Contributions to Sutured Monopole and Sutured Instanton Floer

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Contributions to sutured monopole and sutured instanton Floer homology theories by Zhenkun Li Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 ○c Massachusetts Institute of Technology 2020. All rights reserved.

Author...... Department of Mathematics March 31st, 2020

Certified by ...... Tomasz S. Mrowka Professor of Mathematics Thesis Supervisor

Accepted by...... Davesh Maulik Chairman, Department Committee on Graduate Theses 2 Contributions to sutured monopole and sutured instanton Floer homology theories by Zhenkun Li

Submitted to the Department of Mathematics on March 31st, 2020, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics

Abstract In this thesis, we present the development of some aspects of sutured monopole and sutured instanton Floer homology theories. Sutured monopole and instanton Floer homologies were introduced by Kronheimer and Mrowka. They are the adaption of monopole and instanton Floer theories to the case of balanced sutured manifolds, which are compact oriented 3-manifolds together with some special data on the boundary called the suture. We construct the gluing and cobordism maps in these theories, construct gradings associated to properly embedded surfaces inside the balanced sutured manifolds, and use these tools to further construct minus versions of knot Floer homologies in monopole and instanton theories. These constructions contribute to laying down a solid basis in sutured monopole and sutured instanton Floer homology theories, upon which we could develop further applications.

Thesis Supervisor: Tomasz S. Mrowka Title: Professor of Mathematics

3 4 Acknowledgments

It was a wonderful journey to MIT. The lovely city, the amazing people, and the friendly and active atmosphere here all make it a paradise to study and research in math. Looking back, I cannot achieve thus far without supports and encouragement from my advisor, my friends, my family, and other people around. It is my pleasure to dedicate these acknowledgments to them. First, I wish to express my sincere appreciation to my advisor, Tomasz S. Mrowka, who leads me to the field of Gauge theory and offers enormous encouragement and guidance. It was always a great pleasure to talk with him when I can always hear about new math, inspiring ideas, and even the philosophy of life. Second, I would like to warmly thank the geometry and topology community at MIT and in Boston. People from across Boston communicate quite often, and it feels as if every day, there is some news happening around me. In particular, I am grateful to John Baldwin, Mariano Echeverria, Larry Guth, Jianfeng Lin, Yu Pan, Paul Seidel, Matthew Stoffregen, Yi Xie, Boyu Zhang, for countless helpful conversations and discussions. Third, I would like to thank my friends in and outside Boston, with whom I can share my excitement and happiness in highlight times and from whom I receive companions and relieves in hard times. Finally, I want to thank my parents for their endless love and encouragement. Nothing would be possible without their support.

5 6 This doctoral thesis has been examined by a Committee of the Department of Mathematics as follows:

Professor Tomasz S. Mrowka ...... Chairman, Thesis Committee, Thesis Supervisor Professor of Mathematics

Professor Paul Seidel ...... Member, Thesis Committee Levinson Professor of Mathematics

Doctor Matthew Stoffregen...... Member, Thesis Committee C L E Moore Instructor 8 Contents

1 Introduction 11

2 Preliminaries and backgrounds 19 2.1 Balanced sutured manifolds and sutured manifold hierarchy . . . . . 19 2.2 Closures and monopole Floer homology ...... 25 2.3 Instanton theory and balanced sutured manifolds ...... 30 2.4 Marked closures and naturality ...... 39 2.5 Floer’s Excisions ...... 46 2.6 Basic properties of SHM and SHI ...... 51 2.7 Contact structures and contact elements ...... 62 2.8 Applications and other discussions ...... 79

3 Gluing and cobordism maps 91 3.1 Contact elements and excisions ...... 91 3.2 Another interpretation of contact handle gluing maps ...... 102 3.3 Basic properties of handle attaching maps ...... 112 3.4 Contact cell decompositions ...... 133 3.5 The construction of the gluing maps ...... 136 3.6 The construction of cobordism maps ...... 141

9 3.7 Duality and turning cobordism around ...... 144

4 Gradings on Sutured monopole and instanton Floer homologies 153 4.1 The construction of the grading ...... 154 4.2 A reformulation of Canonical maps ...... 160 4.3 Pairing of the intersection points ...... 167 4.4 A naive version of grading shifting property ...... 176 4.5 Supporting spin푐 structures and eigenvalue functions ...... 179 4.6 The grading shifting property ...... 187 4.7 Floer homologies on a sutured solid torus ...... 190 4.8 The connected sum formula ...... 199

5 Applications to knot theory 207 5.1 The construction ...... 207 5.2 Basic properties of the minus version ...... 216 5.3 Knots representing torsion classes ...... 222

5.4 Concordance invariance of 휏퐺...... 229 5.5 Computations for twisted knots ...... 245

10 Chapter 1

Introduction

The celebrated papers [14, 15] by Donaldson around 1983 marked the birth of (mathematical) gauge theory. In these papers, he studied the moduli spaces of a particular set of partial differential equations, called the anti-self-dual equation, which has its origin in physics, on some special bundle over a closed oriented 4-manifold. These moduli spaces lead to an invariant, which now people call the Donaldson invariant, and encode some information of the topology and the smooth structure of the underlying 4-manifold that could hardly be captured by classical tools prior to the introduction of gauge theory. Later, in 1988, Floer observed in [21] a wonderful relation between dimensions 3 and 4: For a closed oriented 3-manifold 푌 , the negative gradient flow equation of the Chern-Simons functional is exactly the anti-self-dual equation on the infinite cylinder R ˆ 푌 . This observation led to the construction of instanton Floer homology on 3-manifolds. Roughly speaking, the instanton Floer homology on a closed oriented 3-manifold 푌 is an infinite-dimensional Morse theory build upon the Chern Simons functional that is defined on the space of connections on a suitable bundle over 푌 , module the action of gauge transformations. This infinite-

11 dimensional Morse theory gives rise to a module over suitable coefficient rings, such as Z and C, and serves as a topological invariant of 푌 . For a cobordism between two closed oriented 3-manifolds, one can also study the anti-self-dual equation on the cobordism and obtains a homomorphism between the modules associated to the two boundary-3-manifolds. This nature of instanton Floer homology makes it be a p3 ` 1q-TQFT. In 2007 and 2004, monopole Floer homology and Heegaard Floer homology were introduced. Monopole Floer homology was introduced by Kronheimer and Mrowka in [52]. The construction followed the same line as Floer’s construction but used another set of differential equations, called the Seiberg-Witten equation introduced by Seiberg and Witten [83]. Heegaard Floer homology was introduced by Ozsváth and Szabó and is built on Heegaard splittings, a somewhat combinatorial description of 3-manifolds. Later, many other variances and related constructions were made, including the singular instanton Floer homology introduced by Kronheimer and Mrowka [48], Pinp2q-monopole Floer homology introduced by Lin [60], the knot Floer homologies introduced by Ozsváth and Szabó in [74], and Embedded contact homology introduced by Hutchings [37], among others. Floer homology theory has become a very powerful tool in the study of 3-dimensional topology and has many remarkable applications, such as the approval of the Property P conjecture (see Kron- heimer and Mrowka [46]), the disapproval of triangulation conjecture (see Manolescu [62]), and the study of cosmetic surgeries (see Ni and Wu [70]) and knot concordance (see [79, 31, 88, 13]). Different versions of Floer homologies have their own advantages. Instanton Floer homology is most closely related to the representations of the fundamental groups of 3-manifolds; Monopole Floer homology has simpler analytical inputs and better compactness properties in dimension 4 than instanton theory, and is most closely

12 related to some geometric properties of the underlying manifolds; Heegaard Floer homology is the most computable one among all Floer homology theories. Besides, cobordism maps in monopole and instanton Floer homologies are more naturally defined and their naturality is easier to treat than the one in Heegaard Floertheory, while in Heegaard Floer homology, there are more algebraic structures associated to knots and links inside 3-manifolds, which leads to a more fruitful knot Floer homology theory than its correspondences in the monopole and instanton settings. So, it will be great to compare and combine the merits of different Floer theories when solving an actual problem. Sutured manifold theory is another powerful tool in the study of 3-dimensional topology, which was introduced in 1983 by Gabai in [23] and subsequent papers. A sutured manifold is a compact oriented 3-manifold, together with some special data, called the sutures, on the boundary of the 3-manifold. The core of the sutured manifold theory is the sutured manifold hierarchy. It enables us to decompose any taut sutured manifold, in finitely many steps, into product ones, which are the simplest possible sutured manifolds. Along the decomposition, some previously un-attackable problems break down into more approachable pieces. For instance, in [23, 24, 25], Gabai used these techniques to construct taut foliations for 3-manifolds that are not rational homology spheres and approved the famous property R conjecture. The first combination of Floer homology theory and sutured manifold theory was made by Juhász in [38], where he introduced the sutured (Heegaard) Floer homology, on a special class of sutured manifolds called the balanced sutured manifolds. Later, in [53], Kronheimer and Mrowka introduced sutured monopole and instanton Floer homologies. The original monopole and instanton Floer homologies were defined only on closed 3-manifolds, so, to adapt them to the case of balanced sutured manifolds, Kronheimer and Mrowka first constructed some closed 3-manifolds, which they called

13 the closures, out of the sutured data, and apply the usual construction of monopole and instanton Floer homologies. They also proved some non-vanishing results for those invariants, which leads to a new and simpler proof of the Property P conjecture, and, combined with their later establishment of a spectral sequence from Khovanov homology to a suitable version of instanton knot Floer homology, proves the milestone result that Khovanov homology detects the unknots (see [47]). Furthermore, recently, Kronheimer and Mrowka proposed in [50] a possible human checkable approach to the proof of the four-color theorem, in which sutured instanton Floer homology also plays an important role. Despite many significant applications already established, lots of basic aspects of the sutured monopole and instanton Floer homologies remain unknown. For example, here are a few questions to be answered:

∙ It is the nature of monopole and instanton Floer homology theories on closed 3- manifolds that they serve as a (3+1)-TQFT. Do we also have a similar property for sutured monopole and instanton Floer homologies?

∙ To construct the sutured monopole and instanton Floer homologies, Kron- heimer and Mrowka introduced the concept of closures, while those two Floer homology theories are invariants under different choices of closures. To what extent can we say that the sutured monopole and instanton Floer homologies are independent of the closures? Is there a canonical choice of a closure for a fixed balanced sutured manifold?

∙ The monopole Floer homology of a closed 3-manifold decomposes along spin푐 structures and the instanton Floer homology of a closed 3-manifold decomposes as the direct sum of the generalized eigenspaces of an action induced by any

14 properly embedded surface inside the 3-manifold. Do sutured monopole and instanton Floer homologies admit similar decompositions?

∙ How do the sutured monopole and instanton Floer homologies tell us topological information about the balanced sutured manifolds? In particular, how are they related to the depth of the balanced sutured manifolds, the minimal depth of all possible taut foliations, and the Thurston norms of the balanced sutured manifolds?

∙ How can one possibly compute the sutured monopole and instanton Floer homology of some families of balanced sutured manifolds?

The goal of the thesis is to present some of the work the author did towards answering the above questions as well as other related constructions and applications. The thesis is organized as follows: In Chapter 2, we give a detailed introduction of balanced sutured manifolds and sutured monopole and sutured instanton Floer homologies. We also give an account of the development of the two theories up to date. In Chapter 3, we constructed cobordism maps associated to a special type of cobordisms, which are called the sutured cobordisms and were introduced by Juhász in [41]. We also proved that a cobordism map associated to a product sutured cobordism is the identity, and such cobordism maps are functorial under the composition of sutured cobordisms. This makes sutured monopole and instanton Floer homologies functors from the sutured cobordism category, which are formed by balanced sutured manifolds and sutured cobordisms, to some single category of modules, and answers question 1 as above. We also give a new interpretation of the sutured cobordism maps just defined, indicating that the whole construction of a suture cobordism map can simply be interpreted as being induced by a suitable cobordism between two

15 well-chosen closures of the balanced sutured manifolds. Thus it is possible to make use of the good naturality of the cobordism maps between monopole and instanton Floer homologies on closed 3-manifolds. As a direct corollary to this new interpretation, we obtain a duality result for such sutured cobordism maps. Furthermore, along with the construction of the cobordism maps, we also constructed a special type of maps, called the gluing maps, which were first introduced to the context of Heegaard Floer homology by Honda, Kazez, and Matić [35]. The construction of gluing maps is interesting on its own, as we will see in later chapters. In Chapter 4, we construct a grading on sutured monopole and instanton Floer homologies associated to a properly embedded surface inside the balanced sutured manifold and prove an important grading shifting property in the case when the balanced sutured manifold has a connected toroidal boundary, and the suture has two components. This grading shifting property directly leads to the computation of sutured monopole and instanton Floer homologies of an arbitrary sutured solid torus, and will also be useful in later chapters. In Ghosh and Li [27], the restriction on the sutured manifold was removed, and the authors offered a proof for a general balanced sutured manifold. This directly leads to an algorithm that is capable of computing at least some families of sutured handle bodies, which partially answers question 5. Question 2,3, and 4 are also studied extensively in [27], and the main results of the paper are summarized in Chapter 2. In Chapter 5, we used the gluing maps and the gradings already constructed to build minus versions of monopole and instanton knot Floer homologies for null- homologous knots, which are the counterparts of the minus version of knot Floer homology in Heegaard Floer theory introduced by Ozsváth and Szabó [74]. We also used the grading shifting property to derive some basic properties of the minus versions already constructed. What’s more, we further extend the construction of

16 minus versions to the case of rational knots and prove a surgery type formula relating the minus version of a knot inside 푆3 with the minus version of the dual knot inside the 3-manifold obtained from 푆3 by performing Dehn surgeries along 퐾 of large enough slopes. Finally, we define tau invariants in the monopole and instanton settings, in correspondence to the one defined in Heegaard Floer theory by Ozsváth and Szabó in [73] and prove that they are concordance invariants.

17 18 Chapter 2

Preliminaries and backgrounds

In this chapter, we give a detailed introduction to sutured monopole and instanton Floer homology theories. We also give an account of the development of these two theories up to date.

2.1 Balanced sutured manifolds and sutured manifold hierarchy

Definition 2.1.1. A balanced sutured manifold p푀, 훾q consists of a compact oriented 3-manifold 푀 with non-empty boundary, together with a closed oriented 1- submanifold 훾 on B푀. Let 퐴p훾q “ r´1, 1sˆ훾 be an annular neighborhood of 훾 ĂB푀 and let 푅p훾q “ B푀zintp퐴p훾qq. They satisfy the following properties. (1) Neither 푀 nor 푅p훾q have a closed component. (2) If we orient B퐴p훾q “ B푅p훾q in the same way as 훾, then we require that this orientation of B푅p훾q induces one on 푅p훾q. The induced orientation on 푅p훾q is called the canonical orientation.

19 (3) Let 푅`p훾q be the part of 푅p훾q so that the canonical orientation coincides with the induced boundary orientation on B푀, and let 푅´p훾q “ 푅p훾qz푅`p훾q, then we require that

휒p푅`p훾qq “ 휒p푅´p훾qq.

Remark 2.1.2. The requirement (1) in Definition 2.1.1 implies that the canonical orientation on 푅p훾q, if exists, is unique. However, the existence of a canonical orientation is a non-trivial requirement for being a balanced sutured manifold. See Figure 2-1. In Heegaard Floer theory, when Juhász first defined sutured (Heegaard) Floer homology in [38], he required that the balanced sutured manifold to be strongly balanced (i.e., each component of the sutured manifold is balanced). In the monopole and the instanton setups, being strongly balanced is not required to build the Floer homologies. However, it follows from Corollary 2.6.16 that if the balanced sutured manifold is not strongly balanced, then the sutured monopole and instanton Floer homologies both vanish.

Figure 2-1: Left: The (red) curves with arrows denote the oriented 훾. The orientations on the two components of 훾 are not compatible with inducing an orientation on 푅p훾q, so this is a counterexample of being a balanced sutured manifold. Right: After reversing the orientation of one component of 훾, we end up with a balanced sutured manifold.

20 Example 2.1.3. Let 퐹 be a compact connected oriented surface with non-empty boundary. Let p푀, 훾q “ pr´1, 1s ˆ 퐹, t0u ˆ B퐹 q, then this gives us the simplest example of a balanced sutured manifold, where

퐴p훾q “ r´1, 1s ˆ B퐹 and 푅˘p훾q “ t˘1u ˆ 퐹.

Definition 2.1.4. A balanced sutured manifold p푀, 훾q arising in the way as in Example 2.1.3 is called a product balanced sutured manifold.

Next, we recall sutured manifold decompositions and sutured manifold hierarchies, which were introduced by Gabai in [23] and subsequent papers.

Definition 2.1.5. Suppose 푀 is a compact 3-manifold. 푀 is called irreducible if every embedded 2-sphere 푆2 Ă 푀 bounds an embedded 3-ball inside 푀.

Definition 2.1.6. Suppose 푀 is a compact 3-manifold and 푅 Ă 푀 is an embedded surface. 푅 is called compressible if there is a simple closed curve 훼 Ă 푅 so that 훼 does not bound a disk on 푅 but bounds an embedded disk 퐷 Ă 푀 with 퐷 X 푅 “ 훼. 푅 is called incompressible if it is not compressible. A 3-manifold is called boundary- incompressible if its boundary is incompressible.

Definition 2.1.7 (Thurston norm, Thurston [85]). Suppose 푀 is a compact 3- manifold and 푈 ĂB푀 is a submanifold of B푀. Suppose further that 푆 is a properly embedded surface inside 푀 so that B푆 Ă 푈. If 푆 is connected, then define the norm of 푆 to be 푥p푆q “ maxt´휒p푆q, 0u.

In general, suppose the components of 푆 are

푆 “ 푆1 Y ... Y 푆푛,

21 then define the norm of 푆 to be

푥p푆q “ 푥p푆1q ` ... ` 푥p푆푛q.

Moreover, suppose 훼 P 퐻2p푀, 푈q is a non-trivial second relative homology class, then define the norm of 훼 to be

푥p훼q “ mint푥p푆q | p푆, B푆q Ă p푀, 푈q, r푆, B푆s “ 훼 P 퐻2p푀, 푈qu.

Definition 2.1.8 (Thurston [85]). Suppose 푀 is a compact 3-manifold, and 푆 Ă 푀 is a properly embedded surface. 푆 is called norm-minimizing if

푥p푆q “ 푥p훼q,

where 훼 “ r푆, B푆s P 퐻2p푀, 푁pB푆qq. Here, 푁pB푆q is a neighborhood of B푆 ĂB푀.

Definition 2.1.9 (Gabai [23]). A balanced sutured manifold p푀, 훾q is called taut if the following is true. (1) 푀 is irreducible.

(2) 푅`p훾q and 푅´p훾q are both incompressible.

(3) 푅`p훾q and 푅´p훾q are both norm-minimizing.

Remark 2.1.10. Condition (3) in Definition 2.1.9 does not imply condition (2). Ifwe impose a new condition that every component of 푅˘p훾q has negative Euler characteristics, then (3) does. However, in Figure 2-1, the balanced sutured manifold on the right is norm-minimizing (since 푅˘p훾q are both annuli which have zero norms) but compressible.

22 Definition 2.1.11 (Gabai [23]). Let p푀, 훾q be a balanced sutured manifold. A product annulus 퐴 in p푀, 훾q is an annulus properly embedded in 푀 such that B퐴 Ă

푅p훾q and B퐴 X 푅˘p훾q ‰ H. A product disk is a disk 퐷 properly embedded in 푀 such that B퐷 X 퐴p훾q consists of two essential arcs in 퐴p훾q.

Definition 2.1.12 (Juhász [40]). A balanced sutured manifold p푀, 훾q is called reduced if any product annulus 퐴 Ă 푀 either bounds a cylinder r0, 1s ˆ 퐷2 so that r0, 1s ˆ B퐷2 “ 퐴, or is isotopic to a component of 퐴p훾q inside 푀.

Remark 2.1.13. The two terms ’reduced’ and ’irreducible’ (See Definition 2.1.5) are not inverse to each other.

Definition 2.1.14 (Gabai [23]). Let p푀, 훾q be a taut balanced sutured manifold. A properly embedded surface 푆 Ă 푀 is called horizontal if the following four properties hold. (1) 푆 has no closed components and is incompressible.

(2) B푆 Ă 퐴p훾q, and B푆 is parallel to B푅`p훾q inside 퐴p훾q.

(3) r푆s “ r푅`p훾qs in 퐻2p푀, 퐴p훾qq.

(4) 휒p푆q “ 휒p푅`p훾qq. We say that p푀, 훾q is horizontally prime if every horizontal surface in p푀, 훾q is parallel to either 푅`p훾q or 푅´p훾q.

Definition 2.1.15 (Gabai [23]). Suppose p푀, 훾q is a balanced sutured manifold, and 푆 Ă 푀 is a properly embedded surface so that B푆 is non-empty and intersects 훾 transversely. Let 푀 1 “ 푀zintp푀q. Note

1 1 B푀 “ pB푀 X 푀 q Y p´푆`q Y 푆´,

23 where 푆` and 푆´ are parallel copies of 푆 which are oriented in the same way as 푆. We can define a new suture 훾1 on B푀 1 as follows:

1 1 훾 “ p훾 X 푀 q Y r푅`p훾q X 푆`s Y r푅´p훾q X 푆´s.

We call the process of obtaining p푀 1, 훾1q a sutured manifold decomposition along 푆 and write

푆 1 1 p푀, 훾q p푀 , 훾 q.

Remark 2.1.16. It is straightforward to chech that

1 1 푅`p훾 q “ r푅`p훾q X 푀 s Y 푆´, and

1 1 푅´p훾 q “ r푅´p훾q X 푀 s Y 푆`.

So, after a sutured manifold decomposition, the pair p푀 1, 훾1q is still a balanced sutured manifold, provided that 푆 has no closed components. In [82], Scharlemann

1 introduced the concept of double curve surgeries. Then, 푅˘p훾 q can be thought of obtained by performing a double curve surgery on 푅˘p훾q and 푆.

Definition 2.1.17. We say a sutured manifold decomposition

푆 1 1 p푀, 훾q p푀 , 훾 q is taut if both p푀, 훾q and p푀 1, 훾1q are taut in the sense of Definition 2.1.9.

Definition 2.1.18 (Gabai [23]). Suppose p푀, 훾q is a taut balanced sutured manifold.

24 A sutured manifold hierarchy is a sequence of sutured manifold decompositions

푆0 푆1 푆푛 p푀0, 훾0q p푀1, 훾1q ... p푀푛`1, 훾푛`1q, (2.1)

so that each p푀푖, 훾푖q is taut, p푀0, 훾0q “ p푀, 훾q, and p푀푛`1, 훾푛`1q is a product sutured manifold.

Definition 2.1.19 (Gabai [23]). Suppose p푀, 훾q is a taut balanced sutured manifold. The depth of p푀, 훾q, which we write 푑p푀, 훾q, is the minimal integer 푛 so that there exists a sutured manifold hierarchy as in (2.1).

Theorem 2.1.20 (Gabai [23]). Any taut, balanced sutured manifold has a finite depth.

2.2 Closures and monopole Floer homology

The classical monopole and instanton Floer homologies are defined on closed 3- manifolds. So, to construct sutured monopole and instanton Floer homologies, we need first to construct a closed 3-manifold out of the sutured data. Such construction was introduced by Kronheimer and Mrowka in [53]. Let p푀, 훾q be a balanced sutured manifold and let 푇 be a compact oriented surface-with-boundary so that the following is true. (A-1) There is an orientation reversing diffeomorphism 푓 : B푇 Ñ 훾.

(A-2) We have 휒p푇 q ` 휒p푅`p훾qq ă 0. (A-3) 푇 is not a disk. (A-4) 푇 is connected.

25 Once such a 푇 is chosen, we can glue r´1, 1s ˆ 푇 to 푀 and form

푀 “ 푀 Y r´1, 1s ˆ 푇 푖푑ˆ푓 Ă via the map 푖푑 ˆ 푓 : r´1, 1s ˆ B푇 Ñ 퐴p훾q “ r´1, 1s ˆ 훾.

It is straightforward to check that 푀 has two boundary components, since 푇 is connected: Ă B푀 “ 푅` Y p´푅´q, where 푅˘ “ 푅˘p훾q Y t˘1u ˆ 푇.

A straightforwardĂ computation shows that

휒p푅`q “ 휒p푅`p훾qq ` 휒p푇 q “ 휒p푅´p훾qq ` 휒p푇 q “ 휒p푅´q.

Thus 푅` and 푅´ are diffeomorphic. Let ℎ : 푅` Ñ 푅´ be an orientation preserving diffeomorphism, then we can glue 푅` ĂB푀 to 푅´ ĂB푀 and get a closed 3-manifold. Equivalently, let Ă Ă 푌 “ 푀 Y r´1, 1s ˆ 푅`, ℎY푖푑 where the two parts are glued via theĂ map

ℎ Y 푖푑 : 푅` Y 푅´ Ñ t1u ˆ 푅` Y t´1u ˆ 푅`.

Also, let 푅 “ t0u ˆ 푅` Ă 푌 .

Remark 2.2.1. In general, we could drop condition (A-4) when choosing an auxiliary surface, and the usage of disconnected auxiliary surfaces has some critical applications in the sutured monopole and instanton Floer homology theories. For more

26 details, readers are referred to [53].

Definition 2.2.2 (Kronheimer and Mrowka [53]). The pair p푌, 푅q is called a closure of the balanced sutured manifold p푀, 훾q. Sometimes we simply call 푌 a closure. The surface 푅 is called a distinguishing surface and its genus is called the genus of the closure. The surface 푇 is called an auxiliary surface and the diffeomorphism ℎ is called a gluing diffeomorphism. The manifold 푀 is called a pre-closure.

Remark 2.2.3. The definition of the genusĂ of a closure is from Baldwin and Sivek [4].

Definition 2.2.4 (Kronheimer and Mrowka [53]). Suppose 푌 is a closed connected oriented 3-manifold. Suppose 푅 Ă 푌 is an embedded closed connected oriented surface so that its genus is at least two. Then, we define the set of top spin푐 structures to be 푐 Sp푌 |푅q “ ts Spin structures on 푌 | 푐1psqr푅s “ 2푔p푅q ´ 2u.

If in general 푅 is disconnected, then we require that each component of 푅 has genus at least two. Assume that

푅 “ 푅1 Y ... Y 푅푛,

where 푅푖 are the component of 푅, then we define

푛

Sp푌 |푅q “ Sp푌 |푅푖q. 푖“1 č If 푌 is disconnected, we require that each component of 푌 contains at least one component of 푅. With the above notations, we define

퐻푀p푌 |푅q “ 퐻푀 ‚p푌, sq, sPSp푌,푅q à ~ 27 where the notation 퐻푀 ‚p푌, sq is the 푡표-version of monopole Floer homology of the pair p푌, sq, introduced by Kronheimer and Mrowka in [52]. ~

Remark 2.2.5. We want 푔p푅q ě 2 (condition (A-2) in the choice of 푇 ) in Definition 2.2.6 for two reasons. The first is that the inequality 푔p푅q ě 2 together with the 푐 requirement that 푐1psqr푅s “ 2푔p푅q ´ 2 makes sure that the spin structures in question is non-torsion and hence 푆퐻푀p푀, 훾q is finitely generated. The second reason is that for non-torsion spin푐 structures, the from- and to-versions of monopole Floer homology are canonically isomorphic to each other, so there is no difference on which to use. A more precise statement is that we are using the reduced monopole Floer homology. However, when the spin푐 structure is non-torsion, the reduced monopole Floer homology is naturally isomorphic to both from- and to-versions.

Definition 2.2.6 (Kronheimer and Mrowka [53]). Suppose p푀, 훾q is a balanced sutured manifold and p푌, 푅q is a closure of p푀, 훾q. Then, we can define the sutured monopole Floer homology of p푀, 훾q as

푆퐻푀p푀, 훾q “ 퐻푀p푌 |푅q

For later convenience, we also make the following definition.

Definition 2.2.7. For a pair p푌, 푅q, we define the set of supporting spin푐 structures as

˚ S p푌 |푅q “ ts P Sp푌 |푅q | 퐻푀 ‚p푌, sq ‰ 0u.

~ We can also use Q, C and Z2 coefficients. The usage of the last one is more subtle than the others and is valid due to Sivek [84]. Besides this, we can also use some particular local coefficients.

28 Definition 2.2.8. The mod 2 Novikov ring ℛ is defined to be

훼 ℛ “ t 푐훼푡 | 훼 P R, 푐훼 P Z2, 7t훽 ă 푛 | 푐훽 ‰ 0u ă 8 for all 푛 P Zu. 훼 ÿ Remark 2.2.9. We can also use the usual Novikov ring to construct sutured monopole Floer homology, which has Euler characteristics 0. The convenience for using mod 2 coefficients is that the surgery exact triangle in monopole theory is currently only proved in characteristics 2. See Kronheimer, Mrowka, Ozsváth, and Szabó [44].

Suppose p푀, 훾q is a balanced sutured manifold and choose an auxiliary surface 푇 as above. Let p푌, 푅q be a closure of p푀, 훾q arising from 푇 and some gluing diffeomorphism ℎ. Choose a non-separating simple closed curve 휂 Ă 푅.

Definition 2.2.10 (Kronheimer and Mrowka [53]). Under the settings as above, we define

푆퐻푀p푀, 훾;Γ휂q “ 퐻푀p푌 |푅;Γ휂q “ 퐻푀 ‚p푌, s;Γ휂q. sPSp푌,푅q à ~ The notation Γ휂 denotes the use of local coefficients. The Floer homology 푆퐻푀p푀, 훾;Γ휂q is in fact a finitely generated module over the ring ℛ.

Remark 2.2.11. We can define accordingly

˚ S p푌 |푅;Γ휂q “ ts P Sp푌 |푅q | 퐻푀 ‚p푌, s;Γ휂q ‰ 0u.

~ This could potentially be different from the set S˚p푌 |푅q as in Definition 2.2.7. By

˚ ˚ abusing notations, we write S p푌 |푅;Γ휂q simply as S p푌 |푅q when the usage of local coefficients is clear in the context. Accordingly, we will also simply write 푆퐻푀p푀, 훾q for 푆퐻푀p푀, 훾;Γ휂q

29 Definition 2.2.12 (Kronheimer and Mrowka [53]). All the choices made in the above definition, p푇, 푓, ℎ, 휂q, are called the auxiliary data.

There are many choices made in the construction of sutured monopole Floer homology. So, a natural question to ask is whether it is well-defined. The first answer to this question is the following theorem. More discussion will be presented in Section 2.4.

Theorem 2.2.13 (Kronheimer and Mrowka [53]). Suppose p푀, 훾q is a balanced sutured manifold. Then, the isomorphism classes of 푆퐻푀p푀, 훾q and 푆퐻푀p푀, 훾;Γ휂q are independent of all auxiliary data.

