ⊂ S 1 , and the two end points of β ∗ lie in one of D 1 and D 2 . (2) D β ⊂ V . Proof. Since β is an outermost component of B ∩ ( D 1 ∪ D 2 ) on B, the lemma follows immediately.
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Fig. 3. Disks bounded by ∂ S 1 and β ∗ .
Lemma 2.6. Each outermost arc of ∂ B ∩ S 1 is strongly essential on S 1 . Proof. Suppose, otherwise, that one outermost arc of ∂ B ∩ S 1 , say β ∗ , related to an outermost arc of B ∩ ( D 1 ∪ D 2 ), say β , is not strongly essential on S 1 . By Lemma 2.5, we may assume that ∂β = ∂β ∗ ⊂ ∂ D 1 . By Lemma 2.3(2), β ∗ , together with one component of ∂ D 1 − β ∗ , bounds a disk D ∗ in S 1 such that D 2 ⊂ int D ∗ . See Fig. 3. Then each component of ∂ B ∩ S 1 incident to ∂ D 2 has the other end point lying in ∂ D 1 . We let m be the number of end points of the components of ∂ B ∩ S 1 lying in ∂ D 2 , and n the number of end points of the components of ∂ B ∩ S 1 lying in ∂ D 2 . Since the two end points of β ∗ lie in ∂ D 1 , n m + 2. On the other side, by Lemma 2.4(3), n = ∂ B ∩ ∂ D 1 = ∂ B ∩ ∂ D 2 = m, a contradiction. 2 Suppose now that E ∩ ( F − int D ) = ∅. Let β be an outermost component of E ∩ ( F − int D ) on E. This means that γ , together with an arc γ ∗ in S 1 , and two arcs in (α1 ∪ α2 ) × ∂ D, bounds a disk in E, say E γ , such that E γ intersects F − int D in γ . In this case, E γ ⊂ W where W is the closure of W − N (α1 ∪ α2 ). Now E γ is called an outermost disk related to γ , and γ ∗ is called an outermost arc related to γ . Lemma 2.7. (1) The two end points of γ ∗ lie in one of ∂ D 1 and ∂ D 2 . (2) E γ ⊂ W . Proof. Since int E γ is disjoint from F − int D, the lemma holds.
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Similar to the proof of Lemma 2.6, we have the following lemma. Lemma 2.8. Each outermost arc of ∂ E ∩ S 1 is strongly essential on S 1 . Proof of Theorem 1. Recall the stabilizing pair of disks ( B , E ) and the assumption that V ∪ S W has distance at least 3. There are three possibilities: Case 1. E ∩ ( F − int D ) = ∅. By Lemma 2.4, ∂ E ⊂ S 1 . Hence x = ∂ E ∩ ∂ B ⊂ S 1 . Now let B ∗ be the component of B ∩ V such that x ∈ B ∗ . Then ( B ∗ , E ) is a stabilizing pair of disks of V ∪ S W , a contradiction. Case 2. B ∩ D = ∅. In this case, we may assume that ∂ B ⊂ S 1 and B ⊂ V . We first prove that ∂ B is essential on S. Suppose that ∂ B is inessential on S. Then either ∂ B is isotopic to one of ∂ D 1 and ∂ D 2 , say ∂ D 1 , or ∂ E bounds a pair of pants together ∂ D 1 and ∂ D 2 . In the first sub-case, ∂ E intersects ∂ D 1 in one point. In the second sub-case, ∂ B is separating in S ∗ , and ∂ E intersects ∂ B in even number of points. All are impossible. Let γ be an outermost component of E ∩ ( F − int D ) on E, E γ be the outermost disk related to γ , and γ ∗ be the outermost arc related to γ . Since B intersects E in only one point, and E ∩ ( F − int D ) contains at least two outermost components. We may assume that E γ ∩ B = ∅. By Lemma 2.7, we may assume that ∂ γ ∗ ⊂ ∂ D = ∂ D 1 . Then ∂ E γ is disjoint from α2 × ∂ D. Now let W = W ∪ α2 × D. Then E γ is a properly embedded disk in W such that ∂ E γ intersects F 1 − int D in γ , α1 × D in two arcs, and S 1 in γ ∗ . There are two sub-cases: Case 2.1.
