0 efficient triangulations of Haken three-manifolds

Transcription

1 The University of Melbourne Doctoral Thesis 0 efficient triangulations of Haken three-manifolds Shanil Ramanayake A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics and Statistics March 2016

2 Declaration of Authorship This is to certify that The thesis comprises of original work towards the degree of Doctor of Philosophy, Due acknowledgement has been made in the text to all other material used, The thesis is fewer than 100, 000 words in length exclusive of tables, maps, appendices and bibliography. Signed: Date: ii

3 Abstract The thesis constructs 0 efficient triangulations for compact, irreducible, an-annular, orientable, atoroidal, Haken 3 manifolds that are closed or have torus boundary. The triangulations are dual to a hierarchy of the manifold, which is characterised as a special spine. By crushing the dual triangulations we obtain 1 vertex and ideal triangulations. An algorithmic technique is used to prove 0 efficiency. Efficient triangulations exhibit geometric properties of the underlying manifold. We hope the thesis will help elucidate the relationship between geometric structures and triangulations.

4 Acknowledgements I would like to thank my supervisors Hyam Rubinstein and Craig Hodgson for all their support and help. I would also like to thank Kirsten Hoak for all her support and understanding during the submission process. On a personal note, there have been many people over the years who have helped me throughout this time This would not have been achieved without your help and support. Finally, I thank my Lord and saviour Jesus Christ, without whom, this would not have been possible. iv

5 Contents Declaration of Authorship ii Abstract iii Acknowledgements iv List of Figures ix 1 Introduction Overview Triangulations Efficient triangulations Haken 3 manifolds and boundary patterns Hierarchies Boundary patterns and hierarchies Motivation for the thesis Relationship between Efficient triangulations and Angle structures Brief introduction to the thesis Statement of results Outline for proving the main results Preliminaries Introduction Notation and assumptions Triangulations and spines Triangulations Spines of 3 manifolds Duality of special spines and triangulations Normal Surfaces Normal surfaces and incompressibility Special Hierarchies Introduction v

6 Contents vi 3.2 General results Complete system of disks Complexity of Haken 3 manifolds Algebraic results Boundary patterns induced by hierarchies Constructing non-separating incompressible surfaces Constructing minimally separating hierarchies Separating surfaces in mimimally separating hierarchies Postponing cuts along compressing disks Minimal separating hierarchies whose complement is a single 3 ball or only collars Minimal separating hierarchies with two 3 ball complements or collars and 3 ball complements Special hierarchies Dual Triangulations Introduction Haken 2-complexes and dual triangulations Crushing triangulations Crushing closed manifolds Crushing manifolds with boundary An example Borromean rings complement An example The figure eight knot complement Homotoping spheres off the hierarchy Introduction Graph of a hierarchy on a surface Labelling hierarchies Reduction disks Regular neighbourhoods of reduction disks Complexity for the graph of the hierarchy Simplification of the graph of a hierarchy Homotoping spheres off the hierarchy Normal homotopies and reduction moves Introduction Some results on vertex linking normal spheres Characterising reduction disks Reduction moves as normal homotopies Product structures Reduction moves as normal homotopies Classifying embedded normal spheres Efficient Triangulations Introduction

7 Contents vii efficient triangulations for closed Haken 3 manifolds efficient triangulations for Haken 3 manifolds with boundary Conclusion and Further research Summary Further research Bibliography 109

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9 List of Figures 1.1 Local structure of special polyhedra Duality and truncation Elementary disks Angle structures Example of edge folding Local structure of simple polyhedron Gluing faces via special spine Normal disks Example of a non-disk component gon, 3 gon and 4 gon disks in i Good components Boundary pattern induced by hierarchy Constructing hierarchies three torus case Local structure of a special hierarchy Duality between hierarchy and tetrahedra Hierarchy components in a tetrahedron Cell decomposition Trapezoids Crushing cells Face with identified edges Obstruction to trivial product regions closed manifolds Crushing triangulations bounded case E dual to an edge occuring twice in a face Exceptional Case Obstruction to crushing A regular neighborhood of the Borromean rings Cutting along boundary parallel surfaces for the Borromean rings Initial hierarchy surface Borromean rings Schematic of hierarchy surfaces Borromean rings Boundary pattern Borromean rings Generators for punctured torus Boundary pattern for figure eight knot Reduction disks ix

10 List of Figures x 5.2 Regular neighbourhoods of Reduction disks Reduction Normal arcs, sub-arcs and T-junctions Decomposition of tetrahedron by hierarchy Product structures Good normal homotopy Reduction disk that pushes through a vertex Normal disk exchange Preparing for a normal disk exchange Regular exchange

11 Chapter 1 Introduction The thesis constructs 0 efficient triangulations for a class of 3 manifolds called Haken 3 manifolds. The introduction will provide background and motivation for the construction. We start by examining the notion of efficient triangulations and Haken 3 manifolds with boundary patterns. Next we provide an overview of the relationship between efficient triangulations and geometric triangulations the main motivation for the construction. 1.1 Overview In this section we introduce and provide a brief survey of the relevant ideas in the thesis. Many of the topics discussed here will be re-introduced in the subsequent chapters. However we hope that the overview will make the next section detailing the motivation of the thesis accessible, and also provide a useful survey of the literature Triangulations Triangulations are an example of a combinatorial structure on a 3 manifold. In general, a combinatorial structure refers to a presentation of a 3 manifold by finite data. Typically one might also include a collection of moves which act on the 1

