Trisections in three and four dimensions By DALE R. KOENIG B.S. (University of California, Davis) 2011 DISSERTATION Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS in the OFFICE OF GRADUATE STUDIES of the UNIVERSITY OF CALIFORNIA DAVIS Approved: Abigail Thompson, Chair Joel Hass Misha Kapovich Committee in Charge 2017 -i-
c Dale R. Koenig, 2017. All rights reserved.
To my parents -ii-
Contents Abstract Acknowledgments v vi Chapter 1. Introduction 1 Chapter 2. Heegaard Splittings of 3-Manifolds 2 2.1. Notation and basic definitions 2 2.2. Stabilization and the Reidemeister-Singer theorem 6 2.3. Reducible and weakly reducible Heegaard splittings 7 2.4. Hempel distance and the curve complex 8 2.5. Heegaard splittings of some special 3-manifolds 9 Chapter 3. Trisections of 4-Manifolds 14 3.1. Basic Definitions 14 3.2. Trisection Diagrams 15 3.3. Connect Sums and Stabilizations of Trisections 16 3.4. Trisections of Low Genus 17 Chapter 4. Trisections of 3-Manifolds 19 4.1. Overview 19 4.2. Examples 21 4.3. Balancing Trisections 28 4.4. Trisections, Heegaard Splittings, and Trisection Genus 28 4.5. Trisections With Disconnected Pairwise Intersections 32 4.6. The Stabilization Theorem 32 4.7. Trisections of S 3 36 Chapter 5. Classes of Trisections in Four Dimensions 39 -iii-
5.1. Balanced and unbalanced trisecton genus of four dimensional manifolds 39 5.2. Some remarks on Heegaard diagrams and sets of curves on surfaces 40 5.3. Trisections of 3-manifold bundles 42 5.4. Some destabilizations 56 5.5. Summary 57 Bibliography 59 -iv-
Dale R. Koenig June 2017 Mathematics Trisections in three and four dimensions Abstract Every closed orientable three dimensional manifold has a Heegaard splitting, a decomposition into two handlebodies. Any two Heegaard splittings of the same manifold can be made isotopic after a finite number of stabilization operations. The notion of trisections, developed by Gay and Kirby, provided an analogue in four dimensions. They showed that any closed smooth orientable four dimensional manifold can be broken into three four dimensional handlebodies, with niceness conditions on their intersections, and showed that any two trisections are isotopic after stabilizations. In this thesis we investigate the notion of trisections in both three and four dimensions. In dimension three we define trisections of 3-manifolds and stabilization on these trisections. We use this to define the trisection genus of a 3-manifold. We then present several examples, showing among other things that the trisection genus is not additive under connect sum. We prove a stable equivalence theorem for trisections of 3-manifolds, showing that any two trisections of the same three-manifold can be made isotopic after stabilizations. We also show that trisections of S 3 can be very complicated, so there is no analogue of Waldhausen s theorem for trisections of three manifolds. We then move on to trisections in four dimensions. We first show that if there exist four manifolds with unbalanced trisection genus lower than their balanced trisection genus, then trisection genus as defined by Gay and Kirby is not additive under connect sum. We produce several new classes of trisections, including several likely such examples. We include a class of examples that are provably minimal genus. We provide trisection diagrams for many of these trisections, and summarize some methods for quickly checking that these diagrams produce valid trisections. -v-
Acknowledgments I would like to thank everyone who helped me get where I am now. Special thanks to my fantastic advisor Abigail Thompson, who has been extremely supportive since the first time I came to her office as a clueless college graduate. I would also like to thank Joel Hass and Misha Kapovich for their advice and comments. I thank the UC Davis math department for their generous support during my time in grad school and their work to make our research possible. I d also like to give special thanks to the other students and researchers who contributed to our group s small seminars both for the math I learned through them and for the friendly and fun atmosphere of the meetings. Lastly I want to thank my parents for supporting me and my love of learning from a young age. -vi-
CHAPTER 1 Introduction A Heegaard splitting of a closed, orientable 3-manifold can be thought of as a bisection of the 3-manifold into two handlebodies. Gay and Kirby [GK12] introduced trisections of smooth, orientable 4-manifolds to create an analogous construction in the higher dimension, defining the trisection genus of a 4-manifold and proving that all trisections of a 4-manifold are stably equivalent. Here we develop a theory of trisections of 3-manifolds and present several examples of such trisections. We define a notion of stabilization and prove that all trisections of a given closed, orientable 3-manifold are stably equivalent. We then define trisection genus for 3-manifolds and investigate connections between the trisection genus and Heegaard genus of a 3-manifold. We will also show that there is no reasonable analogue of Waldhausen s theorem for trisections of 3-manifolds. After investigating trisections of 3-manifolds, we switch focus to trisections of 4-manifolds. As of yet, there are very few examples of trisection diagrams in the literature. Specifically, all such diagrams are either connect-sums of lower genus diagrams or are genus 2. We describe several classes of nontrivial examples of trisections, along with diagrams for such trisections. We will also describe how these examples might apply to the question of whether trisection genus is additive. In the chapters 2 and 3 we discuss the known results that this work is based upon. Chapter 2 will give an overview of the theory of Heegaard splittings, including the classification of Heegaard splittings of many simple classes of manifolds. Chapter 3 will cover the basic theory of trisections of 4-manifolds as developed by Gay and Kirby. In Chapter 4 we begin the new material, describing the theory of trisections of 3-manifolds, including the main result on stable equivalence. Chapter 5 presents several classes of trisections of 4-manifolds along with figures showing what the diagrams of these trisections look like. 1
CHAPTER 2 Heegaard Splittings of 3-Manifolds 2.1. Notation and basic definitions All manifolds we work with are assumed smooth, closed, and orientable unless stated otherwise, and all statements and theorems are assumed to only apply to such manifolds. M or M 3 will be used to denote an arbitrary 3-manifold. Σ g will denote a closed surface of genus g, and Σ g,b will denote a compact surface of genus g with b boundary components. It is useful to view 3-manifolds in terms of their handle decompositions. A three dimensional k-handle is a 3-ball which we view as B k B 3 k. The k-ball B k {0} is the core of the k-handle, and the (3 k)-ball {0} B 3 k is the cocore. A 3-manifold can be constructed by attaching handles, one at a time, where a k-handle is attached along the portion of its boundary S k 1 B 3 k. A 0-handle is then a copy of B 3 with no attachment necessary. A 1-handle can be viewed as a solid tube attached along its two end disks. A 2-handle is a thickened disk attached along an annulus. A 3-handle is a copy of B 3 attached along its entire boundary. k-handles correspond to index k critical points in Morse theory [Mil63]. It follows that every closed 3-manifold has a handle decomposition with a single 0-handle and a single 3-handle. Every compact connected 3-manifold with nonempty boundary has a handle decomposition with a single 0-handle and no 3-handles. The handle decomposition can be reversed, switching all k-handles with (3 k)-handles, so for example a 3-handle, if viewed from above is just a 0-handle. Definition 2.1.1. A 3-dimensional (orientable) handlebody is a 3-manifold diffeomorphic to the result of attaching some number g of 1-handles to a single 0-handle. The number g of 1-handles needed is called the genus of the handlebody. If H is a genus g handlebody, the boundary of H is a genus g surface. We can also think of H as the boundary connect sum of g copies of S 1 B 2. Heegaard splittings, introduced by Poul Heegaard in 1898 [Hee98], are a useful technique for analyzing 3-manifolds. A Heegaard splitting of M is a decomposition M = H 1 H 2 where both 2
H 1 and H 2 are handlebodies, necessarily of equal genus, glued together along their boundaries by a homeomorphism between their boundaries. Every 3-manifold has such a splitting. To see this, choose a Morse functon h: M R such that h has a single critical point of index 0 and of index 3. We can also choose h to be self-indexing, so all index i critical points lie in h 1 ((i ɛ, i + ɛ)) for small ɛ. Then the sublevel set h 1 ([0, 3/2]) contains one 0-handle and some number of 1-handles, and is therefore a handlebody which we denote by H 1. The region h 1 ([3/2, 3]) contains one 3- handle and some number of 2-handles, but we can view this region from the top to switch i-handles and (3 i)-handles, after which we see that this region is also a handlebody, which we denote by H 2. Since H 1 and H 2 share a common boundary h 1 (3/2), they must be of equal genus. It follows that M = H 1 H 2 is a Heegaard splitting of M. Since M was arbitrary, every 3-manifold has such a Heegaard splitting. We use the notation (Σ g ; H 1, H 2 ) to denote the genus g Heegaard splitting M = H 1 H 2 where Σ g = H 1 = H 2. Two such Heegaard splittings (Σ; H 1, H 2 ) and (Σ ; H 1, H 2 ) of the same manifold M are called isotopic if there is an isotopy of M taking Σ to Σ. They are called oriented isotopic if there is an isotopy that also takes H 1 to H 1 and H 2 to H 2. A Heegaard splitting (Σ; H 1, H 2 ) is called flippable if it is oriented isotopic to (Σ; H 2, H 1 ). We distinguish Heegaard splittings only up to oriented isotopy. In order to represent these Heegaard splittings in a simple, combinatorial manner, we first need to analyze handlebodies more carefully. If D 1,..., D g is a set of disjoint essential disks in a handlebody H such that cutting H open along all of the D i leaves a ball, then {D i } is a minimal system of disks for H. Note that the number of disks in a minimal system is equal to the genus of H. If we recall that the handlebody H can be constructed by attaching g 1-handles to a ball, then the disks D i can be thought of as the cocore disks of the 1-handles. If g > 1 there will be many non-isotopic minimal systems of disks. However, any two such systems are related by a sequence of simple moves called disk slides [Joh06]. To perform a disk slide, let α be an oriented arc embedded in H such that the interior of γ is disjoint from the system of disks, γ D i, and + α D j. Then D i N(γ) D j is an embedded disk in H. Isotope the interior into H, and call this disk D i. Replacing D i with D i gives a new minimal system of disks as desired. This operation corresponds to performing a handle slide, where the 1-handle of which D j is a cocore is slid over the handle of which D i is a cocore. Since the system of disks {D i } is uniquely determined by the set of curves { D i } and vice versa, we will often think in terms of systems of 3