We have the following question.

Question 1. Is there a Pin(2) version of sutured monopole Floer homology? For Pin(2)-monopole Floer homology, see Lin [60].

2.3 Instanton theory and balanced sutured manifolds

In this section, we introduce some basic constructions of instanton Floer homology on closed 3-manifolds and adapt it to construct sutured instanton Floer homology.

Definition 2.3.1 (Donaldson [16]). Suppose 푌 is a closed connected oriented 3- manifold and 휔 is an embedded curve. We call the pair p푌, 훼q admissible if there is an embedded closed connected oriented surface Σ Ă 푌 so that 푔pΣq ą 0 and Σ ¨ 휔 is odd.

30 Suppose p푌, 휔q is an admissible pairs. We can construct instanton Floer homology on p푌, 휔q as follows: Let 퐸 be an 푈p2q-bundle on 푌 so that

푐1pdetp퐸qq “ 푃.퐷.r휔s.

Let g퐸 be the bundle of traceless skew-hermitian endomorphisms of 퐸, and let 풜퐸 be the space of 푆푂p3q-connections on g퐸. Let 풢퐸 be the group of determinant-one gauge transformations and let ℬ퐸 “ 풜퐸{풢퐸. Then, we can use the Chern-Simons functional to construct a well defined 푆푂p3q instanton Floer homology over C, which we denote by 퐼휔p푌 q.

Remark 2.3.2. Though from the construction, 퐼휔p푌 q seems to depend on the homology class r휔s P 퐻2p푌 ; Zq, it is, in fact, an 푆푂p3q theory and 퐼휔p푌 q depends only on the homology class r휔s P 퐻2p푌 ; Z{2Zq, being the second Stiefel-Whitney class of the bundle g퐸.

Notation. If 푥 P 푌 is a point, then there is an action 휇p푥q on 퐼휔p푌 q. The action 휇p푥q has eigenvalues 2 and ´2. By slightly abusing the notations, from now on we use 퐼휔p푌 q to denote only the generalized eigenspace of 휇p푥q corresponding to eigenvalue 2. Suppose Σ Ă 푌 is a closed oriented embedded surface inside 푌 . Then, there is also an action 휇pΣq on 퐼휔p푌 q. If there are two embedded surfaces Σ and Σ1 inside 푌 then the actions 휇pΣq and 휇pΣ1q commute and 휇pΣ Y Σ1q “ 휇pΣq ` 휇pΣ1q. We have some results on the eigenvalues of the action 휇pΣq. The first discussion is due to Muñoz[65], but the following one is more useful.

Proposition 2.3.3 (Kronheimer and Mrowka [53]). Suppose p푌, 휔q is an admissible pair, and Σ Ă 푌 is an embedded surface with 푔pΣq ą 0 and Σ ¨ 휔 being odd. Then,

31 the eigenvalues of the action 휇pΣq on 퐼휔p푌 q belongs to the set of even integers ranged from 2 ´ 2푔pΣq to 2푔pΣq ´ 2.

In Proposition 2.3.3, the condition that Σ ¨ 휔 is odd can be removed, as in the following Proposition.

Proposition 2.3.4. Suppose p푌, 휔q is an admissible pair, and Σ Ă 푌 is an embedded surface with 푔pΣq ą 0. Then, the eigenvalues of the action 휇pΣq on 퐼휔p푌 q belongs to the set of even integers ranged from 2 ´ 2푔pΣq to 2푔pΣq ´ 2.

Remark 2.3.5. The difference between Proposition 2.3.3 and 2.3.4 is that the latter one does not have the requirement that Σ ¨ 휔 is odd. Kronheimer and Mrowka in [53] also proved the later proposition when 퐼휔p푌 q is replaced by 퐼휔p푌 |푅q (See Definition 2.3.7). The proof of Proposition 2.3.4 here is a generalization of their ideas.

Proof of Proposition 2.3.4. Since p푌, 휔q is admissible, there exists a surface Σ0 so that 푔pΣ0q ą 0 and Σ0 ¨ 휔 is odd. For the moment, let

휔 휔 퐼 p푌, 2푖q “ p2푖q generalized eigenspace of 휇pΣ0q acting on 퐼 p푌 q.

We know from Proposition 2.3.3 that

푔pΣ0q´1 퐼휔p푌 q “ 퐼휔p푌, 2푖q. (2.2)

푖“1´푔pΣ0q à

휔 It is a basic fact that the action 휇pΣ0q and 휇pΣq on 퐼 p푌 q commutes, so we know that 휇pΣq preserves the spaces 퐼휔p푌, 2푖q. Claim 1. The eigenvalues of 휇pΣq acting on 퐼휔p푌, 2푖q are all in the interval r2 ´ 2푔pΣq, 2푔pΣq ´ 2s.

32 To prove this claim, suppose there is an eigenvalue 휆 of 휇pΣq acting on 퐼휔p푌, 2푖q, which is not in the interval r2 ´ 2푔pΣq, 2푔pΣq ´ 2s. We prove the case when 휆 ą 2푔pΣq ´ 2 and the other case is exactly the same. Pick an 푛 P 2Z so that

푛p휆 ` 2 ´ 2푔pΣqq ą 2푔pΣ0q ´ 2,

휔 1 and pick 푣 P 퐼 p푌, 2푖q so that 휇pΣqp푣q “ 휆 ¨ 푣. Let Σ “ Σ0 Y 푛Σ, we know that

1 푁 푁 r휇pΣ q ´ 푛휆 ´ 2푖s “ 푛r휇pΣq ´ 휆s ` r휇pΣ0q ´ 2푖s . ` ˘ Hence, we know that r휇pΣ1q ´ 푛휆 ´ 2푖s푁 p푣q “ 0, and thus 푛휆 ` 2푖 is an eigenvalue of 휇pΣ1q on 퐼휔p푌 q. Our choice of 푛 makes sure that

푛휆 ` 2푖 ą 2푔pΣ1q ´ 2 and Σ1 ¨ 휔 is odd. This violates Proposition 2.3.3 and thus concludes the proof of Claim 1.

Claim 2. The eigenvalues of 휇pΣq acting on 퐼휔p푌, 2푖q are all even integers.

If Σ¨휔 is odd, this follows directly from Proposition 2.3.3. If Σ¨휔 is even, then let

1 휔 휔 Σ “ Σ`Σ0. Suppose 휆 is an eigenvalue of 휇pΣq acting on 퐼 p푌, 2푖q, and 푣 P 퐼 p푌, 2푖q is an eigenvector: 휇pΣqp푣q “ 휆 ¨ 푣.

Then, we know that r휆pΣ1q ´ 휆 ´ 2푖s푁 p푣q “ 0.

33 Since Σ1 ¨ 휔 is odd, 휆 must be even by Proposition 2.3.3.

Claim 3. The eigenvalue of 휇pΣq acting on 퐼휔p푌 q belongs to the interval r2 ´ 2푔pΣq, 2푔pΣq ´ 2s.

Suppose the contrary, there is an eigenvalue 휆 R r2 ´ 2푔pΣq, 2푔pΣq ´ 2s and an eigenvector 푣 P 퐼휔p푌 q so that 휇pΣqp푣q “ 휆 ¨ 푣.

From the decomposition (2.2) we can write

푔pΣ0q´1 휔 푣 “ 푣푖, 푣푖 P 퐼 p푌, 2푖q. 푖“1´푔pΣ q ÿ 0 Then we know that 푔pΣ0q´1

0 “ p휇pΣq ´ 휆q푣푖 푖“1´푔pΣ q ÿ 0 which is absurd by Claim 1. This concludes the proof of Claim 3.

Claim 4. The eigenvalues of 휇pΣq acting on 퐼휔p푌 q are all even integers. This follows from the same argument as in the proof of Claim 3 and concludes the proof of Proposition 2.3.4.

Question 2. How can we see the action 휇pΣq and its eigenvalues on the level of the representation variety of the fundamental groups?

Similar to Corollary 7.6 in Kronheimer and Mrowka [53], we can make the following definition.

Definition 2.3.6. Suppose p푌, 휔q is an admissible pair. Suppose further that 휆 :

34 퐻2p푌 ; Zq Ñ 2Z is a linear function, then we can define

퐼휔p푌, 휆q “ 푘푒푟p휇p휎q ´ 휆p휎qq푁 . 휎P퐻 p푌 ;푍q 푁ě0 č2 ď Such a function 휆 is a called an eigenvalue function.

We can lift 휆 to a linear map (which we will use the same notation to denote)

휆 : 퐻2p푌 ; Qq Ñ Q.

Thus, from now on, we regard 휆 as an element in 퐻2p푌 ; Qq. We then have a decomposition 퐼휔p푌 q “ 퐼휔p푌, 휆q. 휆P퐻2p푌 ;Qq à As we did in Definition 2.2.4, we can make the following definition.

Definition 2.3.7. Suppose p푌, 휔q is an admissible pair and 푅 Ă 푌 is an embedded closed oriented surface so that 푔p푅q ą 0 and 푅 ¨ 휔 being odd. If 푅 is connected, we can define the set of top eigenvalue functions:

2 Hp푌 |푅q “ t휆 P 퐻 p푌 ; Qq|휆pr푅sq “ 2푔p푅q ´ 2u.

If 푅 is disconnected and its components are

푅 “ 푅1 Y ... Y 푅푛, then define 푛

Hp푌 |푅q “ Hp푌 |푅푖q. 푖“1 č 35 Define 퐼휔p푌 |푅q “ 퐼휔p푌, 휆q. 휆PHp푌 |푅q à Define the set of supporting eigenvalue functions as

H˚p푌 |푅q “ t휆 P Hp푌 |푅q|퐼휔p푌, 휆q ‰ 0.u

Remark 2.3.8. Unlike the monopole case in Section 2.2, where we require that 푔p푅q ą 1, in the instanton settings, we only require that 푔p푅q ą 0.

We have the following lemma which will be useful later.

Lemma 2.3.9. Suppose p푊, 휈q is a cobordism between p푌, 휔q and p푌 1, 휔1q. Suppose further that 휆 P 퐻2p푌 ; Qq and 휆1 P 퐻2p푌 1; Qq are two eigenvalue functions. Let 푖 : 푌 Ñ 푊 and 푖1 : 푌 1 Ñ 푊 are the inclusion map. If

1 퐼p푊, 휈qp퐼휔p푌, 휆qq X 퐼휔 p푌 1, 휆1q ‰ t0u, then there must be an element 휏 P 퐻2p푊 ; Qq so that 푖˚p휏q “ 휆 and p푖1q˚p휏q “ 휆1.

Proof. For a second homology class 휎 and a rational number 푟 P Q we can define

퐼휔p푌, 휎, 푟q “ 푘푒푟p휇p휎q ´ 푟q푁 . 푁ě0 ď By definition, we know that

퐼휔p푌, 휆q “ 퐼휔p푌, 휎, 휆p휎qq. 휎P퐻 p푌 ; q č2 Q

Similarly, we can define 퐼휔1 p푌 1, 휎1, 푟1q.

36 Note we can regard an element 휏 P 퐻2p푊 ; Qq as a map

휏 : 퐻2p푊 ; Qq Ñ Q.

Suppose there are no such 휏 as in the statement of the lemma, then there is a class

1 1 휎0 P 퐻2p푌 ; Qq and a class 휎0 P 퐻2p푌 ; Qq so that

1 1 푖˚p휎0q “ 푖˚p휎0q P 퐻2p푊 q, while

1 1 휆p휎0q ‰ 휆 p휎0q.

Thus, we know that

휔 휔 휔1 1 1 퐼p푊, 휈qp퐼 p푌, 휆qq Ă 퐼p푊, 휈qp퐼 p푌, 휎0, 휆p휎0qqq Ă 퐼 p푌 , 휎0, 휆p휎0qq.

The last inclusion follows from Lemma 2.6 in [8]. However, 휆p휎q ‰ 휆1p휎1q so

휔1 1 1 휔1 1 1 1 1 퐼 p푌 , 휎0, 휆p휎0qq X 퐼 p푌 , 휎0, 휆 p휎0qq “ t0u.

Hence, we conclude

1 퐼p푊, 휈qp퐼휔p푌, 휆qq X 퐼휔 p푌 1, 휆1q “ t0u, which is a contradiction. Thus, Lemma 2.3.9 follows.

Now we explain the construction of sutured instanton Floer homology. Suppose p푀, 훾q is a balanced sutured manifold. We pick an auxiliary surface 푇 that satisfies

37 the conditions (A-1) to (A-4) as in Section 2.2. We require a new input: (A-5) There is a marked point 푝 P intp푇 q. Form the pre-closure 푀 “ 푀 Y r´1, 1s ˆ 푇 and let

Ă B푀 “ 푅` Y 푅´.

Ă When picking the gluing diffeomorphism ℎ : 푅` Ñ 푅´, we require the following. (A-6) We have ℎpt1u ˆ t푝uq “ t´1u ˆ t푝u. Hence, we obtain a closure p푌, 푅q of p푀, 훾q and the arc r´1, 1s ˆ t푝u is glued to becomes a simple closed curve 휔 Ă 푌 with 푅 ¨ 휔 “ ˘1. Thus, p푌, 휔q is an admissible pair. We make the following definition.

Remark 2.3.10. In the instanton case, we call the triple p푌, 푅, 휔q a closure

Definition 2.3.11 (Kronheimer and Mrowka [53]). Suppose p푀, 훾q is a balanced sutured manifold, and p푌, 푅, 휔q is a closure obtained as above. Then, we define the sutured instanton Floer homology of p푀, 훾q as

푆퐻퐼p푀, 훾q “ 퐼휔p푌 |푅q.

There is also a twisted version of sutured instanton Floer homology, in analog to the local coefficients in sutured monopole Floer homology.

Definition 2.3.12 (Kronheimer and Mrowka [53]). Suppose p푀, 훾q is a balanced sutured manifold, and p푌, 푅, 휔q is a closure obtained as above. Let 휂 Ă 푅 be a non-separating simple closed curve. Then, we define the sutured instanton Floer homology of p푀, 훾q as 푆퐻퐼p푀, 훾; 휂q “ 퐼휔Y휂p푌 |푅q.

38 Question 3. For a balanced sutured manifold p푀, 훾q, how can 푆퐻퐼p푀, 훾q be related to the representations of the fundamental groups of 푀?

Remark 2.3.13. When 휂 is clear in the context, we simply write 푆퐻퐼p푀, 훾; 휂q as 푆퐻퐼p푀, 훾q.

Theorem 2.3.14 (Kronheimer and Mrowka [53]). For a given balanced sutured manifold and any closure p푌, 푅, 휔q, we have

푆퐻퐼p푀, 훾q – 푆퐻퐼p푀, 훾; 휂q.

Moreover, the isomorphic type of 푆퐻퐼p푀, 훾q is an invariant of the balanced suture manifold.

Notation. To simplify the discussions, for a balanced sutured manifold p푀, 훾q,

휔Y휂 we will use 퐻퐺p푌 |푅q to denote both 퐻푀p푌 |푅;Γ휂q and 퐼 p푌 |푅q and use 푆퐻퐺p푀, 훾q to denote both 푆퐻푀p푀, 훾;Γ휂q and 푆퐻퐼p푀, 훾; 휂q. Also, we will use ℛ to denote the coefficient ring. In the monopole settings, it is usually the Novikov ringormod 2 Novikov ring. In the instanton settings, it is the field of complex number.

2.4 Marked closures and naturality

In theorem 2.2.13 and theorem 2.3.14, only the isomorphism classes of the Floer homologies are well defined. However, for purposes such as defining a contact invariant, only having invariance on isomorphism classes is not enough. This leads to the work of Baldwin and Sivek [4] on the naturality of 푆퐻푀 and 푆퐻퐼. The definitions and constructions presented in this section are mainly based on that paper.

39 Definition 2.4.1 (Baldwin and Sivek [4]). Suppose p푀, 훾q is a balanced sutured manifold, then an odd marked closure of p푀, 훾q is a tuple 풟 “ p푌, 푅, 푟, 푚, 휂, 휔q, where (1) 푌 is a closed connected oriented 3-manifold. (2) 푅 is a closed connected oriented surface of genus at least 2. (3) We have an orientation preserving embedding

푟 : 푅 ˆ r´1, 1s Ñ 푌.

(4) We have an orientation preserving embedding

푚 : 푀 Ñ 푌 zintpimp푟qq that satisfies following properties. (a) We have that 푚 extends to a diffeomorphism

푚 : 푀 Y r´1, 1s ˆ 푇 Ñ 푌 zintpimp푟qq 푖푑ˆ푓 for some data 퐴p훾q, 푇, 푓. (b) We have that 푚 restricts to an orientation preserving embedding

푚 : 푅`p훾qz퐴p훾q ãÑ 푟pt´1u ˆ 푅q.

(5) We have that 휂 is an oriented non-separating simple closed curve on 푅. (6) We have that 휔 is a simple closed curve inside 푌 so that there is a point 푝 P 푅 with 휔 X imp푟q “ 푟pr´1, 1s ˆ t푝uq.

40 Define the genus of 풟, which is denoted by 푔p풟q, to be the genus of the surface 푅. Define the sutured monopole or instanton Floer homology of the oddmarked closure 풟 to be 푆퐻퐺p풟q “ 퐻퐺p푌 |푟pt0u ˆ 푅qq.

In [4], Baldwin and Sivek also constructed canonical isomorphisms between the Floer homologies of two different odd marked closures of a fixed balance sutured manifold.

Proposition 2.4.2 (Baldwin and Sivek [4]). Suppose p푀, 훾q is a balanced sutured manifold. Then, for any two odd marked closures 풟 and 풟1 of p푀, 훾q, there is a canonical map

1 Φ풟,풟1 : 푆퐻퐺p풟q Ñ 푆퐻퐺p풟 q, which is well defined up to multiplication by a unit of ℛ, such that the following is true. (1) If 풟 “ 풟1, then . Φ풟,풟1 “ 푖푑. . Here, “ means equal up to multiplication by a unit. (2) If there are 3 marked closures 풟, 풟1 and 풟2, then we have

. Φ풟1,풟2 ˝ Φ풟,풟1 “ Φ풟,풟2 .

Hence, the Floer homologies and the canonical maps fit into what is called a projective transitive system.

Definition 2.4.3 (Baldwin and Sivek [4]). A projective transitive system of ℛ- modules consists of an index set 풜 together with the following data.

41 (1) A collection of ℛ-modules t푀훼u훼P풜.

(2) A collection of equivalence classes of ℛ-modules homomorphisms trℎ훼,훽su훼,훽P풜, such that the following is true. (a) Two morphisms are called equivalent if they differ by multiplication by a unit.

(b) For all 훼, 훽 P 풜, ℎ훼,훽 is an isomorphism from 푀훼 to 푀훽. . (c) If 훼 “ 훽, then ℎ훼,훽 “ 푖푑. (d) For all 훼, 훽, 훾 P 풜, we have

. ℎ훽,훾 ˝ ℎ훼,훽 “ ℎ훼,훾.

With a projective transitive system, we can construct a canonical projective module out of it:

Definition 2.4.4 (Baldwin and Sivek [4]). Suppose p풜, t푀훼u, tℎ훼,훽uq is a projective transitive system, then we can define a canonical projective module or simply a canonical module:

푀 “ 푀훼{„, 훼P퐴 ž where, if we have 푚훼 P 푀훼 and 푚훽 P 푀훽, then 푚훼 „ 푚훽 if and only if

ℎ훼,훽p푚훼q “ 푢 ¨ 푚훽.

Here, 푢 P ℛˆ is a unit.

We can also define maps between two projective transitive systems.

Definition 2.4.5 (Baldwin and Sivek [4]). Suppose we have two projective transitive

1 1 1 systems p풜, t푀훼u, tℎ훼,훽uq and p풜 , t푀훾u, tℎ훾,훿uq.A morphism between them is a collection of equivalent classes of maps tr푓훼,훾su훼P풜,훾P풜1 , where two maps are called

42 equivalent if and only if they differ by multiplication by a unit, such that

. 1 푓훽,훿 ˝ ℎ훼,훽 “ ℎ훾,훿 ˝ 푓훼,훾 for all indices 훼, 훽 P 풜 and 훾, 훿 P 풜1 Such a morphism defines a map between the canonical projective modules

푓 : 푀 Ñ 푀 1 by choosing any 훼 P 풜, 훾 P 풜1 and defining

푓pr푚훼sq “ r푓훼,훾p푚훼qs.

We say that the map 푓 between canonical modules is induced by any map 푓훼훾.

There is a straightforward lemma on how to compare and identify two such morphisms.

1 Lemma 2.4.6. Suppose t푓훼,훾u and t푓훼,훾u are two morphisms between two projective 1 1 1 transitive systems p풜, t푀훼u, tℎ훼,훽uq and p풜 , t푀훾u, tℎ훾,훿uq, then the following three conditions are equivalent. (1) The induced maps are equal:

푓 “ 푓 1 : 푀 Ñ 푀 1.

(2) There exist 훼, 훽 P 풜 and 훾 P 풜1 so that

. 1 푓훽,훾 ˝ ℎ훼,훽 “ 푓훼,훾.

43 (3) There exist 훼 P 풜 and 훾, 훿 P 풜1 so that

. 1 1 푓훼,훿 “ ℎ훾훿 ˝ 푓훼,훾.

From the above discussion, we know that the marked closures t푆퐻퐺p풟qu and the canonical maps trΦ풟,풟1 su together form a projective transitive system, and, hence, we have a canonical projective module

SHGp푀, 훾q associated to it.

Remark 2.4.7. throughout the thesis, we will use 푆퐻퐺 to denote the sutured monopole or instanton Floer homology of a particular marked closure 풟. The notation SHG will be used to denote the canonical module coming from the projective transitive system over p푀, 훾q. This usage of notations might be slightly different from Baldwin and Sivek’s original paper.

Notation. All the discussions in the rest of the thesis are irrelevant with the choice of the curve 휔 so that we will omit it from all notations and discussions. We simply write an odd marked closure as 풟 “ p푌, 푅, 푟, 푚, 휂q and call it a marked closure. As one might have already observed, the data 휔 is not used in the monopole setups, and the phrase ’marked closure’ is exactly the one Baldwin and Sivek introduce their refinement of the closures in the monopole setups.

There is a new ambiguity when dealing with knots in 3-manifolds. Let 퐾 Ă 푌 be a knot. The new ambiguity comes from the choices of tubular neighborhoods of

44 퐾 Ă 푌 to remove to obtain a knot complement. Fix a point 푝 P 퐾. Suppose

휙 : 푆1 ˆ 퐷2 ãÑ 푌 is an embedding, where 퐷2 is the unit sphere in the complex plane, and 푆1 “B퐷2. We require that 휙p푆1 ˆ t0uq “ 퐾, and 휙pt1u ˆ t0uq “ 푝.

2 Let 푌휙 “ 푌 zintpimp휑qq, and let 훾휙 “ 휙pt˘1u ˆ B퐷 q, with opposite orientations on two components. For each fixed 휙, we have a well defined canonical module

SHMp푌 p휙q, 훾휙q, and we want to relate different choices of 휙. Suppose 휙1 is another embedding 푆1 ˆ 퐷2 ãÑ 푌 , satisfying the same conditions as 휙. Pick a tubular neighborhood 푁 of 퐾 Ă 푌 such that imp휙q, imp휙1q Ă 푁. Also, pick an ambient isotopy

푓푡 : 푌 Ñ 푌, 푡 P r0, 1s such that the following is true.

(1) For any 푡 P r0, 1s, 푓푡p푝q “ 푝.

(2) For any 푡 P r0, 1s, 푓푡 restricts to identity outside 푁 Ă 푌 .

1 (3) We have 푓1pimp휙qq “ imp휙 q.

2 1 2 (4) We have 푓1p휙pt˘1u ˆ B퐷 qq “ 휙 pt˘1u ˆ 퐷 q.

It is clear that 푓1 : p푌휙, 훾휙q Ñ p푌휙1 , 훾휙1 q is a diffeomorphism between balanced sutured manifolds. Hence, we can define

Ψ휙,휙1 “ SHMp푓1q : SHMp푌휙, 훾휙q Ñ SHMp푌휙1 , 훾휙1 q.

Theorem 2.4.8. (Baldwin and Sivek [4]) The map Ψ휙,휙1 is well defined, i.e., is

45 independent of the choices of the tubular neighborhood 푁 and the ambient isotopy 푓푡. Also, it has the following properties.

(1) We have Ψ휙,휙 “ 푖푑. (2) If there is a third embedding 휙2, then

Ψ휙,휙2 “ Ψ휙1,휙2 ˝ Ψ휙,휙1 .

Thus, we know that tSHGp푌휙, 훾휙qu and tΨ휙,휙1 u form a transitive system of projective transitive systems that can be viewed as a larger projective transitive system, and, hence, the monopole knot Floer homology KHGp푌, 퐾, 푝q is well defined. We have the following two questions.

Question 4. Can we resolve the ambiguity of multiplication by a unit in the naturality of sutured monopole and instanton Floer homologies?

Question 5. For a fixed balanced sutured manifold p푀, 훾q, is there a canonical closure p푌, 푅q associated to p푀, 훾q?

2.5 Floer’s Excisions

There are many choices made in the construction of sutured monopole and instanton Floer homologies. To show that these choices do give rise to a well-defined invariant, we need to use Floer’s excisions. Floer’s excision is a powerful tool that was originally produced by Floer in instanton theory in [22] and was introduced by Kronheimer and Mrowka into the context of sutured monopole and instanton Floer homologies in [53]. There are various versions of Floer’s excisions that we will explain in detail.

Suppose 푌1 and 푌2 are two closed connected oriented 3-manifolds. For 푖 “ 1, 2,

46 let Σ푖 Ă 푌푖 be a connected oriented homologically essential surface so that

푔pΣ1q “ 푔pΣ2q ě 1. (2.3)

1 For 푖 “ 1, 2, let 푌푖 “ 푌푖zintp푁pΣ푖qq, then we know that

1 B푌푖 “ Σ푖,` Y ´Σ푖,´,

where Σ푖,˘ are parallel copies of Σ푖. Pick an orientation preserving diffeomorphism

ℎ :Σ1 Ñ Σ2,

then we can use ℎ to glue Σ1,` to Σ2,´ and glue Σ1,´ to Σ2,`. The result is a closed 3-manifold 푌 so that

1 1 푌 “ 푌1 Y 푌2 . ℎYℎ

There is a surface Σ “ Σ1,` Y Σ1,´ Ă 푌 and a 4-dimensional cobordism 푊 from

푌1 \ 푌2 to 푌 as follows. Let 푈 be the surface as depicted in Figure 2-2. Four parts of the boundary of 푈, which we denote by 휈1, ..., 휈4, are each identified with the interval

1 1 r0, 1s. We can glue three parts r0, 1s ˆ 푌1 , 푈 ˆ Σ`, and r0, 1s ˆ 푌2 together to get 푊 :

1 1 푊 “ r0, 1s ˆ 푌1 Y 푈 ˆ Σ1,` Yr0, 1s ˆ 푌2 , 휑 휓 where

휑 “ 푖푑r0,1s ˆ p푖푑Σ1 Y 푖푑Σ1 q : r0, 1s ˆ pΣ1,` Y Σ1,´q Ñ p휈1 Y 휈4q ˆ Σ1,`,

47 and

휓 “ 푖푑r0,1s ˆ pℎ Y ℎq : p휈2 Y 휈3q ˆ Σ1,` Ñ r0, 1s ˆ pΣ2,` Y Σ2,´q.

There are various settings regarding the choices of surfaces and local coefficients. We summarize them as follows.

(FEM-1) We have 푔pΣ1q “ 푔pΣ2q ą 1. In this case Floer’s excision works with Z coefficients. However, to make a uniform statement in Theorem 2.5.1, we still write Γ휂 but for this case 휂1 “ 휂2 “ 휂 “H. Also, let 푅푖 “ Σ푖 Ă 푌푖 and 푅 “ Σ1,` Y Σ2,` Ă 푌 . Take 휈 “H.

(FEM-2) We have 푔pΣ1q “ 푔pΣ2q ą 1. Let 휂1 “H and 휂2 Ă Σ2 be a non- separating simple closed curve. Inside 푊 , there is naturally a cylinder 휈 “ r0, 1s ˆ 휂2 and we pick

휂 “ r0, 1s ˆ 휂2 X 푌 “ t1u ˆ 휂2.

Also, let 푅푖 “ Σ푖 Ă 푌푖 and 푅 “ Σ1,` Y Σ2,` Ă 푌 .

(FEM-3) We have 푔pΣ1q “ 푔pΣ2q “ 1. For 푖 “ 1, 2, Let 푅푖 Ă 푌푖 be a connected closed oriented surface which intersects Σ푖 transversely along a circle. Pick 휂푖 a non- separating simple closed curve on 푅푖, which intersects Σ푖 transversely at one point. In this case we also assume that

ℎp푅1 X Σ1q “ 푅2 X Σ2, and

ℎp휂1 X Σ1q “ 휂2 X Σ2.

When cutting and re-gluing Σ1 and Σ2, the surfaces 푅1 and 푅2 are cut and re- glued to form a surface 푅 Ă 푌 and the curves 휂1 and 휂2 are also cut and re-glued to become a simple closed curve 휂 Ă 푅. There is a pair of pants 휈 inside 푊 whose

48 boundary is 휂1 Y 휂2 Y 휂3. 푖푑 ℎ

푖푑 ℎ 휇2 휇3

휇1 휇4

1 1 r0, 1s ˆ 푌1 푈 ˆ Σ1,` r0, 1s ˆ 푌2

Figure 2-2: Gluing three parts together to get 푊 . The middle part is 푈 ˆΣ1,`, while the Σ1,` directions shrink to a point in the figure.

Theorem 2.5.1 (Kronheimer and Mrowka [53]). In all the three cases, (FE-1), (FE-2), and (FE-3), the cobordism p푊, 휈q induces an isomorphism

– 퐻푀p푊 ;Γ휈q : 퐻푀p푌1 \ 푌2|푅1 Y 푅2;Γ휂1Y휂2 q ÝÑ 퐻푀p푌 |푅;Γ휂q

In instanton theory, we need to modify the setups described above:

(FEI-1) We have 푔pΣ1q “ 푔pΣ2q ą 0. Take 휂1 “ 휂2 “ 휂 “H. Let 푅푖 “ Σ푖 Ă 푌푖 and 푅 “ Σ1,` Y Σ2,` Ă 푌 . For 푖 “ 1, 2, take 휔푖 Ă 푌푖 a simple closed curve that intersects Σ푖 transversely once. We require that

ℎp휔1 X Σ1q “ 휔2 X Σ2.

When cutting and re-gluing Σ1 and Σ2, the curves 휔1 and 휔2 are also cut and re- glued to become a simple closed curve 휔 Ă 푅. There is a pair of pants 휈 Ă 푊 whose boundary is 휔1 Y 휔2 Y 휔.