12 Chapter 1. Introduction 2 combinatorial structure to change from one presentation to another and invariants that distinguish between two presentations of distinct manifolds [30, 35, 38]. Informally, the data associated to a triangulation of a 3 manifold consists of a collection of disjoint tetrahedra, and a collection of maps, that pair faces of the tetrahedra to each other. If the complex obtained as a result of this face pairing is homeomorphic to the underlying manifold, then the complex is called a triangulation. It is possible that the face pairing maps self identify tetrahedra for example two faces of the same tetrahedron could be identified to each other. Such triangulations are called singular triangulations [30]. It is an old result of Moise [32] that any compact 3 manifold has a finite triangulation that is, the collection of disjoint tetrahedra is finite. The class of triangulation referred to by Moise provides a piecewise linear structure on the underlying manifold. Singular triangulations are strictly not in this category. However a singular triangulation after subdivision can be transformed into a triangulation that provides a piecewise linear structure on the underlying manifold [38]. For a 3 manifold with boundary, a singular triangulation is an ideal triangulation if the manifold is triangulated as described above, and in addition, the link of each vertex is a surface distinct from a 2 sphere. The point of view taken in the thesis is that triangulations are dual to a combinatorial object known as a special spine (see Matveev [30]). A spine is a polyhedron in the interior of the 3 manifold such that the manifold collapses simplicially onto it. A special polyhedron is a polyhedron that has at least one vertex, each neighbourhood of a point takes one of the forms shown in Figure 1.1 and all its 2 components are disks. If the polyhedron to which the manifold with boundary collapses is special, then the polyhedron is called a special spine. For a closed manifold, a boundary is produced by removing an open ball and then a special spine has the same interpretation. To observe the duality between triangulations and special spines take a triangulation of a 3 manifold and consider the 2 skeleton of the subdivision dual to the triangulation (see Figure 1.2(a)).

13 Chapter 1. Introduction 3 Figure 1.1: Local structure of special polyhedra (a) Duality (b) Truncated tetrahedron Figure 1.2: Duality and truncation for a tetrahedron To understand the topology of a 3 manifold it is useful to study surfaces that it contains. We are particularly interested in a class of surfaces distinguished by a triangulation called normal surfaces. Normal surfaces were initially introduced by Kneser [24], and later developed by Haken [14]. A normal surface is defined by gluing together a collection of disks called normal disks. There are two types of normal disks in each tetrahedron triangular and quadrilateral disks (see Figure 1.3). For normal surfaces to be a useful construct it is important to know which surfaces are represented as normal surfaces. A surface is π 1 injective if the map of fundamental groups between the surface and the manifold induced by inclusion is injective.

14 Chapter 1. Introduction 4 Triangle disks Quadrilateral disks Figure 1.3: Elementary disks Moreover a properly embedded surface in a 3 manifold is an essential surface if it is not a 2 sphere or disk, is π 1 injective, and cannot be properly homotoped into the boundary of the 3 manifold. Importantly, if S is an essential surface then the surface S is realised as a normal surface [13, 14, 24]. See the lecture notes by Gordon [11] for details of normal surface theory Efficient triangulations Jaco and Rubinstein [18] introduced a class of triangulations that exhibit the geometric properties of the underlying manifold called efficient triangulations. Efficient triangulations restrict classes of normal surfaces with non-negative Euler characteristic. To be more precise we note the following: 1. A 0 efficient triangulation of a compact orientable 3-manifold M is an ideal triangulation that does not contain normal embedded 2-spheres if the boundary of M is non-empty. If M is closed and has a triangulation with a single vertex, then the triangulation is 0 efficient if there is only one connected embedded normal 2-sphere which is vertex linking, i.e the boundary of a small regular neighbourhood of the vertex.

15 Chapter 1. Introduction 5 2. A 1 efficient triangulation is a triangulation of a compact, orientable 3 manifold M that is 0 efficient and in addition, if the triangulation is ideal then there are no embedded normal tori except for ideal vertex linking tori. If the boundary of M is empty then the only embedded normal tori are edge linking or thick, i.e. boundaries of a regular neighbourhood of a sub-complex which is a layered solid torus. 3. A strongly 1 efficient ideal triangulation of a compact, orientable 3 manifold M, is 1 efficient triangulation with no immersed normal 2-spheres or tori except for coverings of ideal vertex linking tori Haken 3 manifolds and boundary patterns Thus far, we have discussed the class of triangulations investigated in the thesis. This section will introduce the class of 3 manifolds that are of interest to us Haken 3 manifolds with boundary patterns. Haken manifolds were introduced by Haken in the 1960 s. A defining property of Haken 3 manifolds is that they contain incompressible surfaces. Intuitively, an incompressible surface is a surface that cannot be simplified within the 3 manifold by compressing handles or deleting 2 spheres [16]. Algebraically for 2 sided surfaces, incompressibility is equivalent to the surface being π 1 injective [17]. If we restrict to orientable 3 manifolds, then a Haken 3 manifold is a compact, orientable, irreducible (i.e. every 2 sphere bounds a 3 ball) 3 manifold which contains a 2 sided incompressible surface distinct from a 2 sphere or disk, and whose boundary if non-empty, is incompressible. Well-known examples of Haken 3 manifolds include knot complements and surface bundles over the circle. Furthermore any compact irreducible 3 manifold with non-empty incompressible boundary is a Haken 3 manifold [17] Hierarchies A Haken 3 manifold can be decomposed along a sequence of 2 sided incompressible surfaces 1 and compressing disks into a collection of 3 balls or a simpler Haken 3 manifold. Such a sequence is called a hierarchy for the manifold, and a surface