Figure 2.1. The two green curves give a cut system. γ can be used to slide one curve over the other to get the red curve. This is the same as performing a disk slide on the disks in H bounded by the green curves. curves rather than systems of disks. Given a collectively nonseparating system of g curves α 1,..., α g on a genus g surface Σ, called a cut system, we can recover a handlebody by attaching thickened disks along regular neighborhoods of the α i, and filling in the resulting S 2 boundary component with a 3-ball. We can slide these curves over each other just as if we were performing disk slides on the disks that they bound, and two sets of curves related by such slides determine the same handlebody. See Figure 2.1. We can now define a combinatorial representation of a Heegaard splitting called a Heegaard diagram. If (Σ; H 1, H 2 ) is a Heegaard splitting of M, let α 1,..., α g Σ be the boundaries of a minimal system of disks for H 1, and β 1,..., β g Σ the boundaries of a minimal system of disks for H 2. A Heegard diagram for the splitting consists of Σ together with the collections of curves {α i } and {β i }. Since the α and β curves determine how to attach the handlebodies on each side, the Heegaard splitting can be recovered from a Heegaard diagram. Performing slides on either set of curves produces a Heegaard diagram for the same splitting, as does applying any orientation preserving diffeomorphism to the entire diagram. Often it is convenient to draw Σ as the boundary of a standardly embedded handlebody of S 3, and assume that this handlebody is H 1. In this case, we only need to draw one set of curves, bounding disks in H 2, in order to recover M. See Figure 2.2 for an example. The definition of Heegaard splittings has been generalized to 3 manifolds with boundary [CG87]. Definition 2.1.2. Let Σ g be a closed surface. A compression body is the result of attaching some number of 2-handles to Σ g {0} Σ [0, 1] and possibly filling in some of the resulting 4
(a) A Heegaard diagram. (b) A Heegaard diagram for the same 3-manifold Figure 2.2. The three red curves in (a) bound disks in the handlebody bounded by the surface in the given embedding into R 3. If we assume that the surface bounds a handlebody to the inside as pictured then we do not need to draw the red curves, as in (b). inner 2-sphere boundary components with 3-balls. A compression body is called trivial if it is diffeomorphic to Σ g [0, 1]. If C is such a compression body, define + C to be the outside boundary component Σ {1}, and define C to be the union of the inner boundary components, C + C. If M has boundary, a Heegaard splitting of M is a triple (Σ g ; C 1, C 2 ) where M = C 1 C 2 and C 1 C 2 = Σ g = + C 1 = + C 2. In what follows we do not use 3-manifolds with more than one boundary component, so when M has boundary we will assume that the boundary component lies in C 1 for convenience. In this case we can also think of C 1 as being constructed by attaching some number of 1-handles to a regular neighborhood of M. Then M = C 1 and C 2 is a handlebody. If Σ is a Heegaard surface for M, then Σ is also a Heegaard surface for any 3-manifold obtained by filling boundary components of M by attaching 2-cells to C 1 and filling in S 2 boundary components with 3-balls. Heegaard splittings lead to a natural invariant of 3-manifolds. Given a 3-manifold M, the Heegaard genus of M is the least g such that M has a genus g Heegaard splitting. S 3 is therefore the only 3-manifold with Heegaard genus equal to 0. S 1 S 2 has Heegaard genus 1. If M is a 3-manifold of Heegaard genus 1 that is not S 1 S 2, then M is called a lens space. There is particular interest in Heegaard splittings of minimal genus. For most 3-manifolds the minimal genus Heegaard splitting is unique; See 2.4.2. However, there exist 3-manifolds with infinitely many minimal genus Heegaard splittings [BD06]. 5
Figure 2.3. The standard genus 1 diagram for S 3. 2.2. Stabilization and the Reidemeister-Singer theorem Suppose M 1 is a 3-manifold with a Heegaard diagram given by Σ g1 and curves {α i } 1 i g1 and {β i } 1 i g1, and M 2 is another 3-manifold with a Heegaard diagram given by Σ g2 and curves {α i } 1 i g2 and {β i } 1 i g2. Choose disks D 1 Σ g1 and D 2 Σ g2 disjoint from the curves of the Heegaard diagrams in those surfaces. Remove both disks, and glue the two now-punctured surfaces together along their S 1 boundary components. The resulting surface together with the images of all the α and β curves is a Heegaard diagram for the manifold M 1 #M 2. This operation is a connect sum of Heegaard diagrams. We can now define a stabilization of a Heegaard splitting. Definition 2.2.1. Let M be a manifold with Heegaard surface Σ. Let α be an arc in one of the handlebodies, say H 2, such that α is parallel into the boundary of H 2. That is, there is an arc β H 2 such that α = β and α β bounds a disk in H 2. Then we choose a regular neighborhood of α in H 2, and set H 1 = H 1 N(α) and H 2 = H 2 N(α). The Heegaard splitting M = H 1 H 2 is called a stabilization of M = H 1 H 2. This operation is unique up to isotopy. In terms of Heegaard diagrams, this is the result of taking the connect sum of a diagram for (H 1, H 2 ; Σ) with the standard genus 1 diagram of S 3 in Figure 2.3. We say one Heegaard splitting is a stabilization of another if it is the result of performing some number of stabilization operations. A given 3-manifold may have many different non-isotopic Heegaard splittings of a given genus. However, the Reidemeister-Singer theorem tells us that any two splittings of the same manifold have a common stabilization. 6
Theorem 2.2.1 ( [Rei33], [Sin33]). Suppose (Σ; H 1, H 2 ) and (Σ ; H 1, H 2 ) are two Heegaard splittings of the same 3-manifold M. Then there is a third Heegaard splitting (Σ ; H a stabilization of both (Σ; H 1, H 2 ) and (Σ ; H 1, H 2 ). 1, H 2 ) that is Recall that when M has a single boundary component, we assume in a Heegaard splitting M = C 1 C 2 that M = C1. In this case, the definition of stabilization and the Reidemeister- Singer theorem hold without any changes. 2.3. Reducible and weakly reducible Heegaard splittings A Heegaard splitting is called reducible if there exists a 2-sphere intersecting the Heegaard surface in a single essential curve. In this case, the Heegaard splitting decomposes as a connect sum of two splittings of lower genus. Haken proved: Theorem 2.3.1 ( [Hak68]). If M is a reducible 3-manifold, then any Heegaard splitting of M is reducible. This can be used to show: Corollary 2.3.1 ( [Hak68]). If M 1 and M 2 are two manifolds with Heegaard genus g 1 and g 2 respectively, then M 1 #M 2 has Heegaard genus g 1 + g 2. Every stabilized splitting is reducible except for the genus 1 splitting of S 3 shown in Figure 2.3. Given a diagram of a reducible Heegaard splitting, Theorem 2.3.1 guarantees that we can perform slides on the curves of the diagram to get a diagram that obviously splits. That is, after slides there exists some essential curve in the diagram that is disjoint from both of the two sets of curves determining H 1 and H 2. It will not usually be obvious how to perform these slides in order to simplify the diagram. A Heegaard splitting that is not reducible is called irreducible. Additionally, if M is irreducible then every reducible Heegaard splitting of M is stabilized. Reducibility is therefore a very strong condition. Casson and Gordon revolutionized the theory of Heegaard splittings in 1987 with their notion of weakly reducible and strongly irreducible Heegaard splittings. Definition 2.3.1. A Heegaard splitting (Σ; H 1, H 2 ) is weakly reducible if there exist disjoint curves α, β Σ such that α bounds an essential disk in H 1 and β bounds an essential disk in H 2. A Heegaard splitting that is not weakly reducible is called strongly irreducible. 7
The importance of this definition comes from the following result: Theorem 2.3.2 ( [CG87]). Suppose (Σ; H 1, H 2 ) is a weakly reducible Heegaard splitting of M. Then either the splitting is in fact reducible, or M contains an essential surface. For M with boundary, Casson and Gordon also proved the following important result: Theorem 2.3.3 ( [CG87]). Suppose M is a compact 3-manifold with a properly embedded essential disk. That is, there is an embedded disk D such that D M and D does not bound a disk in M. Then for any Heegaard splitting (Σ; C 1, C 2 ), there exists a properly embedded essential disk such that D Σ is a single essential simple closed curve. 2.4. Hempel distance and the curve complex Given a surface Σ g with g 2, the curve complex is the simplicial complex where each n-cell corresponds to an isotopy class of n + 1 disjoint essential curves. Thus, vertices correspond to essential curves in Σ, and edges correspond to non-isotopic pairs of such curves that have disjoint representatives. The natural metric on the 1-skeleton of this complex lets us talk about the distance between two curves. This notion leads to a measure of the complexity of a Heegaard splitting: Definition 2.4.1 ( [Hem01]). Suppose (Σ; H 1, H 2 ) is a Heegaard splitting. The distance or Hempel distance of the Heegaard splitting is defined to be the minimum distance in the curve complex between two curves α and β such that α bounds an essential disk in H 1 and β bounds an essential disk in H 2. It follows from the definitions that reducible Heegaard splittings are distance 0 and weakly reducible Heegaard splittings are distance 1. Hempel proved the following about distance of Heegaard splittings. Theorem 2.4.1 ( [Hem01]). For integers g 2 there exist Heegaard splittings of genus g and of arbitrarily large distance. A result of Scharlemann and Tomova tells us that a Heegaard splitting that has high distance relative to its genus is unique. 8
Theorem 2.4.2 ( [ST06]). Suppose (Σ g ; H 1, H 2 ) is a distance d splitting of M and (Σ g ; H 1, H 2 ) is another Heegaard splitting of M. Then either (Σ ; H 1, H 2 ) is a stabilization of (Σ g; H 1, H 2 ) or g d/2. In particular, if d > 2g then (Σ g ; H 1, H 2 ) is the unique genus g Heegaard splitting of M. 2.5. Heegaard splittings of some special 3-manifolds For many simple manifolds, there is a unique minimal genus Heegaard splitting, of which every higher genus splitting is a stabilization. We discuss known results for S 3, compression bodies, surface bundles, connect sums of S 1 S 2, and lens spaces. 2.5.1. Heegaard splittings of S 3. The first and most famous result of this kind is Waldhausen s theorem. Theorem 2.5.1 ( [Wal68]). Every Heegaard splitting of S 3 is a stabilization of the trivial genus zero Heegaard splitting into two balls. It follows that every Heegaard splitting of S 3 has a Heegaard diagram with sets of curves α 1,..., α g and β 1,..., β g where α i β j = δ ij. We call such a diagram the standard genus g diagram of S 3, and we say that Heegaard splittings of S 3 are standard. 2.5.2. Heegaard splittings of handlebodies. If H is a handlebody, then H has a Heegaard splitting where the Heegaard surface is parallel to H. We call this Heegaard splitting and any splitting stabilized from it a standard Heegaard splitting of H. Using the tools we have introduced so far, it is in fact not difficult to show that every Heegaard splitting of H is standard. Since the proof is short, we reproduce it here as in [ST93]. Proposition 2.5.1. Heegaard splittings of handlebodies are standard. Proof. The proof is by induction. If H is a ball this follows from Theorem 2.5.1. Indeed, taking a Heegaard splitting of H and filling in the 2-sphere boundary component gives a Heegaard splitting of S 3, which must be standard, so the splitting of H must also be standard. Now suppose H is a handlebody of genus g with genus k Heegaard splitting (Σ; C 1, C 2 ), g k. Then H certainly contains an essential disk, so by Theorem 2.3.3, there is an essential disk D such that D Σ is a single simple closed curve. Let Σ be the result of compressing Σ along the disk component of 9