49 (FEI-2) We have 푔pΣ1q “ 푔pΣ2q ą 0. Let 휂1 “H and 휂2 Ă Σ2 be a non-separating simple closed curve. Inside 푊 , there is a natural cylinder 휈1r0, 1s ˆ 휂2 and we pick

휂 “ r0, 1s ˆ 휂2 X 푌 “ t1u ˆ 휂2.

Also, let 푅푖 “ Σ푖 Ă 푌푖 and 푅 “ Σ1,` Y Σ2,` Ă 푌 .

For 푖 “ 1, 2, take 휔푖 Ă 푌푖 a simple closed curve which intersects Σ푖 transversely once and which is disjoint from 휂푖. We require that

ℎp휔1 X Σ1q “ 휔2 X Σ2.

When cutting and re-gluing Σ1 and Σ2, the curves 휔1 and 휔2 are also cut and re-glued to become a simple closed curve 휔 Ă 푅. There is a pair of pants 휈2 Ă 푊 whose boundary is 휔1 Y 휔2 Y 휔. Take 휈 “ 휈1 Y 휈2.

(FEI-3) We have 푔pΣ1q “ 푔pΣ2q “ 1. For 푖 “ 1, 2, Let 푅푖 Ă 푌푖 be a connected closed oriented surface that intersects Σ푖 transversely along a circle. Pick 휂푖 a non- separating simple closed curve on 푅푖, which intersects Σ푖 transversely at one point.

Pick 휔푖 Ă 푌푖 a simple closed curve which intersects 푅푖 transversely at one point and which is disjoint from 휂푖. In this case we also assume that

ℎp푅1 X Σ1q “ 푅2 X Σ2, and

ℎp휂1 X Σ1q “ 휂2 X Σ2.

When cutting and re-gluing Σ1 and Σ2, the surfaces 푅1 and 푅2 are cut and re- glued to form a surface 푅 Ă 푌 and the curves 휂1 and 휂2 are also cut and re-glued

50 to become a simple closed curve 휂 Ă 푅. There is a pair of pants 휈1 inside 푊 whose boundary is 휂1 Y 휂2 Y 휂3. There is a disjoint union of two annuli

휈2 “ r0, 1s ˆ p휔1 Y 휔2q Ă 푊.

Take 휔 “ 휈2 X 푌 and 휈 “ 휈1 Y 휈2.

Theorem 2.5.2 (Kronheimer and Mrowka [53]). Under all the three cases, (FEI-1), (FEI-2), and (FEI-3), the cobordism p푊, 휈q induces an isomorphism

휔1Y휂1 휔2Y휂2 – 휔Y휂 퐼p푊, 휈q : 퐼 p푌1|Σ1q b 퐼 p푌2|Σ2q ÝÑ 퐼 p푌 |Σq.

2.6 Basic properties of SHM and SHI

In this section, we summarize the basic properties of the sutured monopole and instanton Floer homologies. First, in the monopole settings, it is most useful to use local coefficients and the mod 2 Novikov ring ℛ. In this setup, we can only look at the rank of the homology since it is always free by the following lemma.

Lemma 2.6.1. For any balanced sutured manifold p푀, 훾q, SHMp푀, 훾q is a free ℛ module.

Proof. As in Kronheimer and Mrowka [53], we know that for any closure p푌, 푅q of p푀, 훾q and any non-separating simple closed curve 휂, we have

퐻푀p푌 |푅;Γ휂q “ 퐻푀p푌 |푅; Z2q bZ2 ℛ.

This is a simple application of the Floer’s excision in Theorem 2.5.1 and the following lemma.

51 Lemma 2.6.2 (Kronheimer and Mrowka). Suppose 푅 is a connected closed oriented surface of genus at least 2 in the monopole setup and at least 1 in the instanton setups. Suppose 휂 is a collection of non-separating simple closed curves on 푅 (possibly empty). Let ℎ : 푅 Ñ 푅 be an orientation preserving diffeomorphism. Let 푌 be the mapping torus of 푅, then 푅 naturally embeds into 푌 . We have

퐻푀p푌 |푅;Γ휂q – ℛ.

In the instanton settings, we only need to assume that 푔p푅q ą 0 but further require that ℎ has a fixed point 푝. Let 휔 “ 푝 ˆ 푆1 Ă 푌 . Then, we have

휔 휂 퐼 Y p푌 |푅q – C.

Corollary 2.6.3 (Kronheimer and Mrowka [53]). If p푀, 훾q is a product sutured manifold as in Definition 2.1.4, then SHGp푀, 훾q has rank one.

The inverse of Lemma 2.6.2 is almost true, as we have the following theorems.

Theorem 2.6.4. Suppose 푌 is a closed connected oriented 3-manifold and 푅 Ă 푌 is a closed connected oriented surface of genus at least 2. If 푅 represents a non-trivial homology class and 퐻푀p푌 |푅q has rank one, then 푌 is a fibration over 푆1 with fiber 푅.

Remark 2.6.5. The same statement in Heegaard Floer theory was proved by Ni in [68]. The same conclusion in monopole theory follows from the isomorphism 퐻푀 – 퐻퐹 ` proved by Kutluhan, Lee, and Taubes in [55] and subsequent papers. ~ Theorem 2.6.6 (Baldwin and Sivek [9]). Suppose p푌, 휔q is an admissible pair and 푅 Ă 푌 is a closed connected oriented surface with 푅 ¨ 휔 odd. Suppose further that

52 푌 z푁p푅q is a homology product and 퐼휔p푌 |푅q has rank one, then 푌 is a fibration over 푆1 with fibre 푅.

Question 6. Can we drop the condition that 푌 z푁p푅q is a homology product in Theorem 2.6.6?

We have introduced the operation of sutured manifold decompositions in Defi- nition 2.1.15. In [53], Kronheimer and Mrowka related the sutured monopole and instanton Floer homologies of balanced sutured manifolds before and after a sutured manifold decomposition along with a special type of surfaces.

Definition 2.6.7 (Juhasz [38]). Suppose 퐹 is a compact oriented surface. A simple closed curve 훾 Ă 퐹 is called boundary coherent if it is either homologically essential in

1 퐻1p퐹 q or the oriented boundary of a compact subsurface 퐹 Ă 퐹 with the orientation of 퐹 1 induced by the one on 퐹 .

Theorem 2.6.8 (Kronheimer and Mrowka [53]). Suppose p푀, 훾q is a balanced sutured manifold and 푆 Ă 푀 is a properly embedded surface so that it has no closed components, and that for every component 푉 of 푅p훾q, the set of closed components of 푆 X 푉 consists of parallel oriented boundary-coherent simple closed curves. Suppose p푀 1, 훾1q is obtained from p푀, 훾q by a sutured manifold decomposition along 푆. Then, 푆퐻퐺p푀 1, 훾1q is a direct summand of 푆퐻퐺p푀, 훾q.

There is also a slightly refined version of Gabai’s theorem on sutured manifold hierarchies.

Theorem 2.6.9 (Juhasz [38]). If p푀, 훾q is a taut balanced sutured manifold, then there exists a sutured manifold hierarchy as in Definition 2.1.18 so that each decomposition surface 푆푖, for 푖 “ 1, ..., 푛, satisfies the hypothesis in Theorem 2.6.8.

53 Corollary 2.6.10 (Kronheimer and Mrowka [53]). Suppose p푀, 훾q is a taut balanced sutured manifold then SHGp푀, 훾q ‰ 0.

The inverse of this corollary is almost true.

Corollary 2.6.11. If p푀, 훾q is a balanced sutured manifold and 푀 is irreducible, then p푀, 훾q is taut if and only if SHGp푀, 훾q ‰ 0.

Proof. The if part is proved by corollary 2.6.10. For the only if part, if p푀, 훾q is irreducible but non-taut, then by Definition 2.1.9, 푅˘p훾q must be either not norm- minimizing or compressible. Then, the corollary follows from Definition 2.2.10, Def- inition 2.3.12, Proposition 2.3.4, and the following lemma known as the adjunction inequality.

Lemma 2.6.12 (Adjunction inequality, Kronheimer and Mrowka [53]). Suppose 푌 is a closed oriented 3-manifold and s is a spin푐 structure on 푌 . Suppose there is a closed connected oriented surface 푅 of genus at least one, so that

|푐1psqr푅s| ą 2푔p푅q ´ 2, then

퐻푀 ‚p푌, s;Γ휂q “ 0 for any possible local coefficients~휂 ( can be empty).

One might be wondering what happens if 푀 is not irreducible. In this case, we can write

푀 “ 푀17푀2 for some compact oriented 3-manifolds 푀1 and 푀2 which are both not 3-spheres.

54 Since the connected sum is done in the interior, we know that

B푀 “B푀1 YB푀2.

The suture splits into two parts 훾 “ 훾1 Y 훾2, 훾푖 ĂB푀푖. Then, there are two different cases.

Case 1. One of 푀1 and 푀2 is closed. Suppose, without loss of generality, 푀1 is closed. Let p푀1p1q, 훿q be the balanced sutured manifold obtained form 푀1 by removing a 3-ball in the interior and put one connected simple closed curve on the boundary as the suture. Then, we have

Proposition 2.6.13 (Li [56]). Under the above settings, we have

SHGp푀, 훾q – SHGp푀1p1q, 훿q b SHGp푀2, 훾2q.

In the monopole setups, the Floer homology 푆퐻푀p푀1p1q, 훿q is isomorphic to the tilde-version of monopole Floer homology of 푀1, denoted by 퐻푀p푀1q, which was introduced by Kutluhan, Lee, and Taubes in [55]. In the instanton setups, the Floer Ć homology 푆퐻퐼p푀1p1q, 훿q is isomorphic to the framed instanton Floer homology of

7 푀1, denoted by 퐼 p푀1q, which was introduced by Kronheimer and Mrowka in [48].

Case 2. Both 푀1 and 푀2 has non-trivial boundaries and from Definition 2.1.1, we know that both 훾1 and 훾2 are non-empty. We have the following basic lemma.

Lemma 2.6.14. If SHGp푀, 훾q ‰ 0, then both p푀1, 훾1q and p푀2, 훾2q are balanced sutured manifolds.

Proof. It is straightforward to check that condition (1) and (2) in Definition 2.1.1 are automatically satisfied. If condition (3) does not hold, it is straightforward to

55 apply the adjunction inequality in Lemma 2.6.12 or Proposition 2.3.4 to obtain a contradiction.

Proposition 2.6.15 (Li [56]). When the conclusion of Lemma 2.6.14 holds, we have

3 SHGp푀, 훾q “ SHGp푀1, 훾1q b SHGp푀2, 훾2q b SHGp푆 p2q, 훿q, where p푆3p2q, 훿q is the balanced sutured manifold obtained from 푆3 by removing two disjoint 3-balls and put one connected simple closed curve on each component of the boundary as the suture. Its homology SHGp푆3p2q, 훿q has rank two.

Corollary 2.6.16. Suppose p푀, 훾q is a balanced sutured manifold. Then, SHGp푀, 훾q ‰ 0 if and only if every irreducible connected summand of p푀, 훾q is either a closed 3- manifold having non-trivial Floer homology 퐻푀 or 퐼7, or a taut balanced sutured manifold. Ć It is a basic fact that the sutured Floer homology (SFH) defined by Juhasz [38] decomposes along spin푐 structures. So one could ask whether the same property holds for 푆퐻퐺. However, the case of sutured monopole and instanton Floer homologies is more subtle, since different closures may have different sets of (supporting)푐 spin structures or eigenvalue functions. Nevertheless, if we choose a properly embedded surface 푆 Ă 푀, we can decompose SHG with respect to 푆 by looking at the evaluations of the first Chern classes of푐 spin structures, or the eigenvalue functions, on the fundamental class of a proper extension of the surface 푆 and obtain a well-defined splitting of SHG. This leads to the following discussion.

Definition 2.6.17. Suppose p푀, 훾q is a balanced sutured manifold and 푆 Ă 푀 is a properly embedded oriented surface. 푆 is called admissible if the following holds. (1) Any component of B푆 intersects 훾 transversely and non-trivially.

56 (2) Suppose B푆 intersects 훾 at altogether 2푛 points, then we require that 푛´휒p푆q is even.

Theorem 2.6.18 (Li [58]). Suppose p푀, 훾q is a balanced sutured manifold and 푆 Ă 푀 is an admissible surface. Then, 푆 induces a Z-grading on SHGp푀, 훾q:

SHGp푀, 훾q “ SHGp푀, 훾, 푆, 푖q. 푖PZ à Remark 2.6.19. Strictly speaking, the discussion in Li [58] does not cover the case of all admissible surfaces. However, it has already contained all essential ingredients in constructing a well-defined grading for an admissible surface, except for one problem that is treated by Kavi in [43]. So, as a combination of Kavi’s and the author’s work, the construction of the grading is complete. This issue is also stressed in Ghosh and Li [27]

Using Theorem 2.6.18, we can reformulate Theorem 2.6.8 as follows.

Theorem 2.6.20. Suppose p푀, 훾q is a balanced sutured manifold and 푆 Ă 푀 is an admissible surface, with |푆 X 훾| “ 2푛. Suppose p푀 1, 훾1q is obtained from p푀, 훾q be performing a sutured manifold decomposition along 푆. Then,

1 1 SHGp푀, 훾, 푆, 푔푐p푆qq – SHGp푀 , 훾 q, where 푛 ´ 휒p푆q 푔 p푆q “ . 푐 2

Furthermore, SHGp푀, 훾, 푆, 푖q “ 0 for all |푖| ą 푔푐p푆q.

Remark 2.6.21. Since 훾 can be viewed as the boundary of 푅`p훾q, we know that r훾s “ 0 P 퐻1pB푀q. Hence, the number of intersection points of 푆 X 훾 is always

57 even. However, it is a non-trivial requirement that 푛 ´ 휒p푆q is even. This will be explained in detail in Section 4.1 when we present the construction of the grading in more detail. Also, we can deform 푆 with the intersection 푆 X 훾 fixed. Under such deformations, the grading is unchanged. However, if the deformation changes 푆 X 훾, then the gradings might not be the same. With further discussions, the difference is shown to be only a possible overall grading shifting, as suggested by the following proposition.

Theorem 2.6.22 (Ghosh and Li [27]). Suppose p푀, 훾q is a balanced sutured manifold and 푆1 and 푆2 are two admissible surfaces so that

r푆1, B푆1s “ r푆2, B푆2s P 퐻2p푀, B푀q.

Then, there is an integer 푗퐺 so that for any 푖 P Z, we have

SHGp푀, 훾, 푆1, 푖q “ SHGp푀, 훾, 푆2, 푖 ` 푗퐺q.

Remark 2.6.23. We write the constant 푗퐺 because, a priori, it depends on whether we work in monopole theory or instanton theory. Though it is expected that 푗퐼 is always the same as 푗푀 . Under some conditions, we could fix the number of 푗퐼 for special choices of pairs of surfaces 푆1 and 푆2. For example, if the sutured manifold decompositions of p푀, 훾q along 푆1 and 푆2 are both taut, then we can fix the value of 푗퐺 by looking at the top non-vanishing gradings with the help of Corollary 2.6.10 and Theorem 2.6.20.

Question 7. Can we describe 푗퐺 explicitly?

Theorem 2.6.22 implies that a second relative homology class 훼 P 퐻2p푀, B푀q induces a grading on SHGp푀, 훾q that is well-defined up to an overall grading shifting.

58 To do this, we can choose any admissible surface 푆 that represents the class 훼 and look at the grading induced by 푆. This makes it possible to characterize a special class of elements inside SHG.

Definition 2.6.24 (Ghosh and Li [27]). Suppose p푀, 훾q is a balanced sutured manifold. An element 푎 P SHGp푀, 훾q is called homogenous if it is non-zero and, for any class 훼 P 퐻2p푀, B푀q, the element 푎 is homogenous with respect to the grading associated to the class 훼.

With the help of homogenous elements, we could obtain a relatively canonical decomposition of SHG.

Lemma 2.6.25 (Ghosh and Li [27]). For a taut, balanced sutured manifold, there exists a homogenous element.

Proposition 2.6.26 (Ghosh and Li [27]). Suppose p푀, 훾q is a balanced sutured manifold and 푎 P SHGp푀, 훾q is a homogenous element. Then, for any 휌 P 퐻2p푀, B푀; Qq, there is a sub-module SHG푎p푀, 훾, 휌q, so that we have a direct sum decomposition

SHGp푀, 훾q “ SHG푎p푀, 훾, 휌q. 휌P퐻2p푀,B푀;Qq à The canonical decomposition helps us to define a polytope in sutured monopole or instanton theory, similar to the one introduced by Juhász in [40].

Definition 2.6.27 (Ghosh and Li [27]). Suppose p푀, 훾q is a balanced sutured manifold and 푎 P SHGp푀, 훾q be a homogenous element. Define the polytope 푃 퐺푎p푀, 훾q to be the convex hull of the classes 휌 P 퐻2p푀, B푀; Qq so that

SHG푎p푀, 훾, 휌q ‰ 0.

59 Theorem 2.6.28 (Ghosh and Li [27]). Suppose p푀, 훾q is a taut balanced sutured manifold with 퐻2p푀q “ 0. Suppose further that p푀, 훾q is reduced, horizontally prime, and free of essential product disks. Then, the polytope 푃 퐺푎p푀, 훾q has maximal possible dimension 푏1p푀q. In particular, we conclude that

rkℛpSHGp푀, 훾qq ě 푏1p푀q ` 1.

The proof of Theorem 2.6.28 can also be used to prove the following theorem.

Theorem 2.6.29 (Ghosh and Li [27]). Suppose p푀, 훾q is a taut balanced sutured manifold with 퐻2p푀q “ 0, and

rkpSHGp푀, 훾qq ă 2푘`1.

Then, 푑p푀, 훾q ď 2푘.

Corollary 2.6.30 (Ghosh and Li [27]). Suppose p푀, 훾q is a taut balanced sutured manifold with 퐻2p푀q “ 0, and

SHGp푀, 훾q – ℛ, then p푀, 훾q is a product sutured manifold.

Remark 2.6.31. Corollary 2.6.30 slightly generalizes Theorem 6.1 and Theorem 7.18 in Kronheimer and Mrowka [53]. This type of product detection results were first studied by Ni in [67].

Question 8. Can we drop the condition that 퐻2p푀q “ 0 in the hypothesis of Theo- rem 2.6.29?

60 Question 9. Can sutured monopole or instanton Floer homology be used to bound the minimal depth of all possible taut foliations on a balanced sutured manifold? See also Question 9.14 in Juhász [39] and Conjecture 7.24 in Kronheimer and Mrowka [53].

The canonical decomposition in Proposition 2.6.26also helps us to study the Thurston norm on 푀.

Definition 2.6.32 (Ghosh and Li [27]). Suppose p푀, 훾q is a balanced sutured manifold. For a class 훼 P 퐻2p푀, B푀q, define

푦p훼q “ max t휌p훼qu ´ min t휌p훼qu 2 2 휌P퐻 p푀,B푀;Qq 휌P퐻 p푀,B푀;Qq SHM푎p푀,훾,휌q‰0 SHM푎p푀,훾,휌q‰0

Theorem 2.6.33 (Ghosh and Li [27]). Suppose p푀, 훾q is a taut balanced sutured manifold so that B푀 is a disjoint union of tori:

B푀 “ 푇1 Y ... Y 푇푛.

For 푖 “ 1, ..., 푛, let 2푚푖 be the number of component of 훾 X푇푖 and write 훾푖 an oriented component of 훾 X 푇푖. Then, we have

푟

푥p훼q ` 푚푖 ¨ |x훼, 훾푖y| “ 푦p훼q. 푖“1 ÿ Here, 푥p¨q is the Thurston-norm defined in Definition 2.1.7. x, y is to take the algebraic intersection number of a class 훼 P 퐻2p푀, B푀q with a class r훾푖s P 퐻1p푀q.

Remark 2.6.34. When 푀 is a link complement and 훾 consists of a pair of meridians on each boundary component of 푀, then Theorem 2.6.33 is essentially proved in

61 Ozsváth and Szabó [77].

Question 10. What can we say about a general balanced sutured manifold?

2.7 Contact structures and contact elements

Definition 2.7.1. A contact sutured manifold p푀, 훾, 휉q is a triple where p푀, 훾q is a balanced sutured manifold and 휉 is a contact structure on 휉 so that B푀 is convex and 훾 is (isotopic to) the dividing set. Such a contact structure is called a compatible contact structure.

We want to define a contact invariant associated to a contact structure on abal- anced sutured manifold, as did in sutured Floer homology (SFH) by Honda, Kazez, and Matić [36]. The constructions for SHM and SHI were done by Baldwin and Sivek in [5, 6]. The constructions for the monopole and the instanton settings are different, as is because we have had a contact invariant defined in the monopole settings fora contact structure on a closed 3-manifold, by Kronheimer and Mrowka [45], but not for the instanton settings. We first present the construction of a contact structure in sutured monopole Floer homology theory, based on Baldwin and Sivek [5].

Definition 2.7.2. Suppose 푇 is a connected compact oriented surface with boundary. An arc configuration 풜 on 푇 consists of the following data.

(1) A finite collection of pairwise disjoint simple closed curves t푐1, ...푐푚u so that for any 푗, r푐푗s ‰ 0 P 퐻1p푇 q.

(2) A finite collection of pairwise disjoint simple arcs t푎1, ..., 푎푛u so that

(a) For any 푖, 푗, intp푎푖q X 푐푗 “H.

(b) For each 푖, one end point of 푎푖 lies on B푇 and the other on some 푐푗.

62 See Figure 2-3. It is called reduced if there is only one simple closed curve.

푐1

푐2 푎1 푎3

푎2

Figure 2-3: Above: an arc configuration on 푇 . Below: the shaded region corresponds to the negative region on t푡u ˆ 푇 Ă r´1, 1s ˆ 푇 with respect to the contact structure induced by the arc configuration. Its boundary is the dividing set on t푡u ˆ 푇 .

Now let p푀, 훾, 휉q be a balanced sutured manifold with a compatible contact structure. Suppose 푇 is a connected auxiliary surface of p푀, 훾q and 풜 is a reduced arc configuration on 푇 . Baldwin and Sivek constructed a suitable contact structure 휉 on 푀 “ 푀 Y r´1, 1s ˆ 푇 r as follows. First the arc configurationĂ 풜 gave rise to an r´1, 1s-invariant contact structure on r´1, 1s ˆ 푇 . The negative region on any piece t푡u ˆ 푇 is shown as in

63 Figure 2-3. Then, they perturbed the contact structure on 푀 in a neighborhood of 훾 Ă 푀 so that the dividing set in 퐴p훾q can be identified with that on r´1, 1s ˆ B푇 . So they were able to choose a diffeomorphism 푓 : r´1, 1s ˆ B푇 Ñ 퐴p훾q which also identifies the contact structures. After rounding the corners, they derived 휉 on 푀. Suppose r Ă B푀 “ 푅` Y 푅´, then 푅˘ are convex and the dividingĂ set on each of 푅˘ consists of two parallel non- separating simple closed curves. Finally they chose a diffeomorphism ℎ : 푅` Ñ 푅´ preserving the contact structures to get a closure p푌, 푅q with a contact structure 휉¯, so that 푅 is convex and the negative region on 푅 is just an annulus. They also chose a simple closed curve 휂 Ă 푅 intersecting each dividing set transversely once to support the local coefficients. From the construction

¯ 푐1p휉qr푅s “ 2 ´ 2푔p푅q, and by work of Kronheimer, Mrowka, Ozsváth, and Szabó [44], there is a contact element

휑휉¯ P 퐻푀p´푌, s휉¯;Γ´휂q Ă 푆퐻푀p´푀, ´훾q.

~ Remark 2.7.3. In [5], Baldwin and Sivek only used reduced arc configurations to construct contact elements. However, the same construction on 푀 can be made with a general arc configuration as defined in Definition 2.7.2. The new dividing Ă set on 푅˘ consists of 푚 many pairs of parallel non-separating simple closed curves, where 푚 is the number of simple closed curves in that arc configuration. However, in this case, the diffeomorphism ℎ preserving contact structures may not always exist (as it must identify the dividing sets). The reason why we want to make this more

64 general definition is that we will see in the Section 3.1 that a general arc configuration does exist when performing Floer’s excisions, and the diffeomorphism ℎ can indeed be chosen so that we can construct a contact structure on the closure 푌 .

To make the contact invariant well defined, we need to deal with the projective transitive system over p푀, 훾q. Baldwin and Sivek [5] proved that the contact element is indeed well defined, as well as some basic properties forit.

Theorem 2.7.4 (Baldwin and Sivek [5]). Suppose p푀, 훾, 휉q is a contact sutured manifold. Then, the contact element is a well defined element in the corresponding canonical module: 휑p휉q P SHMp´푀, ´훾q.

Furthermore, it satisfies the following basic properties.

(1) If 휉 is overtwisted, then 휑p휉q “ 0.

(2) If p푀, 훾q “ pr´1, 1sˆ퐹, t0uˆB퐹 q is the product sutured manifold and 휉 is the r´1, 1s invariant contact structure so that t푡u ˆ 퐹 is convex with boundary parallel dividing set, then 휑p휉q P SHMp´푀, ´훾q – ℛ is a generator.

(3) If in general p푀, 휉q embeds into a Stein fillable closed contact 3-manifold, then 휑p휉q is a primitive class in SHMp´푀, ´훾q.

(4) If p푀 1, 훾1, 휉1q is obtained from p푀, 훾, 휉q by performing a contact `1 surgery along a Legendrian curve 훼 Ă intp푀q, then there is a map

1 1 퐹훼 : SHMp´푀, ´훾q Ñ SHMp´푀 , ´훾 q

65 which is induced by the corresponding 2-handle cobordism and

. 1 퐹훼p휑p휉qq “ 휑p휉 q.

In the above construction of the contact element, we always use a connected auxiliary surface to construct closures and extend the contact structures. One might be interested in what if we use a disconnected auxiliary surface to construct closures. The case of using disconnected auxiliary surfaces might happen, especially when we want to look at the disjoint union of balanced sutured manifolds. In [53], Kronheimer and Mrowka used Floer’s excisions to relate the closures arising from connected and disconnected closures. So it is valuable to study how the contact element defined above behaves under performing Floer’s excisions. This will be the main topic of Section 3.1 Given a sutured manifold, we can attach a contact handle to it. The following definition is from Juhász and Zemke [42].

Definition 2.7.5. Suppose p푀, 훾q is a balanced sutured manifold. A 3-dimensional contact handle attachment of index 푘, where 푘 P t0, 1, 2, 3u, is a quadruple ℎ “ p휑, 푆, 퐷3, 훿q. Here, 퐷3 is a standard tight contact 3-ball with 훿 being the dividing set on B퐷. Also, 푆 ĂB퐷3 is a 2-submanifold of B퐷3 and

휑 : 푆 ÑB푀 is the embedding that specifies the gluing. The surface 푆 has different descriptions due to different indices 푘:

∙ When 푘 “ 0, 푆 “H.

66 ∙ When 푘 “ 1, 푆 is the disjoint union of two disks. Each disk intersects the dividing set 훿 in a simple arc.

∙ When 푘 “ 2, 푆 is an annulus on B퐷3 and it intersects the dividing set 훿 in two

simple arcs, with each representing a non-trivial class in 퐻1p푆, B푆q.

∙ When 푘 “ 3, 푆 “B퐷3.

Furthermore, if we set 훿1 “ 훿 X 푆 and 훿2 “ 훿z훿1, then in all cases we require that

1 휑p훿1q Ă 훾 ĂB푀. We can thus obtain a new dividing set 훾 on the new manifold

1 푀 “ 푀 Y휑 퐵: 1 훾 “ p훾z휑p훿1qq Y p훿2q.

See Figure 2-4.

S S S S 훿 푆

훿

푆 “H

훿 푆 훿

푆 “B퐷3 Figure 2-4: The contact handle attachments. Top left: 0-handle. Top right: 1- handle. Bottom right: 2-handle. Bottom left: 3-handle.

We now quickly recall the construction of contact handle attaching maps 퐶ℎ,

67 associated to a contact handle ℎ, introduced by Baldwin and Sivek in [5]. We summarize them in the following three propositions. Note the construction of contact handle attaching maps is the same in both monopole and instanton settings, so we use the unified notation 퐻퐺 and 푆퐻퐺 to present the construction.

Proposition 2.7.6 (Baldwin and Sivek [5]). Suppose p푀, 훾q is a balanced sutured manifold and p푀 1, 훾1q is obtained from p푀, 훾q by attaching a contact 0- or 1-handle ℎ. Then, any pre-closure (see Definition 2.2.2) 푀 of p푀, 훾q with 푔pB푀q large enough is a pre-closure of p푀 1, 훾1q and vise versa. As a result, any closure p푌, 푅q of p푀, 훾q Ă Ă with 푔p푅q large enough is a closure of p푀 1, 훾1q and vise versa. Fix any common closure p푌, 푅q for both p푀, 훾q and p푀 1, 훾1q, the contact handle attaching map

1 1 퐶ℎ : SHGp´푀, ´훾q Ñ SHGp´푀 , ´훾 q is induced by the identity map

푖푑 : 퐻퐺p´푌 | ´ 푅q Ñ 퐻퐺p´푌 | ´ 푅q.

Furthermore, suppose 푇 and 푇 1 are the auxiliary surfaces so that

푀 “ 푀 Y r´1, 1s ˆ 푇 “ 푀 1 Y r´1, 1s ˆ 푇 1,

Ă Then, 푇 1 naturally embeds into 푇 .

Remark 2.7.7. Strictly speaking, when constructing 퐶ℎ, we need to construct a map between canonical modules (see Definition 2.4.4). In Proposition 2.7.6, we only define 퐶ℎ by using the map 푖푑 between the 푆퐻퐺 of two particular marked closures p´푌, ´푅, 푟, 푚, 휂q and p´푌, ´푅, 푟1, 푚1, 휂q, of p푀, 훾q and p푀 1, 훾1q, respectively. Here,

68 p푌, 푅q is the closure as described in Proposition 2.7.6, 푟, 푚, 푟1, 푚1 are all inclusions and 휂 Ă 푅 is any non-separating simple closed curves. It is Baldwin and Sivek’s work proving that the map 푖푑 between the 푆퐻퐺 of these two particular marked closures induces a well define map 퐶ℎ between two canonical modules SHGp´푀, ´훾q and SHGp´푀 1, ´훾1q. In the statement of the proposition, we also omit the notations of marked closures for simplicity since the maps 푟, 푚, 푟1, 푚1 are canonical, and the choice of 휂 is arbitrary. The same issues will happen in the statements of the following two propositions.