γ is inessential on F 1 − int D.
Now after an isotopy on ∂ E γ in S ∗ , ∂ E γ ⊂ S 1 . By Lemma 2.8, ∂ E γ is essential in S. Since ∂ E γ ∩ ∂ B = ∅, d( S ) 1. A contradiction.
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Fig. 4. Outermost disk and outermost arc in E γ .
Fig. 5. Possible orders of endpoints.
Case 2.2.
γ is essential on F 1 − int D.
By the construction of V ∗ ∪ S ∗ W ∗ , there is an essential disk E ∗ in W such that E ∗ is disjoint from α1 × D ∪ α2 × D. Furthermore, we assume that | E ∗ ∩ E γ | is minimal among all such disks. Now we claim that E ∗ ∩ E γ = ∅. Suppose that E ∗ ∩ E γ = ∅. By the minimality of | E ∗ ∩ E γ |, each component of E ∗ ∩ E γ is a properly embedded arc on both E ∗ and E γ . Since E ∗ is disjoint from α1 × D ∪ α2 × D, each component of E ∗ ∩ E γ has its two end points lying in γ ∗ . See Fig. 4. Let b be an outermost arc of E ∗ ∩ E γ on E γ such that b, together with a sub-arc of γ ∗ , bounds a disk E b with int E b ∩ E ∗ = ∅, where ∂ E b ∩ ∂ D = ∅. See Fig. 4. We denote by E 1∗ and E 2∗ the two components of E ∗ − b. Then E 1∗ ∪ E b and E 2∗ ∪ E b are two properly embedded disks in W disjoint from α1 × D ∪ α2 × D. Since E ∗ is essential in W , one of E 1∗ ∪ E b and E 2∗ ∪ E b , say E ∗∗ , is essential. But E ∗∗ can be pushed slightly so that | E ∗∗ ∩ E γ | = | E ∗ ∩ E γ | − 1. This contradicts the minimality of | E ∗ ∩ E γ |. Now γ ∗ , together with one component of ∂ D 1 − γ ∗ , forms a simple closed curve, say γ . By Lemma 2.8, γ is essential on S. Note that ∂ B ∩ γ = ∅ and γ ∩ ∂ E ∗ = ∅. Hence d( S ) 2, a contradiction. Case 3. B ∩ D = ∅, and E ∩ ( F − int D ) = ∅. Since B intersects E in only one point, and B ∩ D contains at least two outermost arcs on B, we may assume that β is an outermost arc of B ∩ D such that B β ∩ E = ∅, where B β is the outermost disk related to β . By Lemma 2.6, B β is an essential disk in V . Similarly, let γ be an outermost arc of E ∩ ( F − int D ) such that E γ ∩ B = ∅. Hence B β ∩ E γ = ∅. Now we consider the outermost arc β ∗ related to β and the outermost arc γ ∗ related to γ . By Lemmas 2.5 and 2.7, there are two sub-cases: Case 3.1. ∂β ∗ , ∂ γ ∗ ⊂ ∂ D j , where j = 1 or 2. We may assume that j = 1. Now β is an arc on F 1 − int D in W , where W is the closure of W − α1 × D ∪ α2 × D. Furthermore, E γ is disjoint from α2 × ∂ D. Now W = W ∪ α2 × D. Then E γ is a properly embedded disk in W . We first suppose that γ is inessential on F 1 − int D. Then ∂ E γ can be isotoped so that ∂ E γ ⊂ S 1 . Furthermore, either ∂ E γ ∩ ∂ B β = ∅ as in Fig. 5(a), or ∂ E γ intersects ∂ B β in one point as in Fig. 5(b) according to the order of the end points of β ∗ and γ ∗ . By Lemma 2.8, ∂ E γ is essential on S. Hence V ∪ S W is weakly reducible or stabilized, a contradiction. Suppose now that γ is essential on F 1 − int D. By the argument in Case 2.2, there is an essential disk E ∗ in W such that E ∗ is disjoint from α1 × D ∪ α2 × D and γ ∗ . Let N (γ ∗ ∪ ∂ D 1 ) be a regular neighborhood of γ ∗ ∪ ∂ D 1 on S 1 disjoint