D D Σ. Then Σ can be isotoped to be disjoint from D. Let H be the genus g 1 handlebody resulting from cutting H along D. Then Σ is actually a Heegaard surface for H ; Indeed, C 2 is still a handlebody since it is the result of cutting the handlebody C 2 along a disk, and C 1 is the result of adding a 2-handle to C 1 and then cutting the resulting compression body along an essential disk. By induction, we assume that the splitting (Σ ; C 1, C 2 ) is standard, so by reversing the cut we can see that the original splitting was also standard. 2.5.3. Heegaard splittings of surface bundles. Now consider Σ I/(ϕ(x), 0) (x, 1), the surface bundle with monodromy ϕ. Take Σ [0, 1/2], drill a vertical hole out of it, and attach a vertical 1-handle to it, where by vertical we mean transverse to the fibers Σ x. Call this H 1, and the complement H 2. Both H 1 and H 2 are the result of thickening a punctured surface and then attaching a 1-handle, so they are both genus 2g + 1 handlebodies. We call this the standard genus 2g + 1 Heegaard splitting of the surface bundle. It is not true that all Heegaard splittings of surface bundles are standard. However, we have the following two useful results. Theorem 2.5.2 ( [Sch93]). If M = Σ S 1, then any Heegaard splitting of M is stabilized from the standard genus 2g + 1 splitting. Theorem 2.5.3 ( [BS05]). Suppose M = Σ g I/(ϕ(x), 0) (x, 1) is a surface bundle. Define the translation distance d(ϕ) of the bundle to be the minimum distance in the curve complex between an essential curve γ and ϕ(γ). If d(ϕ) > 4g, then the standard genus 2g + 1 splitting is the unique minimal genus Heegaard splitting of M. 2.5.4. Heegaard splittings of connect sums of S 1 S 2. If M = # k S 1 S 2, then M has a genus k Heegaard splitting in which α i β i for all i. A Heegaard splitting of M is called standard if it is a stabilization of this splitting. See Figure 2.4. Combining 2.5.2 and [Hak68] gives Corollary 2.5.1. Heegaard splittings of # k S 1 S 2 are standard. 2.5.5. Heegaard splittings of lens spaces. If M is a 3-manifold with a genus-1 Heegaard splitting that is not S 3 or S 1 S 2, then M is called a lens space. If one of the handlebodies H 1 is embedded standardly in S 3, then an embedded 10
Figure 2.4. The standard genus 1 diagram for S 3. curve determining a meridian in the second handlebody H 2 is completely determined by its slope p/q on H 1, where p and q are nonzero relatively prime integers. p is the number of times the curve goes around H 1 meridionally and q the number of times it goes around longitudinally. We denote the 3-manifold with such a Heegaard diagram by L p/q. Not all choices of p/q give distinct 3-manifolds. For example, performing a Dehn twist around the meridian of the Heegaard surface changes p by ±q. The complete classification, first done by Reidemeister, is as follows: Theorem 2.5.4 ( [Rei35], [Bro60]). L p/q is homeomorphic to L p /q iff q = q and p = ±p ±1 (mod q). When p = ±p (mod q) the Heegaard splittings determined by p/q and p /q give the same Heegaard splitting up to oriented isotopy. When p = ±p 1 (mod q) the Heegaard splittings determined by p/q and p /q still give the same Heegaard splitting up to isotopy, but not in general up to oriented isotopy. Thus, two genus one Heegaard splittings of a lens space M are always isotopic. We therefore call a Heegaard splitting of a lens space M standard when it is isotopic to a stabilization of the unique genus one Heegaard splitting of M. Theorem 2.5.5 ( [BO83]). Heegaard splittings of lens spaces are standard. We now analyze which of these Heegaard splittings are flippable. That is, in which genus 1 Heegaard splittings (Σ; H 1, H 2 ) are H 1 and H 2 isotopic. Consider the lens space L p/q. By Theorem 2.5.4 we may assume that 1 p q/2. We state the following simple fact without proof: Remark 2.5.1. Let (a, b) and (p, q) be the slopes of two curves on a torus. Then these curves have geometric intersection number ±1 iff aq bp = ±1. 11
Figure 2.5. A Heegaard diagram for L 1/3. The curves determining the handlebodies H 1 and H 2 are shown red and green respectively. Since a (0, 1) curve intersects the red and green curve in one point each, it is isotopic to a spine of both H 1 and of H 2. We can then show that the lens spaces with flippable genus 1 Heegaard splittings are precisely the L 1/q spaces. Proposition 2.5.2. Let M be a lens space L p/q, q > 1 and 0 < p q/2 relatively prime, and consider the standard genus 1 Heegaard splitting. If p = 1 then a (0, 1) curve in the Heegaard surface is isotopic to a spine of both handlebodies. If p 1 then the spines of the two handlebodies are not isotopic, so the Heegaard splitting is not flippable. Proof. The first statement follows from the fact that a (1,0) curve intersects each curve of the Heegaard diagram a single time. For the second statement, consider a spine of the outer handlebody. This spine is isotopic to a curve (a, b) on the genus one Heegaard surface that intersects the (p, q) curve once. Therefore, aq bp = ±1. Moreover we may assume that 0 < b < q. If this spine does not generate the same element of π 1 (M) as a spine of the inner handlebody, then the spines cannot be isotopic. Thus, we must have b = 1. The condition on the determinant then becomes aq p = ±1. However, since 0 < p < q this can only be true if a = 1 and p = q 1 or if a = 0 and p = 1. The case where p = q 1 produces nothing new because of our assumption that p q/2. The lemma follows. A Heegaard diagram for L 1/3 is shown in Figure 2.5. 12
CHAPTER 3 Trisections of 4-Manifolds 3.1. Basic Definitions We now give an introduction to the theory of trisections of 4-manifolds developed by David Gay and Robion Kirby in 2012 [GK12]. A 4-dimensional handlebody is an object obtained by attaching some number of four dimensional 1-handles B 1 B 3 to B 4, where the 1-handles are attached using the pair of 3-balls S 0 B 3. The genus is the number of 1-handles attached. A genus k handlebody can also be thought of as a boundary connect sum k S 1 B 3 of k genus 1 handlebodies, where we take the result to be B 4 if k = 0. The boundary of such a handlebody is the 3-manifold # k S 1 S 2. We can now define a trisection of a 4-manifold. Rather than using the original definition of [GK12], we use a slightly more general definition introduced by Trent Schirmer in [Sch15]. Let M be a smooth, closed, orientable 4-manifold. Definition 3.1.1. A (g; k 1, k 2, k 3 )-trisection of M is a quadruple (Σ; X 1, X 2, X 3 ) satisfying the following conditions: M = X 1 X 2 X 3 Each X i is a 4-dimensional handlebody of genus k i The triple intersection surface X 1 X 2 X 3 = Σ is an orientable closed surface of genus g Σ induces a Heegaard splitting of each X i into the two 3-dimensional handlebodies X i X j and X i X k, i j k If k 1 = k 2 = k 3 then the definition restricts to that of Gay and Kirby and the trisection is called balanced. Otherwise it is unbalanced. When referring to a balanced (g; k 1, k 2, k 3 ) trisection we often leave out k 2 and k 3 to simplify notation. The genus of a trisection is defined to be g. The trisection genus of a 4-manifold M, defined in [GK12], is the minimum value of g over all balanced trisections of M. One can also ask about the unbalanced trisection genus of M, defined as the minimum value of g over all trisections, balanced and unbalanced, of M. The simplest trisection is the genus 0 trisection of S 4, splitting it into three copies of B 4 each pair of which intersect in a 13
Figure 3.1. A trisection of a 4-manifold. Σ is represented by the center point, and the three 3 dimensional handlebodies X i X j, i j by the three radial lines. 3-ball, and this is the only trisection of genus 0. The Euler characteristic can be calculated from the genera of the surface and the 4-dimensional handlebodies, giving the following important remark: Remark 3.1.1. If M has a (g; k 1, k 2, k 3 )-trisection then χ(m) = 2 + g k 1 k 2 k 2 A trisection intuitively looks like Figure 3.1. 3.2. Trisection Diagrams Let Y ij, i j be the 3-dimensional handlebody X i X j. The complement of Y 12 Y 13 Y 23 in M is a union of three 4-dimensional handlebodies. Unlike 3-dimensional handlebodies, 4-dimensional handlebodies are determined by their boundary, in the sense that any orientation preserving diffeomorphism of the boundary can be extended to an orientation preserving diffeomorphism of the handlebody [LP72]. If follows that after fixing Y 12 Y 13 Y 23, there is a unique way to fill in the three handlebodies. Thus, the entire trisection is determined by the object Y 12 Y 13 Y 23. This gives a method of representing trisections by diagrams: on Σ, choose sets of curves {α i } 1 i g,{β i } 1 i g, and {γ i } 1 i g bounding minimal systems of disks in Y 12, Y 13, and Y 23 respectively. These sets of curves determine how the Y ij attach to Σ. Since the complement of the Y ij is filled in uniquely, the three sets of curves therefore determine the entire trisection. The surface Σ together with the sets of α, β and γ curves is called a trisection diagram for the trisection. Unlike the situation for Heegaard splittings, the sets of curves {α i } 1 i g, {β i } 1 i g, and {γ i } 1 i g have a restriction on them: any two sets of curves must give a Heegaard diagram for 14
a connected sum of some number of copies of S 1 S 2. By Corollary 2.5.1 such splittings are standard. Thus, for any given two sets of curves, say {α i } 1 i g and {β i } 1 i g, there must be a sequence of handle slides after which the two sets give a standard Heegaard diagram for one of the X i = # k i S 1 S 2. It is not in general possible for all three pairs to look standard at the same time. 3.3. Connect Sums and Stabilizations of Trisections There is a natural notion of the connect sum of two trisections, and a diagram of the trisection obained by connect summing is given by taking the connect sum of diagrams of the summands in the obvious way. We also have a notion of stabilization, defined as follows: Definition 3.3.1. Let (Σ; X 1, X 2, X 3 ) be a trisection of a 4-manifold M. Let i, j, k be some permutation of 1, 2, 3. Let α be an unknotted arc properly embedded in Y jk, and note that α is unique up to isotopy. Fix a regular neighborhood N(α) and let X i = X i N(α) X j = X j X j N(α) X k = X k X k N(α) Σ = X i X j X k The trisection (Σ ; X 1, X 2, X 3 ) is called an i-stabilization of (Σ; X 1, X 2, X 3 ). The effect of an i-stabilization on the trisection diagram is to connect sum with a genus 1 trisection of S 3 as shown in Figure 3.2. A trisection is stabilized if it is isotopic to an i-stabilization of a lower genus trisection for any i. Note that since we focus on unbalanced trisections, this definition is different than that found in Gay and Kirby s paper. Gay and Kirby s main results are the following: Theorem 3.3.1 ( [GK12]). Every orientable 4-manifold has a balanced trisection. Theorem 3.3.2 ( [GK12]). Any two trisections of a fixed 4-manifold are stably equivalent. That is, if (Σ; X 1, X 2, X 3 ) and (Σ ; X 1, X 2, X 3 ) are two trisections of a 4-manifold M, then there is another trisection of M isotopic to a stabilization of both (Σ; X 1, X 2, X 3 ) and (Σ ; X 1, X 2, X 3 ). 15