16 Chapter 1. Introduction 6 in this sequence will be referred to as a hierarchy surface or a surface of the hierarchy. Note however that a hierarchy does not provide a unique decomposition. The books by Jaco [17], Hempel [15] and the survey by Aitchinson and Rubinstein[3] provide a good reference for much of the material related to hierarchies used in the thesis. Many results for Haken 3 manifolds are obtained by induction along a hierarchy. Examples include the algorithmic classification of Haken 3 manifolds [30], solution of the word problem for Haken 3 manifolds [50] and Thurston s Hyperbolization theorem [34]. As a technical point we note that when considering Haken 3 manifolds with boundary the hierarchy surfaces are required to be boundary incompressible as well Boundary patterns and hierarchies In the thesis, we consider Haken 3 manifolds equipped with an additional structure called a boundary pattern. Informally, a boundary pattern for a 3 manifold M, is a finite collection of compact, connected 2 manifolds in the boundary of M such that, the intersection of any three such 2 manifolds is a 0 manifold; the intersection of any two such 2 manifolds is a 1 manifold; and the intersection of more than three such 2 manifolds is empty [19]. Historically, manifolds with boundary patterns were introduced in the work of Waldhausen [50] and Johannson [19]. A boundary pattern on a 2-manifold S is a collection of embedded arcs and circles called elements of the boundary pattern. The elements can meet only at end points of the arcs which lie in the interior of an arc or circle. Of particular interest are i-faced disks. An i-faced disk is a 2-disk D which meets the boundary pattern in D only. Moreover D consists of i arc elements of the boundary pattern, which makes D into an i-polygonal disk 2. 1 In the thesis we require that in addition, the surfaces are embedded, and if a surface has boundary then it is properly embedded. As a technical point we note that when considering Haken 3-manifolds, the hierarchy surfaces with non-empty boundary are required to be boundary incompressible in the manifold obtained by splitting along previous surfaces in the hierarchy. Namely for a properly embedded surface S in a manifold M with boundary, the induced map π 1 (S, S) π 1 (M, M) must be one-to-one. 2 Using the language of Johannson, a small disk is a 0, 2 or 3-faced disk. If the boundary pattern is useful then each small disk can be admissibly homotoped to a point.

17 Chapter 1. Introduction 7 A boundary pattern that does not have small disks is called a useful boundary pattern [19]. Hierarchies that induce useful boundary patterns play a pivotal role in the work of Waldhausen [50] and Johannson [19]. If M has a boundary pattern, we can associate a tri-valent graph to the boundary of M. In the thesis, we abuse notation slightly, and call this graph the boundary pattern for M in Johannson [19] this graph is called the graph of the boundary pattern. We are particularly interested in boundary patterns induced by cutting along hierarchy surfaces. In this case we say the boundary pattern is induced by the hierarchy. Consider a hierarchy surface with some boundary curves. When we cut along this surface, copies of its boundary curves become part of the boundary pattern. After cutting along such a hierarchy surface the manifold obtained is either a three ball(s) or a simpler Haken 3 manifold with boundary pattern. In Section of the introduction we will discuss the properties of the hierarchies and boundary patterns constructed in the thesis. While the approach to Haken 3 manifolds and boundary patterns in the thesis is similar to the one taken by Waldhausen and Johannson the viewpoint we take is different. We translate the notion of a hierarchy for Haken 3 manifolds to the combinatorial setting of triangulations and spines. 1.2 Motivation for the thesis The main motivation in constructing 0 efficient triangulations is to investigate triangulations that exhibit geometric properties of the underlying manifold. Thurston s initial work and recent results of Perelman show that 3 manifolds can be canonically decomposed into pieces which can be classied depending on the geometric structure they carry 3 [5, 36, 37, 44]. Furthermore 3 manifolds that carry a hyperbolic structure are by far the largest class of geometric examples [42, 44]. In the 1970 s, Thurston established a far reaching relationship between Haken 3 manifolds and geometric structures on the underlying manifold. The theorem 3 Informally, a geometric structure on the manifold can be viewed as a complete locally homogeneous Riemannian metric on the 3 manifold.