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from E ∗ . By Lemma 2.8, one component of ∂ N (γ ∗ ∪ ∂ D 1 ) is an essential simple closed curve in S, say ∂1 . If the order of the endpoints of β ∗ and γ ∗ is as in Fig. 5(a), ∂1 is disjoint from B β and ∂ E ∗ . Hence d( S ) 2. If the order of the endpoints of β ∗ and γ ∗ is as in Fig. 5(b), then ∂1 intersects ∂ B β in one point. Let N (∂ B β ∪ ∂1 ) be a regular neighborhood of ∂ B β ∪ ∂1 in S. Since g ( S ) 2, ∂ N (∂ B β ∪ ∂1 ) is essential on S. Note that ∂ N (∂ B β ∪ ∂1 ) bounds a disk in V . Hence d( S ) 2, a contradiction. Case 3.2. ∂β ∗ ⊂ ∂ D 2 , ∂ γ ∗ ⊂ ∂ D 1 . By Lemma 2.6, B β is an essential disk in V which is disjoint from D 1 . Let W = W ∪ α2 × D. Then E γ is a properly embedded disk in W . By the above argument, d( S ) 2, a contradiction. 2 Acknowledgements The authors thank Tsuyoshi Kobayashi, Saul Schleimer and Tao Li for helpful communications. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
David Bachman, Connected sums of unstabilized Heegaard splittings are unstabilized, Geom. Topol. 12 (2008) 2327–2378. A. Casson, C.McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275–283. Kun Du, Xutao Gao, A note on Heegaard splittings of amalgamated 3-manifolds, Chin. Ann. Math. Ser. B 32 (2011) 475–482. Kun Du, Ruifeng Qiu, The self-amalgamation of high distance Heegaard splittings is always efficient, Topology Appl. 157 (2010) 1136–1141. W. Haken, Some results on surfaces in 3-manifolds, in: Studies in Modern Topology, Math. Assoc. Amer., 1968, pp. 34–98. J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001) 631–657. Tsuyoshi Kobayashi, Ruifeng Qiu, Yo’av Rieck, Shicheng Wang, Separating incompressible surfaces and stabilizations of Heegaard splittings, Math. Proc. Cambridge Philos. Soc. 137 (2004) 633–643. Tsuyoshi Kobayashi, Ruifeng Qiu, The amalgamation of high distance Heegaard splittings is always efficient, Math. Ann. 341 (2008) 707–715. Marc Lackenby, The Heegaard genus of amalgamated 3-manifolds, Geom. Dedicata 109 (2004) 139–145. Tao Li, On the Heegaard splittings of amalgamated 3-manifolds, in: Geom. Topol. Monogr., vol. 12, Geom. Topol. Publ., Coventry, 2007. Tao Li, Heegaard surfaces and the distance of amalgamation, Geom. Topol. 14 (2010) 1871–1919. Ruifeng Qiu, Martin Scharlemann, A proof of the Gordon conjecture, Adv. Math. 222 (6) (2009) 2085–2106. Juan Souto, The Heegaard genus and distances in the curve complex, preprint, http://www.math.ubc.ca/~jsouto/papers/Heeg-genus.pdf, 2003. Jennifer Schultens, Richard Weidmann, Destabilizing amalgamated Heegaard splittings, Geom. Topol. Monogr. 12 (2007) 319–334. G. Yang, F. Lei, Amalgamations of Heegaard splittings in 3-manifolds without essential surfaces, Algebr. Geom. Topol. 9 (2009) 2041–2054.