Proposition 2.7.8 (Baldwin and Sivek [5]). Suppose p푀, 훾q is a balanced sutured manifold and p푀 1, 훾1q is obtained from p푀, 훾q by attaching a contact 2-handle ℎ. Suppose 훼 ĂB푀 is the attaching curve for the 2-handle. Push 훼 slightly into the interior of 푀 to a curve that we still call 훼, and equip it with a framing induced by B푀. Pick any closure p푌, 푅q of p푀, 훾q with 푔p푅q large enough, perform a 0-surgery along the curve 훼 Ă intp푀q Ă 푌 and let 푌 1 be the resulting 3-manifold. Then, p푌 1, 푅q is a closure of p푀 1, 훾1q. Furthermore, there is a natural 2-handle cobordism 푊 from 푌 to 푌 1 associated to the surgery along 훼. This cobordism 푊 induces a map

퐻퐺p´푊 q : 퐻퐺p´푌 | ´ 푅q Ñ 퐻퐺p´푌 1| ´ 푅q.

Then, the contact 2-handle attaching map

1 1 퐶ℎ : SHGp´푀, ´훾q Ñ SHGp´푀 , ´훾 q is induced by 퐻퐺p´푊 q. Moreover, suppose p푇, 푓, ℎq is the auxiliary data that gives rise to 푌 and p푇 1, 푓 1, ℎ1q is the auxiliary data that gives rise to 푌 1, then there is a natural way to embed 푇

69 into 푇 1.

Remark 2.7.9. Here, we explain the map 퐻퐺p´푊 q a little bit more. Recall we also need to pick a non-separating curve 휂 Ă 푅 to form marked closures. This curve is disjoint from the surgery curve 훼. In the monopole settings, take 휈 “ r0, 1s ˆ 휂, and take 퐻퐺p´푊 q “ 퐻푀p´푊, ´휈q. In the instanton settings, there is another curve 휔 Ă 푌 which is also disjoint from 훼. Take 휈 “ r0, 1s ˆ p휔 Y 휂q and take ~ 퐻퐺p´푊 q “ 퐼p´푊, ´휈q. For the rest of the thesis, we will always use 퐻퐺p푊 q to denote both 퐻푀p푊, 휈q and 퐼p푊, 휈q, for appropriate choices of 휈.

Proposition~ 2.7.10 (Baldwin and Sivek [5]). Suppose p푀, 훾q is a balanced sutured manifold and p푀 1, 훾1q is obtained from p푀, 훾q by attaching a contact 3-handle ℎ. Let 푆2 be a component of B푀 along which ℎ is attached. Push it into the interior of 푀 to a surface that we still denote by 푆2. Pick any closure p푌, 푅q of p푀, 훾q with 푔p푅q large enough, then we can un-do a connected sum operation along 푆2 Ă int푀 Ă 푌 : Cut 푌 open along 푆2 and glue back two 3-balls. Let 푌 1 be the resulting 3-manifold. Then, p푌 1, 푅q is a closure of p푀 1, 훾1q. There is naturally a 3-handle cobordism 푊 from 푌 to 푌 1 associated to the inverse connected sum operation along 푆2. This cobordism 푊 induces a map 퐻퐺p´푊 q : 퐻퐺p´푌 | ´ 푅q Ñ 퐻퐺p´푌 1| ´ 푅q.

Then, the contact 3-handle attaching map

70 into 푇 1.

A basic property for the contact handle gluing map in the monopole settings is the following.

Theorem 2.7.11 (Baldwin and Sivek [5]). Suppose p푀, 훾, 휉q is a contact sutured manifold and p푀 1, 훾1, 휉1q is obtained from p푀, 훾, 휉q by attaching a contact handle. Then, in the monopole settings, we have

. 1 퐶ℎp휑p휉qq “ 휑p휉 q.

Inspired by theorem 2.7.11, we can make the following definition.

Definition 2.7.12 (Baldwin and Sivek [6]). Suppose p푀, 훾, 휉q is a contact sutured manifold. The contact invariant is defined as follows. Suppose 풞 “ tℎ0, ..., ℎ푛u is a contact handle decomposition of p푀, 훾, 휉q without contact 3-handles. Suppose ℎ0 is a zero handle. Then, it is topologically a standard contact 3-ball. The underlining sutured manifold is just p푆3p1q, 훿q defined as in Proposition 2.6.13. We know that

3 SHIp´푆 p1q, ´훿q – C, so let the contact invariant of 휉 be

휑p휉q “ 퐶ℎ푛 ˝ ... ˝ 퐶ℎ1 p1q P SHIp´푀, ´훾q.

Question 11. Is there an analytical definition of the contact element in instanton theory?

Theorem 2.7.13 (Baldwin and Sivek [5]). Suppose p푀, 훾, 휉q is a contact sutured

71 manifold. Then, the contact invariant 휑p휉q P SHIp´푀, ´훾q is well defined. Further- more, it has the following properties. (1) If 휉 is overtwisted, then 휑p휉q “ 0. (2) If p푀, 휉q embeds into a Stein fillable closed contact 3-manifold, then 휑p휉q is a primitive class in SHMp´푀, ´훾q. (3) If p푀 1, 훾1, 휉1q is obtained from p푀, 훾, 휉q by attaching a contact 0, 1 or 2-handle, then

1 퐶ℎp휑p휉qq “ 휑p휉 q.

(4) If p푀 1, 훾1, 휉1q is obtained from p푀, 훾, 휉q by performing a contact `1 surgery along a Legendrian curve 훼 Ă intp푀q, then there is a map

1 1 퐹훼 : SHMp´푀, ´훾q Ñ SHMp´푀 , ´훾 q which is induced by the corresponding 2-handle cobordism and

. 1 퐹훼p휑p휉qq “ 휑p휉 q.

Remark 2.7.14. In (3), contact 3-handle attaching maps also preserve contact elements. This is proved in Corollary 3.5.9

It is worth mentioning that in [1], Baldwin and Sivek used the contact element to study the relation between Stein fillings and the representations of the fundamental groups. In particular, they obtained the following theorem.

Theorem 2.7.15. If 푌 is the boundary of a Stein domain, which is not an integer homology ball, then there is a non-trivial homomorphism 휋1p푌 q Ñ 푆푈p2q.

One might be interested in what if we glue a more general contact 3-manifold

72 (with boundary) instead of just a contact handle. In sutured Floer homology (SFH), this was constructed by Honda, Kazez, and Matić [35] and was re-visited by Juhász and Zemke [42].

Theorem 2.7.16 (Li [57]). Suppose p푀, 훾q and p푀 1, 훾1q are balanced sutured manifolds and 푀 Ă intp푀 1q. Suppose there is a contact structure 휉 on 푍 “ 푀 1zintp푀q so that B푍 is convex with 훾 Y 훾1 being the dividing set, then there is a contact gluing map

1 1 Φ휉 : SHGp´푀, ´훾q Ñ SHGp´푀 , ´훾 q, which is well defined up to the multiplication by a unit. Furthermore, the gluing map

Φ휉 satisfies the following properties. (1) If 푍 – r0, 1s ˆ B푀, then there exists a diffeomorphism

휑 : 푀 Ñ 푀 1, which restricts to the identity outside a collar of B푀 Ă 푀 and is isotopic to the inclusion 푀 ãÑ 푀 1, so that . Φ휉 “ SHGp휑q. . Here, “ means equal up to multiplication by a unit. (2) Suppose p푀 2, 훾2q is another balanced sutured manifold. Let 푀 1 Ă intp푀 2q, and suppose 푍1 “ 푀 2zintp푀 1q is equipped with a contact structure 휉2 so that B푍1 is convex with dividing set being 훾1 Y 훾2, then we have

. Φ휉Y휉1 “ Φ휉1 ˝ Φ휉.

(3) Suppose p푀 1, 훾1q admits a contact structure 휉1 so that B푀 1 is convex and 훾1

73 is the dividing set, then we have

1 1 1 Φ휉 |푍 p휑p휉 |푀 qq “ 휑p휉 q.

Here, 휑p휉1q P SHGp´푀 1, ´훾1q is the contact element associated to the contact struc-

1 1 ture 휉 , and 휑p휉 |푀 q is similar. (4) If 푍 – r0, 1s ˆ B푀 Y ℎ where t0u ˆ B푀 is identified with B푀 Ă 푀 and ℎ is a contact handle attached to 푀 Y r0, 1s ˆ B푀 along t1u ˆ B푀, then there is a suitable diffeomorphism 휑 : 푀 Ñ 푀 Y 푍, which restricts to the identity outside a collar of B푀 Ă 푀 and is isotopic to the inclusion 푀 ãÑ 푀 Y 푍, so that

. Φ휉 “ 퐶ℎ ˝ SHGp휑q,

where 퐶ℎ is the contact handle attaching map associated to ℎ.

Using gluing maps, we can also construct cobordism maps in sutured monopole and instanton Floer homology theories.

Definition 2.7.17. Suppose p푀, 훾q is a balanced sutured manifold and 휉0, 휉1 are two compatible contact structures. We say that 휉0 and 휉1 are equivalent if there is a

1-parameter family 휉푡 so that for any 푡 P r0, 1s, 휉푡 is a contact structure on 푀 with convex boundary B푀.

Definition 2.7.18 (Juhász [41]). Suppose p푀0, 훾0q and p푀1, 훾1q are two balanced sutured manifolds. A sutured cobordism from p푀0, 훾0q to p푀1, 훾1q is a triple 풲 “ p푊, 푍, r휉sq so that the following is true.

74 (1) 푊 is a compact 4-dimensional smooth oriented manifold with boundary

(2) 푍 is a compact oriented 3-manifold so that B푊 zintp푍q “ ´푀1 Y 푀2.

(3) We have that r휉s is an equivalent class of oriented and co-oriented contact structures on 푍 so that B푍 “ ´B푀1 Y ´B푀2 is a convex surface with dividing set

훾0 Y 훾1.

Remark 2.7.19. We need the equivalent class of 휉 instead of an actual contact structure 휉 to deal with the composition of sutured cobordisms. For more details, readers are referred to [41].

Theorem 2.7.20 (Li [57]). Suppose 풲 “ p푊, 푍, r휉sq is a sutured cobordism between two balanced sutured manifolds p푀1, 훾1q and p푀2, 훾2q then 풲 induces a cobordism map

SHGp풲q : SHGp푀1, 훾1q Ñ SHGp푀2, 훾2q, which is well defined up to multiplication by a unit and satisfies the following properties.

(1) Suppose 풲 “ p푀 ˆ r0, 1s, B푀 ˆ r0, 1s, r휉0sq so that 휉0 is r0, 1s-invariant, then

. SHGp풲q “ 푖푑.

1 1 1 1 (2) Suppose 풲 “ p푊 , 푍 , r휉 sq is another sutured cobordism from p푀2, 훾2q to

2 1 1 1 p푀3, 훾3q, then we can compose them to get a cobordism 풲 “ p푊 Y푊 , 푍Y푍 , r휉Y휉 sq from p푀1, 훾1q to p푀3, 훾3q and there is an equality

. SHGp풲2q “ SHGp풲1q ˝ SHGp풲q.

75 (3) For any balanced sutured manifold there is a canonical pairing

x¨, ¨y : SHGp푀, 훾q ˆ SHGp´푀, 훾q Ñ ℛ 표푟 C, which is well defined up to multiplication by a unit. Here ℛ is the coefficient ring.

Furthermore, let 풲 “ p푊, 푍, r휉sq be a cobordism from p푀1, 훾1q to p푀2, 훾2q, and let

_ 풲 “ p푊, 푍, r휉sq be the cobordism with same data but viewed as from p´푀2, 훾2q to

_ p´푀1, 훾1q. Then, 풲 and 풲 induce cobordism maps which are dual to each other under the canonical pairings.

There is another very important tool called the by-pass exact triangle. By-pass is a very important tool in the context of contact geometry introduced by Honda in [33]. Later its relation with sutured (Heegaard Floer homology) was studied by

Honda in [32]. Suppose we have three balanced sutured manifolds p푀, 훾1q, p푀, 훾2q, and p푀, 훾3q so that the underlining 3-manifolds are the same, but the sutures are different. Suppose further that p푀, 훾1q, p푀, 훾2q and p푀, 훾3q are only different within a disk 퐷 ĂB푀, and within the disk 퐷, they are depicted as in Figure 2-5. We say that p푀, 훾2q is obtained from p푀, 훾1q by a by-pass attachment along the arc 훼.

Similarly, p푀, 훾3q is obtained from a by-pass attachment from p푀, 훾2q and p푀, 훾1q from p푀, 훾3q. Then, we have the following theorem.

Theorem 2.7.21 (Baldwin and Sivek [5, 6]). There is an exact triangle relating the sutured monopole Floer homologies of the three balanced sutured manifolds as follows:

휓12 SHGp´푀, ´훾1q / SHGp´푀, ´훾2q i

휓 휓 31 u 23 SHGp´푀, ´훾3q

76 훼 -

@I@

p푀, 훾 q@ p푀, 훾 q 1 @ © 2

p푀, 훾3q Figure 2-5: The by-pass exact triangle.

In contact geometry, a by-pass is a half-disk carrying some particular contact structure attached along a Legendrian arc to a convex surface. For details, see Honda [33]. There is a description of the maps in the above by-pass exact triangle as follows.

We deal with the map 휓12, and the other two are the same. Let 푍 “B푀 ˆ r0, 1s and we can pick the suture 훾1 on B푀 ˆ t0u as well as the suture 훾2 on B푀 ˆ t1u. Then, there is a particular contact structure 휉12 on 푍, which corresponds to the by-pass attachment and makes p푍, 훾1 Y 훾2q a contact sutured manifold. Hence we can attach

푍 to 푀 by the identification B푀 ˆ t0u “ B푀 Ă 푀. The result p푀 Y 푍, 훾2q is just diffeomorphic to p푀, 훾2q and we have

휓12 “ Φ휉12 .

77 Here, Φ휉12 is the gluing map associated to 휉12 as in Theorem 2.7.16.

There is a second way to interpret the maps 휓˘ associated to by-pass attachments by Ozbagci [72]. He proved that a by-pass attachment could be realized by attaching a contact 1-handle followed by a contact 2-handle. In sutured monopoles, we have maps associated to the contact handle attachments due to Baldwin and Sivek [5] so we can composite those contact handle attaching maps to define 휓˘. This is the original way Baldwin and Sivek constructed the by-pass maps (when they define by-pass maps, there was no construction of gluing maps) and proved the existence of the exact triangle. The two interpretations are the same because of the functoriality of the gluing maps, and their relation with the contact handle attaching maps.

The by-pass exact triangle, as presented in Theorem 2.7.21, is a very powerful tool in computing the sutured monopole and instanton Floer homologies of some particular sutured manifolds. In particular, we have the following result.

Proposition 2.7.22 (Li [58]). Suppose p푉, 훾q is a balanced sutured manifold so that 푝 푉 is a solid torus and 훾 has 2푛 components with slope 푞 . Then, we know that

푛´1 SHGp푉, 훾q – ℛ2 ¨|푝|.

This result can be compared with Juhász [40] and Golovko [29]. In [27], similar computation techniques were adapted to the case of sutured handle bodies. Though only one particular example was studied in that paper, the author believes that at least a family is computable. This leads to the following question.

Question 12. For what sutured handle handle body p퐻, 훾q, can we compute SHGp퐻, 훾q?

78 2.8 Applications and other discussions

The non-vanishing property in Corollary 2.6.10 is a very important property of sutured monopole and instanton Floer homologies. In [53], a non-vanishing result on closed 3-manifolds was also proved.

Theorem 2.8.1 (Kronheimer and Mrowka [53], Theorem 7.21). Suppose p푌, 휔q is an admissible pair in the sense of Definition 2.3.1. Suppose 푅 Ă 푌 is a closed connected oriented surface of positive genus, so that 푅 ¨ 휔 is odd. Then, 퐼휔p푌 |푅q ‰ 0.

Follow the discussion in Kronheimer, Mrowka, Ozsváth, and Szabó [44], we have a corollary to Theorem 2.8.1 as follows, which provides a new proof of the famous Property P conjecture.

Corollary 2.8.2 (Kronheimer and Mrowka [53], Corollary 7.23). Let 푌1 be obtained 3 3 from 푆 by performing a `1 surgery along a non-trivial knot 퐾 Ă 푆 , then 휋1p푌 q admits a non-trivial 푆푈p2q-representation.

Representations of the fundamental groups of surgery 3-manifolds have been studied extensively beyond Corollary 2.8.2. In [46], Kronheimer and Mrowka proved the following.

3 3 Theorem 2.8.3. Suppose 퐾 Ă 푆 is a non-trivial knot, then 휋1p푆푟 p퐾qq admits an 3 irreducible 푆푈p2q-representation for all rational 푟 so that |푟| ă 2. Here, 푆푟 p퐾q is the 3-manifold obtained from 푆3 by performing an 푟-surgery along 퐾.

Recently, Baldwin and Sivek studied 3- and 4- surgeries in [9] and prove the following theorem.

Theorem 2.8.4. Suppose 퐾 Ă 푆3 is a non-trivial knot.

79 3 (1) There is always an irreducible 푆푈p2q representation of 휋1p푆4 p퐾qq. 3 (2) If 휋1p푆3 p퐾qq admits no irreducible 푆푈p2q representations, then 퐾 is fibred and strongly quasi-positive (See Hedden [30]) and has genus 2.

Remark 2.8.5. Note the 5-surgery of a right-handed trefoil is a Lens space whose fundamental group does not admit an irreducible 푆푈p2q-representations.

3 Question 13. Can we find a knot 퐾 so that 휋1p푆3 p퐾qq does not admits any irreducible 푆푈p2q representations?

Monopole and instanton Floer homology theories can be applied to the study of knots and links. The first construction of knot homology was introduced by Floer in [22] in instanton theory. In [53], Kronheimer and Mrowka used sutured monopole and instanton Floer homologies to construct knot homologies that are counterparts to the hat version of knot Floer homology in Heegaard Floer theory. To present their construction, let 퐿 Ă 푆3 be a link, 푆3p퐿q “ 푆3z푁p퐿q be the link complement, and 3 Γ휇 be the suture on B푆 p퐿q which consists of a pair of meridians on each boundary components of 푆3p퐿q. Then, define

3 3 퐾퐻퐺p푆 , 퐿q “ 푆퐻퐺p푆 p퐿q, Γ휇q.

From the discussion in Section 2.4, we know that there is a projective transitive system associated to the pair p푆3, 퐿q, which gives rise to a canonical module KHGp푆3, 퐿q. For a knot 퐾 Ă 푆3, Theorem 2.6.18 implies that there is a Z grading on KHGp푆3, 퐾q, induced by a Seifert surface of 퐾. We write KHGp푆3, 퐾, 푖q and call this grading an Alexander grading. This grading has some basic properties.

Proposition 2.8.6 (Kronheimer and Mrowka [53]). Suppose 퐾 Ă 푆3 is a knot, then the following is true.

80 (1) We have KHGp푆3, 퐾, 푖q – KHGp푆3, 퐾, ´푖q for all 푖. (2) The rank of KHGp푆3, 퐾, 0q is odd.

KHGp푆3, 퐾q detects the genus and fibredness of a knot. This is a combination of Lemma 2.6.12, Proposition 2.3.4, Theorem 2.6.20, Corollary 2.6.10, and Corollary 2.6.30. In particular, we have the following theorem.

Theorem 2.8.7 (Kronheimer and Mrowka [53]). Suppose 퐾 Ă 푆3 is a knot, then the following is true. (1) We have KHGp푆3, 퐾, 푖q “ 0 for |푖| ą 푔p퐾q, where 푔p퐾q is the genus of the knot 퐾. (2) We have KHGp푆3, 퐾, 푖q ‰ 0 for |푖| “ 푔p퐾q. (3) We have KHGp푆3, 퐾, 푖q – ℛ for |푖| “ 푔p퐾q if and only if 퐾 is a fibred knot.

Corollary 2.8.8. The rank of KHGp푆3, 퐾q detects unknots.

In [8], Baldwin and Sivek proved the following proposition.

Proposition 2.8.9. Suppose 퐾 Ă 푆3 is a fibred knot and is not the unknot. Then

KHGp푆3, 퐾, 푖q ‰ 0

81 for |푖| “ 푔p퐾q ´ 1.

Corollary 2.8.10. The rank of KHGp푆3, 퐾q detects trefoils.

Question 14. For a fibred knot 퐾 Ă 푆3, is it true that

KHGp푆3, 퐾, 푖q ‰ 0 for all 푖 such that |푖| “ 푔p퐾q ´ 1?

For links, the Floer homology 퐾퐻퐺p푆3, 퐿q satisfies an oriented skein relation. The instanton case was proved by Kronheimer and Mrowka in [51], and the monopole case is proved in Proposition 4.8.1. The oriented skein relation further implies the following, which was proved by Kronheimer and Mrowka in the monopole settings in [53] and in the instanton settings in [51].

Theorem 2.8.11. For a link 퐿 Ă 푆3, 퐾퐻퐺p푆3, 퐿q categorifies the single-variable Alexander polynomial.

Remark 2.8.12. In [53], the proof of Theorem 2.8.11 does not rely on the oriented skein relation, but was due to results from Fintushel and Stern [20] or Meng and Taubes [63].

For a link 퐿 Ă 푆3 of 푟 components, Theorem 2.6.22 in fact guarantees a Z푟 grading on KHGp푆3, 퐿q. Thus, one could also ask the following question in correspondence to Ozsváth and Szabó’s results in [76].

Question 15. Does 퐾퐻퐺p푆3, 퐿q categorifies the multi-variable Alexander polynomial?

82 In [48, 47], Kronheimer and Mrowka defined a new invariant for knots and links, denoted by 퐼6, using singular instanton Floer homology (as well as some of its variances). It is a Z2 graded vector space over C, and we have the following proposition.

Proposition 2.8.13 (Kronheimer and Mrowka [47]). For a knot 퐾 Ă 푆3, we have

퐼6p푆3, 퐾q – 퐾퐻퐼p푆3, 퐾q.

Note originally there is not a Z-grading on 퐼6, though we can pull back the Alexander grading on 퐾퐻퐼p푆3, 퐾q, through the above isomorphism. This leads to the following question.

Question 16. Can we construct an Alexander grading on 퐼6p푆3, 퐾q without using Proposition 2.8.13?

One important property of 퐼6 is that it satisfies an un-oriented skein relation. This is proved by Kronheimer and Mrowka in [47].

3 Theorem 2.8.14. Suppose 퐿0, 퐿1, and 퐿2 are three links in 푆 which only differ in a 3-ball 퐵, and, inside the 3-ball 퐵, the three links are depicted as in Figure 2-6, then there is an exact triangle

6 3 6 3 퐼 p푆 , 퐿2q / 퐼 p푆 , 퐿1q f

x 6 3 퐼 p푆 , 퐿0q

Question 17. Does 퐾퐻퐺 satisfy a suitable version of un-oriented skein relation?

It has been known that an unoriented skein relation as in Theorem 2.8.14 can result in a spectral sequence relating the Khovanov homology with the targeting

83 퐿2 퐿1 퐿0

Figure 2-6: Unoriented skein relation. The dashed circle indicates the 3-ball 퐵 knot homology. See Ozsváth and Szabó [75]. Accordingly, Kronheimer and Mrowka proved in [47] the following theorem.

Theorem 2.8.15. For any link 퐿 Ă 푆3, there is a spectral sequence whose second page is the reduced Khovanov homology and which abuts to 퐽 6p퐿¯q, where 퐾¯ is the mirror image of 퐿.

Remark 2.8.16. Recently, in [18], Dowlin constructed a spectral sequence from Kho- vanov homology to a suitable version of knot Floer homology in Heegaard Floer theory. He used a totally different approach, called the cube of resolutions, which was studied by Ozsváth and Szabó in [78] and himself in [17].

As a corollary to Theorem 2.8.15, Corollary 2.8.8, and Corollary 2.8.10, we have the following.

Theorem 2.8.17. The rank of the reduced Khovanov homology detects unknots and trefoils.

Following this line, Baldwin, Sivek, and Xie proved the following theorem in [2].

Theorem 2.8.18. The Khovanov homology detects Hopf links.

84 Along this direction, Xie and Zhang recently classify all possible links whose Khovanov homology has minimal ranks in [86], answering a question asked by Batson and Seed in [10]. To achieve this goal, they first constructed a version of instanton Floer homology for sutured manifold with tangles.

Definition 2.8.19 (Xie and Zhang [87]). Suppose p푀, 훾q is a balanced sutured manifold. A tangle 푇 is a properly embedded compact 1-manifold 푇 Ă 푀 so that B푇 Ă 푅p훾q. A tangle is called balanced if

|B푇 X 푅`p훾q| “ |B푇 X 푅´p훾q|.

Here, | ¨ | means the number of points inside a finite set. A tangle is called vertical if each component of 푇 either intersects both 푅`p훾q and 푅´p훾q or intersects neither

푅`p훾q nor 푅´p훾q.

Theorem 2.8.20 (Xie and Zhang [87]). Suppose p푀, 훾, 푇 q is a balanced sutured manifold with tangles. Then, there is a well-defined instanton Floer homology, which we write 푆퐻퐼p푀, 훾, 푇 q associated to the triple p푀, 훾, 푇 q.

The instanton Floer homology for sutured manifold with tangles, 푆퐻퐼p푀, 훾, 푇 q has some desired non-vanishing and fibredness detection properties. Those properties further contribute to the establishment of the following theorem.

Theorem 2.8.21 (Xie and Zhang [86]). If 퐿 is an 푛 component link so that

푛 rkZ2 퐾ℎp퐿; Z2q “ 2 , then 퐿 is a forest of unknots, i.e., obtained from some unknots and Hopf links by performing disjoint union and connected sums.

85 It is worth mentioning that Corollary 2.6.30 can also be applied to the case of a sutured manifold with tangles, and prove the following.

Corollary 2.8.22 (Ghosh and Li [27]). Suppose p푀, 훾q is a balanced sutured manifold equipped with a vertical tangle 푇 . Suppose further that 퐻2p푀z푇 q “ 0 and 푆퐻퐼p푀, 훾, 푇 q – C. Then, p푀, 훾, 푇 q is diffeomorphic to a product sutured manifold equipped with a product tangle, i.e.,

p푀, 훾, 푇 q – pr´1, 1s ˆ 퐹, t0u ˆ B퐹, r´1, 1s ˆ t푝1, ..., 푝푛uq.

Here, 퐹 is a compact oriented surface-with-boundary, and 푝1, ..., 푝푛 are distinct points on 퐹 .

This result is slightly sharper than the one proved in Xie and Zhang [87].

Question 18. Can we use the instanton Floer homology for sutured manifold with tangles to construct some new invariants for contact structures and transverse knots?

Using tools from sutured manifold theory, and an idea originates from Etnyre, Vela-Vick, and Zarev in [19], the author constructed a minus version of knot Floer homology in monopole and instanton theories, in correspondence to the minus version of knot Floer homology in Heegaard Floer homology introduced by Ozsváth and Szabó.

Theorem 2.8.23 (Li [58]). Suppose 푌 is a closed connected oriented 3-manifold and 퐾 Ă 푌 is an oriented null-homologous knot. Suppose further that 푆 is a Seifert surface of 퐾, and 푝 P 퐾 is a base point. Then, we can associate the triple p푌, 퐾, 푝q a module KHG´p푌, 퐾, 푝q over the mod 2 Novikov Ring ℛ. It is well defined up to multiplication by a unit in ℛ. The Seifert surface 푆 induces a Z grading on

86 KHG´p푌, 퐾, 푝q, which we denote by KHG´p푌, 퐾, 푃, 푆, 푖q. Moreover, the following properties hold. (1) For 푖 ą 푔 “ 푔p푆q, KHG´p푌, 퐾, 푝, 푆, 푖q “ 0. (2) There is a map

푈 : KHG´p푌, 퐾, 푝q Ñ KHG´p푌, 퐾, 푝q that is of degree ´1.

(3) There exists an 푁0 P Z such that if 푖 ă 푁0, then

푈 : KHG´p푌, 퐾, 푝, 푆, 푖q – KHG´p푌, 퐾, 푝, 푆, 푖 ´ 1q.

(4) There exists an exact triangle

푈 KHG´p푌, 퐾, 푝q / KHG´p푌, 퐾, 푝q h

휓1 휓 v KHGp푌, 퐾, 푝q

(5) If 푌 “ 푆3 and 푆 realizes the genus of the knot, then we have

KHG´p푌, 퐾, 푝, 푆, 푖q ‰ 0 for 푖 “ 푔p푆q. (6) (Li [59]) For any knot 퐾 Ă 푆3, there is a unique infinite 푈-tower in KHG´p푆3, 퐾, 푝q.

Statement (6) in Theorem 2.8.23 motivates the following definition, in correspondence to the 휏-invariant defined in Heegaard Floer theory by Ozsváth and Szabóin [73].

87 Definition 2.8.24 (Li [58]). Suppose 퐾 Ă 푆3 is a knot. Define

´ 3 푗 휏퐺p퐾q “ maxt푖 : D 푥 P KHG p푆 , 퐾, 푝, 푆, 푖q 푠.푡. 푈 푥 ‰ 0 푓표푟 푎푙푙 푗 P Zě0u

Here, 푝 P 퐾 is an arbitrary base point and 푆 is a Seifert surface of 퐾.

Remark 2.8.25. Here the subscript 퐺 represents either 푀 (for monopole) or 퐼 (for instanton).

Proposition 2.8.26 (Li [59]). The invariant 휏퐺 is a concordance invariant.

Remark 2.8.27. As a project working in progress, the author and his collaborators are working on proving that 휏퐺 is additive under connected sum and bounding four-ball genus.

Question 19. How is 휏퐺 related to the 휏 invariant defined in Heegaard Floer theory?

Question 20. How is 휏퐼 possibly be related to the representations of the fundamental group of the knots?

7 Question 21. How is 휏퐼 related to the 푠 invariant defined by Kronheimer and Mrowka in [49]?

Recently, in [12], Daemi and Scaduto made use of the unique reducible flat con- nection in a suitable representation arising from the fundamental group of the knot complements, and used an equivariant method to constructed four different flavors ˆ ˇ ¯ ˜ of instanton knot Floer homology, which they denote by 퐼˚, 퐼˚, 퐼˚, and 퐼˚, associated to a knot 퐾 inside an integral rational homology sphere 푌 . They are also closely related to some versions of singular instanton Floer homology with special choices of local coefficients constructed by Kronheimer and Mrowka [47, 54]. The following questions might be interesting.

88 ˆ ´ Question 22. How could 퐼˚ and KHI be related to each other?

Question 23. How could the instanton knot Floer homologies introduced above (퐼6, ˆ ¯ ´ 퐼˚, 퐼˚, KHI , etc.) be related to the instanton Floer homology of closed 3-manifolds (say, the framed version studied by Scaduto in [81] or the equivariant versions constructed by Miller in [64])? In particular, can we prove a surgery-type formula as done in Heegaard Floer theory by Ozsváth and Szabó in [74] and subsequent papers?