18 Chapter 1. Introduction 8 known as Thurston s hyperbolization theorem states that the interior of a compact, atoroidal, Haken 3 manifold with zero Euler characteristic admits a complete hyperbolic metric of finite volume [23]. Thus Haken 3 manifolds that carry finite volume hyperbolic structures are completely characterised. Remark 1. The class of manifolds discussed in the thesis are part of the class of Haken 3 manifolds described by Thurston s hyperbolization theorem. It is important to note that we do not make any assumptions regarding the underlying geometry of the manifolds. However the topological conditions we choose are consistent with the conditions of Thurston s hyperbolization theorem. From a computational and combinatorial perspective it is an important question whether the underlying geometric structure of a 3 manifold can be represented by a triangulation. A triangulation that has this property is called a geometric triangulation. In particular, a triangulation for a hyperbolic 3 manifold is a hyperbolic triangulation if each tetrahedron in the triangulation is an ideal hyperbolic tetrahedron. An ideal hyperbolic tetrahedron is obtained by taking the convex hull of the four vertices on the boundary of hyperbolic space. Furthermore after pairing faces of hyperbolic tetrahedra, the resulting complex carries a hyperbolic metric that agrees with the metric on the underlying manifold. In fact, Thurston [44] in his lecture notes, provided a beautiful example of a hyperbolic triangulation for the figure eight knot complement. He showed that the figure eight knot complement has a complete hyperbolic structure obtained by gluing two ideal hyperbolic tetrahedra. Experimental evidence obtained from computer programs such as SnapPea [52] suggests that hyperbolic triangulations of hyperbolic 3 manifolds can always be constructed. However, it is still an open question whether finite volume hyperbolic 3 manifolds have a hyperbolic triangulation. While there are numerous examples that support this view (see for example [12]) the question remains unanswered [41]. To construct finite volume hyperbolic structures using triangulations Thurston [44] defined a system of non-linear equations called Thurston s gluing equations by associating a complex number to each tetrahedron in the triangulation. The complex number stipulates the shape of the tetrahedron.

19 Chapter 1. Introduction 9 A solution to the equations implies the existence of a hyperbolic structure where each tetrahedron in the triangulation is hyperbolic. After gluing the tetrahedra together the underlying manifold will have a complete hyperbolic metric (hyperbolic structure)[44]. In general, however, Thurston s gluing equations are hard to solve, and even proving the existence of a solution where all the tetrahedron are positively oriented is difficult to obtain [10]. Another approach (assuming the manifold is hyperbolic and is the interior of a compact 3-manifold with non-empty boundary and zero Euler characteristic) is to construct ideal hyperbolic triangulations from its Epstein-Penner decomposition into ideal hyperbolic polyhedra [7]. While it is possible to further subdivide the polyhedra into ideal hyperbolic tetrahedra, in order to match the faces of the polyhedra, we need to include flat tetrahedra. Thus the resulting complex is not a geometric triangulation in the strict sense of the word. It is still an open question whether hyperbolic manifolds admit geometric triangulations without flat or inverted tetrahedra [39]. Casson and Rivin [40] introduced the notion of angle structures which represent a weak hyperbolic structure described via a triangulation. In particular, an angle structure represents a solution to the linear part of Thurston s gluing equations. An angle structure is a choice of dihedral angles satisfying conditions associated with hyperbolic geometry to each ideal tetrahedron in a triangulation. Finding an angle structure for a triangulation and maximising hyperbolic volume is one approach to solving Thurston s gluing equation (see Casson and Rivin [40]). Recent results with respect to angle structures are found in the survey papers of Luo [28], Futer and Guéritaud [10]. If T denotes a triangulation of a 3 manifold then an angle structure associated with the triangulation T is obtained by assigning a real number (angle) in the interval (0, π) to each edge of a tetrahedron in the triangulation so that the following conditions are satisfied (see Figure 1.4): 1. Around each vertex of a tetrahedron the angles sum to π. 2. Opposite edges of a tetrahedron are assigned the same angle. 3. Around each edge of T the angles sum to 2π.

20 Chapter 1. Introduction 10 α i + β i + γ i = π 0 < α i, β i, γ i < π α i γ i β i β i γ i αi Figure 1.4: Angles in a single tetrahedron The existence of an angle structure with all angles strictly positive implies the existence of a complete hyperbolic metric and the same is true with only nonnegative angles provided certain weak topological conditions are satisfied (see Lackenby [26]) Relationship between Efficient triangulations and Angle structures While efficient triangulations need not be hyperbolic, a deeper relationship between efficient triangulations and hyperbolic triangulations exists. Necessary conditions for the existence of angle structures (for the purposes of this thesis) are categorized into topological conditions on the manifold and conditions or properties of the triangulation the conditions are listed as follows: 1. Topological: The manifold is compact, orientable, irreducible and atoroidal and does not contain an essential annulus. 2. Triangulation: The triangulation is strongly 1 efficient. Remark 2. A manifold is irreducible if each 2 sphere in the manifold bounds a 3 ball. A manifold is atoroidal if there are no essential tori all π 1 injective tori are homotopic to boundary components.