Contents lists available at SciVerse ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Unstabilized self-amalgamation of a Heegaard splitting ✩ Yanqing Zou a,∗ , Kun Du b , Qilong Guo a , Ruifeng Qiu c a b c
School of Mathematical Sciences, Dalian University of Technology, Dalian 116023, China School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China Department of Mathematics, East China Normal University, Shanghai 200241, China
a r t i c l e
i n f o
Article history: Received 22 April 2012 Accepted 28 November 2012 Keywords: Heegaard splitting Distance Stabilization
a b s t r a c t Let M be a compact orientable 3-manifold, M = V ∪ S W be a Heegaard splitting of M, and F 1 , F 2 be two homeomorphic components of ∂ M lying in the minus boundary of W . Let M ∗ be the manifold obtained from M by gluing F 1 and F 2 together. Then M ∗ has a natural Heegaard splitting called the self-amalgamation of V ∪ S W . In this paper, we prove that the self-amalgamation of a distance at least 3 Heegaard splitting is unstabilized. There are some examples to show that the lower bound 3 is the best. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Let M be a compact orientable 3-manifold. If there is a closed surface S which cuts M into two compression bodies V and W so that S = ∂+ V = ∂+ W , then we say that M has a Heegaard splitting, denoted by M = V ∪ S W . In this case, S is called a Heegaard surface, and g ( S ) is called the genus of the Heegaard splitting. A Heegaard splitting M = V ∪ S W is said to be stabilized if there are essential disks D in V and E in W such that D intersects E in just one point; otherwise, it is said to be unstabilized. M = V ∪ S W is said to be reducible if there is an essential simple closed curve on S which bounds disks in both V and W ; otherwise, it is said to be irreducible. M = V ∪ S W is said to be weakly reducible if there is an essential disk D in V and E in W such that D ∩ E = ∅; otherwise, it is said to be strongly irreducible. For any two essential simple closed curves α and β on S, d(α , β) is defined to be the smallest integer n 0 so that there is a sequence of essential simple closed curves α0 = α , . . . , αn = β on X such that αi −1 is disjoint from αi for 1 i n. The distance of the Heegaard surface S, denoted by d( S ), is defined to be min{d(α , β)}, where α bounds an essential disk in V and β bounds an essential disk in W . See [2,5,6]. Let F be a properly embedded surface in a compact orientable 3-manifold M. F is said to be compressible if either F is a 2-sphere which bounds a 3-ball or there is an essential simple closed curve on F which bounds a disk in F ; otherwise, F is said to be incompressible. An incompressible surface not parallel to ∂ M is said to be essential. Let M be a compact orientable 3-manifold which contains an essential closed surface F . We first assume that F is a 2-sphere. Then M is said to be reducible. By Kneser–Milnor’s theorem, M is the connected sum of n manifolds M 1 , . . . , M n , where M i is either irreducible or S 2 × S 1 . By Haken’s lemma, any Heegaard splitting of M is reducible. See [5]. Now if M i admits a Heegaard splitting V i ∪ S i W i for each 1 i n, then M admits a natural Heegaard splitting V ∪ S W called a connect sum of n Heegaard splittings V i ∪ S i W i . It’s well known that the connect sum of unstabilized V i ∪ S i W i is unstabilized. See [1] and [12].
✩ This work is supported by NSFC (11171108). The first author and the third author are also supported by a grant (No. 11271058/a010402) of NSFC. The second author is also supported by the Fundamental Research Funds for the Central Universities (No. lzujbky-2011-124). Corresponding author. E-mail addresses: [email protected] (Y. Zou), [email protected] (K. Du), [email protected] (Q. Guo), [email protected] (R. Qiu).
*
0166-8641/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2012.11.020
Y. Zou et al. / Topology and its Applications 160 (2013) 406–411
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Fig. 1. The way of gluing.