89 90 Chapter 3

Gluing and cobordism maps

In this chapter, we present the work done by the author on constructing the gluing and cobordism maps in sutured monopole and instanton Floer homology theories.

3.1 Contact elements and excisions

As explained in Section 2.7, the definition of a contact invariant in sutured monopole Floer homology uses connected auxiliary surface, and to incorporate the usage of disconnected auxiliary surface, one need to study how the contact element behave under performing Floer’s excisions. Though the argument is this section is carried out in the monopole settings, the same idea can be adapted to the instanton settings. Also, the techniques used in this section can also be applied in the proof of Proposition 3.3.13, which is important in the construction of gluing maps in the monopole settings.

Suppose, for 푖 “ 1, 2, p푀푖, 훾푖q is a balanced sutured manifold. Suppose further that p푇푖, 푓푖, ℎ푖q is the auxiliary data for constructing a closure p푌푖, 푅푖q of p푀푖, 훾푖q, as in Definition 2.2.2. In addition to the requirement of choosing an auxiliary surface and

91 a gluing diffeomorphism as mentioned in Section 2.2, we also choose a non-separating simple closed curve 푐푖 Ă 푇푖. Let

푀푖 “ 푀푖 Y r´1, 1s ˆ 푇푖

Ă be the pre-closure and B푀푖 “ 푅푖,` Y 푅푖,´ be its boundary. There are curves 푐푖,˘ Ă 푅 coming from the curve 푐 Ă 푇 . We further require that ℎ p푐 q “ 푐 . Thus, 푖,˘ Ă 푖 푖 푖 푖,` 푖,´ inside the closure, there is a curve 푅푖 coming from 푐푖 Ă 푇푖, which, by abusing the notations, we still write 푐푖. Also, we pick a simple closed curve 휂푖 Ă 푅푖 so that 휂푖 intersects 푐푖 Ă 푅푖 transversely once.

Let 푀 “ 푀1 \ 푀2 and 훾 “ 훾1 Y 훾2. Then, p푀, 훾q is also a balanced sutured manifold. We can cut 푇푖 along 푐푖 and re-glue the newly created boundary with respect to the orientation. Then, 푇1 and 푇2 become a connected surface 푇 so that

푔p푇 q “ 푔p푇1q ` 푔p푇2q ´ 1, B푇 “B푇1 YB푇2.

Choose 푓 “ 푓1 Y 푓2 and ℎ “ ℎ1 Y ℎ2. When cutting and re-gluing along 푐1 and 푐2, the two curves 휂1 and 휂2 can also be cut and glued together to become a curve 휂. We can use the auxiliary data p푇, 푓, ℎ, 휂q to close up p푀, 훾q and then obtain a closure p푌, 푅q of p푀, 훾q. See Figure 3-1. As in Section 2.5, we can construct an excision map:

퐹 : 퐻푀p´p푌1 \ 푌2q| ´ p푅1 Y 푅2qq Ñ 퐻푀p´푌 | ´ 푅q. (3.1)

We have the following~ theorem. ~

Theorem 3.1.1. Suppose that the genus of 푇1 and 푇2 are large enough, and, for

푖 “ 1, 2, p푀푖, 훾푖q is equipped with a compatible contact structure 휉푖. Suppose further that 푡 is obtained from 푇1 and 푇2 but cutting and re-gluing as described above. Then,

92 we can find suitable reduced arc configurations (see Definition 2.7.2) 풜1, 풜2, and 풜 ¯ ¯ ¯ on 푇1, 푇2, and 푇 , respectively to construct contact structures 휉1, 휉2, and 휉 on suitable closures 푌1, 푌2, and 푌 . Then, the map 퐹 in (3.1) preserves contact elements:

. 퐹 p휑휉¯q “ 휑휉¯1Y휉¯2 .

. Here, “ means equal up to multiplication by a unit.

푐1 푐2

휂1 휂2

푇1 푇2

휂 푇

Figure 3-1: Top, the two auxiliary surfaces 푇1 and 푇2. Bottom, the connected auxiliary surface 푇 .

Proof. We first choose the special arc configurations 풜1 and 풜2. For 푖 “ 1, 2, assume that we have a reduced arc configuration 풜푖 on 푇푖 so that the simple closed curve is

푐푖, and all arcs are attached to only one side of 푐푖 Ă 푇푖. See Figure 3-2.

To show that such special arc configurations 풜1 and 풜2 do exist, we work only with p푇1, 푐1q, and the argument for p푇2, 푐2q is similar. Cut 푇1 open along 푐1 and

93 let 푐1,` 푐1,´ be the two newly created boundary components. Now, it is enough to find a set of pair-wise disjoint properly embedded arcs 푎1, ..., 푎푛 on 푇1z푐1 so that the following is true.

(1) For any 푖 P t1, ..., 푛u, one end point of 푎푖 is on B푇푖, and the other end point of 푎푖 is on 푐1,`.

(2) Each component of B푇푖 intersects with some 푎푖.

We can pick the set of arcs t푎1, ..., 푎푛u one by one. First, pick any 푎1 that satisfies

(1) and is non-separating on 푇1z푐1. Then, we can pick an arc 푎2 that connects a different component of B푇1 to 푐1,` and is non-separating in 푇1zp푐1 Y 푎1q. Keep running this process until all boundary components of B푇1 have been connected to

푐1,` by an arc. Note that we can make the genus of 푇1 as large as we want, which means that it is always possible to find the desired set of arcs. To obtain thearc configuration 풜1, we glue 푐1,` to 푐1,´ to recover 푇1. Since all arcs are chosen to be attached to 푐1,`, they appear on the same side of 푐1 on 푇1.

Since the arcs 푎푗 are attached to the same side of 푐푖, when looking at the induced contact structure on r´1, 1s ˆ 푇푖, the boundary of the negative region on t푡u ˆ 푇푖 consists of a few arcs, whose end points are on t푡uˆB푇푖, and one simple closed curve, which is a parallel copy of 푐푖 Ă 푇푖. Abusing the notation, we still use 푐푖 to denote this closed component of the boundary of the negative region on t푡u ˆ 푇푖. We then pick a gluing diffeomorphism ℎ푖 that identifies the contact structures on the boundary of

푀푖 “ 푀푖 Y r´1, 1s ˆ 푇푖 and which also preserves 푐푖.

¯ Ă When we extend 휉푖 to 휉푖, which is defined on all of 푌푖, the new contact structure ¯ 1 휉푖 will be 푆 -invariant in a neighborhood of 푐푖. To describe this contact structure in 1 coordinates, let 퐴푖 Ă 푇푖 be a neighborhood of 푐푖 Ă 푇푖. In 푌푖, 퐴푖ˆ푆 is a neighborhood

94 푐2

푐1

푇1 푇2

푇

Figure 3-2: Top, The two reduced arc configurations on 푇1 and 푇2. Bottom, the resulting arc configuration on 푇 from slicing. It has two simple closed curves instead of one.

of 푐푖 Ă 푌푖. In this neighborhood, we can write the contact form as

훼푖 “ 훽푖 ` 푢푖 ¨ 푑휙푖,

where 훽푖 is a 1-form on 퐴푖, 푢푖 is a function on 퐴푖 with

푐푖 “ t푝 P 퐴푖|푢푖p푝q “ 0u,

1 and 휙푖 is the coordinate for 푆 direction. See Geiges [26]. The non-degeneracy condition reads

0 ‰ 훼푖 ^ 푑훼푖 “ p푢푖 ¨ 푑훽푖 ` 훽푖 ^ 푑푢푖q ^ 푑휙푖.

Along 푐푖, we have 훽푖 ^ 푑푢푖 ‰ 0. Hence, along 푐푖, the 푑휃푖 component of 훽 is always

95 non-zero. Here, 휃푖 is a coordinate for 푐푖, and p푢, 휃푖q can serve as local coordinates in a small neighborhood r´휀, 휀s ˆ 푐푖 Ă 퐴푖. Then, the slicing operation defined by 1 Niederkrüger and Wendl in [71] can be described as follows: Let 퐿푖 “ 푐푖 ˆ 푆 be a pre-Lagrangian torus, (for the definition of pre-Lagrangian tori, see Ma [61]) andlet

1 푁푖 “ r´휀, 휀s ˆ 푐푖 ˆ 푆 be a neighborhood of 퐿푖 with the coordinates p푢푖, 휃푖, 휙푖q. Note the coordinate 푢푖 corresponds to 푟 in [71], and the other two coordinates are the same as in that paper. We can cut 푁푖 open along 퐿푖 so that 푁푖 is cut into two parts

푁푖,` and 푁푖,´, which correspond to where 푢푖 ě 0 and 푢푖 ě 0, respectively. Then, re-glue 푁1,` to 푁2,´ and 푁1,´ to 푁2,` by identifying 퐿1 with 퐿2 so that p휃1, 휙1q is identified with p휃2, 휙2q. Suppose that the resulting 3-manifold is 푌 , then 푌 has a distinguishing surface 푅 obtained by cutting and re-gluing 푅1 and 푅2 along 푐1 and

푐2. Recall that there is a simple closed curve 휂푖 Ă 푅푖, which intersects 푐푖 transversely once. After a suitable isotopy, we can assume that, under the above identification of 퐿1 with 퐿2, we can also identify 휂1 X 푐1 with 휂2 X 푐2. Hence, 휂1 and 휂2 are also cut and re-glued to become a curve 휂 Ă 푅. This is exactly the same procedure of performing a Floer’s excision along the tori 퐿1 and 퐿2. Thus, p푌, 푅q is a closure of p푀1 \ 푀2, 훾1 Y 훾2q. Also, by Theorem 2.5.1, there is an isomorphism

퐹 : 퐻푀p´p푌1 \ 푌2q| ´ p푅1 Y 푅2q;Γ´p휂1Y휂2qq Ñ 퐻푀p´푌 | ´ 푅;Γ´휂q.

¯ The process of slicing also cuts and re-glues the contact structures 휉푖 on 푌푖 to obtain a contact structure 휉¯1 on 푌 , as explained in [71]. The contact structure 휉¯1, however, arises from an arc configuration 풜1 that is not reduced in the sense of Definition 2.7.2. This is because, with respect to 휉¯1, the dividing set on 푅 consists of two pairs of parallel non-separating simple closed curves rather than just one pair. See Figure 3-2. Let 휉¯ be a contact structure on 푌 , which is obtained by extending

96 휉푖 on 푀푖 using a reduced arc configuration 풜. Here, 풜 is obtained by “merging” the two simple closed curves of 풜1 into one as depicted in Figure 3-3. The proof of Theorem 3.1.1 is clearly the combination of the following two lemmas.

푇

Figure 3-3: Top, the arc configuration on 푇 obtained from slicing. Bottom, the reduced arc configuration after merging the two simple closed curves.

푐 Lemma 3.1.2. If the genus of 푇1 and 푇2 are large enough, then the spin structures associated to 휉¯ and 휉¯1 are the same. Furthermore, if we denote that spin푐 structure by s0, then we have . 휑휉¯ “ 휑휉¯1 P 퐻푀p´푌, s0;Γ´휂q.

~ Lemma 3.1.3. If the genus of 푇1 and 푇2 are large enough, then we have

. 1 퐹 p휑휉¯1 Y 휑휉¯2 q “ 휑휉¯ .

97 To prove the above two lemmas, we first need some preliminaries.

Lemma 3.1.4 (Baldwin and Sivek [5]). Suppose p푀, 훾q is a balanced sutured manifold equipped with a compatible contact structure 휉 and 푇 is a connected auxiliary surface with a large enough genus. Suppose further that we use an arc configuration (not necessarily reduced) on 푇 to extend 휉 to a contact structure 휉¯ on a suitable

1 closure p푌, 푅q of p푀, 훾q. Then, there is a contact structure 휉푅 on 푅 ˆ 푆 and a set of pair-wise disjoint simple closed curves t훼1, ..., 훼푛u so that the following is true. 1 (1) The contact structure 휉푅 is 푆 -invariant so that each 푅 ˆ t푡u is convex with the dividing set being some pairs of parallel non-separating simple closed curves.

(2) Each 훼푖 is Legendrian and is disjoint from the pre-Lagrangian tori of the form

pDividing set on 푅q ˆ 푆1.

1 (3) The result of performing `1 contact surgeries along all 푎푖 Ă 푅 ˆ 푆 is contactomorphic to p푌, 휉¯q.

Lemma 3.1.5 (Niederkrüger and Wendl [71]). Suppose 푅 is the surface as described

1 1 in Lemma 3.1.4, and 휉푅 is an 푆 -invariant contact structure on 푅 ˆ 푆 so that each 푅 ˆ 푡 is convex with the dividing set being a few pairs of non-separating simple closed curves. Suppose further that there is a curve 휂 Ă 푅, which intersects every 1 component of the dividing set transversely once. Then, p푅 ˆ 푆 , 휉푅q is weakly fillable by some p푊, 휔q so that 휂 is dual to 휔|푅ˆ푆1 up to a (non-zero) scalar.

Lemma 3.1.6 (Kronheimer, Mrowka, Ozsváth, and Szabó [44]). In Lemma 3.1.5, the contact element

1 휑휉푅 P 퐻푀p´푅 ˆ 푆 , s휉푅 ;Γ´휂q is primitive. ~

98 Proof of lemma 3.1.2. As in the settings of Theorem 3.1.1, 휉¯ and 휉¯1 are contact structures on 푌 , which are obtained from the contact structures 휉1 Y휉2 on p푀1, 훾1qY

1 p푀2, 훾2q and some particular arc configurations 풜 and 풜 on 푇 . From Lemma 3.1.4,

1 1 we know that there are contact structures 휉푅 and 휉푅 on 푅 ˆ 푆 and a set of pair-wise 1 disjoint curves 훼1, ..., 훼푛 Ă 푅 ˆ 푆 so that the following is true.

1 1 1 (1) Both 휉푅 and 휉푅 are 푆 -invariant, and any 푅 ˆ t푡u, for 푡 P 푆 , is convex.

1 (2) We have 휉푅 “ 휉푅 near a neighborhood of each 훼푖.

(3) All 훼푖 are disjoint from the pre-Lagrangian tori of the form

pDividing set on 푅q ˆ 푆1,

1 for the dividing sets with respect to both 휉푅 and 휉푅.

1 (4) If we perform contact `1 surgery along all of 훼푖, then p푅 ˆ 푆 , 휉푅q [or p푅 ˆ 1 1 ¯ ¯1 푆 , 휉푅q] will become a contact manifold that is contactomorphic to p푌, 휉q [or p푌, 휉 q]. Condition (2) relies on the proof of Lemma 3.1.4 (of the current paper) in [5]. The essential reason is that 휉¯ and 휉¯1 are only different in the part of 푌 that comes from gluing auxiliary surfaces, while the curves 훼푖 are contained in the interior of the original balanced sutured manifold.

Via Lemmas 3.1.5 and 3.1.6, we know that the contact invariants 휑 and 휑 1 are 휉푅 휉푅 both primitive in the same monopole Floer homology. Then, Lemma 2.6.2 makes

1 푐 1 sure that 휉푅 and 휉푅 correspond to the same spin structure s0 on 푅 ˆ 푆 (because there is only one candidate for the spin푐 structures). Thus, we have

. 1 휑 휑 1 퐻푀 푅 푆 , s ;Γ , (3.2) 휉푅 “ 휉푅 P p´ ˆ 0 ´휂q

~ for a suitable choice of local coefficients.

99 The surgery description in condition (4) makes sure that, on 푌 , 휉¯ and 휉¯1 correspond to the same spin푐 structure. This fact, together with Theorem 2.7.4 and equality (3.2), implies Lemma 3.1.2.

¯ Proof of Lemma 3.1.3. First, by applying Lemma 3.1.4 to p푌푖, 휉푖q, for 푖 “ 1, 2, we 1 obtain a contact structure 휉푅푖 on 푅푖ˆ푆 and a set of Legendrian curves t훼푖,1, ..., 훼푖,푛푖 u satisfying the conclusion of the lemma. In particular, if we perform contact `1 ¯ surgery along all of 훼푖,푗, we will obtain p푌푖, 휉푖q. If we pick a suitable connected component 푐푖 of the dividing set on 푅푖 ˆ 푡 and perform the slicing operation on 1 1 1 1 푅1 ˆ 푆 and 푅2 ˆ 푆 , along the two pre-Lagrangian tori 푐1 ˆ 푆 and 푐2 ˆ 푆 , then the

1 1 result is the 3-manifold 푅ˆ푆 with the contact structure 휉푅, as in the proof of Lemma

3.1.2. Also, the two sets of curves t훼1,1, ..., 훼1,푛1 u and t훼2,1, ..., 훼2,푛2 u together form the set of curves t훼1, ..., 훼푛u, as in the proof of Lemma 3.1.2. There is a cobordism associated to the slicing operation or, equivalently, performing a Floer’s excision on

1 1 1 1 푅1 ˆ푆 and 푅2 ˆ푆 . We call this cobordism 푊푒, and it is from p푅1 ˆ푆 q\p푅2 ˆ푆 q 1 to 푅 ˆ 푆 . There is a second cobordism 푊푠, associated to the surgeries along all 1 of 훼푖, as in Theorem 2.7.4, from 푅 ˆ 푆 to 푌 . Finally, there is a third one, 푊퐹 , corresponding to the map 퐹 (which is also obtained from a Floer’s excision again), from 푌 to 푌1 \ 푌2. As usual, we choose suitable surfaces and local coefficients to make the cobordism map precise, but we omit them from the notation. The map 퐻푀p´푊푒q preserves contact elements because it is an isomorphism between two copies of ℛ, and the two contact elements are both units in the corresponding copy of ℛ. Furthermore, the map 퐻푀p´푊푠q preserves contact elements by Theorem 2.7.4. So, if we could prove that the composition 퐻푀p´p푊푒 Y 푊푠 Y 푊퐹 qq preserves the contact elements, then so does 퐻푀p´푊퐹 q “ 퐹 , and, thus, Lemma 3.1.3 follows.

100 To show that 퐻푀p´p푊푒 Y 푊푠 Y 푊퐹 qq preserves contact elements, we observe 1 that, when we cut the cobordism 푊푒 Y 푊푠 Y 푊퐹 open along 푇1,` ˆ 푆 and glue back 2 two copies of 푇1,` ˆ 퐷 , the result is the disjoint union of two cobordisms, which we 1 call 푊1 and 푊2, respectively. See Figure 3-4. For 푖 “ 1, 2, 푊푖 is from 푅푖 ˆ 푆 to

푌푖 and is associated to the surgeries along 훼푖,푗, as in Theorem 2.7.4. Hence, by that lemma, 퐻푀p´p푊1 Y푊2qq would preserve contact elements. Finally, by Lemma 2.10 in Kronheimer and Mrowka [53], we know that

. 퐻푀p´p푊푒 Y 푊푠 Y 푊퐹 qq “ 퐻푀p´p푊1 Y 푊2qq.

Thus, we conclude the proof of Lemma 3.1.3.

1 1 1 1 푅1 ˆ 푆 푅2 ˆ 푆 푅1 ˆ 푆 푅2 ˆ 푆

훼1,푖 훼2,푗 푊푒

훼1,푖 훼2,푗 푇 푆1 푊 1,` ˆ 푊1 푊2

푊퐹

푌1 푌2 푌1 푌2 1 Figure 3-4: Left, the union of the three cobordisms, cut along the 3-torus 푇1,` ˆ 푆 . Right, the two disjoint cobordisms resulting from the cutting and pasting.

Question 24. Suppose p푀1, 훾1q and p푀2, 훾2q are two balanced sutured manifold as in Theorem 3.1.1. Instead of taking 푀 “ 푀1 \푀2, we can pick a connected balanced

101 sutured manifold p푀, 훾q so that B푀 “B푀1 YB푀2 and 훾 “ 훾1 Y 훾2. Then, does Theorem 3.1.1 still hold?

3.2 Another interpretation of contact handle gluing maps

Proposition 3.2.1. Suppose p푀0, 훾q and p푀1, 훾q are two balanced sutured manifolds so that B푀0 “B푀1 and the sutures are also identical. Suppose 푊 is a smooth compact oriented 4-manifold so that 푊 can also be viewed as a manifold with corners: the boundary B푊 consists of two horizontal parts ´푀0 and 푀1 as well as a vertical part r0, 1sˆB푀0. The two parts ´푀0 and r0, 1sˆB푀0 meet in the corner t0uˆB푀0.

The two parts 푀1 and r0, 1s ˆ B푀0 meet in the corner t1u ˆ B푀0. See Figure 3-5. Then, we can define a morphism between canonical modules:

퐹푊 : SHGp푀0, 훾q Ñ SHGp푀1, 훾q.

Proof. Suppose 푇 is an auxiliary surface for p푀0, 훾q and 푓 : B푇 Ñ 훾 is the map gluing 푇 to p푀0, 훾q. Let

푀0 “ 푀0 Y r´1, 1s ˆ 푇 푖푑ˆ푓 Ă be the pre-closure and B푀0 “ 푅` Y 푅´. Suppose ℎ : 푅` Ñ 푅´ is a diffeomorphism, we can use ℎ to glue r´1, 1s ˆ 푅 to 푀 and get a closure 푌 . Suppose 휂 is a non- Ă ` 0 0 separating curve on 푅 “ t0u ˆ 푅 , we get a marked closure 풟 “ p푌 , 푅, 푟, 푚 , 휂q ` Ă 0 0 0 for p푀0, 훾q. Since the boundaries of 푀0 and 푀1 are identified, we can use the same auxiliary data p푇, 푓, ℎ, 휂q to get a marked closure 풟1 “ p푌1, 푅, 푟, 푚1, 휂q.

There is a natural way to construct a cobordism from 푌0 to 푌1 out of 푊 . Use

102 푀0

B푀0 푠

푊 푡

푀1

Figure 3-5: The special type of cobordism 푊 .

푖푑 ˆ 푖푑 ˆ 푓 to glue r0, 1s ˆ r´1, 1s ˆ 푓 to r0, 1s ˆ 퐴p훾q Ă r0, 1s ˆ B푀0 ĂB푊 , and use

푖푑 ˆ p푖푑 Y ℎq to glue r0, 1s ˆ pt0, 1u ˆ 푅`q to the result of the first gluing. Finally, we get a cobordism 푊 from 푌0 to 푌1. We have a map

x 퐻퐺p푊 q : 푆퐻퐺p풟0q Ñ 푆퐻퐺p풟1q.

x Definition 3.2.2. The process of obtaining 푊 from 푊 is called parallel closing up.

x We claim that this map induces a morphism between canonical modules. We only prove here that the cobordism map constructed above commutes with canonical maps

1 Φ풟,풟1 , where 푔p풟q “ 푔p풟 q, and the commutativity with general canonical maps

1 1 1 1 1 1 follows from a similar argument. To proceed, suppose 풟0 “ p푌0 , 푅 , 푟 , 푚0, 휂 q is 1 another marked closure for p푀0, 훾q, with 푔p푅 q “ 푔p푅q, and is obtained in a similar

1 1 1 1 1 1 way as 풟. Let 풟1 “ p푌1 , 푅 , 푟 , 푚1, 휂 q be the corresponding marked closure for 1 1 1 p푀1, 훾q, and 푊 be the corresponding cobordism from 푌0 to 푌1 . Then, we need to

x 103 show that the following diagram commutes up to multiplication by a unit:

Φ풟 ,풟1 0 0 1 푆퐻퐺p풟0q / 푆퐻퐺p풟0q (3.3)

퐻퐺p푊 q 퐻퐺p푊 1q

x x Φ풟 ,풟1 1 1 1 푆퐻퐺p풟1q / 푆퐻퐺p풟1q

The definitions of Φ 1 and Φ 1 are from Baldwin and Sivek [4], and we 풟0,풟0 풟1,풟1 sketch their constructions as follows: There is a diffeomorphism

휑 : 푅 Ñ 푅

1 so that if we cut 푌0 open along 푅 and re-glue along 휑, then we obtain 푌0 . We can decompose 휑 into Dehn twists:

푒1 푒푛 휑 „ 퐷푎1 ˝ ... ˝ 퐷푎푛 .

Here 푎푖 Ă 푅 is a non-separating simple closed curve, and 푒푖 P t˘1u with 푒푖 “ 1 indicating a positive twists and ´1 indicating a negative twist.

For simplicity, we assume here that all 푒푖 “ 1 (the general case follows from a similar argument). The map Φ 1 is then induced by a cobordism 푊 obtained from 풟0,풟0 0 r0, 1s ˆ 푌0 by attaching 4-dimensional 2-handles along curves 푎1, ..., 푎푛 Ă t1u ˆ 푌0.

Since the two manifolds 푀0 and 푀1 have identical boundary: B푀 “B푀1, from

1 푔 the construction of 풟1 and 풟1, we know that the canonical map Φ 1 can be 풟1,풟1 described as follows: we have the same set of curves t푎1, ..., 푎푛u on 푅 Ă 푌1. Let 푊1

1 be the cobordism from 푌1 to 푌1 obtained from r0, 1s ˆ 푌1 by attaching 4-dimensional

104 푔 2-handles along 푎1, ..., 푎푛, then Φ 1 is induced by the cobordism map 퐻퐺p푊1q. 풟1,풟1 Now, the commutativity of the diagram in (3.3) is equivalent to

1 퐻퐺p푊1q ˝ 퐻퐺p푊 q “ 퐻퐺p푊 q ˝ 퐻퐺p푊0q. (3.4)

x x From Lemma 3.2.3, we can view 푊 as obtained from r0, 1s ˆ 푀0 by attaching 4- dimensional handles ℎ4, ..., ℎ4 to t1uˆintp푀 q, and then 푊 1 is obtained from r0, 1sˆ 1 푚 x 0 푌 by attaching the same set of 4-dimensional handles ℎ4, ..., ℎ4 to t1u ˆ intp푀 q Ă 0 1x 푚 0 1 t1u ˆ 푌0 . So, to prove the equality (3.4), it is enough to prove that the set of handles 4 4 ℎ1, ..., ℎ푚 and the set of 2-handles attached along 푎1, ..., 푎푛 Ă t1u ˆ 푌0, which come from the construction of canonical maps between marked closures, commute with

4 4 each other. This is obvious: ℎ1, ..., ℎ푚 are attached to t1u ˆ intp푀0q Ă t1u ˆ 푌0, the curves 푎1, ..., 푎푛 are inside t1u ˆ intpimp푟qq Ă t1u ˆ 푌0, and

intp푀0q X intpimp푟qq “ H.

Hence, we conclude the proof of Proposition 3.2.1.

Lemma 3.2.3. Suppose p푀0, 훾q, p푀1, 훾q and 푊 are defined as in Proposition 3.2.1.

Then, 푊 is diffeomorphic to a 4-manifold obtained from r0, 1s ˆ 푀0 by attaching some 4-dimensional handles to t1u ˆ intp푀0q.

Proof. We can assume a neighborhood 푁 of the vertical boundary part r0, 1s ˆ B푀1 of 푊 is identified with r0, 1s푡 ˆ r´1, 0s푠 ˆ B푀0 so that the vertical boundary part is r0, 1s ˆ t0u ˆ B푀0. We can choose a smooth function 푓 : 푊 Ñ r0, 1s so that

푓p´푀0q “ 0, 푓p푀1q “ 1, 푓pt푡u ˆ r´1, 0s ˆ B푀0q “ 푡.

105 Perturb 푓 a little bit so that 푓 is Morse and there is no critical points of 푓 near B푊 Ă 푊 . Such perturbation exists since the set of Morse functions is dense in the space of smooth functions and 푓 has already been Morse near the boundary B푊 Ă 푊 , and having no critical points there. Then, 푓 induces the desired handle decomposition.

Suppose we are given a smooth compact oriented 4-manifold-with-boundary 푊 , and let 푆 ĂB푊 be a closed oriented surface surface which separates B푊 into two parts, 푀1 and 푀2, with orientations chosen so that

B푀1 “B푀2 “ 푆, ´ 푀1 Y 푀2 “B푊.

Suppose 훾 Ă 푆 is a collection of disjoint oriented simple closed curves so that p푀1, 훾q and p푀2, 훾q are both balanced sutured manifolds. We can view 푊 as a cobordism from p푀1, 훾q to p푀2, 훾q. An adaption of Lemma 3.2.3 shows that 푊 is diffeomorphic to the 4-manifolds obtained from r0, 1sˆ푀1 by attaching some 4-dimensional handles along t1u ˆ intp푀1q. Hence, just as in the proof of Proposition 3.2.1, we have a map

퐹푊 : SHGp푀1, 훾q Ñ SHGp푀2, 훾q.

Definition 3.2.4. Under the above settings, we call 푊 a cobordism with a sutured surface p푆, 훿q, from p푀1, 훾1q to p푀2, 훾2q. The collection of disjoint oriented simple curves 훾 on 푆 is called a suture.

Lemma 3.2.5. Suppose p푀, 훾q is a balanced sutured manifold. Let

푀 1 “ 푀 Y r0, 1s ˆ B푀 and 훾1 “ t1u ˆ 훾. B푀“t0uˆB푀

106 Then, there is a canonical isomorphism

Ψ : SHGp푀, 훾q Ñ SHGp푀 1, 훾1q.

Proof. When closing up p푀 1, 훾1q, we choose an auxiliary surface 푇 1 and glue r´1, 1sˆ

1 1 1 1 1 푇 to p푀1, 훾1q along 퐴p훾 q “ r´1, 1s ˆ 훾 by a map

푓 : B푇 1 Ñ 훾1.

Let 푀 1 “ 푀 1 Y r´1, 1s ˆ 푇 1 and suppose

Ă 1 1 1 B푀 “ 푅` Y 푅´

Ă 1 1 so that 푅˘ contains 푅˘p훾 q. If we choose an orientation preserving diffeomorphism 1 1 1 1 1 ℎ : 푅` Ñ 푅´, then we can glue r0, 1s ˆ 푅` to 푀 and get a marked closure 풟 “ p푌 1, 푅1 , 푟1, 푚1, 휂1q for p푀 1, 훾1q. ` Ă

Next, we want to show that there is a canonical way to view 풟1 as a closure of p푀, 훾q. Recall the annular neighborhood 퐴p훾q of 훾 is identified with r´1, 1s ˆ 훾.