21 Chapter 1. Introduction 11 More concretely, the relationship between 0 efficient triangulations and geometric structures is better illustrated if we consider taut angle structures and semi-angle structures. A taut angle structure is a solution to the angle structure equations where the values of the angles are either 0 or π; a semi-angle structure is a solution where the angles are non-negative. The following theorem by Kang and Rubinstein [22] relates taut structures to efficient triangulations. The result also extends to semi-angle structures. Theorem 1. Let M be an irreducible compact 3 manifold with torus or Klein bottle boundary components, and T an ideal triangulation of M. If T has a taut structure, then T is 0 efficient. Further if M is atoroidal, then T is strongly 1 efficient. Thus, the condition of 0 efficiency is a necessary condition for taut structures. Under certain conditions taut structures can be deformed to angle structures [8, 22]. To summarise, given the difficulty of constructing hyperbolic triangulations in a systematic manner, an angle structure for a triangulation is one method of constructing a weak hyperbolic triangulation that may be strengthened to a hyperbolic triangulation. From the discussion above, some form of efficiency of the triangulation is a necessary condition for the triangulation to admit an angle structure. Thus the motivation for the thesis is clear. We construct 0 efficient triangulations for the class of Haken 3 manifolds that admit complete hyperbolic structures. Hopefully the condition of 0 efficiency can be strengthened to strong 1 efficiency and the triangulations will admit an angle structure. While such a plan can be characterised as future work, these ideas are discussed further in the Conclusion (see Chapter 8). 1.3 Brief introduction to the thesis In this section we state the main results of the thesis, and provide a broad outline of the techniques used in the proofs.

22 Chapter 1. Introduction Statement of results The thesis proves the following result that relates special spines with special hierarchies of Haken 3 manifolds. Theorem 2. Let M be a compact, irreducible, orientable Haken 3 manifold that is atoroidal and does not contain an essential annulus. Assume the boundary of M is empty or consists of tori. Then there exists a hierarchy which is a special spine for M. Denote by H the hierarchy constructed in Theorem 2. Theorem 3. For a 3 manifold M and hierarchy H as described in Theorem 2, the triangulation T obtained as the dual to H is a 0 efficient triangulation (possibly after crushing T along an edge or a face). The condition that M is orientable is primarily for simplicity. Throughout the construction we consider only 2 sided incompressible surfaces, and it seems likely that Theorem 2 holds without the condition of orientability, provided the manifold is P 2 -irreducible as well as irreducible. Remark 3. Here P 2 irreducible means that the 3 manifold does not contain a 2 sided projective plane. The hypothesis of being irreducible is necessary for the interior of M to admit a hyperbolic metric [47]. Similarly the condition that M consists of torus boundary components and that M does not contain an essential annulus is included because we wish to consider Haken 3 manifolds that admit a hyperbolic structure with finite volume [47]. The atoroidal condition is needed to construct the hierarchies of Theorem 2. Similarly we need the condition that there are no essential annuli to prove that the hierarchy is a special spine. However, both conditions are not restrictive since Thurston shows that these conditions are necessary for hyperbolic structure on Haken 3 manifolds [45, 46, 48] Outline for proving the main results Broadly the thesis consists of the following three tasks:

23 Chapter 1. Introduction 13 (i) Construct a special class of triangulations (Chapter 3 4). (ii) Classify all normal 2 spheres with respect to these triangulations (Chapter 5 6). (iii) Prove 0 efficiency of these triangulations (Chapter 6 7). The key idea in task (i) is to relate the topological construction of a hierarchy to a combinatorial construction of a special spine. We start by constructing what we call a special hierarchy for Haken 3 manifolds in Chapter 3. Special hierarchies described in the thesis have the following properties. Property A The hierarchy induces a useful boundary pattern. Property B The surfaces of the hierarchy separate the manifold minimally. Property A is needed to ensure that a special hierarchy is a special spine. In particular if the boundary pattern induced by the hierarchy is useful then, locally, the intersection of hierarchy surfaces corresponds to exactly the local structure of a special polyhedron described in Section Remark 4. More precisely, using the language of Matveev [30], if the boundary pattern induced by the hierarchy is useful then each vertex of the boundary pattern is a true vertex; assuming that there are no essential annuli, the hierarchy can be viewed as a cellular 2-complex onto which the manifold, with at most two balls removed, collapses. The boundary pattern is then the singular graph of the special spine. If M is a closed manifold and H is a special hierarchy then Property B means that the manifold M \H (the complement of the hierarchy) is at most two 3 balls. Similarly if M has boundary then M \ H is either a collar of the boundary or a collar of the boundary and at most a single 3 ball. An explicit construction of a hierarchy with these properties is carried out in Chapter 3. Depending on homology conditionss on the underlying manifold, if the hierarchy contains surfaces that separate the manifold, these are constructed first. The remaining hierarchy surfaces are non-separating, incompressible, boundary incompressible and are constructed using a method of Stallings [43]. In particular,