Figure 3.2. One of the three possible genus 1 trisections of S 3. The other two are obtained by switching around the colors. The former theorem is proved using the theory of Morse 2-functions [GK11]. The latter is then proved using more standard methods, viewing a trisection as a handle decomposition. It is shown that X 1 can be viewed as the 0-handle and 1-handles of a handle decomposition, X 2 as the collection of 2-handles, and X 3 as the 3-handles and 4-handle. From this we see that π 1 (M). Remark 3.3.1. The genus of each of the handlebodies X i is bounded below by the rank of 3.4. Trisections of Low Genus An obvious problem is that of the classification of unstabilized trisections. This is done for g 2. Balanced trisections for g 2 are classified in [MZ14]. If g = 1 then the trisection is one of the three shown in 3.3. If g = 2 then either the trisection is a connect sum of genus 1 trisections or is the trisection of S 2 S 2 shown in Figure 3.4. The fact that there are no unstabilized unbalanced trisections for g 2 follows from the following theorem. Theorem 3.4.1 ( [MSZ15]). Suppose that X admits a (g; k 1, k 2, k 3 )-trisection T with k 1 g 1. Let k = max{k 2, k 3 }. Then X is diffeomorphic to either # k (S 1 S 3 ) or to the connected sum of # k (S 1 S 3 ) with one of CP 2 or CP 2, and T is the connected sum of genus one trisections. Indeed, if g = 2 and all k i = 0, then the trisection is balanced and covered by the classification of balanced trisections of g 2. If some k i > 0, then by the above theorem the trisection must break down as a connect sum of genus 1 trisections. 16
(a) The (1; 0) trisection of CP 2 (b) The (1; 0) trisection of CP 2 (c) The (1; 1) trisection of S 1 S 3 Figure 3.3. The three possible unstabilized genus 1 trisections. The only other genus 1 trisections are the stabilized trisections of S 4 Figure 3.4. The (2; 0) trisection of S 2 S 2 17
CHAPTER 4 Trisections of 3-Manifolds 4.1. Overview We can now introduce the notion of a trisection of a closed, orientable 3-manifold. We present several examples of these trisections. We define the trisection genus of a 3-manifold, and relate it to the Heegaard genus of the manifold. We analyze the behavior of trisection genus under connect sum, showing that if M is the connect sum of two manifolds of Heegaard genus g, M has trisection genus equal to half its Heegaard genus. We then prove that, with only one exception, all trisections of a closed, orientable 3-manifold M can be made equivalent by stabilization. We conclude by discussing trisections of S 3, showing that there are infinitely many distinct unstabilized trisections. We begin with the definition of a trisection. Let M be a closed, orientable 3-manifold. Definition 4.1.1. A (h 1, h 2, h 3 ; b)-trisection of M is a quadruple (H 1, H 2, H 3 ; B) such that M = H 1 H 2 H 3 H i is a handlebody of genus h i Each S ij = H i H j is a compact connected surface with boundary K B = H 1 H 2 H 3 is a b-component link If h 1 = h 2 = h 3 = h, the trisection is balanced, and we call it an (h; b)-trisection. Otherwise, it is unbalanced. If (H 1, H 2, H 3 ; B) is a balanced (h; b)-trisection, we define the genus of the trisection to be h. Note that, in contrast to trisections of 4-manifolds where the genus refers to the complexity of the triple intersection, here it refers to the genus of the handlebodies H 1, H 2, H 3. The simplest trisection is the trisection of S 3 into three balls, with each pair of balls intersecting in a disk. We refer to this as the trivial trisection of S 3. Many more examples of trisections will be covered in the next section. 18
Definition 4.1.2. Let i, j, k be the indices 1,2,3 in any order. Suppose that S jk is not a disk, and let α be a nonseparating arc in S jk. Define a new trisection (H 1, H 2, H 3 ; B ) by H i = H i N(α) H j = H j H j N(α) H k = H k H k N(α) B = H i H j H k This results in a new trisection of M where h i is increased by 1, and b is changed by ±1. We call this operation a stabilization. This operation depends on the choice of α, so it is not generally unique even after fixing a choice of i, j, k. Two trisections (H 1, H 2, H 3 ; B) and (H 1, H 2, H 3 ; B ) are isotopic if there is an isotopy of M taking each H i to the corresponding H i and taking B to B. We say that one trisection of M is a stabilization of another if it can be obtained by some sequence of stabilizations, up to isotopy. Notice that a relabelling of the handlebodies does not necessarily produce an isotopic trisection. So, for example, (H 1, H 2, H 3 ; B) and (H 2, H 1, H 3 ; B) may be distinct trisections. In some settings, the order of the handlebodies may be unimportant. In the examples in the following section, we provide one possible order of the handlebodies, and implicitly treat all reorderings as part of the same class of examples. We can now state the main theorem. Theorem 4.1.1. Let M be a closed, orientable 3-manifold with two trisections. If M = S 3, assume that neither of the two trisections is the trivial trisection into three balls. Then there exists a third trisection isotopic to a stabilization of each of the original two trisections. In the following section, we will discuss several examples of trisections, and methods to obtain interesting trisections for many classes of manifolds. Section 3 we discuss how to get a balanced trisection from an unbalanced one. In section 4 we will define the trisection genus of a manifold and relate the trisection genus and Heegaard genus of M. Section 5 will present the proof of Theorem 4.1.1. 19
Figure 4.1. A (2, 2, 0, 1) trisection is constructed from a genus-2 trisection of S 3 by the construction of Example 4.2.1. A balanced trisection can be obtained by stabilizing H 3 twice. 4.2. Examples We begin with a few ways to get trisections of any 3-manifold, and then present some more interesting trisections of specific classes of 3-manifolds. The order of the handlebodies is unimportant for producing examples, so we will use whatever order is convenient, usually ordering from largest to smallest genus. Example 4.2.1. Let M be any closed orientable 3-manifold, and let V Σ W be a genus g Heegaard splitting of M. Let D be a disk in Σ. Define H 3 to be a regular neighborhood of D, and let H 1 = V V H 3, H 2 = W W H 3. Then this defines a (g, g, 0; 1)-trisection of M. We can stabilize H 3 g times to produce a balanced genus g trisection of M. See Figure 4.1. The trivial (0, 0, 0; 1)-trisection of S 3 is a special case of this construction. 20
Example 4.2.2. Suppose (K, φ) is an open book decomposition of M with binding circle K and φ: M K S 1. Then we can define H 1 = K φ 1 ([0, 1/3]) H 2 = K φ 1 ([1/3, 2/3]) H 3 = K φ 1 ([2/3, 1]) Since each H i is a thickening of a once punctured surface, each is indeed a handlebody. This gives a (2g, 2g, 2g; 1)-trisection of M where g is the genus of the fiber surface. In fact, this is a special case of Example 4.2.1. If we set V = K φ 1 ([0, 1/2]) and W = K φ 1 ([1/2, 1]) we get a Heegaard splitting of M. Applying the technique of Example 4.2.1 gives a (g, g, 0; 1)-trisection which can be stabilized to the (g; 1)-trisection described in this example. See Figure 4.2. It is known that any two open book decompositions of S 3 are related by plumbing and deplumbing Hopf bands [GG06]. Hopf plumbings gives a different notion of stabilization from that used here, but it is worth noting that plumbing and deplumbing of Hopf bands is also not a unique operation. Example 4.2.3. We can generalize the construction of Example 4.2.1 as follows. Let V Σ W be a genus g Heegaard splitting of M. Let H 1 = W. Choose some disk properly embedded in V that cuts it into two handlebodies H 2 and H 3 of genus h and g h respectively, where 0 h g. Then this gives a (g, h, g h; 1)-trisection. When h = 0 or h = g this construction reduces to the construction of Example 4.2.1, possibly after relabelling the handlebodies. Example 4.2.4. We describe a specific instance of the construction of Example 4.2.3. Let K be some knot in S 3. Set H 1 = N(K). Let D be a disk in H 1, and α 1,..., α m be a tunnel system for K, with the endpoints of each α i lying in D. Then we can set H 2 = N(D i α i) and H 3 = M H 1 H 2. This gives a (1, m, m + 1; 1) trisection of S 3. Here H 1 H 2 is a handlebody and H 1 H 2 a disk. See Figure 4.4 for an example. 21
Figure 4.2. We see a (2, 2, 0; 1) trisection where the first two handlebodies are neighborhoods of Seifert surfaces of the trefoil knot. If we stabilize H 3 along the two red arcs, we get a trisection where each handlebody is a neighborhood of a Seifert surface, and the triple intersection curve B is a trefoil. Figure 4.3. Here we construct a (4, 2, 2, 1)-trisection of S 3 using the technique of Example 4.2.3. When H 2 or H 3 has genus 0 then we can relabel handlebodies and perform an isotopy to get the upper trisection in Figure 4.1. Although the disk H 2 H 3 here cuts the complement of H 1 into two standard handlebodies, it is possible that H 2 and H 3 are knotted. An example of this is described in Example 4.2.4. All examples so far have been stabilizations of the construction in Example 4.2.3. To provide some different classes of examples, we present some trisections where all three handlebodies have genus lower than the Heegaard genus of the manifold. Example 4.2.5. Suppose M is the connect sum of two 3-manifolds of Heegaard genus 1. That is, each connect summand is either S 1 S 2 or a lens space. M has Heegaard genus 2 by 2.3.1, so 22
Figure 4.4. A trisection constructed from a trefoil knot and a tunnel. we can fix a genus 2 Heegaard splitting X 1 X 2. We produce a (1, 1, 1; 2)-trisection of M. See Figure 4.5 for the case where M is the connect sum of two copies of S 1 S 2. We describe this case in detail. Let S be the reducing sphere splitting M into the two copies of S 1 S 2. Begin by splitting X 1 along the disk X 1 S, resulting in two genus 1-handlebodies H 1 and H 2. Then both H 1 and H 2 contain essential curves α 1, α 2 respectively bounding disks in X 2. Let γ be an arc connecting α 1 and α 2 such that the intersection of γ with H 1 H 2 is only a single point. Let α now denote the result of performing a handle slide of α 1 across α 2 using the arc γ. α intersects both H 1 and H 2 in a single arc, and bounds a disk D in X 2. Therefore, we can isotope H 2 to add a neighbourhood of D. We continue to call the result of this isotopy H 2. H 1 and H 2 now intersect in an annulus essential in both H 1 and H 2, and H 3 = M (H 1 H 2 ) is a genus 1-handlebody. If follows that each intersection S ij is an annulus, and so all pairwise intersections are connected. We therefore have a (1, 1, 1; 2)-trisection as desired. An identical argument can be applied when one or both of the connect summands are replaced with lens spaces. Example 4.2.6. Now consider a more general connect sum M = M 1 #M 2 where both of M 1, M 2 have Heegaard genus g. M then has Heegaard genus 2g by 2.3.1. Fix genus g Heegaard splittings (X 1, X 2, Σ) and (X1, X 2, Σ ) for M 1 and M 2 respectively. Let α 1,, α g (resp. β 1,, β g ) be a 23