1 1 1 There is a new annular neighborhood 퐴 p훾q “ r´ 2 , 2 s ˆ 훾 Ă 퐴p훾q. Now, let

푇 “ 푇 1 Yr0, 1s ˆ 훾, 푓 where 푓 : B푇 1 Ñ 훾1 “ t1u ˆ 훾 is the map defined above when choosing the auxiliary surface 푇 1, and the annuli glued to 푇 1 via 푓 are chosen to be r0, 1s ˆ 훾 Ă r0, 1s ˆ B푀. 1 1 Then, we can view r´ 2 , 2 sˆ푇 as attached to p푀, 훾q along the annular neighborhood 1 1 1 퐴 p훾q of 훾. Let 푀 “ 푀 Y r´ 2 , 2 s ˆ 푇 , and we want to show that there is a canonical

Ă 107 way to identify

1 1 1 푀 zintp푀q – r´1, 1s ˆ p푅` \ 푅´q. (3.5)

See Figure 3-6. Ă Ă

1 1 1 1 푇 푇 r´ 2 , 2 s ˆ 푇

r´1, 1s ˆ 푇 1 1 6 r0, 1s ˆ B푀 0 intp푀q Figure 3-6: The collar r0, 1s ˆ B푀 and the auxiliary surfaces 푇 1 and 푇 .

1 Suppose B푀 “ 푅` Y 푅´, then there is a canonical way to identify 푅˘ with 푅˘: The surface 푅 p훾1q is identified with 푅 p훾1 q, and 푇 1 is identified with itself. The Ă˘ ˘ 1 rest of 푅˘ is the product annuli r0, 1sˆ훾. Hence, there is a canonical way to identify

1 1 1 Bp푀 zintp푀qq “ t´1, 1u ˆ p푅` \ 푅´q. (3.6)

Ă Ă Note it is obvious that

1 1 1 p푀 zintp푀qq – r´1, 1s ˆ p푅` \ 푅´q,

Ă Ă so, by Proposition A3 in Baldwin and Sivek [4], there is a unique isotopy class of diffeomorphisms which restrict to the canonical identification (3.6) on the boundary.

108 Hence, there is a well defined map

Ψ : SHGp푀, 훾q Ñ SHGp푀 1, 훾1q induced by the identity on 퐻퐺p푌 1|푅1q.

Now, we want to introduce a new interpretation of the contact 2- and 3-handle attaching maps. Suppose p푀, 훾q is a balanced sutured manifold and p푀 1, 훾1q is obtained from p푀, 훾q by attaching a contact 2- or 3-handle ℎ “ p휑, 푆, 퐷3, 훿q. Let 푍 “ 푀 1zintp푀q. The idea is that, when we turn ℎ up-side down, we get a 1- or 0- handle as a result. To turn ℎ up-side down, we consider the manifold 푊 “ r0, 1sˆ푀 1. Choose the surface 푆 “ t0uˆB푀 Ă t0uˆ푀 1 ĂB푊 and the suture 훾 “ t0uˆ훾 Ă 푆. Then, 푊 can be viewed as a cobordism with sutured surface p푆, 훾q, from p푀, 훾q to

1 1 p푀1, 훾q “ p푀 Y r0, 1s ˆ B푀 Y 푍, 훾q.

In this case, 푍 is attached to t1u ˆ B푀 1 and can be viewed as a 1- or 0-handle ℎ_.

1 1 1 1 1 1 Let 푀1 “ 푀 Y r0, 1s ˆ B푀 and 훾1 “ t1u ˆ 훾 ĂB푀1. See Figure 3-7.

푀 푍 0 1 푀2 푀1 푀1

3 푊1 “ r0, 1s ˆ 푀 r0, 1s ˆ 퐷

? 1 푊 “ r0, 1s ˆ 푀 1 1 Figure 3-7: The product 푊 and 푊1, and the sutured manifolds 푀1, 푀2, and 푀1.

109 Now, we have a handle attaching map

1 1 퐶ℎ_ : SHGp´푀1, ´훾1q Ñ SHGp´푀1, ´훾q which is an isomorphism by Proposition 2.7.6. By Proposition 3.2.1, 푊 induces a map

퐹´푊 : SHGp´푀, ´훾q Ñ SHGp´푀1, ´훾q.

Finally, Lemma 3.2.5 leads to a map

1 1 1 1 Ψ : SHGp´푀 , ´훾 q Ñ SHGp´푀1, ´훾1q.

We have the following property.

Proposition 3.2.6. Suppose ℎ is a 2- or 3-handle attached to p푀, 훾q, and p푀 1, 훾1q is the resulting balanced sutured manifold. Suppose 퐶ℎ_ , 퐹´푊 and Ψ are defined as above, then the contact 2- or 3-handle attaching map can be re-interpreted as

´1 ´1 1 1 퐶ℎ “ Ψ ˝ 퐶ℎ_ ˝ 퐹푊 : SHGp´푀, ´훾q Ñ SHGp´푀 , ´훾 q.

Proof. Suppose p푀, 훾q is the original sutured manifold and p푀 1, 훾1q is the result of

3 1 1 attaching a contact 2- or 3-handle ℎ “ p휑, 푆, 퐷 , 훿q. Suppose p푀1, 훾q and p푀1, 훾1q are constructed as above. Recall 푊 “ r0, 1s ˆ 푀 1 is the product which can be viewed as a cobordism with a sutured surface p푆 “ t0u ˆ B푀, 훾q. When performing parallel closing up along p푆, 훾q, we get two marked closures 풟 “ p푌, 푅, 푟, 푚, 휂q and 풟1 “ p푌1, 푅, 푟, 푚1, 휂q of p푀, 훾q and p푀1, 훾q, respectively, and a cobordism 푊 from 푌 to 푌1 that induces the map 퐹 . From Proposition 2.7.6 and Lemma 3.2.5, we can see that 풟 is also a ´푊 x 1 110 1 1 ´1 ´1 marked closure of p푀 , 훾 q. Thus, the composition Ψ ˝ 퐶ℎ_ ˝ 퐹푊 is simply induced by the cobordism 푊 .

Let 푊 0, 1 푀 푊 , and we can view 푊 as a cobordism with sutured 1 “ r xs ˆ Ă 1 surface p푆1, 훾q. Let 푀2 “ t1u ˆ 푀 Y r0, 1s ˆ B푀. See Figure 3-7. By performing a suitable parallel closing up along p푆1, 훾q, we get two marked closures 풟 (the same as the one obtained in above paragraph) and 풟2 “ p푌2, 푅, 푟, 푚2, 휂q for p푀, 훾q and p푀2, 훾q, respectively, and a cobordism 푊1 from 푌 to 푌2. Recall ℎ 휑, 푆, 퐷3, 훿 is the contact handle attached to 푀. Then, 푊 can be “ p q x 3 viewed as obtained from 푊1 by attaching r0, 1s ˆ 퐷 through the map

푖푑 ˆ 휑 : r0, 1s ˆ 푆 Ñ r0, 1s ˆ B푀 Ă 푀2 ĂB푊1.

3 Accordingly, 푊 can be viewed as obtained from 푊1 by attaching r0, 1s ˆ 퐷 through the map x x 푖푑 ˆ 휑 : r0, 1s ˆ 푆 Ñ r0, 1s ˆ B푀 Ă 푀2 Ă 푌2 ĂB푊1.

This r0, 1s ˆ 퐷3 now becomes a 4-dimensional handle which wex write ℎ4 “ p푖푑 ˆ 휑, r0, 1s ˆ 푆, r0, 1s ˆ 퐷3q. If ℎ is a contact 2-handle, then ℎ4 is a 4-dimensional 2-handle. Recall, for a 2-handle ℎ, 푆 is an annulus. Suppose 훼 Ă intp푆q is the core of 푆, i.e.,

r훼s “ ˘1 Ă 퐻1p푆q – Z, then we can view ℎ as being attached along the curve 휑p훼q Ă B푀, and view ℎ4 as being attached along the curve

1 t u ˆ 휑p훼q Ă 푌 ĂB푊 . 2 2 1 111 x Note that 푊1 is diffeomorphic to r0, 1s ˆ 푌 , so such a 4-dimensional 2-handle attachment corresponds to a Dehn surgery on 푌 along the curve 휑p훼q Ă B푀 Ă 푌 . The x slope of the Dehn surgery can be compute from the framing of the 4-dimensional

1 2-handle attached and it is a 0-Dehn surgery with respect to the t 2 u ˆ B푀-surface framing. This description of 푊 coincides with the one in Proposition 2.7.8, and, hence, we can conclude that x

´1 ´1 퐶ℎ “ Ψ ˝ 퐶ℎ_ ˝ 퐹푊 .

If ℎ is a 3-handle, then ℎ4 is also a 3-handle. Now 푊 is diffeomorphic to the result of attaching a 4-dimensional 3-handle to t1u ˆ 푌 Ă r0, 1s ˆ 푌 . This also x coincides with the description in Proposition 2.7.10. Thus, we conclude the proof of Proposition 3.2.6.

3.3 Basic properties of handle attaching maps

For some special pairs of handles, we can cancel them both on the level of topology and the level of handle gluing maps.

Lemma 3.3.1. Suppose p푀, 훾q is a balanced sutured manifold, ℎ “ p휑, 푆, 퐷3, 훿q is a 0-handle, and ℎ1 “ p휑1, 푆1, 퐷31, 훿1q is a 1-handle such that the attaching map 휑1 maps one component of 푆1 to B푀 and the other component to B퐷3. Let p푀 1, 훾1q be the resulting balanced sutured manifold, then there is a diffeomorphism 휓01 : p푀, 훾q Ñ p푀 1, 훾1q so that the following is true.

1 1 (1) The map 휓01 restricts to identity outside a neighborhood of 휑 p푆 q X B푀.

1 (2) The map 휓01 is isotopic to the inclusion map 푀 ãÑ 푀

112 (3) We have

1 1 퐶ℎ1 ˝ 퐶ℎ “ SHGp휓01q : SHGp´푀, ´훾q Ñ SHGp´푀 , ´훾 q.

Proof. The two handles ℎ and ℎ1 can be canceled topologically, so it is straightforward

1 1 to find such a diffeomorphism 휓01 : p푀, 훾q Ñ p푀 , 훾 q satisfying the two conditions (1) and (2) above.

From Proposition 2.7.6, we know that a marked closure 풟1 “ p푌, 푅, 푟, 푚1, 휂q of

1 1 1 p푀 , 훾 q can also be thought of as a marked closure 풟 “ p푌, 푅, 푟, 푚 “ 푚 |푀 , 휂q of p푀, 훾q and the composition 퐶ℎ1 ˝ 퐶ℎ is induced by the identity map

푖푑 : 푆퐻퐺p´풟q Ñ 푆퐻퐺p´풟1q.

1 Let us now describe the map SHGp휓01q. If we start with the same closure 풟 “ p푌, 푅, 푟, 푚1, 휂q, then we can get a closure

1 풟 “ p푌, 푅, 푟, 푚 ˝ 휓01, 휂q

r of p푀, 훾q and the map SHGp휓01q is the induced by the map

푖푑 : 푆퐻퐺p´풟q Ñ 푆퐻퐺p´풟1q.

1 Since 푚 and 푚 ˝ 휓01 are isotopic in 푌 so that the isotopy is identity outside a

2 2 neighborhood of 휑 p푆 q Ă intp푌 zintpimp푟qqq, the canonical map from Φ´풟,´풟 is just

113 r the identity map. So, we have a commutative diagram

Φ푔 ´풟,´풟 푆퐻푀p´풟q / 푆퐻푀p´풟q r r 푖푑“SHGp휓0,1q 푖푑“퐶ℎ1 ˝퐶ℎ

% 푆퐻푀p´풟1q

By Lemma 2.4.6, the above commutative diagram implies that

1 1 퐶ℎ1 ˝ 퐶ℎ “ SHGp휓01q : SHGp´푀, ´훾q Ñ SHGp´푀 , ´훾 q.

Lemma 3.3.2. Suppose p푀, 훾q is a balanced sutured manifold, ℎ “ p휑, 푆, 퐷3, 훿q is a 1-handle attached to p푀, 훾q, and ℎ1 “ p휑1, 푆1, 퐷31, 훿1q is a 2-handle attached to p푀, 훾q Y ℎ. Suppose 훼1 Ă 푆1 is the core of the annulus 푆1, i.e., a simple closed curve

1 1 which generates 퐻1p푆q. Suppose the attaching map 휑 maps the core 훼 to a curve which is the union of two arcs

1 1 1 1 3 휑 p훼 q “ 푐1 Y 푐2 Ă Bp푀 Y 퐷 q. 휑

1 1 Here, 푐1 is an arc on B푀 disjoint with the suture 훾, and 푐2 intersects each of the two components of 훿z푆 once.

1 Suppose p푀2, 훾2q is the resulting manifold of attaching ℎ and ℎ , then there is a diffeomorphism 휓12 : p푀, 훾q Ñ p푀2, 훾2q so that the following is true.

1 1 (1) The map 휓12 restricts to identity outside a neighborhood of p휑p푆q Y 휑 p푆 qq X B푀.

114 1 (2) The map 휓12 is isotopic to the inclusion 푀 ãÑ 푀 . (3) We have

퐶ℎ1 ˝ 퐶ℎ “ SHGp휓12q : SHGp´푀, ´훾q Ñ SHGp´푀2, ´훾2q.

푅˘p훾1q

훿 푆 1 푆 푐2 훿

푅¯p훾1q 1-handle ℎ

푅-framing B푀1-framing

훽

A neighborhood of 훽 1 1 Figure 3-8: Top: the 1-handle ℎ and the part 푐2 of the core of the 2-handle ℎ . Bottom: in a neighborhood of the curve 훽, the longitudes of the two surface framings

Proof. The two handles can be canceled topologically, so the map 휓12 is straightforward to find.

Suppose p푀1, 훾1q is the result of attaching the 1-handle ℎ. By Proposition 2.7.6, we know that a marked closure 풟1 “ p푌, 푅, 푟, 푚1, 휂q of p푀1, 훾1q can also be viewed as a marked closure 풟 “ p푌, 푅, 푟, 푚 “ 푚1|푀 , 휂q of p푀, 훾q, and the map 퐶ℎ is induced by the identity map

푖푑 : 푆퐻퐺p´풟q Ñ 푆퐻퐺p´풟1q.

By Proposition 2.7.8, there is a curve 훽 Ă 푌 isotopic to 푚1p휑p훼qq Ă 푚1pB푀1q Ă 푌 so that if we perform a 0-Dehn surgery with respect to 푚1pB푀1q-surface framing,

115 then the resulting manifold 푌2 is a closure of p푀2, 훾2q. Note 훽 Ă intp푚1p푀1qq Ă 푌 , and the Dehn surgery can be supported in an arbitrarily small tubular neighborhood of 훽. Hence, the data 푟, 푅, 휂 in 풟1 is not influenced by the Dehn surgery along 훽, and, hence, we get a marked closure 풟2 “ p푌2, 푅, 푟, 푚2, 휂q of p푀2, 훾2q. By Proposition

2.7.8, the Dehn surgery corresponds to a cobordism 푊 from 푌1 to 푌2, obtained from r0, 1s ˆ 푌 by attaching a 0-framed 4-dimensional 2-handle along t1u ˆ 훽 Ă t1u ˆ 푌 . 1 x 1 So, we have a cobordism map

퐻퐺p´푊 q : 푆퐻퐺p´풟1q Ñ 푆퐻퐺p´풟2q

x that induces 퐶ℎ1 .

Let us now describe the map SHGp휓12q. If we fix the same closure 풟2 “ p푌2, 푅, 푟, 푚2, 휂q, then we can obtain a closure

풟 “ p푌2, 푅, 푟, 푚2 ˝ 휓12, 휂q

r for p푀, 훾q, and the map SHGp휓12q is induced by the map

푖푑 : 푆퐻퐺p´풟q Ñ 푆퐻퐺p´풟2q.

r By Lemma 2.4.6, to finish the proof, it suffices to show the commutativity, upto

116 multiplication by a unit, of the following diagram:

Φ´풟,´풟 푆퐻푀p´풟q / 푆퐻푀p´풟q (3.7) r

퐻푀p´푊 q˝푖푑 r 푖푑 x % 푆퐻푀p´풟2q

Now, let us describe Φ´풟,´풟 in details. The key observation is that we can isotope 1 훽 into 푅`p훾1q or 푅´p훾1q andr then to a curve 훽 Ă 푟1pt푡u ˆ 푅q for any 푡 P p´1, 1q. 1 1 1 1 The reason is that, from the hypothesis of the lemma, we know that 휑 p훼 q “ 푐1 Y 푐2, 1 1 with 푐1 having been contained in one of 푅`p훾1q and 푅´p훾1q, and 푐2 being possible to be isotoped into the same part of 푅p훾1q within the 1-handle ℎ. The surgery on 훽 has slope p`1q with respect to the contact framing and 0 with respect to the B푀1-surface framing. It is straightforward to see that, after the isotopy,

1 the surgery becomes a p˘1q-surgery along 훽 with respect to the surface 푟1p푅 ˆ t푡uq. See Figure 3-8. We can go through contact framing again to fix the sign: Since 훽1 does not intersect the dividing set on 푟1p푅 ˆ t푡uq, we see that it is a p`1q-surgery. When reversing the orientation to deal with ´풟 and ´풟, it becomes a p´1q-surgery and corresponds to a positive Dehn twist on 푅. Hence, from the definition of the r canonical map Φ´풟,´풟 in Baldwin and Sivek [4], we know that this canonical map 1 is induced by a cobordismr 푊 that is obtained from r0, 1s ˆ 푌 by attaching a 4- dimensional 2-handle to along 훽1 Ă t1u ˆ 푌 . Yet 푊 and 푊 1 are diffeomorphic since x 훽 and 훽1 are isotopic and the framing of the handle glued are also the same. Hence, x x diagram (3.7) indeed commutes and we are done.

Lemma 3.3.3. Suppose p푀, 훾q is a balanced sutured manifold, ℎ “ p휑, 푆, 퐷3, 훿q is a 2-handle and ℎ1 “ p휑1, 푆1, 퐷31, 훿1q is a 3-handle. Suppose 훼 Ă intp푆q is a curve

117 which represents a generator of 퐻1p푆q, then we require that 훼 is mapped to a curve on B푀 which intersects the suture 훾 twice and bounds a disk 퐷 on B푀. Hence, a retraction of this disk 퐷 union with one component of B퐷3z푆 will become a new 2 spherical boundary 푆 of the resulting manifold p푀1, 훾1q of attaching ℎ to p푀, 훾q. We further require that the attaching map 휑1 maps 푆1 “B퐷31 to 푆2. See Figure 3-9.

1 Suppose p푀2, 훾2q is the resulting manifold of attaching ℎ and ℎ , then there is a diffeomorphism 휓23 : p푀, 훾q Ñ p푀2, 훾2q so that the following is true.

(1) The map 휓23 restricts to an identity outside a neighborhood of 퐷 Ă 푀.

1 (2) The map 휓23 is isotopic to the inclusion 푀 ãÑ 푀 . (3) We have

퐶ℎ1 ˝ 퐶ℎ “ SHMp휓23q : SHMp´푀, ´훾q Ñ SHMp´푀2, ´훾2q.

the 2-handle ℎ

푆2 “ 휑1p푆1q

훾 B푀 퐷 @ @ 휑p훼q Figure 3-9: The 2-handle ℎ and the sphere 푆2 along which ℎ1 is attached.

Proof. The two handles can be canceled topologically, so the map 휓23 is straightforward to find.

Suppose p푀0, 훾0q “ p푀, 훾q. By Proposition 2.7.8 and Proposition 2.7.10, for

푖 “ 0, 1, 2, there are suitable marked closures 퐷푖 “ p푌푖, 푅푖, 푟푖, 푚푖, 휂푖q for p푀푖, 훾푖q. The

118 map 퐶ℎ is induced by a cobordism ´푊 from ´푌0 to ´푌1 so that 푊 is obtained from r0, 1sˆ푌 by attaching a 4-dimensional 2-handle r0, 1sˆ퐷3 along a curve 훽 Ă t1uˆ푌 . 0 x x 0 1 1 The map 퐶ℎ1 is induced by a cobordism ´푊 from ´푌1 to ´푌2 so that 푊 is obtained from r0, 1s ˆ 푌 by attaching a 4-dimensional 3-handle r0, 1s ˆ 퐷31 along a sphere in 1 x x 3 31 t1u ˆ 푌1. The 3-dimensional handles 퐷 and 퐷 are a pair of handles which can be canceled topologically, so the corresponding pair of 4-dimensional handles will also be canceled topologically. Hence the composition of the cobordism 푊 Y 푊 1 is diffeomorphic to the product cobordism r0, 2sˆ푌 . We can think of the identification 0 x x t2u ˆ 푌0 with 푌2 as being induced by the diffeomorphism 휓23, and, hence, by Lemma 2.4.6, we have an equality

퐶ℎ1 ˝ 퐶ℎ “ SHMp휓23q

Lemma 3.3.4. Suppose p푀, 훾q is a balanced sutured manifold. Suppose further that ℎ “ p휑, 푆, 퐷3, 훿q and ℎ1 “ p휑1, 푆1, 퐷31, 훿1q are two contact handles attached to p푀, 훾q which give rise to p푀 1, 훾1q. If the two attaching maps have disjoint images:

휑p푆q X 휑1p푆1q “ H, then the two handle attaching maps commute:

1 1 퐶ℎ ˝ 퐶ℎ1 “ 퐶ℎ1 ˝ 퐶ℎ : SHGp´푀, ´훾q Ñ SHGp´푀 , ´훾 q.

Proof. From Proposition 2.7.6, 2.7.8, and 2.7.10, we know that the handle attaching maps between canonical modules are basically induced by cobordism maps from cobordisms which are either products or those obtained from products by adding a 4-dimensional 2- or 3-handle. The hypothesis in the lemma implies that the at-

119 tachments of those 4-dimensional handles can be moved apart, and, hence, the corresponding cobordism maps commute with each other. As a result, the inducing handle attaching maps between canonical modules also commute.

Remark 3.3.5. Suppose we first attach ℎ and then attach ℎ1 so that the index of ℎ is no smaller than that of ℎ1, then, by an isotopy, we can always move them apart. Hence, such handle attachments always commute.

Lemma 3.3.6. Suppose p푀, 훾q is a balanced sutured manifold with a local contact structure defined in a collar of B푀. Suppose p푀 1, 훾1q is another balanced sutured manifold. Suppose 푓 : p푀, 훾q Ñ p푀 1, 훾1q is a diffeomorphism and 푓 ´1 pulls back the local contact structure. Suppose ℎ1 is a contact handle attached to p푀 1, 훾1q, and, via 푓, we can regard ℎ1 as a contact handle ℎ attached to p푀, 훾q. Suppose further that the map 푓 extends to a diffeomorphism

푓˜ : 푀 Y ℎ Ñ 푀 1 Y ℎ1 which preserves the contact structure on where it exits. Then, we have an equality:

˜ 퐶ℎ1 ˝ SHGp푓q “ SHGp푓q ˝ 퐶ℎ.

Proof. The instanton version is proved in Baldwin and Sivek [6]. The monopole version is the same.

As explained in the introduction, we are also interested in the case when p푀, 훾q is included into a disjoint union p푀, 훾q \ p푁, 훿q where p푁, 훿q is equipped with a contact structure 휉 so that B푁 is a convex surface and 훿 is the dividing set. Then,

120 from Ozbagci [72], we know that p푁, 휉q admits a contact handle decomposition

p푁, 훿q “ ℎ1 Y ... Y ℎ푛.

We can regard those contact handles as attached to p푀, 훾q but all attaching maps have images disjoint from B푀. From this point of view, there is a map:

퐶ℎ푛 ˝ 퐶ℎ푛´1 ˝ ... ˝ 퐶ℎ1 : SHGp´푀, ´훾q Ñ SHGp´p푀 \ 푁q, ´p훾 Y 훿qq.

We want to prove that this map is independent of the contact handle decompositions of p푁, 훿q. The idea is that, essentially, this map is the identity on 푀 tensoring with the contact element of 푁. The proof will become easier if we require that there is no 3-handles existing in the handle decomposition of p푁, 훿q, as such decompositions are related to partial open book decompositions of p푁, 훿q. We will not introduce the basic definitions of partial open book decompositions or positive stabilizations, and readers are referred to [5, 6, 42].

Lemma 3.3.7 (Juhász and Zemke [42], Section 4.1). Suppose p푁, 훿q is a balanced sutured manifold and 휉 is a contact structure on 푁 so that B푁 is convex and 훿 is the dividing set. The the following two objects are in one-to-one correspondence to each other: (1). A partial open book decomposition of p푁, 훿, 휉q. (2). A handle decomposition of p푁, 훿, 휉q with no 3-handles.

Lemma 3.3.8 (Honda, Kazez, and Matić [36], Theorem 1.3). Suppose p푀, 훾q is a balanced sutured manifold, and 휉 is a positive contact structure on p푀, 훾q so that B푀 is a convex surface and 훾 is the dividing set. Then, p푀, 훾q admits a partial open book decomposition. Furthermore, for any two partial open book decompositions of

121 p푀, 훾q, one can perform positive stabilizations on each finitely many times so that the two resulting partial open book decompositions are isotopic.

Remark 3.3.9. Lemma 3.3.8 is also known as the relative Giroux correspondence (RGC).

Lemma 3.3.10 (Juhász and Zemke [42], Lemma 4.7). Suppose p푁, 훿q is a balanced sutured manifold and 휉 is a contact structure on 푁 so that B푁 is convex and 훿 is the dividing set. Suppose p푆, 푃, ℎq is a partial open book decomposition of p푁, 훿, 휉q, and p푆1, 푃 1, ℎ1q is a positive stabilization of p푆, 푃, ℎq. Suppose ℋ and ℋ1 are two contact handle decompositions of p푁, 훿, 휉q arising from p푆, 푃, ℎq and p푆1, 푃 1, ℎ1q, respectively. Then, ℋ1 can be obtained from ℋ by adding pairs of canceling index 1- and 2-handles (See Lemma 3.3.2).

Lemma 3.3.11. Suppose p푀, 훾q is a balanced sutured manifold and p푁, 훿q is a balanced sutured manifold equipped with a contact structure 휉 so that B푁 is convex and 훿 is the dividing set. Suppose we have two different ways to decompose p푁, 훿, 휉q into contact handles:

1 1 p푁, 훿q “ ℎ1 Y ... Y ℎ푛 “ ℎ1 Y ... Y ℎ푚 so that neither decompositions involve 3-handles. Then, we can regard those handles as attached to 푀 and have an equality

. 퐶 ... 퐶 퐶 1 ... 퐶 1 : SHG 푀, 훾 SHG 푀 푁 , 훾 훿 . ℎ푛 ˝ ˝ ℎ1 “ ℎ푚 ˝ ˝ ℎ1 p´ ´ q Ñ p´p \ q ´p Y qq

Proof. The proof is a combination of Lemmas 3.3.7, 3.3.8, 3.3.10, 3.3.2, and 3.3.6.

Remark 3.3.12. This is essentially the way Baldwin and Sivek defined a contact invariant in sutured instanton Floer theory in [6].

122 As explained in Baldwin and Sivek [7], one advantage one can take is that, in the monopole theory, the construction of contact elements is independent of the relative Giroux correspondence as in Lemma 3.3.8. Following this principle, in the monopole settings, we also want to reproduce Lemma 3.3.11 without using Lemma 3.3.8. In the following proof, the requirement that there is no 3-handle is unnecessary, so we also remove it from the hypothesis.

Proposition 3.3.13. In the monopole settings, in Lemma 3.3.11, if we allow 3- handles to exist in both of the decompositions, then the same conclusion still holds.

Proof. Recall we have two different contact handle decompositions:

1 1 p푁, 훿q “ ℎ푛 Y ... Y ℎ1 “ ℎ푚 Y ... Y ℎ1.

Note both contact handle decompositions must have at least one 0-handle. Let p푁0, 훿0q be a 0-handle or a 3-ball with one simple closed curve being the suture on

1 its boundary, we can view all other handles ℎ푖 or ℎ푗 as being attached to 푁0. By Lemma 3.3.4, we can assume that the handles are ordered so that their indices are non-decreasing. Suppose p푁1, 훾1q is obtained from p푁0, 훿0q by attaching all 0- and 1-handles in tℎ푖u, and p푁2, 훿2q is obtained from attaching all remaining 2- and

3-handles to p푁1, 훿1q. There is a contactomorphism 푔 : 푁2 Ñ 푁. As in Definition

2.2.2, we can use some auxiliary data p푇, 푓q to form a pre-closure 푁1 of p푁1, 훿1q. We require the following to be true. r (1) 푇 is connected and has large enough genus.

(2) The contact structure 휉1 on 푁1 arising from the handle description extends ˜ to a contact structure 휉1 by using an arc configuration on the auxiliary surface 푇 , as explained in Section 2.7.

123 By Proposition 2.7.6, 푁1 is also a pre-closure of p푁0, 훿0q. From Section 2.7, we know that 휉˜ also extends the canonical contact structure 휉 on 푁 . Write 1 r 0 0

B푁1 “ 푅` Y 푅´.

r By Proposition 2.7.8 and Proposition 2.7.10, there are curves 훽1, ..., 훽푠 Ă intp푁1q Ă intp푁1q Ă 푁1 and spheres 푆1, ..., 푆푢 Ă intp푁1q Ă 푁1 so that if we perform 0-surgeries along all 훽 and perform inverse connected sum operations along all 푆 , then we will 푖r r 푗 get a pre-closure 푁2 for p푁2, 훿2q so that B푁2 “B푁1. As discussed in Baldwin and Sivek [5], the Dehn surgeries can be made to be contact `1 surgeries and the inverse r r r connected sum operations can also be performed to preserve contact structures. So, ˜ there is a contact structure 휉2 on 푁2 extending the contact structure 휉2 on 푁2 coming from its handle description. r 1 1 1 1 1 1 We can similarly form p푁1, 훿1q, p푁2, 훿2q and a contactomorphism 푔 : 푁2 Ñ 푁. We further require the following.

1 1 1 (3) The pre-closure 푁1 “ 푁1 is also a pre-closure for p푁1, 훿1q, and the contact structure 휉˜ also extends the contact structure 휉1 on 푁 1 coming from its handle 1 r r 1 1 description. This requirement can be achieved by choosing an auxiliary surface 푇 with a large enough genus.

1 1 Similarly, there are curves 훽푖 Ă 푁1 and spheres 푆푗 Ă 푁1 so that performing suitable surgeries along these objects will result in a pre-closure 푁 1 carrying a contact r r 2 structure 휉˜ which extends the contact structure 휉1 on 푁 1 coming from its handle 2 2 2 r description. Note the boundary of all pre-closures are identified to be 푅` Y 푅´, and ˜ ˜ ˜ all contact structures 휉1, 휉1, and 휉 are identified in a collar of 푅` Y 푅´. For later convenience of describing the canonical map Φ 1 in (3.8), we need a diffeomor- ´풟2,´풟2

124 p p phism

1 퐶 : 푁2 Ñ 푁2 so that r r

1 ´1 1 퐶|p푁2q “ p푔 q ˝ 푔 : 푁2 Ñ 푁2.