We assume now that F is a separating surface with g ( F ) 1. Then F cuts M into two manifolds M 1 and M 2 . In other words, M is obtained by gluing M 1 and M 2 along a homeomorphism f from F 1 to F 2 . Now if M i admits a Heegaard splitting V i ∪ S i W i for each 1 i 2, M has a natural Heegaard splitting called the amalgamation of V 1 ∪ S 1 W 1 and V 2 ∪ S 2 W 2 . A question on amalgamation is: Question 1. Is the amalgamation of two unstabilized Heegaard splittings unstabilized? It is possible that the amalgamation of two unstabilized Heegaard splittings can be destabilized many times simultaneously, see [7] and [14]. On the other side, (1) the amalgamation of two high distance Heegaard splittings is unstabilized, see [8,15,3]; and (2) if the gluing map f is complicated enough, then the amalgamation of two minimal Heegaard splittings of M 1 and M 2 is unstabilized. See [9–11,13]. Finally we assume that F is a non-separating surface M with g ( F ) 1. Now let F × I be the regular neighborhood of F in M and M 1 = M − F × (0, 1). Then M is obtained from M 1 by gluing F × {0} and F × {1} along a homeomorphism f from F × {0} to F × {1}. Let M 1 = V 1 ∪ S 1 W 1 be a Heegaard splitting. Then M has a natural Heegaard splitting called the self-amalgamation of V 1 ∪ S 1 W 1 , see Section 2. A question on self-amalgamation is the following: Question 2. Is the self-amalgamation of an unstabilized Heegaard splitting unstabilized? It is possible that the self-amalgamation of an unstabilized Heegaard splitting is stabilized. On the other side, the selfamalgamation of a high distance Heegaard splitting is unstabilized, see [4]. However, all the known examples on stabilized amalgamations and self-amalgamations of Heegaard splittings have one of its summands with Heegaard distance at most 2. All the known results on unstabilized amalgamations and selfamalgamations of Heegaard splittings have the complexity (high distance or complicated gluing map) strongly depending on manifolds. Hence a natural question arises: Question 3. Is there a constant k such that: (1) the amalgamation of two distance at least k Heegaard splittings is unstabilized? (2) the self-amalgamation of a distance at least k Heegaard splitting is unstabilized? In this paper, we give a positive answer to Question 3(2). The main result is the following: Theorem 1. The self-amalgamation of a distance at least 3 Heegaard splitting is unstabilized. It is possible that the self-amalgamation of a strongly irreducible Heegaard splitting is stabilized, see [4]. Hence the lower bound 3 in Theorem 1 is the best. 2. The proof of Theorem 1 Let M be a compact orientable 3-manifold with homeomorphic boundary components F 1 and F 2 , and M = V ∪ S W be a Heegaard splitting such that F 1 , F 2 ⊂ ∂− W . Let M ∗ be the manifold obtained from M by gluing F 1 and F 2 along a homeomorphism f from F 1 to F 2 . Then M ∗ has a Heegaard splitting called the self-amalgamation of V ∗ ∪ S ∗ W ∗ as follows: Let p i be a point on F i such that f ( p 1 ) = p 2 . Note that W is obtained by attaching 1-handles h1 , . . . , hm to ∂− W × I . Let αi = p i × I , αi × D be the regular neighborhood of αi for i = 1, 2. We may assume that αi × D is disjoint from the 2 1-handles h1 , . . . , hm , and f ( p 1 × D ) = p 2 × D. Let W be the closure of W − i =1 αi × D. Now, in the closure of M ∗ − V , the arc α = α1 ∪ α2 has a regular neighborhood α × D which intersects ∂+ V = S in two disks D 1 and D 2 . We denote by p the point p i , D the disk p × D ⊂ α × D, and F the surface F i in M ∗ . See Fig. 1 and Fig. 2. Let V ∗ = V ∪ α × D and W ∗ be the closure of M ∗ − V . Then V ∗ and W ∗ are compression bodies. Hence M ∗ = V ∗ ∪ S ∗ W ∗ is a Heegaard splitting called the self-amalgamation of V ∪ S W . Let S 1 be the surface S − int D 1 ∪ int D 2 . Then S 1 is a sub-surface of S with two boundary components ∂ D 1 and ∂ D 2 .
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Fig. 2. Construction of V ∗ from V .