Figure 4.5. A trisection of the connect sum of two copies of S 1 S 2, viewed as the union of two standard genus 2 handlebodies glued by the identity map on their boundary. Each row shows how one of the three handlebodies lies in the manifold. collectively nonseparating set of g disjoint curves on Σ (resp. Σ ) such that each curve bounds a disk in X 2 (resp. X 2 ). (X 1#X 1, X 2#X 2, Σ#Σ ) is a minimal genus Heegaard splitting for M. Let C be a curve splitting Σ#Σ into the punctured copies of Σ and Σ. We can apply a diffeomorphism to Σ#Σ that fixes C and sends Σ to Σ and Σ to Σ in order to get a Heegaard diagram of the form shown in Figure 4.6. For each i, let γ i denote the result of sliding α i across β i using connecting arcs intersecting C once as shown in Figure 4.6. Let D 1... D g denote the meridian disks in X 2 #X 2 bounded by the γ i. We can now define a trisection. Let H 1 = X 1, so it is a genus g handlebody. Define H 2 to be the union of X 1 and the collection of all N(D i). Removing these disks N(D i ) from X 2 #X 2 leaves another genus g handlebody, which we define to be X 3. H 2 as defined is isotopic to X1, because we defined it by attaching disks that intersected X 1 in a single arc each. Thus, attaching the disk D i is equivalent to isotoping X 2 to extend from the arc D i Σ across the disk D i to the arc D i Σ. We can also observe that each such attachment introduces a new curve component of H 1 H 2, so the resulting trisection is a (g, g, g; g + 1) trisection. 24
Figure 4.6. The standard picture of what a connect sum Heegaard diagram looks like after forgetting about which curves on Σ and Σ bound disks in the inner handlebody. The purple arcs are used to guide handle slides. Example 4.2.7. Let Σ be a closed orientable genus-g surface. Let M be a surface bundle Σ [0, 1]/(x, 0) (φ(x), 1). M has a genus 2g + 1 Heegaard splitting. If the surface bundle is a product bundle Σ S 1 or if the translation distance of the monodromy map is sufficiently high relative to the genus of Σ, then the genus 2g + 1 Heegaard splitting is minimal genus by 2.5.2 and 2.5.3. We produce a (2g, g + 1, g + 1; b) trisection of M, where b is either 1 or 3 depending on whether g is even or odd. Case 1. g is even. See Figure 4.7. There is a curve C Σ cutting Σ into two punctured genus g/2 surfaces. Let α be a path in Σ such that α(0) lies on φ(c) and α(1) lies on C. Then the path P = {α(2t) t : 0 t 1/2} is transverse to the fibers, so Σ [0, 1/2] N(P ) is homeomorphic to a thickened punctured genus g surface, and is therefore a genus 2g handlebody. Let H 1 be this handlebody. Now, C [1/2, 1] cuts Σ [1/2, 1] into two genus g handlebodies H 2, H 3. Split the tube N(P ) into two halves as in Figure 4.7, and assign half to H 2 and half to H 3 so that they become genus g + 1 handlebodies. Note that performing twists to N(P ) will possibly produce non-isotopic trisections. In the resulting trisection, H 2 H 3 is a punctured torus, and each of H 1 H 2 and H 1 H 3 is the union of two punctured genus g surfaces connected by a band. 25
Figure 4.7. We get a trisection from a surface bundle Case 2. g is odd. The idea is approximately the same. Instead of C we choose two curves C 1, C 2 cutting Σ into two twice punctured genus (g 1)/2 surfaces. Choose the path α to connect a point on C 1 to a point on C 2. Everything else goes through as before, and we end up with a (2g, g, g; 3) trisection. H 2 H 3 is now a thrice punctured planar surface, and each of H 1 H 2 and H 1 H 3 is a thrice punctured genus g 1 surface. 4.3. Balancing Trisections We prove that an unbalanced trisection can be turned into a balanced trisection without increasing the genus of the largest handlebody. 26
Proposition 4.3.1. Let (H 1, H 2, H 3 ; B) be an (h 1, h 2, h 3 ; b) trisection of M. Then there is a balanced (h ; b )-trisection of M that is a stabilization of (H 1, H 2, H 3 ; B), where h = max(h 1, h 2, h 3 ). Additionally, we can ensure that b max(b, 2). That is, either b b or b = 2 and b = 1. Proof. Choose i, j, k to be a permutation of 1, 2, 3 such that h i h j h k. If h i = h j = h k then we are done. Otherwise, we know that h i > h k. Now if S ij were a disk, then H i H j would be a genus h i + h j handlebody with complement H k. This would give a Heegaard splitting and would imply that h i + h j = h k, which contradicts h i > h k. Therefore S ij is not a disk. Hence there exists some nonseparating arc α properly embedded in S ij. Moreover, unless b = 1, we can choose α to have its endpoints lie on two distinct components of B. Performing a stabilization with this choice of α gives a (h i, h j, h k + 1; b )-trisection, where b is either b 1 or 2. This operation does not increase the genus of any handlebody beyond h i, and only increases b if b = 1. Therefore, we can repeat the operation until we get a balanced trisection of genus h i. We investigate the surfaces S ij. Again, let i, j, k be some permutation of 1, 2, 3. We can compute the genus of the handlebody H i from b and the genera g(s ij ) and g(s ik ) of S ij and S ik by the formula h i = g(s ij ) + g(s ik ) + b 1. In a balanced trisection, h 1 = h 2 = h 3. Comparing the formula for h 1 and h 2 we see that g(s 13 ) = g(s 23 ). Similarly we can compare the formulas for h 2 and h 3 to see that g(s 12 ) = g(s 13 ). Therefore, in a balanced trisection, all g(s ij ) are the same, and are equal to h+1 b 2. Since this must be an integer, we also get the following: Remark 4.3.1. In a balanced (h, b)-trisection, b and h must have opposite parities. 4.4. Trisections, Heegaard Splittings, and Trisection Genus Just as the Heegaard genus g(m) of a 3-manifold is defined as the smallest g for which M has a genus g Heegaard splitting, we can define the trisection genus t(m) to be the smallest t for which M has a balanced trisection of genus t. Here we state some facts about trisection genus, and about how trisections relate to Heegaard splittings. Proposition 4.4.1. If M is a closed orientable 3-manifold, its Heegaard genus g(m) and trisection genus t(m) are related by t(m) g(m) 2t(M) 27
Proof. First, note that by combining the construction of Example 4.2.1 or 4.2.3 with Proposition 4.3.1, whenever M has a Heegaard splitting of genus g we can also construct balanced trisections of genus g. It follows that t(m) g(m). We can also get a Heegaard splitting from a trisection (H 1, H 2, H 3 ; B) as follows. First choose one of the three handlebodies H i, and let j, k be the indices not chosen. Choose a maximal set of nonseparating arcs in S jk, and stabilize H i along each of these arcs in turn to get a new trisection (H 1, H 2, H 3 ; B ). In this trisection, S jk is now a disk, since if it were not then there would be some nonseparating arc in it, contradicting the maximality of our choice of arcs. Therefore, H j H k is a handlebody. If we started with a balanced trisection of genus h then h stabilizations were required to make S jk a disk, so g(h i ) = 2h. It follows that (H i, H j H k ; H i ) is a genus 2h Heegaard splitting. Applying this construction to a minimal genus balanced trisection of M, we conclude that g(m) 2t(M). Since the construction used in the previous proposition is quite useful, we set it aside as a definition. Definition 4.4.1. Suppose (H 1, H 2, H 3 ; B) is a trisection. If we stabilize H i along a maximal set of arcs in H jk, then we get a trisection (H 1, H 2, H 3 ; B ) where H j H k is a handlebody, so (H i, H j H k ; H i ) is a Heegaard splitting. We call this the Heegaard splitting built from the trisection (H 1, H 2, H 3 ; B) by stabilizing H i. If we do not care which i was chosen, we just say that it is a Heegaard splitting built from the trisection. Remark 4.4.1. For a given trisection (H 1, H 2, H 3 ; B), we do not know that, for example, the Heegaard splitting built by stabilizing H 1 and the Heegaard splitting built by stabilizing H 2 are isotopic. However, after fixing a choice of handlebody H i we are stabilizing along a maximal set of arcs in S jk. Any two such maximal system of arcs in S jk are slide equivalent, so any two choices of arc systems will result in isotopic Heegaard splittings. It follows that there are at most three isotopy classes of Heegaard splittings that can be built from a given trisection, one for each choice of handlebody H i. It is natural to ask how strict these inequalities are. We have already shown that the inequality g(m) 2t(M) is the best general bound possible; Example 4.2.6 demonstrates that if M = M 1 #M 2 is a connect sum with g(m 1 ) = g(m 2 ) then t(m) = g(m 1 ). However, g(m) = 2g(M 1 ) = 2t by 2.3.1. This gives us 28
Proposition 4.4.2. Suppose both M 1 and M 2 are closed orientable 3-manifolds with Heegaard genus g. Let M = M 1 #M 2. Then M has Heegaard genus 2g and trisection genus g. Corollary 4.4.1. For each integer t 0, there exists a 3-manifold with trisection genus t and Heegaard genus 2t. We can also ask whether for every t 0 there exists a 3-manifold M such that both the trisection and Heegaard genus are equal to t. S 3 satisfies this for t = 0, and any lens space satisfies it for t = 1. The fact that there exist examples for t = 2 follows from the following proposition relating Heegaard splittings built from trisections to Hempel distance. Proposition 4.4.3. Suppose (H 1, H 2, H 3 ; B) is a trisection of M such that no H i has genus g(h i ) = 0. Then any Heegaard splitting built from (H 1, H 2, H 3 ; B) has distance at most 2. Proof. Suppose without loss of generalization that we build a Heegaard stabilization by stabilizing H 1. Let (H 1, H 2, H 3 ; B) be the trisection obtained by stabilization so that (H 1, H 2 H 3 ; H 1 ) is the Heegaard splitting. In order to demonstrate the distance bound we find a sequence of 3 curves α, β, γ on H 1 such that α bounds a disk in H 1 and γ bounds a disk in H 2 H 3. See Figure 4.8 for a picture of the case where each handlebody is genus 1, which easily generalizes to higher genus. Let α be a loop enclosing a cocore of one of the stabilizations that took H 1 to H 1. Let β be some curve in H 1 H 3 H 1, and note that the stabilization occured away from β, so we can treat β as also lying in H 1 H 3. Now to find γ, first choose any meridian disk D of H 2. Since H 2 H 3 is a disk, we can isotope D so that D lies in H 1 H 2. Set γ to be D. By construction α H 1 H 1 H 3, β H 1 H 3, and γ H 1 H 2. Thus, α β is empty, as is β γ. Moreover, α bounds a disk in H 1 and γ bounds a disk in the complement of H 1, so this is indeed a distance 2 path. to 2. Corollary 4.4.2. There exist 3-manifolds with both Heegaard genus and trisection genus equal Proof. Suppose M is a 3-manifold with a Heegaard splitting of genus g = 2 and distance at least 5. Such manifolds exist by 2.4.1. When a Heegaard surface has distance d > 2g, it represents the unique minimal genus Heegaard splitting by 2.4.2. It follows that M has no other genus 2 29
Figure 4.8. Here we see a Heegaard splitting built from a balanced genus 1 trisection. H 1 is the yellow torus on the left, and H 2 the union of the blue torus, the blue band, and the blue disk attached in some way along the boundary. H 1 is obtained from H 1 by attaching an arc that intersects a meridian disk of the blue torus once. Therefore, H 1 is isotopic to the union of H 1 and the blue torus. α is then a meridian curve of the blue torus, β the red curve on the left, and γ the attaching curve for the blue disk. Heegaard splittings, and hence has no genus 2 Heegaard splitting of distance 2. M has a genus 2 trisection by Example 4.2.1. However, if M had a genus 1 trisection, it would have a distance 2 Heegaard splitting by Proposition 4.4.3, which would be a contradiction. Therefore, both the trisection genus and Heegaard genus of M must be equal to 2. It would be interesting to know whether there exist higher genus examples with trisection genus equal to their Heegaard genus. 4.5. Trisections With Disconnected Pairwise Intersections Now that we have shown a connection between trisection genus and Heegaard genus, we analyze our initial definition of a trisection a bit further. At first glance, the choice to require the surfaces S ij to be connected may seem arbitrary. Let us define a weak trisection of M to be any decomposition M = H 1 H 2 H 3 into three disjoint handlebodies. Certainly the weak trisection genus, defined in the natural way, must be bounded above by the Heegaard genus since the normal trisection genus is. However, knowing the weak trisection genus does not give any additional information about the Heegaard genus. In particular, Proposition 4.5.1. There exist manifolds that can be decomposed into three tori but have arbitrarily large Heegaard genus. 30