24 Chapter 1. Introduction 14 take an element of the homology group H 1 (M, Z) that has infinite order and choose a homomorphism from H 1 (M, Z) to Z which maps the element to a generator of Z. This defines an element of cohomology; and by Lefschetz duality this defines a non-trivial element of H 2 (M, M, Z). Next we note that there is a non-separating surface whose intersection with the boundary of the manifold is an essential curve, which represents this element. By performing compressions and boundary compressions and choosing a connected component we obtain a non-separating incompressible, boundary incompressible surface. The details of this process are outlined in Chapter 3. Chapter 4 will prove that the special hierarchy is a special spine and hence there exists a triangulation dual to it. For manifolds without boundary we will obtain either 1 vertex or 2 vertex triangulations. For manifolds with boundary we will obtain either ideal triangulations or triangulations with ideal vertices and at most one finite vertex. However, we want to construct only 1 vertex triangulations and ideal triangulations for closed manifolds and manifolds with boundary respectively. To achieve this end we use a technique introduced by Jaco and Rubinstein [18] to crush the triangulation along an embedded normal surface. Crushing will convert a 2 vertex triangulation of a closed manifold into a 1 vertex triangulation. Similarly for a manifold with boundary, if the triangulation has a finite vertex, then after crushing we will have an ideal triangulation. The normal surface we crush along is obtained by normalising a regular neighbourhood of an edge or a face of the triangulation. The technique works in this context because the triangulations do not have certain self identifications (see Chapter 4). After constructing triangulations dual to special spines we turn our attention, in Chapters 5 7, to classifying embedded normal 2 spheres. Recall that to construct a 0 efficient triangulation we need show that for closed, triangulated 3 manifolds, the only normal embedded 2 spheres are vertex linking, or if the manifold has boundary that such a normal 2 sphere does not exist. This is proved by using an algorithmic process due to Waldhausen [50]. The same technique was used by Johannson [21] in his proof of the loop and sphere theorems. The main idea is to show that there exists a homotopy that removes intersections

25 Chapter 1. Introduction 15 between an embedded sphere and the hierarchy 2 complex. realised as a collection of moves on polygonal disks. The homotopy is To translate the algorithm by Waldhausen from a topological context to that of normal surface theory is not straight forward and requires some modification. One of the main obstacles here is that a homotopy of the embedded normal sphere that pushes through a vertex is not a normal homotopy. Due to the duality between the hierarchy and triangulation, it is reasonable that this may happen. Nonetheless, the thesis resolves this issue by allowing homotopies through a vertex but doing so in a controlled manner. Finally 0 efficiency is proved by noting that for a closed manifold, if the normal embedded 2 sphere can be homotoped by a normal homtopy off the hierarchy then by construction it must be vertex linking. Similarly for manifolds with boundary, if the normal embedded sphere can be normally homotoped off the hierarchy then it is parallel to a boundary component. Using the minimal separating condition on the hierarchy and since we assumed that all boundary components are tori we have a contradiction. Thus there are no such embedded normal 2 spheres and the triangulation is 0 efficient. In both cases we use the separating conditions of the special hierarchy (Property B listed above) to prove the result.

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27 Chapter 2 Preliminaries 2.1 Introduction We begin by introducing two methods of presenting 3 manifolds special spines and triangulations. Normal surfaces are also defined, and we end the chapter with a proof of the Haken Kneser finiteness theorem as presented in Hempel [15]. All of these results and definitions are well known (see [15, 17, 30]). 2.2 Notation and assumptions All manifolds and surfaces in the thesis will be assumed to be compact and orientable. Furthermore all 3 manifolds will be irreducible each 2 sphere in the manifold bounds a 3 ball. If X is a 2 or 3 manifold with possible boundary then we will denote the boundary of X by X. Also, for each set X the closure of X will be denoted by X. A surface S embedded in a 3 manifold M with boundary M is properly embedded if S M = S. All surfaces (with boundary) will be assumed to be properly embedded and 2 sided. Suppose T i is a boundary component of M. Then a collar of the boundary component T i, is a closed neighbourhood of T i homeomorphic to T i [0, 1]. We will abuse notation by just writing T i [0, 1] for such a region. 17

28 Chapter 2. Preliminaries 18 v 2 v 3 Edge e Fold edge v 1 v 3 v 1 v 2 Figure 2.1: Example of edge folding 2.3 Triangulations and spines Triangulations Let = { 1,..., t } be a collection of pairwise disjoint geometric tetrahedra in R 3. Let Φ be a collection of affine isometries, pairing faces of the tetrahedra in. If φ ij is in the collection Φ, then φ ij is an affine isomorphism that pairs a face σ i of i with a face σ j of j. We allow face self identifications, in that it is possible that i = j. The collection Φ will be called face identifications for. The pair (, Φ) will be called an identification scheme. A singular triangulation T is the quotient /Φ. At times we refer to the quotient /Φ as a realisation of an identification scheme. A tetrahedron will be denoted i if it is in the collection ; if we wish to identify the corresponding tetrahedron in the quotient /Φ we will denote it as i. An identification scheme does not necessarily define a genuine triangulation of a 3 manifold due to the possible existence of non-manifold points. Given an identification scheme, the non-manifold points of the quotient are called singular points. Such singular points may occur either due to edge folding in the quotient, or at the vertices of the triangulation. Edge folding occurs when an edge in a face of a tetrahedron is folded (around its midpoint) under an identification map (see Figure 2.1). Definition 1. An identification scheme is admissible if a realisation of the scheme does not fold edges.