Lemma 2.1. F − int D is incompressible in W ∗ . from W by gluing F 1 − int D and F 2 − int D along the homeomorphism f , Proof. By the construction, W ∗ is obtained 2 where W is the closure of W − i =1 αi × D. In this case, F − int D = F 2 − int D = f ( F 1 − int D ). Hence we only need to prove that both F 1 − int D and F 2 − int D are incompressible in W . Suppose that one of F 1 − int D and F 2 − int D, say F 1 − int D, is compressible in W . Then there is an essential simple closed curve C on F 1 − int D which bounds a disk in W . Since F 1 is incompressible in W , C is isotopic to ∂ D on F 1 − int D. Hence there is a 2-sphere in W such that S = ∂+ W and F 1 lie in distinct sides of the sphere. This means that W is reducible, a contradiction. 2 Let α be a simple closed curve or a properly embedded arc in S 1 , and N (α ∪ ∂ S 1 ) be the regular neighborhood of α ∪ ∂ S 1 in S 1 . Then α is said to be strongly essential if one component of ∂ N (α ∪ ∂ S 1 ) is essential in S. Since S = S 1 ∪ D 1 ∪ D 2 , we have the following observations: Lemma 2.2. A simple closed curve α on S 1 is strongly essential if and only if α is essential on S. Lemma 2.3. A properly embedded arc α on S 1 is strongly essential if and only if (1) the two end points of α lie in one component of ∂ S 1 , say ∂ D j , and (2) α is an essential arc on S 1 ∪ D k , where { j , k} = {1, 2}. In order to obtain a contradiction, in the following argument, we assume that M ∗ = V ∗ ∪ S ∗ W ∗ is stabilized. This means that there are essential disks B ⊂ V ∗ and E ⊂ W ∗ such that B intersects E in one point x. Furthermore, we may assume that | B ∩ D | + | E ∩ ( F − int D )| is minimal among all stabilizing pairs of disks. By Lemma 2.1 and standard arguments, we have the following observations: Lemma 2.4. (1) Each component of B ∩ D is a properly embedded arc in both B and D, and each component of E ∩ ( F − int D ) is a properly embedded arc in both E and F − int D. (2) Each component of ∂ B ∩ S 1 is essential on S 1 , and each component of ∂ E ∩ S 1 is essential on S 1 . (3) Each component of B ∩ α × ∂ D is an arc with its two end points lying in distinct boundary components of α × ∂ D, and |∂ B ∩ ∂ D | = |∂ B ∩ ∂ D 1 | = |∂ B ∩ ∂ D 2 |. (4) Each component of E ∩ α × ∂ D is an arc with its two end points lying in distinct boundary components of α × ∂ D, and |∂ E ∩ ∂ D | = |∂ E ∩ ∂ D 1 | = |∂ B ∩ ∂ E 2 |. Suppose first that B ∩ ( D 1 ∪ D 2 ) = ∅. Let β be an outermost component of B ∩ ( D 1 ∪ D 2 ) on B. This means that β , together with an arc β ∗ on ∂ B, bounds a disk in B, say B β , such that B β intersects D 1 ∪ D 2 in β . In this case, D β is called an outermost disk related to β , and β ∗ is called an outermost arc related to β . Lemma 2.5. (1) β ∗ ⊂ S 1 , and the two end points of β ∗ lie in one of D 1 and D 2 . (2) D β ⊂ V . Proof. Since β is an outermost component of B ∩ ( D 1 ∪ D 2 ) on B, the lemma follows immediately.
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Fig. 3. Disks bounded by ∂ S 1 and β ∗ .
Lemma 2.6. Each outermost arc of ∂ B ∩ S 1 is strongly essential on S 1 . Proof. Suppose, otherwise, that one outermost arc of ∂ B ∩ S 1 , say β ∗ , related to an outermost arc of B ∩ ( D 1 ∪ D 2 ), say β , is not strongly essential on S 1 . By Lemma 2.5, we may assume that ∂β = ∂β ∗ ⊂ ∂ D 1 . By Lemma 2.3(2), β ∗ , together with one component of ∂ D 1 − β ∗ , bounds a disk D ∗ in S 1 such that D 2 ⊂ int D ∗ . See Fig. 3. Then each component of ∂ B ∩ S 1 incident to ∂ D 2 has the other end point lying in ∂ D 1 . We let m be the number of end points of the components of ∂ B ∩ S 1 lying in ∂ D 2 , and n the number of end points of the components of ∂ B ∩ S 1 lying in ∂ D 2 . Since the two end points of β ∗ lie in ∂ D 1 , n m + 2. On the other side, by Lemma 2.4(3), n = ∂ B ∩ ∂ D 1 = ∂ B ∩ ∂ D 2 = m, a contradiction. 2 Suppose now that E ∩ ( F − int D ) = ∅. Let β be an outermost component of E ∩ ( F − int D ) on E. This means that γ , together with an arc γ ∗ in S 1 , and two arcs in (α1 ∪ α2 ) × ∂ D, bounds a disk in E, say E γ , such that E γ intersects F − int D in γ . In this case, E γ ⊂ W where W is the closure of W − N (α1 ∪ α2 ). Now E γ is called an outermost disk related to γ , and γ ∗ is called an outermost arc related to γ . Lemma 2.7. (1) The two end points of γ ∗ lie in one of ∂ D 1 and ∂ D 2 . (2) E γ ⊂ W . Proof. Since int E γ is disjoint from F − int D, the lemma holds.