Proof. Let M = S 1 Σ g. Then the Heegaard genus of M is 2g + 1 by 2.5.2. However, we can show that M has a decomposition into three tori. First we decompose the surface Σ g as a union of three disks. We do this as in Figure 4.9. More formally, let D 1 be some disk embedded in Σ g. Let α 1,, α 2g be a collection of 2g properly embedded curves cutting Σ g D 1 into a disk. Then let β 1,..., β 2g 1 be a collection of arcs in D 1 connecting the endpoints of the α i curves, such that the union of all α i and β j is a single simply connected region that we call γ. Such a collection of arcs can always be found simply by repeatedly finding a new arc that connects two disconnected subcollections of α curves. We can think of this process as just moving together the endpoints of alpha curves until the alpha curves form the single simply connected region γ. Then set D 2 = N(γ) Σ g D 1 and D 3 = Σ g D 1 N(γ). Now M = S 1 D 1 S 1 D 2 S 1 D 3 is a decomposition into three tori. Figure 4.9. A genus 2 surface decomposed as a union of 3 disks. D 1 is green, D 2 is red, and D 3 is the complement of D 1 D 2 Any weak trisection can be stabilized to get a normal trisection. If S ij is disconnected, then stabilize along some arc connecting the two components, and repeat until all S ij are connected. It follows that Theorem 4.1.1 holds for weak trisections as well. 4.6. The Stabilization Theorem Before proving the theorem, we need one more definition. Definition 4.6.1. Suppose (H 1, H 2, H 3 ; B) is a trisection such that (H 1, H 2 H 3 ; H 1 ) is a Heegaard splitting of M. Suppose moreover that there exists a disk D properly embedded in H 1 such that D consists of a nonseparating arc in S 12 and a nonseparating arc in S 13. We can stabilize H 2 and then H 1 as in Figure 4.10. We call this operation a fake Heegaard stabilization. The effect of this operation is to perform a standard stabilization between H 1 and H 2, as in Figure 4.11. 31
Remark 4.6.1. A disk D as required in the above definition always exists if we have just stabilized H 1. Indeed, stabilizing changes H 1 by attaching a 1-handle to it, and a core disk of this one handle will satisfy the requirements. Once we have found such a disk, we can use parallel copies of it to perform an arbitrary number of fake Heegaard stabilizations. (a) (b) (c) (d) Figure 4.10. We perform a fake Heegaard stabilization by stabilizing H 2 and then H 1 Figure 4.11. A standard stabilization between H 1 and H 2 performed by adding to H 2 a regular neighborhood of an arc parallel into S 12, and removing the corresponding neighborhood from H 1. Now we provide the proof of Theorem 4.1.1, which we restate here for convenience. 32
Theorem 4.1.1. Let M be a closed, orientable 3-manifold with two trisections. If M = S 3, assume that neither of the two trisections is the trivial trisection into three balls. Then there exists a third trisection isotopic to a stabilization of each of the original two trisections. Let (H 1, H 2, H 3 ; B) and (H1, H 2, H 3 ; B ) be two trisections of a closed orientable 3-manifold M. To avoid excessive notation, we use the same notation for both a trisection and the stabilizations that we obtain from that trisection. The basic strategy is as follows: (1) Perform stabilizations until the quadruple (h 1, h 2, h 3 ; b) is the same as the quadruple (h 1, h 2, h 3 ; b ) (2) Stabilize H 1 and H 1 until H 2 H 3 and H 2 H 3 are handlebodies (3) Perform fake Heegaard stabilizations until the Heegaard splittings (H 1, H 2 H 3 ; H 1 ) and (H1, H 2 H 3 ; H 1 ) are isotopic (4) Stabilize H 3 and H 3 until S 12 and S 12 (5) Stabilize H 2 and H 2 until S 13 and S 13 are disks are disks. After these steps, we show that the resulting trisections will be isotopic. Since we will have started with two arbitrary trisections and stabilized both until we have isotopic trisections, the theorem follows. Proof of Theorem 4.1.1. Step 1. First we stabilize so that the genera of the handlebodies in the two trisections are the same. By applying Proposition 4.3.1, we may assume both trisections are balanced. If b > 2 we stabilize the first trisection along an arc connecting two components of B, and then reapply Proposition 4.3.1. Do the same for the second trisection if b > 2. Then we have an (h; b) and an (h ; b ) balanced trisection where both b, b are either 1 or 2. If h < h, perform any stabilization on the first trisection, and then reapply Proposition 4.3.1, and repeat until h = h. Do the same to the second trisection if h < h. So we may assume that h = h and, by the proof of Proposition 4.3.1, b and b must still be 2. By Remark 4.3.1 h and b must have opposite parities, so (h; b) is the same as (h ; b ) as desired. In future steps we perform stabilizations equally to both trisections so as to retain the property that both trisections have the same tuple (h 1, h 2, h 3 ; b). Step 2. Choose a maximal nonseparating set of h properly embedded arcs in S 23. Stabilizing along all arcs of this set results in a trisection where S 23 is a disk. This means the complement of H 1 33
is the union of two handlebodies H 2 H 3 glued along a disk in their boundaries, so it, too, is a handlebody. Do the same thing to the other trisection. Since we began this step with a balanced trisection with h > 0, S 23 was not a disk, so at least one stabilization was required in this step. By Remark 4.6.1, both trisections now satisfy the necessary conditions to apply fake Heegaard stabilizations. Step 3. We now have that (H 1, H 2 H 3 ; H 1 ) and (H1, H 2 H 3 ; H 1 ) are Heegaard splittings of M. By 2.2.1, there exists a common Heegaard stabilization of these two Heegaard splittings. The fake Heegaard stabilization operation affects the Heegaard splitting (H 1, H 2 H 3 ; H 1 ) just as Heegaard stabilization does. Therefore, by repeatedly performing fake Heegaard stabilizations to both trisections we may assume that (H 1, H 2 H 3 ; H 1 ) and (H1, H 2 H 3 ; H 1 ) represent isotopic Heegaard splittings. In particular, H 1 and H 1 are isotopic in M. Step 4. Choose a maximal set of nonseparating arcs properly embedded in S 12 and stabilize H 3 along all of them. Do the same for the other trisection. Since no stabilizations are performed on H 1 or H 1, H 1 and H 1 are still isotopic in M. After doing this, S 12 and S 12 are disks. Step 5. Up to isotopy, we may now assume that H 1 = H 1 and S 12 = S 12. Since S 12 is a disk, H 1 H 2 can be obtained by attaching some 1-handles to H 1, with all attaching points occuring on S 12. We show that if we were allowed to slide the ends of these handles along loops in H 1, we could arrange for them to be a set of small arcs each parallel rel into S 12. Note that sliding the ends around freely will not necessarily correspond to an isotopy of the trisection because the ends may need to slide across S 13 to trivialize the handles. See Figure 4.12. To see that this is true, we define a Heegaard splitting of H 2 H 3. Let W be the union of H 2 with a regular neighborhood of (H 2 H 3 ), so V is a compression body. Let W be the complement of V, so W is a slightly shrunken version of H 3, and is a handlebody. Then V W is a Heegaard splitting of H 2 H 3 as desired. By 2.5.1 this Heegaard splitting must be standard. Therefore, W must topologically be formed by attaching some number of trivial 1-handles to a regular neighborhood of (H 2 H 3 ). These 1-handles are therefore simultaneously parallel into (H 2 H 3 ). To allow us to slide the ends of these 1-handles freely, stabilize H 2 as much as possible until S 13 is a disk. After these stabilizations, H 2 consists of N( H 1 {disk}) {trivial 1-handles}. If we have done the same thing to the other trisection, we now know that there is an isotopy taking H 1 to H1 and H 2 to H2. Note that if we hadn t performed step 1, it would be possible that H 2 34
Figure 4.12. After Step 4, H 2 looks like a set of 1-handles attached to the thickened disk N(S 12 ). The set of such 1-handles is simultaneously isotopic into H 1, so it is possible to slide the ends around on H 1 to get to the lower picture where the set of handles is parallel into S 12. However, performing such slides might require sliding the ends of the handles across S 13, which does not correspond to an isotopy of the trisection. had fewer or more of the trivial 1-handles than H 2. The isotopy must also necessarily take H 3 to H3, so the trisections are in fact isotopic. Therefore, we have constructed a common stabilization of both initial trisections. This concludes the proof. 4.7. Trisections of S 3 Recall 2.5.1 which states that any Heegaard splitting of S 3 is a stabilization of the standard genus 0 splitting. One might hope for a similar result for trisections. Since the genus 0 trisection cannot be stabilized, the simplest form of such a statement can be immediately ruled out. However, one might still hope that there exists some finite list of low genus trisections such that any trisection of S 3 is a stabilization of something in the list. This turns out this too is impossible. Specifically, 35
Proposition 4.7.1. There exists an infinite class of (1, 2, 2; 2) trisections of S 3 that are not stabilizations of any other trisection. Therefore, there is no finite list of trisections of S 3 that can be stabilized to cover all possible trisections of S 3. Proof. First, we must investigate how to detect stabilized trisections. Since a stabilization is performed by adding a neighborhood of an arc in S jk to H i, it follows that we can detect destabilizations as follows: Definition 4.7.1. Suppose M is a closed oriented 3-manifold with trisection (H 1, H 2, H 3 ; B). A destabilizing disk D is an essential nonseparating disk properly embedded in some H i such that D consists of a nonseparating arc of S ij and a nonseparating arc of S ik. Remark 4.7.1. For a destabilization of a weak trisection, D must still be nonseparating but the arcs D S ij and D S ik may be separating. If there exists a destabilizing disk D, we can pinch H j and H k together across D, performing a compression on H i. Since D is nonseparating, H i is still a handlebody, and H j and H k are unaffected topologically, so we still have a decomposition of M into three handlebodies. Since the arcs of D in S ij and S ik were nonseparating, the two surfaces are still connected after the compression. S jk has been affected by attaching a band, so it too is still connected. Thus, all pairwise intersections are still connected. Therefore the result of this operation is indeed still a trisection, and we call this trisection a destabilization of the original trisection. This destabilization operation is the reverse of a stabilizatiom. Now we can demonstrate the class of examples. Koda and Ozawa describe a class of knots whose exteriors contain an incompressible, boundary incompressible twice punctured genus-1 surface Σ cutting the exterior into two handlebodies [KO15]. This surface intersects N(K) in two toroidal curves with nonzero rational slope on N(K). Let H 1 be N(K) and let H 2 and H 3 be the two genus-2 handlebodies resulting from cutting the exterior of K along Σ. Let B be Σ H 1. Then (H 1, H 2, H 3 ; B) is a trisection of S 3. Since the components of B are toroidal curves on H 1, any disk properly embedded in H 1 must intersect each component of B at least twice. Therefore, no such disk can be a destabilizing disk. Any destabilizing disk in H 2 or H 3 would contradict boundary incompressibility, so these also cannot exist. It follows that this (1, 2, 2; 2)-trisection is 36