29 Chapter 2. Preliminaries 19 The singular points of an admissible identification scheme are at the vertices. A triangulation of a closed orientable 3 manifold consists of an admissible identification scheme such that the link of each vertex is a 2 sphere. For a compact 3 manifold M with boundary, a triangulation is an ideal triangulation if each vertex of the triangulation has a neighbourhood that is a cone over a boundary component and the interior of the manifold is triangulated by tetrahedra with their vertices removed. Moreover, the ideal triangulations we consider will have a single vertex in each boundary component. A triangulation T with some vertices in the interior of the manifold and other vertices on the boundary is an ideal triangulation with finite vertices Spines of 3 manifolds This section will introduce the notion of a spine and in particular will define a class of spines called special spines. (See [30] for more details). Definition 2. Let M be a compact connected 3 manifold with boundary M. A sub-polyhedron P in the interior of M is a spine of M if M \ P is homeomorphic to M (0, 1]. If M is a closed 3 manifold then a spine of M is a spine for M \ int B 3 where B 3 is a closed 3 ball in M. We next give a definition for a simple polyhedron. Definition 3. A compact polyhedron is simple if each point has a neighborhood given by one of the following forms (see Figure 2.2). (i) A non-singular point where the link is a circle. (ii) A triple point where the link is a theta-graph. (iii) A true vertex where the link is the 1-skeleton of a tetrahedron. A point in P whose link is not a circle is a singular point. The collection of all singular points of P is called a singular graph of P and denoted by SP. Each connected component in the complement of the graph SP is a 2 manifold without boundary. We refer to such components as 2 components of P. A polyhedron P is cellular if each 2 component is a 2 cell [30].

30 Chapter 2. Preliminaries 20 True Vertex True Vertex Non-Singular point Triple point Figure 2.2: Local structure of simple polyhedron Definition 4. A polyhedron is special (i) if it is a simple polyhedron (ii) it has at least one true vertex, and (iii) each 2 component is a 2 cell. The class of spines of interest to us in the thesis are special spines. Definition 5. A spine of a 3 manifold M is a special spine if the subpolyhedron that defines the spine is a special polyhedron. A special spine of a 3 manifold determines the manifold uniquely. Furthermore a manifold can be reconstructed from its special spine [30, 38]. Thus a special spine provides a combinatorial presentation of the underlying 3 manifold. In general, not every special polyhedron is the spine of a 3 manifold [30] Duality of special spines and triangulations There exists a natural duality between special spines and triangulations (Again, we refer to [30] for the details). This duality can be observed as follows.

31 Chapter 2. Preliminaries 21 σ i σ j Glue G i Gj Figure 2.3: Gluing faces via special spine Let P be a special spine. If v is a true vertex of P, then we associate a tetrahedron to v, by associating each face of P (in a small neighborhood around v) to an edge of the tetrahedron. Thus, for a given special spine P, we can construct a collection of tetrahedra, where each tetrahedron in the collection corresponds to a true vertex of the special spine. To construct a collection of face identification maps Φ, we use the identifications of P. To be more precise, for each tetrahedron i in the collection, let G i = i P. Then there is a gluing map φ ij in Φ if and only if there exists G i and G j that are identified in P. Hence, we have a singular triangulation T with T = /Φ dual to a special spine of a 3 manifold. The triangulation T will be called the triangulation dual to the special spine P (see Figure 2.3). Of course, for such an identification scheme to be a genuine triangulation of a 3 manifold, we need to show that the identification scheme is admissible see Matveev [30] for more details. 2.4 Normal Surfaces Normal surfaces were initially introduced by Kneser [24] and used in proving the prime decomposition theorem for 3 manifolds. Normal surface theory was developed by Haken [14] and has proved to be useful in algorithmic and combinatorial contexts, in 3 manifold topology [11]. Let σ be a 2 face in a tetrahedron. A normal arc in σ is a properly embedded arc in σ with endpoints on distinct edges of σ. A normal disk in is a properly

32 Chapter 2. Preliminaries 22 Triangle disks Quadrilateral disks Figure 2.4: Normal disks embedded disk D in so that the intersection with each 2 face σ of is either a normal arc or the empty set, and the boundary of the normal disk consists of at most four normal arcs. Definition 6. Let the pair (M, T ) denote a compact 3 manifold M and a triangulation T for M given by /Φ. A surface S is a normal surface with respect to T if the pre-image of S in consists of normal disks. A normal isotopy of M is an isotopy that leaves each cell of T invariant. A tetrahedron, contains only seven distinct normal disk types up to normal isotopy. These consist of four normal triangle disks and three normal quadrilateral disks, as shown in Figure 2.4. Each normal triangle cuts off a vertex of and each normal quadrilateral separates a pair of opposite edges in. Normal isotopy classes of normal disks are referred to as normal disk types. A normal surface S is an embedded normal surface if the collection of normal disks in are disjoint. There is a unique way of gluing together such a collection to form an embedded surface. If t denotes the number of tetrahedra in a triangulation T, we have a total of 7t normal disk types. After fixing an ordering, there is a 7t tuple (p 1,..., p 4t, q 1,..., q 3t ) where for each i, p i denotes a triangular disk type; and for each j, q j denotes a quadrilateral disk type.