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Similar to the proof of Lemma 2.6, we have the following lemma. Lemma 2.8. Each outermost arc of ∂ E ∩ S 1 is strongly essential on S 1 . Proof of Theorem 1. Recall the stabilizing pair of disks ( B , E ) and the assumption that V ∪ S W has distance at least 3. There are three possibilities: Case 1. E ∩ ( F − int D ) = ∅. By Lemma 2.4, ∂ E ⊂ S 1 . Hence x = ∂ E ∩ ∂ B ⊂ S 1 . Now let B ∗ be the component of B ∩ V such that x ∈ B ∗ . Then ( B ∗ , E ) is a stabilizing pair of disks of V ∪ S W , a contradiction. Case 2. B ∩ D = ∅. In this case, we may assume that ∂ B ⊂ S 1 and B ⊂ V . We first prove that ∂ B is essential on S. Suppose that ∂ B is inessential on S. Then either ∂ B is isotopic to one of ∂ D 1 and ∂ D 2 , say ∂ D 1 , or ∂ E bounds a pair of pants together ∂ D 1 and ∂ D 2 . In the first sub-case, ∂ E intersects ∂ D 1 in one point. In the second sub-case, ∂ B is separating in S ∗ , and ∂ E intersects ∂ B in even number of points. All are impossible. Let γ be an outermost component of E ∩ ( F − int D ) on E, E γ be the outermost disk related to γ , and γ ∗ be the outermost arc related to γ . Since B intersects E in only one point, and E ∩ ( F − int D ) contains at least two outermost components. We may assume that E γ ∩ B = ∅. By Lemma 2.7, we may assume that ∂ γ ∗ ⊂ ∂ D = ∂ D 1 . Then ∂ E γ is disjoint from α2 × ∂ D. Now let W = W ∪ α2 × D. Then E γ is a properly embedded disk in W such that ∂ E γ intersects F 1 − int D in γ , α1 × D in two arcs, and S 1 in γ ∗ . There are two sub-cases: Case 2.1.
γ is inessential on F 1 − int D.
Now after an isotopy on ∂ E γ in S ∗ , ∂ E γ ⊂ S 1 . By Lemma 2.8, ∂ E γ is essential in S. Since ∂ E γ ∩ ∂ B = ∅, d( S ) 1. A contradiction.
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Fig. 4. Outermost disk and outermost arc in E γ .
Fig. 5. Possible orders of endpoints.
Case 2.2.
γ is essential on F 1 − int D.