If we pick a non-separating simple closed curve 푐 Ă 푇 then 푐 will correspond to two curves 푐` Ă 푅` and 푐´ Ă 푅´. We require that 푐` Y 푐´ is disjoint from all the dividing curves on 푅` Y 푅´. This can be achieved since the dividing set on 푅` consists of only two parallel non-separating simple closed curves in the construction Section 2.7, and, by enlarging the genus of 푇 , there are plenty of candidates for 푐. We further require the following.

(4) 퐶 preserves 푐 ˆ r´1, 1s Ă 푁2. (5) There exists a gluing diffeomorphism ℎ : 푅 Ñ 푅 which preserves the r ` ´ contact structures near 푅` and 푅´ and identifies 푐` with 푐´.

(6) There exists a smooth curve 휂 Ă 푅` intersecting 푐 transversely once and is disjoint from the dividing curves on 푅`. (7) 퐶 preserves 휂. Note condition (6) can be achieved by enlarging the genus of 푇 just as we argued in the existence of 푐. Condition (5) can be achieved since, to preserve the contact structures, we essentially only need to choose an ℎ, which identifies the dividing sets on 푅` and 푅´, and there is still some freedom to make ℎ to preserve 푐. Claim. A diffeomorphism 퐶 satisfying (4) and (7) can be chosen.

1 To prove this claim, first note that 푁2 and 푁2 are pre-closures of p푁2, 훿2q and p푁 1 , 훿1 q, respectively, and p푁 , 훿 q and p푁 1 , 훿1 q are both diffeomorphic to p푁, 훿q. As 2 2 2 2 r2 2 r 1 a result, 푁2 and 푁2 are both pre-closures of p푁, 훿q, and their boundaries are both identified with 푅 Y푅 . As a result, we know that 푁 and 푁 1 must be diffeomorphic. r `r ´ 2 2 125 r r The diffeomorphism 퐶 is chosen to be the ’identity’ on 푁.

Second, from the construction of pre-closures in Kronheimer and Mrowka [53],

1 we know that there are surfaces 푇2 and 푇2 so that

1 1 1 푁2 “ 푁2 Y r´1, 1s ˆ 푇2 and 푁2 “ 푁2 Y r´1, 1s ˆ 푇2.

r r 1 The conclusion we arrived in the above paragraph that 푁2 and 푁2 are diffeomorphic implies that 푇 and 푇 1 are isomorphic. 2 2 r r

Recall 푇 is the auxiliary surface we used to construct a pre-closure 푁1 of p푁1, 훿1q. From Propositions 2.7.6, 2.7.8, and 2.7.10, when 푇 has large enough genus, part of r 1 푇 , which we call 푇0, naturally embeds into both 푇2 and 푇2. We can pick 푐 and 휂 being non-separating simple closed curves inside 푇0.

To prove the claim, we need to show that the diffeomorphism

1 ´1 1 p푔 q ˝ 푔 : 푁2 Ñ 푁2 extends to a diffeomorphism

1 퐶 : 푁2 Ñ 푁2

1 ´1 so that 퐶|r´1,1sˆ푐 “ 푖푑 and 퐶p휂q “ 휂.r Ther diffeomorphism p푔 q ˝ 푔 induces a diffeomorphism

1 ´1 1 p푔 q ˝ 푔|B푇2 : B푇2 ÑB푇2.

1 ´1 1 To obtain 퐶, we need only to extend p푔 q ˝푔|B푇2 to a diffeomorphism 퐶|푇2 : 푇2 Ñ 푇2 so that 퐶|푇2 p푐q “ 푐 and 퐶|푇2 p휂q “ 휂. Such a construction is straightforward.

Back to the proof of Proposition 3.3.13. We can use the same auxiliary data p푅`, ℎ, 휂q to obtain a marked closure 풟0 “ p푌0, 푅`, 푟, 푚0, 휂q of p푁0, 훿0q and two

126 ´1 1 1 1 ´1 marked closures 풟2 “ p푌2, 푅`, 푟, 푔 , 휂q and 풟2 “ p푌2 , 푅`, 푟, p푔 q , 휂q of p푁, 훿q. Note that inside all such three marked closures, the curve 푐 becomes a torus: there are two annuli r´1, 1s ˆ 푐 Ă r´1, 1s ˆ 푇 and r´1, 1s ˆ 푐 Ă r´1, 1s ˆ 푅`, and the way we obtain the marked closures identifies the boundaries of these two annuli

1 1 1 and hence gives rise to a torus 푆 ˆ 푐. Let Σ0 P 푌0, Σ2 Ă 푌2, and Σ2 Ă 푌2 denote the corresponding tori.

Let 푇 be an auxiliary surface for p푀, 훾q and 푓˜ : B푇 Ñ 훾 be the gluing map. Let

r r 1 푀 “ 푀 Yr´1, 1s ˆ 푇 and B푀 “ 푅` Y 푅´. 푓˜ Ă r Ă r r Suppose there is a non-separating simple closed curve 푐˜ Ă 푇 and a diffeomorphism

r ˜ ℎ : 푅` Ñ 푅´

r r so that ℎ˜pt1u ˆ 푐˜q “ t´1u ˆ 푐.˜ With ℎ˜ we obtain a marked closure

˜ 풟 “ p푌, 푅`, 푟,˜ 푚,˜ 휂˜q of p푀, 훾q and there is a torus Σ Ă 푌 corresponding to 푐˜.

We can now form a marked closure of p푀, 훾q \ p푁0, 훿0q. Cut 푌 open along Σ,

2 2 and let 푌 “ 푌 zpintp푁pΣqqq. We have B푌 “ Σ` Y Σ´. Cut 푌0 along Σ0 and let 2 2 푌0 “ 푌0zpintp푁pΣ0qqq with B푌0 “ Σ0,` Y Σ0,´. Let

휏 :Σ Ñ Σ0

be a diffeomorphism so that 휏pΣ X 휂˜q “ Σ0 X 휂. We can use 휏 to glue Σ` to Σ0,´ and Σ´ to Σ0,`. Let 푌 be the resulting manifold. There are corresponding data

p 127 p푅, 푟,ˆ 푚,ˆ 휂ˆq so that 풟 “ p푌 , 푅, 푟,ˆ 푚,ˆ 휂ˆq p

1 is a marked closure of p푀, 훾q \ p푁p 0, 훾0pq. Ifp we use 푌2 or 푌2 and the same 휏, we can construct similarly two marked closures

1 1 1 풟2 “ p푌2, 푅, 푟,ˆ 푚ˆ 2, 휂ˆq and 풟2 “ p푌2 , 푅, 푟,ˆ 푚ˆ 2, 휂ˆq

p p p p p p of p푀, 훾q \ p푁, 훿q. The diffeomorphism 퐶 extends by identity [Using condition (4)] to a diffeomorphism which we also called 퐶:

1 1 퐶 : 푌2zintpimp푟ˆqq Ñ 푌2 zintpimp푟ˆ qq.

p p

There are Legendrian curves and spheres 훽푖, 푆푗 Ă p푌 z푁pΣqq Ă 푌 so that if we perform contact `1-surgeries along 훽 and perform inverse connected sum operations 푖 p along 푆푗, then the resulting manifold will be exactly 푌2. Hence, there is a cobordism 푊 from 푌 to 푌 so that 푊 is obtained obtained from r0, 1s ˆ 푌 by gluing 0-framed 2 p 4-dimensional 2-handles along all 훽 Ă t1u ˆ 푌 and gluing 4-dimensional 3-handles p p 푖 p along all 푆 Ă t1u ˆ 푌 . Pick the product 휈 “ r0, 1s ˆ 휂 Then, the map 푗 p p 퐻푀p´푊 ;Γ´휈q : 퐻푀p´푌 | ´ 푟ˆpt0u ˆ 푅q;Γ´휂^q Ñ 퐻푀p´푌2| ´ 푟ˆpt0u ˆ 푅q;Γ´휂^q

~ p p p p induces the map

퐶ℎ푛 ˝ ... ˝ 퐶ℎ2 : SHMp´p푀 \ 푁0q, ´p훾 Y 훿0qq Ñ SHMp´p푀 \ 푁q, ´p훾 Y 훿qq.

For simplicity, we will write 퐻푀p푊 q instead of 퐻푀p´푊 ;Γ´휈q.

128 ~ We will also need another interpretation of 푊 : Attaching 4-dimensional 2- and 3- handles to r0, 1s ˆ 푌 at t1u ˆ 푌 is equivalent to glue 4-dimensional 2- and 1-handles to r0, 1s ˆ 푌 at t0u ˆ 푌 . Suppose those handles are attached along curves 휃 , which 2 p 2 p 푖 correspond to 훽 , and along pairs of points p푝 , 푞 q, which correspond to 푆 . p 푖 p 푗 푗 푗 1 1 1 There are curves 훽푗 Ă 푌 and spheres 푆푗 Ă 푌2 as well. We can construct similarly a cobordism 푊 1 from ´푌 to ´푌 1 which induces the map p 2 p p p 퐶 1 ... 퐶 1 : SHM 푀 푁 , 훾 훿 SHM 푀 푁 , 훾 훿 . ℎ푚 ˝ ˝ ℎ2 p´p \ 0q ´p Y 0qq Ñ p´p \ q ´p Y qq

1 1 1 Just as for 푊 , there are curves 휃푖 Ă 푌2 corresponding to 훽푖 and pairs of points p푝1 , 푞1 q corresponding to 푆1 Ă 푌 1. 푗 푗 푗 2 p By Lemma 2.4.6, to show thatp

퐶 ... 퐶 퐶 1 ... 퐶 1 , ℎ푛 ˝ ˝ ℎ2 “ ℎ푚 ˝ ˝ ℎ2 it suffices to show that

. 1 Φ 1 ˝ 퐻푀p푊 q “ 퐻푀p푊 q, (3.8) ´풟2,´풟2

p p 1 where Φ 1 is the canonical map between the two marked closures ´풟2 and ´풟 ´풟2,´풟2 2 of p´p푀 \푁q, ´p훾 Y훿qq. The diffeomorphism 퐶 is used to construct such a canonical p p x p map. As in Section 5 of Baldwin and Sivek [4], there is a map

휙퐶 : 푅 Ñ 푅,

p p 퐶 so that if we cut 푌2 open along 푅 and re-glue using 휙 , then the resulting 3-manifold is 푌 1. 2 p p 129 The way we choose 퐶 makes sure that the map 휙퐶 is the identity on the part of 푅 coming from 푅` (which was used to build the marked closure 풟 of p푀, 훾q). Hence, we can decompose 휙퐶 into Dehn twists p r

퐶 푒1 푒푛 휙 „ 퐷푎1 ˝ ... ˝ 퐷푎푛

so that all 푎푘 are disjoint from the part of 푅 coming from 푅`.

In general, there are both positive andp negative Dehn twists.r However, for simplicity, we only deal with the case when all 푒푖 “ ´1. The general case follows from a similar argument. As in Section 5 in Baldwin and Sivek [4], let 푊푐 be the cobordism

1 from 푌2 to 푌2 obtained from r0, 1s ˆ 푌2 by attaching 4-dimensional 2-handles along all the curves 푎1, ..., 푎푞 Ă t1u ˆ 푌2, then the canonical map Φ 풟 , 풟1 is induced by p p p ´ 2 ´ 2 the cobordism 푊 . 푐 p p p Equation (3.8) is then equivalent to

. 1 퐻푀p푊 Y 푊푐q “ 퐻푀p푊 q. (3.9)

Note that the curves 휃푖 and pairs of points t푝푗, 푞푗u used to construct 푊 are all ˚ contained in 푌0z푁pΣ0q Ă 푌 , so intuitively there is nothing happened in the 푌 part of 푌 , and it should be possible to split off a product copy of 푌 . This idea is carried p out explicitly as follows. p Let 푈 be the surface depicted as in Figure 3-10. It has four vertical parts of the boundary which we call 휇1, ..., 휇4. Suppose each of them is parametrized by

2 2 r0, 1s. Recall we have 푌 and 푌0 by cutting open 푌 along Σ and 푌0 along Σ0, respectively. We have the gluing diffeomorphism 휏 to get 푌 . Now let 푊푒 be the cobordism obtained by gluing three parts r0, 1s ˆ 푌 2, 푈 ˆ Σ and r0, 1s ˆ 푌 2 where we p 0 130 푖푑 휏

푖푑 휏 휇2 휇4

휇1 휇3

2 2 r0, 1s ˆ 푌 푈 ˆ Σ r0, 1s ˆ 푌0

Figure 3-10: The three parts of the cobordism ´푊푒. The middle part is Σˆ푈, while the Σ directions shrink to a point in the figure.

2 use 푖푑 ˆ 푖푑 to glue r0, 1s ˆ B푌 to p휇1 Y 휇2q ˆ Σ and use 푖푑 ˆ 휏 to glue p휇3 Y 휇4q ˆ Σ

2 to r0, 1s ˆ B푌0 . The resulting manifold 푊푒 can be thought of as a cobordism from 1 1 1 p푌 \ 푌0q to 푌 . Similarly we can construct a cobordism 푊푒 from ´p푌 \ 푌2 q to ´푌2 and same cobordism is just one from 푌 1 to p푌 \ 푌 1q. From Theorem 2.5.1, we know p 2 2 p that 푊 and 푊 1 induces isomorphisms, so the equality (3.9) is equivalent to 푒 푒 p

1 . 1 1 퐻푀p푊푒 Y 푊 Y 푊푐 Y 푊푒q “ 퐻푀p푊푒 Y 푊 Y 푊푒q. (3.10)

1 1 On 푊푒 Y 푊 Y 푊푐 Y 푊푒 we can cut along the 3-manifold 푆 ˆ Σ as shown in the Figure 3-11, and glue back two copies of 퐷2 ˆ Σ along boundaries. The result is a cobordism 푊 “ 푊1 Y 푊2, where 푊1 – r0, 1s ˆ 푌 , and 푊2 is a cobordism from 푌 to 푌 1 obtained from r0, 1s ˆ 푌 1 by attaching 4-dimensional 1-handles at pairs of 0 2 x x x 2 x x points t푝푗, 푞푗u Ă t1u ˆ 푌2 and then attaching 4-dimensional 2-handles along curves

1 1 휃푖 Ă t1u ˆ 푌2 and 푎푘 Ă t1u ˆ 푌2. We can apply a similar argument to 푊푒 Y 푊 Y 푊푒, 1 1 1 1 1 and get 푊 “ 푊1 Y 푊2, where 푊1 “ 푊1 – r0, 1s ˆ 푌 , and 푊2 is a cobordism from 푌 to 푌 1 obtained from r0, 1s ˆ 푌 1 by attaching 4-dimensional 1-handles at 0 x 2 x x x 2x x 131 푌 푌0 푌 푌0

푊푒

Q @ Q @ @ Q @ 1-handles 푊 푆1 ˆ Σ Q @ 푊 푊 Q @ 1 2 QQ2-handles x x

1 푊푒

1 1 푌 푌2 푌 푌2

Figure 3-11: Cut along Σ ˆ 푆1 and glue back two copies of Σ ˆ 퐷2.

1 1 1 1 t푝푗, 푞푗u Ă t1u ˆ 푌2 and then attaching 4-dimensional 2-handles to 푌2 ˆ r0, 1s along 1 1 휃푖 Ă t1u ˆ 푌2 . There is a commutative diagram from the naturality of Künneth formula: (there is no room for writing down the local coefficients so we omit them from the notations)

˜ 푖 ˜ 퐻푀p´p푌 \ 푌0q| ´ p푅` Y 푅`qΓ´p휂˜q / 퐻푀p´푌 | ´ 푅`q b 퐻푀p´푌0| ´ 푅`q

1 1 1 퐻푀p푊 q 퐻푀p푊 q 퐻푀p푊1qb퐻푀p푊2q 퐻푀p푊1qb퐻푀p푊2q

x x x x x x 1 1 ˜ 푖 ˜ 1 퐻푀p´p푌 \ 푌2 q| ´ p푅` Y 푅`qq / 퐻푀p´푌 | ´ 푅`q b 퐻푀p´푌2 | ´ 푅`q

1 We have an identity 퐻푀p푊1q “ 퐻푀p푊1q since they are both induced by product . cobordisms. We claim that 퐻푀p푊 q “ 퐻푀p푊 1q. Since both cobordisms are exact x 2 x 2 symplectic, they both map contact elements to contact elements by Theorem 2.7.4. x x 1 Yet 푌0 is a surface fibration over 푆 with fibre 푅`, so 퐻푀p´푌0| ´ 푅`;Γ´휂q – ℛ by

132 Lemma 2.6.1, and the contact element in 퐻푀p푌0q is a generator of ℛ by Lemma . 3.1.6. From the Künneth formula, the maps 푖 and 푖1 are injective, so 퐻푀p푊 q “ 퐻푀p푊 1q. Finally, from Corollary 2.10 in Kronheimer and Mrowka [53], we know x that x

1 . . 1 . 1 1 퐻푀p푊푒 Y 푊 Y 푊푐 Y 푊푒q “ 퐻푀p푊 q “ 퐻푀p푊 q “ 퐻푀p푊푒 Y 푊 Y 푊푒q.

x x Hence, we conclude the proof of Proposition 3.3.13.

3.4 Contact cell decompositions

Definition 3.4.1. Suppose p푀 1, 훾1q is a balanced sutured manifold. By a sutured submanifold we mean a balanced sutured manifold p푀, 훾q so that 푀 Ă intp푀 1q.

In [42], Juhász and Zemke used contact cell decompositions to re-construct the gluing map originally introduced by Honda, Kazez, and Matić in [35]. The following definition is from [42], while contact cell decompositions have already been studied before by Giroux in [28] and Honda, Kazaz, and Matić in [35].

Definition 3.4.2. Suppose p푀, 훾q is a sutured submanifold of p푀 1, 훾1q and 휉 is a contact structure on p푍 “ 푀zintp푀q, 훾 Y 훾1q so that B푍 is convex and 훾 Y 훾1 is the dividing set. A contact cell decomposition of p푍, 휉q consists of the following data. (1) A non-vanishing contact vector field 푣 that is defined on a neighborhood of B푍 Ă 푍 with respect to which B푍 is a convex surface with dividing set 훾 Y 훾1. The flow of 푣 induces a diffeomorphism from B푍 ˆ 퐼 to a collar neighborhood of

B B푍. Under this diffeomorphism, 휈 corresponds to the vector field B푡 , B푀 is identified

133 with t0u ˆ B푀, and B푀 1 is identified with t1u ˆ B푀 1. We call

1 휈 “ 푣|B푀ˆ퐼 and 휈 “ 푣|B푀 1ˆ퐼 .

(2) Barrier surfaces

푆 ĂB푀 ˆ p0, 1q and 푆1 ĂB푀 1 ˆ p0, 1q that are isotopic to B푀 and B푀 1, respectively, and are both transverse to 푣. Write 푁 for the collar neighborhood of B푀 bounded by 푆, and 푁 1 for B푀 1 and 푆1, similarly. We call 푍1 “ 푍zintp푁 Y 푁 1q.

Note B푍1 “ 푆 Y 푆1 is a convex surface. (3) A Legendrian graph Γ Ă 푍1 that intersects B푍1 transversely in a finite collection of points along the dividing set on B푍1. Furthermore, Γ is tangent to 푣 in a neighborhood of B푍1 Ă 푍1. (4) A choice of regular neighborhood 푁pΓq Ă 푍1 of Γ such that 휉 is tight on 푁pΓq and B푁pΓqzB푍1 is a convex surface. We also require that 푁pΓq X B푍1 is a collection of disks 퐷 with Legendrian boundary such that each boundary B퐷 has 푡푏p퐷q “ ´1. We also assume that 푁pΓq meets B푍1 tangentially along the Legendrian unknots forming B푁pΓq.

1 (5) A collection of 2-cells 퐷1, ..., 퐷푛 inside 푍 zintp푁pΓqq with Legendrian bound-

1 ary on B푍 YB푁pΓq so that each B퐷푖 has 푡푏pB퐷푖q “ ´1. Furthermore, the following two conditions hold.

1 (a) Each component of 푍 zp푁pΓq Y 퐷1 Y ... Y 퐷푛q is a 3´ball on which 휉 is tight.

1 (b) The disks 푁pΓq X B푍 and the Legendrian arcs B퐷푖 XB푍 induces a sutured

134 cell decomposition (see Definition 3.1 in [42]), with the dividing set induced by 푣.

A fixed p푍, 휉q may have different contact cell decompositions. We have the following proposition to relate them.

Proposition 3.4.3 (Juhasz and Zemke [42], Proposition 3.6). Suppose p푀, 훾q is a contact submanifold of p푀 1, 훾1q, and 휉 is a contact structure on 푍 “ 푀 1zintp푀q so that B푍 is convex and 훾 Y 훾1 is the dividing set. Suppose further that there are two different contact cell decompositions 풞1 and 풞2 of p푍, 휉q, then there is a finite sequence of moves, each chosen from one of the following four types, which relates

풞1 with 풞2.

(1) The isotopy: Replacing a contact cell decomposition 풞 with another one 휑p풞q, where 휑 is a diffeomorphism of 푍 which is the identity on B푍 and is isotopy to the identity on 푍 relative to B푍.

(2) The 0-1 cancelation: Subdividing a Legendrian edge of the graph Γ, or adding a new Legendrian edge that has one endpoint on Γ and the other endpoint disjoint from Γ and all 2-cells in 풞.

(3) The 1-2 cancelation: Adding a Legendrian edge 휆 to the graph Γ, and adding a convex 2-cell 퐷 with 푡푏pB퐷q “ ´1. We also require that the interior of 휆 and 퐷 are disjoint from Γ and all other 2-cells in 풞, and B퐷 “ 푐 Y 푐1, with 푐 being a Legendrian arc on a neighborhood of 휆 and 푐2 ĂB푍1 YB푁pΓq being a Legendrian arc disjoint from the dividing set on B푍1 YB푁pΓq.

(4) The 2-3 cancelation: Adding a convex disk 퐷 with B퐷 Ă 푍1 YB푁pΓq and 푡푏pB퐷q “ ´1. Also 퐷 is disjoint from Γ and all other 2-cells in 풞.

135 3.5 The construction of the gluing maps

Now we are ready to define general gluing maps. Suppose p푀, 훾q is a sutured submanifold of a balanced sutured manifold p푀 1, 훾1q. Suppose 푍 “ 푀 1zintp푀q is equipped with a contact structure 휉 so that B푍 is a convex surface with dividing set 훾 Y 훾1. Suppose 풞 is a contact cell decomposition of p푍, 휉q, in the sense of Definition 3.4.2.

The contact vector field 휈 induces a diffeomorphism 휑휈 : p푀, 훾q Ñ p푀 Y 푁, 훿q where

훿 is the dividing set on 푆 ĂB푁 with respect to 휈. Suppose ℎ1, ..., ℎ푛 is a handle decomposition of p푁 1, 훿1 Y 훾1q with no 3-cells. The existence of such a decomposition is guaranteed by Lemma 3.3.8 and Lemma 3.3.7. A contact cell decomposition gives rise to a set of contact handles: vertices of Γ are 0-handles, edges of Γ are 1-handles, 2-cells 퐷푖 are 2-handles, and the remaining

1 1 is a collection of 3-handles. Suppose we get a sequence of contact handles ℎ1, ..., ℎ푚 from it, then we define the map

1 1 Φ휉 : SHGp´푀, ´훾q Ñ SHGp´푀 , ´훾 q to be

Φ 퐶 1 ... 퐶 1 퐶 ... 퐶 SHG 휑 . 휉 “ ℎ푚 ˝ ˝ ℎ1 ˝ ℎ푛 ˝ ˝ ℎ1 ˝ p 휈q

Definition 3.5.1. The map Φ휉 is called a gluing map.

Proposition 3.5.2. The gluing map Φ휉 is well defined, i.e., independent of the choice of contact cell decompositions.

Proof. By Proposition 3.4.3, it suffices to show that each of the four moves does not influence the gluing map Φ휉. For isotopies, since 휑 is isotoped to identity, it induces the identity map on sutured monopole and instanton Floer homologies. For

136 0-1 cancelations, we apply Lemma 3.3.1; for 1-2 cancelations, we apply Lemma 3.3.2; for 2-3 cancelations, we apply Lemma 3.3.3. This concludes the proof.

Proposition 3.5.3. Suppose p푀, 훾q is a sutured submanifold of p푀 1, 훾1q, and p푀 1, 훾1q is a sutured submanifold of p푀 2, 훾2q. Suppose there are contact structures 휉 on 푍 “ 푀 1zintp푀q and 휉1 on 푍1 “ 푀 2zintp푀 1q, and their union 휉2 “ 휉 Y 휉1 is a contact structure on 푍2 “ 푀 2zintp푀q, so that the boundaries of corresponding manifolds are all convex surfaces and the sutures are all dividing sets. Then, we have an equality:

2 2 Φ휉1 ˝ Φ휉 “ Φ휉2 : SHGp´푀, ´훾q Ñ SHGp´푀 , ´훾 q.

Proof. We follow the idea from Juhász and Zemke [42]. Suppose 풞 is a contact cell decomposition of 푍 and 휈 is defined as in Definition 3.4.2. Suppose ℎ1, ..., ℎ푛 and

1 1 1 ℎ1, ..., ℎ푚 are chosen according to 풞, as explained in Definition 3.5.1. Suppose 풞 is 1 ˜ ˜ ˜1 ˜1 a contact cell decomposition of 푍 with 휈˜, ℎ1, ..., ℎ푠 and ℎ1, ..., ℎ푡 chosen similarly. Then, we have

1 1 1 Φ휉 ˝Φ휉 “ 퐶˜1 ˝...˝퐶˜1 ˝퐶˜ ˝...˝퐶˜ ˝SHGp휑휈˜q˝퐶ℎ ˝...˝퐶ℎ ˝퐶ℎ푛 ˝...˝퐶ℎ ˝SHGp휑휈q. ℎ푡 ℎ1 ℎ푠 ℎ1 푚 1 1

¯ ¯1 1 Suppose ℎ푖 “ 휑휈˜pℎ푖q and ℎ푖 “ 휑휈˜pℎ푖q, then by Lemma 3.3.6, we know that

SHGp휑 q ˝ 퐶 1 ˝ ... ˝ 퐶 1 ˝ 퐶 ˝ ... ˝ 퐶 “ 퐶¯1 ˝ ... ˝ 퐶¯1 ˝ 퐶¯ ˝ ... ˝ 퐶¯ . 휈˜ ℎ푚 ℎ1 ℎ푛 ℎ1 ℎ푚 ℎ1 ℎ푛 ℎ1

If we go through the definition of gluing maps again, we can see that theset ˜ ˜ ¯ ¯ ¯1 ¯1 of handles ℎ1, ..., ℎ푠 and the set of handles ℎ1, ..., ℎ푛, ℎ1, ..., ℎ푚 are attached to dis- ˜ ˜ joint regions, so we can switch their order by Lemma 3.3.4. The handles ℎ1, ..., ℎ푠 correspond to the neighborhood of B푀 2 bounded by B푀 2 and the barrier surface

137 ˜1 2 ˜1 ˜1 ¯1 ¯1 푆 Ă 푍 . The handles ℎ1, ..., ℎ푡 and ℎ1, ..., ℎ푚 correspond to Legendrian graphs and 2-cells and tight 3-balls in 푍 and 푍1. So they are all basic elements to form a contact

1 cell decomposition of 푍 Y 푍 . The remaining handles ℎ1, ..., ℎ푛 correspond to the neighborhood of B푀 1 in 푍 bounded by B푀 1 and the barrier surface 푆1. They consist of only 0-, 1- and 2- handles by Lemma 3.3.8, so we can consider them as Legendrian graphs as well as some 2-cells. Hence, the whole composition

퐶˜1 ˝ ... ˝ 퐶˜1 ˝ 퐶˜ ˝ ... ˝ 퐶˜ ˝ 퐶¯1 ˝ ... ˝ 퐶¯1 ˝ 퐶¯ ˝ ... ˝ 퐶¯ ˝ SHGp휑휈q ℎ푡 ℎ1 ℎ푠 ℎ1 ℎ푚 ℎ1 ℎ푛 ℎ1 can be thought of as arising from some particular contact cell decomposition of 푍Y푍1, and, hence, the proposition follows from Proposition 3.5.2.

Suppose p푀, 훾q is a sutured submanifold of p푀 1, 훾1q. If 푍 “ 푀 1zintp푀q is the product B푀 ˆ r0, 1s equipped with an 퐼-invariant contact structure so that for any 푡 P r0, 1s, B푀 ˆ t푡u is convex with 훾 ˆ t푡u being the dividing set, then we expect the contact gluing map to be the ’identity’. This is made precise by the following proposition.

Proposition 3.5.4. Suppose p푀, 훾q is a sutured submanifold of p푀 1, 훾1q and 휉 is a compatible contact structure on 푍 “ 푀 1zintp푀q. Suppose there is a Morse function 푓 and a contact vector field 휈 on 푍 so that

(1). There are no critical points of 푓 and

푓pB푀q “ 0, 푓pB푀 1q “ 1.

(2). The contact vector field 휈 is gradient like, i.e., 휈p푓q ą 0 everywhere in 푍.

138 Then, we have the equality

1 1 Φ휉 “ SHGp휑휈q : SHGp´푀, ´훾q Ñ SHGp´푀 , ´훾 q.

where 휑휈 is the diffeomorphism induced by 휈.

Proof. With Lemmas 3.3.1, 3.3.2, and 3.3.3, the proof is exactly the same as the proof of Proposition 5.1 in Juhász and Zemke [42].

At the end of the section, we want to relate the general gluing maps with the contact handle attaching maps introduced in Section 2.7. Suppose p푀, 훾q is a balanced sutured manifold and ℎ is a contact handle attached to p푀, 훾q. Let p푀 1, 훾1q be the resulting balanced sutured manifold. Note that p푀, 훾q is not a sutured submanifold of p푀 1, 훾1q in the sense of Definition 3.4.1, where we require 푀 Ă intp푀 1q. The way to resolve this issue is to glue a product region r0, 1s ˆ B푀ˆ to 푀 along t0u ˆ B푀 and glue ℎ to t1uˆB푀. This is made precise by the following definition from Juhász and Zemke [42].

Definition 3.5.5. Suppose p푀, 훾q is a sutured submanifold of p푀 1, 훾1q, and 휉 is a contact structure on 푍 “ 푀 1zintp푀q so that B푍 is convex and 훾 Y 훾1 is the dividing set. Suppose further that there is a contact vector field 휈 on 푍 and a decomposition

푍 “ 푍0 Y ℎ such that the following is true. (1) The contact vector field 휈 points into 푍 on B푀 ĂB푍 and points out of 푍 on B푀 1 ĂB푍.