33 Chapter 2. Preliminaries 23 Thus a normal surface is described by the 7t tuple (x 1,..., x 4t, y 1,..., y 3t ) where x i denotes the number of triangular disks of disk type p i, and y j denotes the number of quadrilateral disks of disk type q j. Given a pair (M, T ) where M is a compact 3 manifold and T a triangulation of M, denote by T (1) the 1 skeleton of T. The weight of a surface is defined as follows. Definition 7. Let (M, T ) denote a 3 manifold with triangulation T. Let S be a surface in M. The weight of S, denoted wt(s), is the number of intersections between S and T (1) and is denoted by wt(s) = S T (1). If S is a normal surface in (M, T ), then S is least weight if for every normal surface S isotopic to S, wt(s) wt(s ). 2.5 Normal surfaces and incompressibility This section will define incompressible surfaces and prove the Haken Kneser finiteness theorem. The proof presented here closely follows the proof by Hempel [15]. Definition 8. Let M be a 3 manifold and F a surface in M. A compressing disk for F is a disk D M such that D F = D and D does not bound a disk in F. A surface is incompressible if it has no compressing disk and no component of the surface is a 2 sphere. Remark 5. For a properly embedded 2 sided surface F with F S 2, the wellknown Loop theorem and Dehn s lemma show that a surface is incompressible if and only if the map π 1 (F ) π 1 (M) is injective [17]. We also consider compressing disks for the boundary of a 3 manifold, and as a consequence define the notion of a 3 manifold with incompressible boundary. Definition 9. Let M be a 3 manifold with boundary M. A compressing disk for the boundary M is a properly embedded disk D in M such that D does not bound a disk in M. A 3 manifold has incompressible boundary if there is no compressing disk for the boundary, and no component of the boundary is a 2 sphere.

34 Chapter 2. Preliminaries 24 Figure 2.5: Example of a non-disk component Next we define what we mean by boundary compressions for the boundary of a properly embedded surface. Definition 10. Let M be a 3 manifold with boundary M and let S be a properly embedded surface in M with non-empty boundary S. The surface S is boundary compressible if 1. S is a disk, or 2. S is not a disk, but there exists a disk D M such that D S = α D and D M = β D where α β = D with α either not separating S, or if α does separate S, then the closure of each component separated by α in S is not a disk. If there is no such disk D in M, then S is boundary incompressible. An incompressible surface after a suitable isotopy may be put into normal form[15]. Theorem 4. Let M be a compact irreducible 3 manifold, with triangulation T. If S is a closed incompressible surface (possibly disconnected) in M, then S can be isotoped to a normal surface with respect to T. Proof. Without loss of generality assume that S is transverse to the edges of T and has minimal weight in its isotopy class that is, wt(s) is minimal in its isotopy

35 Chapter 2. Preliminaries 25 3 gon i 0 gon 4 gon Figure 2.6: 0 gon, 3 gon and 4 gon disks in i class. Denote by i a tetrahedron in the triangulation T. The theorem is proved by successively proving the following claims. Claim (i): S meets each tetrahedron i in the triangulation as a collection of disks Suppose that S meets i in a component which is not a disk. (See Figure 2.5). It is easy to find a curve α of S i which bounds a disk D in i which meets S only in loops which bound disks in i but the component A of S i containing α is not a disk. But since S is incompressible, α must bound a disk D in S. We can isotope D into i to give a disk bounded by α with interior disjoint from S. Since M is irreducible, there is an isotopy of D to this disk which reduces the number of components of intersection of S with the tetrahedra which are not disks, and hence we have a contradiction. Claim (ii): 0 gon intersections can be removed We call a simple closed curve lying in the interior of a single face of i, a 0 gon disk. Suppose there exists a 0 gon disk D where D S i. Then there exists a disk D such that D i, and D = D. By choosing an innermost curve D in i S we can isotope D to D and then off i. This allows the removal of the 0 gon disk D. Successive use of this procedure results in the removal of all 0 gons in i.

36 Chapter 2. Preliminaries 26 Claim (iii): S meets an edge e of the triangulation at most once The proof is by contradiction. Suppose a component of S i meets an edge e of T more than once. Then there exists a loop γ of S i such that e is an edge of i, γ bounds a disk D in i and D e contains an arc β. Since we have already proved that all the intersections of S with i are disks, it follows that γ bounds a disk D which is a component of S i. Since i is a ball, by Alexander s well-known theorem, the disks D and D are isotopic in i. Hence we can find a disk D embedded in i so that D S = δ and D i = β where δ is an arc with D = β δ and β δ = β = δ. Let N(D ) be a small neighbourhood of D. Then pushing across D gives an isotopy of S that is fixed outside N(D ) and decreases the weight wt(s). This is a contradiction. Hence S meets an edge e at most once. The assertion of the theorem follows easily since we have shown above, that there are no 0-gons, and that the intersection of S with each tetrahedron i consists of disks which meet each edge at most once. This completes the proof that the least weight surface in the isotopy class of S is a normal surface. We are now in a position to state and prove the main result of the section the Haken Kneser finiteness theorem for compact, irreducible 3 manifolds. The theorem asserts that the number of non-parallel closed incompressible surfaces in a given 3 manifold is bounded. The result is well-known and the details of the proof are in [15]. Theorem 5. Let M be a compact, irreducible 3 manifold. If S is a closed incompressible surface in M, and if S denotes the number of components of S, then there exists a natural number h(m) such that if S > h(m), then two components of S are parallel. Outline of proof ([15] Lemma 13.2). Let T be a triangulation of M. From Theorem 4 we can isotope S so that S is a normal surface. A component X of M \ S is good if X intersects each tetrahedron in components which lie between two normal disks of the same type. Otherwise X is bad. The good components are I-bundles over closed surfaces; this is easily seen by noting