not a stabilization of any other trisection. Since the class of knots provided by Koda and Ozawa is infinite, the proposition follows. Remark 4.7.2. These examples cannot be stabilizations of weak trisections either, so there is also no analogue of Waldhausen s theorem for weak trisections. 37
CHAPTER 5 Classes of Trisections in Four Dimensions 5.1. Balanced and unbalanced trisecton genus of four dimensional manifolds It is known that for 4-manifolds X 1 and X 2, the diagrams of a genus g 1 trisection of X 1 and a genus g 2 trisection of X 2 can be connect summed to produce a genus g 1 + g 2 trisection of X 1 #X 2. A natural question is whether there is an analogue of Theorem 2.3.1 for trisections of 4-manifolds. Conjecture 5.1.1. If a 4-manifold M can be decomposed as a connect sum X = X 1 #X 2, then any trisection of X can be decomposed as the connect sum of trisections of X 1 and X 2. Specifically. if Σ is the triple intersection surface for the trisection of X, and sets of curves α i, β i, γ i are sets of curves on Σ giving a trisection diagram, then after handleslides there exists a separating curve in Σ that cuts Σ into a trisection diagram of X 1 and a trisection diagram of X 2. If we restrict to balanced trisections, the conjecture is likely false. Indeed, we can show that: Proposition 5.1.1. If there exists a 4-manifold X whose unbalanced trisection genus is less than its balanced trisection genus, then X#X#X has a balanced trisection that cannot be fully decomposed as a connect sum of lower genus balanced trisections. Proof. Suppose X has balanced trisection genus g and unbalanced trisection genus g, where g < g. Let (Σ g ; X 1, X 2, X 3 ) be an unbalanced genus g trisection of X. Then (Σ g ; X 2, X 3, X 1 ) and (Σ g ; X 3, X 1, X 2 ) are also unbalanced genus g trisecitons of X. Taking the connect sum of all three of these trisections gives a balanced genus 3g trisection of X#X#X. If this trisection could be broken up into three balanced trisections of X, one of those three trisections would have to have genus g, contradicting the assumption that g < g. We therefore have good reason to search for 4-manifolds whose unbalanced trisection genus is strictly less than their balanced trisection genus. In 5.3 we produce some possible examples. 38
5.2. Some remarks on Heegaard diagrams and sets of curves on surfaces Before constructing new trisections and trisection diagrams, we first want to describe some tools for modifying and analyzing these diagrams. First we observe that we can find a presentation of π 1 (X) from a trisection diagram for X. Indeed, if X has a trisection (Σ; X 1, X 2, X 3 ) and Y ij = X i X j then π 1 (M) = π 1 (Y 12 Y 13 Y 23 ) since the former is obtained by attaching 3 and 4 dimensional handles to the latter. Y 12 Y 13 Y 23 is obtained by attaching 2 and 3 dimensional handles to Σ, where the 2-handle attachments are given by the trisection diagram. Therefore Remark 5.2.1. π 1 (M) has a presentation with generators the standard generators of Σ, and relators given by the 3g curves of the trisection diagram. Given a genus g surface and three sets of g curves on the surface, we will often want to check that the choice of curves actually does determine a trisection diagram. If {α} 1 i g and {β} 1 i g are two of the sets of curves, we need to check that the α and β curves together give a Heegaard diagram for some # k S 1 S 2. We first make a couple straightforward remarks about simplifying the diagram by removing copies of the standard genus 1 Heegaard splitting of S 1 S 2 or of S 3. Remark 5.2.2. Suppose some α curve and some β curve are isotopic. Assume then that α g = β g without loss of generalization. Let Σ be the surface obtained by cutting Σ along α g and filling the two resulting S 1 boundary components with disks. Then the diagram (Σ; {α} 1 i g, {β} 1 i g ) is a diagram for # k S 1 S 2 iff (Σ ; {α} 1 i g 1, {β} 1 i g 1 ) is a diagram for # k 1 S 1 S 2. Remark 5.2.3. Suppose some α curve intersects the collection of β curves in only a single point. Assume without loss of generalization that α g intersects β g in a single point, and is disjoint from β i for i g. Let Σ be the surface obtained by cutting Σ along α g and filling the two resulting S 1 boundary components with disks. Then the Heegaard diagram (Σ; {α} 1 i g, {β} 1 i g ) is a diagram for # k S 1 S 2 iff (Σ ; {α} 1 i g 1, {β} 1 i g 1 ) is a diagram for # k S 1 S 2. Note that β g might have intersections with the α i for i g curves, but each of these intersections can be removed by sliding α i over α g using the segment of β g between α i β g and α g β g as a guide. Recall that Σ g,1 I is diffeomorphic to a 3 dimensional handlebody. If α Σ g,1 is an essential properly embedded arc, then α I is an essential disk in Σ g,1 I, and α is nonseparating iff the disk is. We call such a disk a (nonseparating) product disk. 39
Now suppose Σ 2g is a genus 2g surface for some g, and let γ be a curve cutting Σ 2g into two punctured genus g surfaces. There is then a diffeomorphism f : Σ 2g (Σ g,1 I) such that f(γ) = Σ g,1 {1/2}. Therefore, by fixing such a diffeomorphism we can think of Σ 2g as the boundary of the handlebody Σ g,1 I and γ as the curve separating Σ g,1 {0} from Σ g,1 {1}. Let us call a simple closed curve in Σ 2g symmetric with respect to the identification Σ 2g = (Σg,1 {0}) if it is the inverse image under f of the boundary of a nonseparating product disk. Similarly, a nonseparating collection of curves is symmetric if it is the inverse image of a nonseparating collection of arcs. A maximal symmetric collection of nonparallel curves then corresponds to a minimal system of disks, all of which are product disks. Since any two minimal systems of disks are related by disk slides, we deduce the following: Remark 5.2.4. Let Σ 2g be a genus 2g surface and fix a diffeomorphism to (Σ g,1 I). Then any two maximal symmetric collections of nonparallel curves are slide-equivalent. By forgetting how f acts on half of the surface Σ 2g we obtain the following: Remark 5.2.5. Let Σ 2g be a genus 2g surface and let γ be a curve cutting Σ 2g into two punctured genus g surfaces Σ g,1 and Σ g,1. Let {α i} 1 i 2g be a maximal nonseparating collection of curves, and assume that # α i γ = 2 for all i. Let {β i } 1 i 2g be a maximal nonseparating collection of properly embedded arcs in Σ g,1. Then there exists a sequence of slides on the α curves after which α i Σ g,1 = β i for all i. We can now use the above remarks to get a useful trick for simplifying certain Heegaard diagrams. Let Σ 2g be a surface equipped with a diffeomorphism to (Σ g,1 I) and let γ, Σ g,1, and Σ g,1 be as before. Let {α i} 1 i 2g be a maximal symmetric nonseparating collection of curves and let {β i } 1 i g be a nonseparating collection of g curves all of which lie on one side of γ, say Σ g,1. The α and β curves then give a partial Heegaard diagram on Σ 2g, which could be made into a full Heegaard diagram by completing the collection of β curves to contain 2g collectively nonseparating curves. Since {β i } 1 i g must cut Σ g,1 into a planar surface, any curves that we might add to the β collection could be slid to be disjoint from Σ g,1. In this situation, we can perform slides on the α curves to get a new set of α curves such that (1) The α curves are still symmetric 40
(2) # α i β j = ij for all 1 i 2g, 1 j g (3) For 1 i g, β i is isotopic to the result of taking the curve obtained by taking α i+g Σ g,1 and connecting the two ends with an arc in γ. From (1) and (3) we see that α i+g is obtained by banding together a copy of β i and a copy of β i reflected across γ. By (2) we can in fact perform g destabilizations on the (partial) Heegaard diagram. The effect on the partial Heegaard diagram is to cut along all of the β curves, removing the corresponding α curves, and filling the resulting S 1 boundary components with disks. We are left with Σ g,1 with g alpha curves on it. Moreover, the resulting α curves are the result of reflecting the old β curves across γ, where by the reflection we mean the result of isotoping from Σ g,1 {0} to Σ g,1 {1} or vice-versa. See Figure 5.1. The techniques of this section can be applied to check that the 4-dimensional trisection diagrams of the next section are valid. In particular, these techniques can be used to show that any two collections of curves in the trisection diagrams give a Heegaard diagram for some # k S 1 S 2. 5.3. Trisections of 3-manifold bundles We begin by presenting the construction of trisections of 3-manifold bundles given by Gay and Kirby [GK12]. Example 5.3.1. Let M be a 3-manifold and suppose X is a 4-manifold that fibers over M with monodromy µ. First, choose some Heegaard surface Σ of M such that Σ is oriented isotopic to µ(σ). This will not be true for an arbitrary choice of Heegaard surface Σ. However, since both Σ and µ(σ) are Heegaard surfaces for M, then by Theorem 2.2.1 there is a stabilization Σ of Σ such that Σ and µ(σ ) are oriented isotopic. Thus, after replacing Σ with Σ we have such a Heegaard surface. We rename Σ as Σ for notational convenience. If (Σ; H 1, H 2 ) is the Heegaard splitting given by Σ, we can think of µ as sending each of Σ, H 1 and H 2 to themselves. We can now construct the trisection. Begin by cutting up I M as in Figure 5.2. Each vertical slice is a copy of M, cut into two handlebodies by the middle line. Along the center line, representing Σ S 1, we see 6 red dots separating the line into six horizontal line segments. Each red dot is therefore a copy of Σ, and each horizontal segment is diffeomorphic to Σ I. Each X i block is a copy of (handlebody) I, and is therefore a 4-dimensional handlebody. Since µ fixes the Heegaard splitting, the left and right side of the diagram glue together in a natural way to get X. 41
(a) A partial Heegaard diagram on a genus 2 surface. α 1 is in dark blue and α 2 in light blue. β 1 is in red. (b) α 1 and α 2 after performing handle slides such that the partial Heegaard diagram satisfies the conditions (1) through (3). (c) The final result after cancelling α 1 and β 1. Figure 5.1. After performing handle slides on the α curves, we see that α 1 and β 1 have geometric intersection number 1, while α 2 and β 1 are disjoint. Therefore, we can perform a destabilization to get a genus 1 surface with a single α curve, and this curve is the reflection of β 1. To turn this into a trisection, we add in tunnels connecting the copies of Σ at each red dot. Consider a horizontal segment connecting two such red dots, and assume the segment lies along X j X k where i, j, k is some permutation of 1, 2, 3. Let be some point in Σ, so that I is a path connecting the surfaces lying at the two red dots on the two sides of the segment. Dig out a regular neighborhood of this path from X j X k and assign it to X i. This affects the triple intersection surface by attaching the two copies of Σ lying at the two red dots with a tube. Perform this operation to every such horizontal segment. The last such operation creates a loop rather than 42