By the construction of V ∗ ∪ S ∗ W ∗ , there is an essential disk E ∗ in W such that E ∗ is disjoint from α1 × D ∪ α2 × D. Furthermore, we assume that | E ∗ ∩ E γ | is minimal among all such disks. Now we claim that E ∗ ∩ E γ = ∅. Suppose that E ∗ ∩ E γ = ∅. By the minimality of | E ∗ ∩ E γ |, each component of E ∗ ∩ E γ is a properly embedded arc on both E ∗ and E γ . Since E ∗ is disjoint from α1 × D ∪ α2 × D, each component of E ∗ ∩ E γ has its two end points lying in γ ∗ . See Fig. 4. Let b be an outermost arc of E ∗ ∩ E γ on E γ such that b, together with a sub-arc of γ ∗ , bounds a disk E b with int E b ∩ E ∗ = ∅, where ∂ E b ∩ ∂ D = ∅. See Fig. 4. We denote by E 1∗ and E 2∗ the two components of E ∗ − b. Then E 1∗ ∪ E b and E 2∗ ∪ E b are two properly embedded disks in W disjoint from α1 × D ∪ α2 × D. Since E ∗ is essential in W , one of E 1∗ ∪ E b and E 2∗ ∪ E b , say E ∗∗ , is essential. But E ∗∗ can be pushed slightly so that | E ∗∗ ∩ E γ | = | E ∗ ∩ E γ | − 1. This contradicts the minimality of | E ∗ ∩ E γ |. Now γ ∗ , together with one component of ∂ D 1 − γ ∗ , forms a simple closed curve, say γ . By Lemma 2.8, γ is essential on S. Note that ∂ B ∩ γ = ∅ and γ ∩ ∂ E ∗ = ∅. Hence d( S ) 2, a contradiction. Case 3. B ∩ D = ∅, and E ∩ ( F − int D ) = ∅. Since B intersects E in only one point, and B ∩ D contains at least two outermost arcs on B, we may assume that β is an outermost arc of B ∩ D such that B β ∩ E = ∅, where B β is the outermost disk related to β . By Lemma 2.6, B β is an essential disk in V . Similarly, let γ be an outermost arc of E ∩ ( F − int D ) such that E γ ∩ B = ∅. Hence B β ∩ E γ = ∅. Now we consider the outermost arc β ∗ related to β and the outermost arc γ ∗ related to γ . By Lemmas 2.5 and 2.7, there are two sub-cases: Case 3.1. ∂β ∗ , ∂ γ ∗ ⊂ ∂ D j , where j = 1 or 2. We may assume that j = 1. Now β is an arc on F 1 − int D in W , where W is the closure of W − α1 × D ∪ α2 × D. Furthermore, E γ is disjoint from α2 × ∂ D. Now W = W ∪ α2 × D. Then E γ is a properly embedded disk in W . We first suppose that γ is inessential on F 1 − int D. Then ∂ E γ can be isotoped so that ∂ E γ ⊂ S 1 . Furthermore, either ∂ E γ ∩ ∂ B β = ∅ as in Fig. 5(a), or ∂ E γ intersects ∂ B β in one point as in Fig. 5(b) according to the order of the end points of β ∗ and γ ∗ . By Lemma 2.8, ∂ E γ is essential on S. Hence V ∪ S W is weakly reducible or stabilized, a contradiction. Suppose now that γ is essential on F 1 − int D. By the argument in Case 2.2, there is an essential disk E ∗ in W such that E ∗ is disjoint from α1 × D ∪ α2 × D and γ ∗ . Let N (γ ∗ ∪ ∂ D 1 ) be a regular neighborhood of γ ∗ ∪ ∂ D 1 on S 1 disjoint
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from E ∗ . By Lemma 2.8, one component of ∂ N (γ ∗ ∪ ∂ D 1 ) is an essential simple closed curve in S, say ∂1 . If the order of the endpoints of β ∗ and γ ∗ is as in Fig. 5(a), ∂1 is disjoint from B β and ∂ E ∗ . Hence d( S ) 2. If the order of the endpoints of β ∗ and γ ∗ is as in Fig. 5(b), then ∂1 intersects ∂ B β in one point. Let N (∂ B β ∪ ∂1 ) be a regular neighborhood of ∂ B β ∪ ∂1 in S. Since g ( S ) 2, ∂ N (∂ B β ∪ ∂1 ) is essential on S. Note that ∂ N (∂ B β ∪ ∂1 ) bounds a disk in V . Hence d( S ) 2, a contradiction. Case 3.2. ∂β ∗ ⊂ ∂ D 2 , ∂ γ ∗ ⊂ ∂ D 1 . By Lemma 2.6, B β is an essential disk in V which is disjoint from D 1 . Let W = W ∪ α2 × D. Then E γ is a properly embedded disk in W . By the above argument, d( S ) 2, a contradiction. 2 Acknowledgements The authors thank Tsuyoshi Kobayashi, Saul Schleimer and Tao Li for helpful communications. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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