connecting two disjoint components, so it increases the total genus by 1. We therefore end up with a genus 6g + 1 surface in the triple intersection, where a factor of g comes from the copy of Σ lying at each of the six red dots, and the last 1 comes from connecting all the copies together in a loop. Figure 5.2. A (6g + 1; 2g + 1) trisection of the bundle over M with monodromy µ, where M has a genus g Heegaard splitting (Σ; H 1, H 2 ) such that µ sends Σ, H 1 and H 2 to themselves. Each vertical slice is a copy of M. The middle horizontal line represents Σ I/(x, 1) (µ(x), 0). Example 5.3.2. We now investigate how to construct diagrams for the trisections of Example 5.3.1 in the case X = S 1 M where the monodromy is trivial. The result when we set the three manifold M = S 1 S 2 shown in Figure 5.3. The case where M is the lens space L 1/4 is shown in Figure 5.4. At each red dot, there lies a copy of Σ g,2. Consider two cyclically adjacent red dots in Figure 5.2 together with the line segment between them. The inverse image of this is a twice punctured genus 2g surface, which we think of as two copies of Σ g,2 connected by a tube connecting one puncture from each copy. The remaining two punctures can be filled to get a genus 2g surface that bounds Σ g,1 I in one of the X i X j. Being the boundary of a Σ g,1 I, symmetric curves are defined as in the discussion from the previous section. Any pair of symmetric curves determines a pair of essential meridian disks in X i X j. When drawing diagrams we draw them such that, whenever possible, the symmetric curves (bounding disks in Σ g,1 I) actually look symmetric in the picture. From Figure 5.2 we see that one of the copies of Σ g,1 bounds a copy of H 2 above it in X j X k, while one bounds a a copy of H 1 below it in X i X k. In the case of M = L 1/4 the contribution of this horizontal segment looks like Figure 5.5. After putting these pieces together to get a genus 6g + 1 surface X 1 X 2 X 3 there are six curves of each color. To get one more curve of each color, recall that along the horizontal line between X i and X j we added a 1-handle D 3 I to X k 43
(where i, j, k is any permutation of 1, 2, 3). The boundary of this tube is S 2 I, which is the union of a copy of D 2 I in X i X k and another copy of D 2 I in X j X k. Therefore, the meridian S 1 {1/2} S 2 I can be used to determine a curve in either of X i X k or X j X k. Only one curve of each color may be chosen in this manner, since multiple such curves would separate the triple intersection surface. Figure 5.3. A diagram for the (7; 3) trisection of S 1 S 1 S 2 that comes from the method of Example 5.3.1. We can see immediately in the diagram the symmetry generated by a 2π/3 rotation in the S 1 component. Example 5.3.3. The trisection in the previous example was balanced. However, the construction can be modifying to produce an unbalanced trisection. Break down the bundle as in Figure 5.6. A possible diagram for the case when M = L 1/3 is shown in Figure 5.7. As with the balanced case, the triple intersection surface consists of a copy of Σ g,2 at each red dot, where the punctured are connected by tubes S 1 I lying in each of the horizontal lines. There are Σ g,1 I components of X i X j coming from the horizontal line segments. Consider one such horizontal line segment, and the two copies of Σ g,2 lying at its endpoints. As with the previous example, when drawing the diagram we embed each such pair of Σ g,2 and the tube between them such that the curves that bound disks in Σ g,1 I are precisely the curves which look symmetric in the picture. Since the 44
Figure 5.4. A diagram for the (7; 3) trisection of S 1 L 1/4 that comes from the method of Example 5.3.1. We can see immediately in the diagram the symmetry generated by doing a 2π/3 rotation in the S 1 component. number of red dots is even, these embeddings are consistent as we go around the first component of S 1 M. Example 5.3.4. If Σ is a flippable Heegaard surface, then we can break down S 1 M as in Figure 5.8. See Figure 5.9 and 5.10 for diagrams of the trisections coming from this method in the cases when X = S 1 S 1 S 2 and X = S 1 RP 3. If M is a lens space or S 1 S 2 as in these examples, then the trisection is necessarily minimal genus. This is because the fundamental group of S 1 M is Z π(m) which has rank 2, so any handle decomposition of M must have at least two 1-handles. It follows that each handlebody in the trisection must be at least genus 2. The same argument shows that this method gives minimal genus trisections whenever M has a flippable genus g Heegaard splitting such that g is equal to the rank of π 1 (M). For example, the (10; 4) trisection of the 4-torus produced by this method must be minimal genus. The fundamental group can be checked from the diagrams to be Z π 1 (M) as required. Note that these diagrams are double covered by the diagrams from Example 5.3.1, with the double cover corresponding to the subgroup 2Z π 1 (M). 45
Figure 5.5. A section coming from one of the horizontal segments when M = L 1/4. With this choice of symmetric blue curves, one of the blue curves cancels the red curve in the Heegaard splitting determined by the blue and red curves. The two symmetric blue curves could have been chosen instead such that one of them cancelled with the green curve in the Heegaard splitting determined by the blue and green curves. Figure 5.6. A (4g + 1; 2g + 1, g + 1, g + 1) trisection of S 1 M 3 where M 3 has a genus g Heegaard splitting. The S 1 coordinate increases in the horizontal direction, so each vertical slice is a copy of M. The middle horizontal line is a copy of (Heegard surface) S 1. In Figure 5.10 the blue curve at the top loops around the opposite way as the curve that loops around the bottom right. We explain this now. The surface is formed by three copies of Σ 1,2, tubed together as in Figure 5.11. Each of these twice puntured tori lies on the boundary of a handlebody in one of the X i X j. The top copy of Σ 1,2 corresponds to the middle red dot bounding a handlebody above itself in Figure 5.8. The other two copies of Σ 1,2 bound handlebodies below. 46
Figure 5.7. An unbalanced (5; 3, 2, 2) trisection of S 1 L 1/3. The two green curves at the top are a (1, 3) curves and its reflection. The diagram for a more general S 1 L(p, q) can be obtained by replacing these two green curves with the appropriate (p, q) curve and its reflection. We can take a spine of one of these handlebodies and isotopy it through the S 1 component comparing how it lies in each copy of Σ 1,2. Suppose the spine looks like a (0, 1) curve, going around clockwise, when embedded in the bottom-right copy of Σ 1,2. Since we have chosen the surface such that the pair of red curves connecting the lower-right and upper Σ 1,2 and the pair of green curves connecting the lower-left and upper Σ 1,2 look nice, this determines how the spine looks when isotoped to lie in the other two copies of Σ 1,2. Specifically, the spine is a (0, 1) curve oriented counterclockwise in the upper Σ 1,2, and is a (0, 1) curve going clockwise in the lower-left Σ 1,2 as shown in Figure 5.11 Figure 5.12 shows how flipping the Heegaard splitting sends a (1, 0) curve to a (1, 2) curve. By performing handleslides and reimbedding the diagram, we can get a more symmetric picture as in Figure 5.13. With no additional work this construction can be generalized to bundles over M with possibly nontrivial monodromy so long as the monodromy interchanges the two handlebodies of a genus g Heegaard splitting of M. 47
Figure 5.8. A (3g + 1; g + 1) balanced trisection of S 1 M 3 where M 3 has a flippable genus g Heegaard splitting. The S 1 coordinate increases in the horizontal direction, so each vertical slice is a copy of M. The middle horizontal line is a copy of (Heegard surface) I. The right side of the rectangle is glued up to the left side by a diffeomorphism realizing the flipping of the Heegaard surface. Although this diffeomorphism is isotopic to the identity on M, it acts nontrivially on the Heegaard surface. Figure 5.9. A minimal genus (4; 2) trisection of S 1 S 1 S 2. Example 5.3.5. Suppose M is a lens space or S 1 S 2 with genus 1 Heegaard splitting (Σ; H 1, H 2 ). There is a canonical orientation preserving involution of H 1 sending a generator of π 1 (H 1 ) to its inverse. If β σ = H 1 bounds a disk in H 2, then this involution actually sends β to itself (with the opposite orientation). Therefore, it extends to an orientation preserving involution of M. Let ϕ: M M be this involution and let X be the bundle over M with monodromy ϕ. We can produce an exceptionally simple trisection diagram for X as follows: 48
Figure 5.10. A minimal genus (4; 2) trisection of S 1 L 1/2. Figure 5.11. Three copies of Σ 1,2 are glued together to form the triple intersection surface. The three oriented pink curves are isotopic in S 1 M. Start with one of the trisection diagrams for S 1 M constructed in the previous examples. A generic M fibre intersects the triple intersection surface in a single simple closed curve. Let C be some such closed curve. C intersects only one of the three sets of curves giving the trisection diagram. Moreover, it intersects precisely two curves from that set, and the intersections with those 49
(a) A Heegaard diagram for RP 2, where the green curve bounds a disk in H 1 and the red curve bounds a disk in H 2. The pink curve is isotopic to a spine of H 1. (b) The same Heegaard diagram embedded in S 3 such that the red curve appears bounds a disk in the inner handlebody in S 3. The red curve and pink curve together determine the embedding of the green curve. Figure 5.12. Two Heegaard diagrams of RP 2 related by a diffeomorphism of the surface. The diagram in (b) is though of as the result of flipping the Heegaard splitting given by the diagram in (a). The green curve in (b) loops around longitudinally in the opposite direction of the red curve in (a). two curves interleave. See Figure 5.14. After cutting along C and gluing back up with a half-twist the number of curves of each color is unchanged. Using the techniques of the previous section it can be checked that each two colors still give a Heegaard diagram for # k S 1 S 2, so these sets of curves produce a valid trisection diagram. If the trisection diagram we started with was a genus 4 diagram of S 1 M for a flippable lens space M, then the diagram already had a half twist in it. In this case C can be chosen at the location of this half-twist, so doing a half-twist around C straightens out the curves passing through it. Alternatively if C is chosen to be one of the other two possible isotopy classes, then after performing a half-twist around C both twists can be removed by a self-diffeomorphism of the diagram. The results when M is S 1 S 2 and RP 3 are shown in Figures 5.15 and 5.16 respectively. It can be checked from the diagrams that the fundamental group of X is the semidirect product of Z and π 1 M such that conjugation by the generator of Z sends each element of π(m) to its inverse. 50