(2) We have 푍0 – r0, 1s ˆ B푀 and B푀 is identified with t0u ˆ B푀 ĂB푍0. Also, 휈 is non-vanishing on 푍0, pointing into 푍0 on t0uˆB푀, pointing out of 푍0 on t1uˆB푀 and each flow line of 휈 on 푍0 is an arc from t0u ˆ B푀 to t1u ˆ B푀.

139 (3) We have that ℎ is a topologically 3-ball with a piece-wise smooth boundary and is tight under 휉.

(4) We can view ℎ as a contact 푘-handle, for 푘 “ 0, 1, 2, 3, attached to 푀 Y 푍0, with corner smoothed. Then, p푍, 휉q is called a Morse-type contact handle of index 푘.

Proposition 3.5.6. Suppose p푀, 훾q is a sutured submanifold of p푀 1, 훾1q p푍 “ 푀 1zintp푀q, 휉q is a Morse-type contact handle of index 푘 for 푘 “ 0, 1, 2, 3. Suppose the contact vector field 휈 and the decomposition 푍 “ 푍0 Y ℎ are given as in Definition 3.5.5, and 훾0 Ă t1u ˆ B푀 ĂB푍0 is the dividing set with respect to 휈. Let 휑휈 : p푀, 훾q Ñ p푀 Y 푍0, 훾0q be the diffeomorphism induced by 휈. Then, we have an equality

1 1 Φ휉 “ 퐶ℎ ˝ SHGp휑휈q : SHGp´푀, ´훾q Ñ SHGp´푀 , ´훾 q.

Proof. The proof is exactly the same as the proof of Proposition 5.6 in [42]. The handle cancelations needed have been proved in Lemmas 3.3.1, 3.3.2, and 3.3.3.

Corollary 3.5.7 (Baldwin and Sivek [5], Conjecture 1.7). Under the above settings, suppose there are two different ways to decompose 푍, both relative to B푀:

1 1 1 푀 “ 푀 Y ℎ1 Y ... Y ℎ푛 “ 푀 Y ℎ1 Y ... Y ℎ푚.

Then, the compositions of the two sets of handle attaching maps are the same:

. 1 1 퐶 ... 퐶 퐶 1 ... 퐶 1 : SHG 푀, 훾 SHG 푀 , 훾 . ℎ푛 ˝ ˝ ℎ1 “ ℎ푚 ˝ ˝ ℎ1 p´ ´ q Ñ p´ ´ q

Proof. We can interpret contact handle attaching maps by a special type of gluing maps by Proposition 3.5.6. Then, Corollary 3.7.4 follows from Proposition 3.5.2 and

140 Proposition 3.5.3.

In the instanton settings, we can also make a new definition of the contact element.

Definition 3.5.8. Suppose p푀, 훾q is a balanced sutured manifold and 휉 is a contact structure on 푀 so that B푀 is convex and 훾 is the dividing set. Suppose 퐷 Ă intp푀q is a Darboux ball with convex boundary and 훿 ĂB퐷 is the dividing set. Then, the contact structure 휉 induces a gluing map

Φ휉 : SHIp´퐷, ´훿q Ñ SHIp´푀, ´훾q.

We define the contact element to be

휑p휉q “ Φ휉p1q, where 1 P SHIp´퐷, ´훿q is the generator of SHIp´퐷, ´훿q – C.

Corollary 3.5.9. The contact elements defined in Definition 3.5.8 coincides with the definition of the contact elements by Baldwin and Sivek in [6]. Furthermore, the contact element is preserved by all types of contact handle attaching maps.

Proof. This follows directly from Propositions 3.5.2, 3.5.3, and 3.5.6.

Question 25. Does 휉 being overtiwsted implies that Φ휉 “ 0?

3.6 The construction of cobordism maps

In this section, we present the construction of cobordism maps in sutured monopole and instanton Floer theories. We have presented the definition of sutured cobordisms

141 in Definition 2.7.18. We construct cobordism maps associated to them in this section, as well as provide some basic properties of them.

Definition 3.6.1. Suppose p푀0, 훾0q and p푀1, 훾1q are two balanced sutured manifolds and 풲 “ p푊, 푍, r휉sq is a suture cobordism between them. We can regard p푀0, 훾0q as a sutured submanifold of p푀0 Y p´푍q, 훾1q, and, by Definition 3.5.1, we have a gluing map

Φ´휉 : SHGp푀0, 훾0q Ñ SHGp푀0 Y p´푍q, 훾1q.

The cobordism 푊 can be thought of as one with a sutured surface pB푀2, 훾2q, from p푀0 Y p´푍q, 훾1q to p푀1, 훾1q, in the sense of Definition 3.2.4. Hence, there isa morphism

퐹푊 : SHGp푀0 Y p´푍q, 훾1q Ñ SHGp푀1, 훾1q.

The sutured cobordism map induced by 풲 “ p푊, 푍, 휉q is defined as the composition

SHGp풲q “ 퐹푊 ˝ Φ´휉 : SHGp푀0, 훾0q Ñ SHGp푀1, 훾1q.

Proposition 3.6.2. Suppose p푀0, 훾0q is a balanced sutured manifold and 풲 “ p푊, 푍, r휉sq is a suture cobordism from p푀0, 훾0q to itself so that 푊 “ r0, 1s ˆ 푀0,

푍 “ r0, 1s ˆ B푀0, and 휉 is 퐼-invariant. Then, we have

SHGp풲q “ 푖푑 : SHGp푀0, 훾0q Ñ SHGp푀0, 훾0q

Proof. Note the map 퐹푊 is induced by a cobordism 푊 as in the proof of Proposition 3.2.1. Under the above settings, however, 푊 is diffeomorphic to a product cobordism. x Hence, the proposition follows from Proposition 3.5.4. x

Proposition 3.6.3. Suppose p푀0, 훾0q, p푀1, 훾1q, and p푀2, 훾2q are three balanced su-

142 tured manifolds. Suppose further that 풲 “ p푊, 푍, r휉sq is a suture cobordism from

1 1 1 1 p푀0, 훾0q to p푀1, 훾1q, and 풲 “ p푊 , 푍 , r휉 sq is a suture cobordism from p푀1, 훾1q to

1 p푀2, 훾2q. The composition of 풲 and 풲 is a suture cobordism

풲2 “ p푊 2 “ 푊 Y 푊 1, 푍2 “ 푍 Y 푍1, r휉2s “ r휉 Y 휉1sq

from p푀0, 훾0q to p푀2, 훾2q. Then, we have the equality

2 1 SHGp풲 q “ SHGp풲 q ˝ SHGp풲q : SHGp푀0, 훾0q Ñ SHGp푀2, 훾2q.

Proof. Suppose we have marked closures

풟0 “ p푌0, 푅0, 푟0, 푚0, 휂0q, 풟1 “ p푌1, 푅1, 푟1, 푚1, 휂1q, and 풟2 “ p푌2, 푅2, 푟2, 푚2, 휂2q

for p푀0, 훾0q, p푀1, 훾1q, and p푀2, 훾2q respectively. The map SHMp풲q is induced by ˜ ˜ a cobordism obtained by attaching 4-dimensional handles ℎ1, ..., ℎ푛 and ℎ1, ..., ℎ푚 to r0, 1s ˆ 푌0 along t1u ˆ 푌0. Here, ℎ1, ..., ℎ푛 correspond to the gluing map Φ´휉, ˜ ˜ 1 and ℎ1, ..., ℎ푚 correspond to the map 퐹푊 . Similarly, for SHMp풲 q, we have han- 1 1 ˜1 ˜1 dles ℎ1, ..., ℎ푠 corresponding to the gluing map Φ´휉1 and ℎ1, ..., ℎ푡 corresponding to 1 the map 퐹푊 1 . Then, the composition SHMp풲 q ˝ SHMp풲q is induced by a cobordism obtained from r0, 1s ˆ 푌0 by attaching four collections of 4-dimensional handles ˜ ˜ 1 1 ˜1 ˜1 ℎ1, ..., ℎ푛, ℎ1, ..., ℎ푚, ℎ1, ..., ℎ푠, ℎ1, ..., ℎ푡 to 푌0 ˆr0, 1s at 푌0 ˆt1u in the order we wrote ˜ ˜ them down. Note the attachments of the two collections of handles ℎ1, ..., ℎ푚 and 1 1 ˜ ˜ ℎ1, ..., ℎ푠 commute with each other. Indeed, the handles ℎ1, ..., ℎ푚 correspond to the

4-manifold 푊 and hence by Proposition 3.2.1 are attached to intp푚1p푀1qq Ă 푌1.

1 1 The handles ℎ1, ..., ℎ푠 correspond to the gluing map Φ휉1 and hence are attached to

푌1 near 푚1pB푀q Ă 푌1, so the two collections of handles are attached disjoint from

143 1 1 each other. Then, the handles ℎ1, ..., ℎ푛 and ℎ1, ..., ℎ푠 are attached firstly and corre- ˜ ˜ spond to the map Φ´휉2 as in the proof of Proposition 3.5.3. The handles ℎ1, ..., ℎ푚 ˜1 ˜1 and ℎ1, ..., ℎ푡 are attached secondly and correspond to the cobordism map 퐹푊 2 as in Proposition 3.2.1. Hence, we get the desired equality:

SHMp풲2q “ SHMp풲1q ˝ SHMp풲q.

Remark 3.6.4. Intuitively, the three types of maps: cobordism maps (associated to cobordisms with a sutured surface), gluing maps, and canonical maps all commute with other types. The reason is that for a suitable marked closure 풟 “ p푌, 푅, 푟, 푚, 휂q, cobordism maps (associated to cobordisms with a sutured surface) correspond to handles attached in 푚pintp푀qq Ă 푌 , gluing maps correspond to handles attached near 푚pB푀q Ă 푌 and canonical maps correspond to handles attached in intpimp푟qq Ă 푌 , and the three regions in 푌 are pair-wise disjoint.

3.7 Duality and turning cobordism around

Suppose 풲 “ p푊, 푍, 휉q is a sutured cobordism from a balanced sutured manifold p푀1, 훾1q to another p푀2, 훾2q. We can turn the cobordism around, to make another

_ cobordism 풲 “ p푊, 푍, 휉q from p´푀2, 훾2q to p´푀1, 훾1q. In this section, we study the relation between the two maps SHGp풲q and SHGp풲_q. Suppose 푌 is a closed oriented 3-manifold, s is a spin푐 structure on 푌 , and 휂 is a simple closed curve in 푌 . Then, we have a pairing (ℛ denoting the Novikov ring)

x¨, ¨y : 퐻푀p푌, s;Γ휂q ˆ 퐻푀p´푌, s;Γ휂q Ñ ℛ.

z 144~ Suppose further that 휔 is a simple closed curve inside 푌 which intersects a closed oriented surface 푅 of genus at least two transversely at one point, then we also have a pairing:

휔Y휂 휔Y휂 x¨, ¨y : 퐼˚ p푌 |푅q ˆ 퐼˚ p´푌 |푅q Ñ C.

Using the notation 퐻퐺, the above two pairings can be written uniformly as

x¨, ¨y : 퐻퐺p푌 |푅q ˆ 퐻퐺p´푌 |푅q Ñ ℛ (3.11)

Note ℛ is the Novikov ring in the monopole settings and the field of complex numbers in the instanton settings. The following Lemma is a built-in property of cobordism maps in monopole and instanton theories for closed 3-manifolds.

Lemma 3.7.1. Suppose 푊 is a cobordism from 푌 to 푌 1. The same manifold 푊 can also be viewed as a cobordism 푊 _ from ´푌 1 to ´푌 . Then, the two maps 퐻퐺p푊 q and 퐻퐺p푊 _q are dual to each other with respect to the pairing in (3.11).

Lemma 3.7.2. Suppose p푀, 훾q is a balanced sutured manifold, then there is a pairing well defined up to multiplication by a unit:

x¨, ¨y : SHGp푀, 훾q ˆ SHGp´푀, 훾q Ñ ℛ. (3.12)

Proof. First, suppose 풟 “ p푌, 푅, 푟, 푚, 휂q and 풟1 “ p푌 1, 푅1, 푟1, 푚1, 휂1q are two marked closures of p푀, 훾q of the same genus. These two marked closures give rise to two marked closures 풟_ “ p´푌, 푅, 푟, 푚, 휂q and 풟1_ “ p´푌 1, 푅1, 푟1, 푚1, 휂1q of p´푀, 훾q. Suppose 푎 P 푆퐻퐺p풟q and 푏 P 푆퐻퐺p풟_q are two elements, then we need to show

145 that

x푎, 푏y “ xΦ풟,풟1 p푎q, Φ풟_,풟1_ p푏qy, (3.13) where the pairing x¨, ¨y is the one in (3.11). We prove here only the case when there is a curve 훼 Ă 푅 so that after performing a p´1q surgery along 푟pt0u ˆ 훼q Ă 푌 with respect to the 푟pt0u ˆ 푅q-framing, we obtain the 푌 1. In general, as in Section 5 in Baldwin and Sivek [4], there might be multiple curves, and it might involve both ´1 and `1 surgeries. However, the general case follows from a similar argument, and the fact that the canonical map Φ is functorial. Suppose there is a curve 훼1 Ă 푅 parallel to 훼 but is disjoint from 훼. Since the Dehn surgery is supported in an arbitrarily small neighborhood of 훼, we can assume that 푟pt0u ˆ 훼1q is not affected by the Dehn surgery and hence is also contained in 푌 1. Now, there is a cobordism 푊 ` from 푌 to 푌 1 obtained by attaching a 4-dimensional 2-handle to r0, 1s ˆ 푌 along t1u ˆ 푟pt0u ˆ 훼q Ă t1u ˆ 푌 , and

` Φ풟,풟1 “ 퐻퐺p푊 q.

On the other hand, ´푌 1 can be thought of as obtained from ´푌 by performing a p`1q-surgery along 훼, and, hence, ´푌 is obtained from ´푌 1 by performing a p´1q- surgery. There is an associated 2-handle cobordism 푊 ´ from ´푌 1 to ´푌 . From Section 5 in Baldwin and Sivek [4], we know that

´ ´1 Φ풟_,풟1_ “ 퐻퐺p푊 q .

In fact, 푊 ´ can simply be viewed as turning 푊 ` up side down. As a result, by

146 Lemma 3.7.1, we have

푔 푔 ` ´ ´1 xΦ풟,풟1 p푎q, Φ풟_,풟1_ p푏qy “ x퐻푀p푊 qp푎q, 퐻푀p푊 q p푏qy “ x푎, 퐻푀p푊 ´q ˝ 퐻푀p푊 ´q´1p푏qy

“ x푎, 푏y.

Hence, (3.13) is proved. Second, we deal with the general case when two marked closures 풟 “ p푌, 푅, 푟, 푚, 휂q and 풟1 “ p푌 1, 푅1, 푟1, 푚1, 휂1q of p푀, 훾q might have different genus. Due to the above argument and the fact that canonical maps are functorial, we only need to deal with

1 the following special case: there are two disjoint oriented embedded tori 푇1, 푇2 Ă 푌 so that the following is true.

1 (1) For 푖 “ 1, 2, 푇푖 X 푚 p푀q “ H.

1 1 1 1 (2) For 푖 “ 1, 2, 푇푖 X 푟 pr´1, 1s ˆ 푅 q “ 푟 pr´1, 1s ˆ 푐푖q where 푐푖 Ă 푅 is an

1 embedded oriented circle, and the two circles 푐1 and 푐2 together cut 푅 into two

1 1 oriented parts 푅1 and 푅2 so that

1 1 1 푐1 Y 푐2 “B푅1 “ ´B푅2 and 푅2 – Σ1,2,

where Σ1,2 is the compact connected oriented surface of genus 1 and having two boundary components.

1 1 1 (3) 푇1 and 푇2 cut 푌 into two parts 푌1 and 푌2 so that

1 1 1 푇1 Y 푇2 “B푌1 “ ´B푌2 and 푚 p푀q Ă 푌1 .

1 1 (4) For 푖 “ 1, 2, 휂 intersects 푅푖 in an oriented, non-boundary-parallel properly 1 embedded arc 휂푖.

147 (5) There is a diffeomorphism ℎ : 푇1 Ñ 푇2 with following properties.

1 1 (a) ℎp푇1 X 휂 q “ 푇2 X 휂 .

1 (b) When using ℎ to glue the two boundary components of 푌1 together, the resulting 3-manifold is exactly 푌 , and the other auxiliary data p푅, 푟, 푚, 휂q are also glued or inherited so that this gluing via ℎ gives rise to the chosen marked closure 풟.

1 (c) When using ℎ to glue the two boundary components of 푌1 together, the 1 resulting 3-manifold is 푌2 that is a fibration over 푆 with fibers diffeomorphic to Σ2, a closed connected oriented surface of genus 2.

Now, we can describe the canonical maps Φ풟,풟1 and Φ풟_,풟1_ as follows. Pick

1 the surface 푈 depicted in Figure 3-10. Glue the three part r0, 1s ˆ 푌1 , 푈 ˆ 푇1 and 1 r0, 1s ˆ 푌2 together using 푖푑 and ℎ just as depicted in Figure 3-10. The result of this ` 1 gluing is a cobordism 푊 from 푌 disjoint union with 푌2 to 푌 . As in Section 5 in

` Baldwin and Sivek [4], this cobordism 푊 induces the canonical map Φ풟,풟1 . The same cobordism, with the reversed orientation, is a cobordism 푊 ´ “ ´푊 ` from

1 ` p´푌 q \ p´푌2q to ´푌 and induces the canonical map Φ풟_,풟1_ . If we turn 푊 up

_ 1 side down, it becomes a cobordism 푊 from ´푌 to ´푌 \ p´푌2q and induces a dual map by Lemma 3.7.1. Then, the equality (3.13) will follow from the fact that the cobordism 푊 _ Y 푊 ´ induces the identity map up to multiplication by a unit, which is proved in the proof of Theorem 3.2 in Kronheimer and Mrowka [53].

We want to re-interpret the gluing maps and cobordism maps defined in Section 3.5 and Section 3.6 to better deal with the duality. Suppose p푀 1, 훾1q is a balanced sutured manifold and p푀, 훾q is a sutured submanifold. Suppose 푍 “ 푀 1zintp푀q and 휉 is a contact structure on 푍 so that B푍 is convex with dividing set 훾 Y훾1. Suppose 푍 has a contact handle decomposition relative to 푀. That is, there are contact handles

148 1 1 ℎ1, ..., ℎ푛 so that if we attach them to p푀, 훾q, then we will get p푀 , 훾 q. Suppose

ℎ1, ..., ℎ푚 are all 0- and 1-handles and ℎ푚`1, ..., ℎ푛 are all 2- and 3-handles. Suppose 1 p푀1, 훾1q is the result of attaching all ℎ1, ..., ℎ푚 to p푀, 훾q. Let 푊 “ r0, 1s ˆ 푀 , and let 푀2 “B푊 zpt0u ˆ 푀1q with suitable orientation. We can view 푊 as a cobordism from p푀1, 훾1q to p푀2, 훾1q with sutured surface p푆 “ t0u ˆ B푀1, 훾1q.

Proposition 3.7.3. Under the above settings, the marked closure 풟1 is also a marked closure for p푀, 훾q so there is a map

Φ : SHMp´푀, ´훾q Ñ SHMp´푀1, ´훾1q.

1 1 The marked closure 풟2 is also a marked closure for p푀 , 훾 q so there is a map

1 1 Ψ : SHMp´푀 , ´훾 q Ñ SHMp´푀2, ´훾2q.

Furthermore, the gluing map can be written as

´1 Φ휉 “ Ψ ˝ 퐹´푊 ˝ Φ.

Proof. Following Proposition 2.7.6, it is straightforward to see that the marked closure 풟1 is indeed a marked closure for p푀, 훾q and the associated map Φ equals the composition of handle attaching maps:

Φ “ 퐶ℎ푚 ˝ ... ˝ 퐶ℎ1 .

If 푛 “ 푚`1, i.e., there is only one 2- or 3-handle, then it follows from Proposition

1 1 3.2.6 that the marked closure 풟2 is indeed a marked closure of p푀 , 훾 q and we have

149 an identification

´1 Ψ ˝ 퐹´푊 “ 퐶ℎ`1 .

Thus the proposition is proved. In general when 푛 ą 푚 ` 1, there are more than one 2- or 3-handle involved, but the proposition still follows from Proposition 3.2.6 and the fact that cobordism maps in monopole and instanton theories are functorial under the composition of cobordisms.

Corollary 3.7.4. Suppose 풲 “ p푊, 푍, r휉sq is a sutured cobordism from p푀1, 훾1q to

1 p푀2, 훾2q. Suppose 푆 Ă 푍 is chosen as in Proposition 3.7.3 and 훾1 correspondingly. 1 1 1 Suppose 푆 separates B푊 into two parts 푀1 and 푀2, so that 푀푖 contains 푀푖 and is 1 1 oriented in the same way as 푀푖. We can view 푊 as a cobordism 푊 from p푀1, 훾1q to p푀 1 , 훾1 q with a sutured surface p푆, 훾1 q and it induces a map 퐹 . Then, there are 2 1 1 푊 Ă isomorphisms Ă 1 1 Φ : SHMp푀1, 훾1q Ñ SHMp푀1, 훾1q, and

1 1 Ψ : SHMp푀2, 훾2q Ñ SHMp푀2, 훾1q. furthermore, the cobordism map can be written as

´1 SHGp풲q “ Ψ ˝ 퐹푊 ˝ Φ.

Ă 1 Proof. When performing a parallel closing up along p푆, 훾1q, we obtain two marked clo- 1 1 1 1 sures 풟1 “ p푌1, 푅1, 푟1, 푚1, 훾1q and 풟2 “ p푌2, 푅2, 푟2, 푚2, 훾2q of p푀1, 훾1q and p푀2, 훾2q, respectively, and a cobordism 푊 from 푌1 to 푌2. The cobordism 푊 can be viewed as obtained from r0, 1s ˆ 푌 by attaching two collections of 4-dimensional handles 1 x x 4 4 ˜4 ˜4 4 4 ℎ1, ..., ℎ푛 and ℎ1, ..., ℎ푚. The first collection of handles ℎ1, ..., ℎ푛 comes from the glu-

150 ing map Φ´휉 as in Proposition 3.7.3 and Proposition 3.2.6, and the second collection ˜4 ˜4 of handles ℎ1, ..., ℎ푚 comes from the the cobordism W as in Proposition 3.2.1. Thus it is straightforward to check that

´1 Ψ ˝ 퐹푊 ˝ Φ “ 퐹푊 ˝ Φ´휉 “ SHGp풲q.

Note 퐹푊 is the map associated to 푊 viewed as a cobordism with a sutured surface p푆, 훾1 q, as in the hypothesis of the corollary, and 퐹 is the map associated to 푊 1 Ă Ă 푊 viewed as a cobordism with a sutured surface pB푀2, 훾2q, as in Definition 3.6.1.

Corollary 3.7.5. Suppose 풲 “ p푀, 푍, r휉sq is a sutured cobordism from p푀1, 훾1q to

_ p푀2, 훾2q. The same cobordism can be also viewed as a cobordism 풲 from p´푀2, 훾2q

_ to p´푀1, 훾1q. Then, the cobordism map SHGp풲q and SHGp풲 q are dual with respect to the pairing (3.12).

Proof. If put the cobordism 푊 up-side-down, then the distinguising surface 푆 is unchanged (since in 푍, an 푖-handle becomes a p3´푖q-handle). Hence, from Corollary 3.7.4, SHGp풲q is induced by a cobordism 푊 , while SHGp풲_q is induced by a cobordism 푊 _ that is obtained by putting 푊 up-side-down. Hence, the conclusion x follows from Lemma 3.7.1. x x

There is a question related to the trace and co-trace cobordism. Suppose p푀, 훾q is a balanced sutured manifold and 풲 “ p푊 “ r0, 1s ˆ 푀, 푍 “ r0, 1s ˆ B푀, r휉sq is the sutured cobordism from p푀 \ p´푀q, 훾 Y p´훾qq to the ’empty balanced sutured manifold’. Here, 휉 is a r0, 1s-invariant contact structure on 푍 so that B푀 is convex

B with respect to B푡 , and 훾 is the corresponding dividing set. Let ℛ be the coefficient ring, then we ask the following question.

151 Question 26. How to describe the cobordism map

SHGp풲q : SHGp푀 \ p´푀q, 훾 Y 훾q Ñ ℛ?

Note from Künneth formula, there is a map

푖 : SHGp푀 \ p´푀q, 훾 Y 훾q Ñ SHGp푀, 훾q b SHGp´푀, 훾q.

Also there is a canonical map

푡푟 : SHGp푀, 훾q b SHGp´푀, 훾q Ñ ℛ defined as 푡푟p푎 b 푏q “ 푏p푎q, since SHGp´푀, 훾q is the dual of SHGp푀, 훾q. We make the following conjecture.

Conjecture 3.7.6. With the above settings, we have

SHGp풲q “ 푡푟 ˝푖. (3.14)

152 Chapter 4

Gradings on Sutured monopole and instanton Floer homologies

In this chapter, we present the construction of the grading on sutured monopole and instanton Floer homologies, associated to a properly embedded surface inside the balanced sutured manifold. Though, in general, the construction works with a general admissible surface in the sense of Definition 2.6.17, in this chapter, we focus on the case when the properly embedded surface has connected boundary, as this special case is enough for the application in the next chapter. For a general construction, see Remark 2.6.19. In the last two sections of this chapter, we will also present how the grading can be applied to carry out some computations in sutured monopole and instanton Floer homology theories.

153 4.1 The construction of the grading

Suppose p푀, 훾q is a balanced sutured manifold, and 푆 Ă 푀 is a properly embedded surface. As pointed out in Remark 2.6.21, the condition of being admissible is non- trivial and is not satisfied by any surfaces. So, for a given 푆, we need to perturb it to become admissible. The following definition indicates two basic ways to perform the perturbations.

Definition 4.1.1. Suppose p푀, 훾q is a balanced sutured manifold, and 푆 is a properly embedded oriented surface. A stabilization of 푆 is an isotopy of 푆 to a surface 푆1, so that the isotopy creates a new pair of intersection points:

1 B푆 X 훾 “ pB푆 X 훾q Y t푝`, 푝´u.

We require that there are arcs 훼 ĂB푆1 and 훽 Ă 훾, oriented in the same way as B푆1 and 훾, respectively, such that the following is true.

(1) We have B훼 “B훽 “ t푝`, 푝´u. (2) The curves 훼 and 훽 cobound a disk 퐷 so that intp퐷q X p훾 YB푆1q “ H. The stabilization is called negative if 퐷 can be oriented so that B퐷 “ 훼 Y 훽 as oriented curves. it is called positive if B퐷 “ p´훼q Y 훽. See Figure 4-1. We denote by 푆˘푘 the result of performing 푘 many positive or negative stabilizations of 푆.

The following lemma is straightforward.

Lemma 4.1.2. Suppose p푀, 훾q is a balanced sutured manifold, and 푆 is a properly embedded oriented surface. Suppose 푆` and 푆´ are the results of doing a positive and negative stabilization on 푆, respectively. Then, we have the following.

154 훼 퐷 훽

negative

훾 ¨¨* ¨ ¨¨ ¨ HH 퐷 H HH 훼 훾 Hj 훽 B푆 positive

Figure 4-1: The positive and negative stabilizations of 푆.

(1) If we decompose p푀, 훾q along 푆 or 푆`, then the resulting two balanced sutured manifolds are diffeomorphic. (2) If we decompose p푀, 훾q along 푆´, then the resulting balanced sutured manifold

1 1 1 p푀 , 훾 q is not taut, as 푅˘p훾 q would both become compressible.

Suppose p푀, 훾q is a balanced sutured manifold, and 푆 is a properly embedded oriented surface. Suppose further that 푆 has precisely one boundary component, and

B푆 intersects 훾 at 2푛 points. Since 훾 is parallel to the boundary of 푅`p훾q, it is null- homologous, so the algebraic intersection number of B푆 with 훾 on B푀 must be zero. We also assume that 푛 “ 2푘 ` 1 is odd, as this can be achieved by a stabilization of 푆 if needed. Suppose the intersection points are 푝1, ..., 푝2푛, and they are indexed according to the orientation of B푆.

155 Now pick a connected auxiliary surface 푇 for p푀, 훾q, which is of large enough

1 genus. Let 푓 : B푇 Ñ 훾 be an orientation reversing diffeomorphism and let 푝푖 “ ´1 푓 p푝푖q. Suppose 훼1, ..., 훼푛 are pair-wise disjoint simple arcs on 푇 , so that the following is true.

(1) The classes r훼1s, ..., r훼푛s are linearly independent in 퐻1p푇, B푇 q.

1 1 (2) We have that B훼1 “ t푝1, 푝2u, and, for all 1 ď 푖 ď 푘, we have

1 1 1 1 B훼2푖 “ t푝4푖´1, 푝4푖`2u, and B훼2푖`1 “ t푝4푖, 푝4푖`1u.

Let 푛

푀 “ 푀 Y r´1, 1s ˆ 푇, and 푆 “ 푆 Y p r´1, 1s ˆ 훼푖q. 푖푑ˆ푓 푖푑ˆ푓 푖“1 ď We know thatĂ r 푘`1

B푀 “ 푅` Y 푅´, and B푆 X 푅˘ “ 퐶푖,˘. 푖“1 ď Here we require that forĂ푖 “ 1, ..., 푘 ` 1, r

훼2푖´1 ˆ t˘1u Ă 퐶푖,˘.

Pick an orientation preserving diffeomorphism ℎ : 푅` Ñ 푅´ so that for 푖 “ 1, ..., 푘`1,

ℎp퐶푖,`q “ 퐶푖,´.

Then, we can use ℎ and 푀 to obtian a closure p푌, 푅q of p푀, 훾q. The boundary components of the surface 푆 are glued with each other under ℎ, so 푆 becomes a Ă closed surface 푆¯ Ă 푌 . From the construction, we know that r r

휒p푆¯q “ 휒p푆q ´ 푛.

156 We pick a non-separating simple closed curve 휂 Ă 푅, so that 휂 is disjoint from 푆¯ X 푅 and represents a class which is linearly independent from the classes represented by ¯ the components of 푆 X 푅 in 퐻1p푅q.

Definition 4.1.3. We say that the surface 푆¯ Ă 푌 is associated to the surface 푆 Ă 푀. We can use 푆¯ to define a grading on 푆퐻퐺p푀, 훾q as follows.

푆퐻푀p푀, 훾, 푆, 푖q “ 퐻푀 ‚p푌, s;Γ휂q, sPSp푌 |푅q ¯ 푐1psàqr푆s“2푖 ~ and

휔Y휂 푆퐻퐼p푀, 훾, 푆, 푖q “ 퐼˚ p푌, 휆q. 휆PHp푌 |푅q 휆rà푆¯s“2푖 We say that this grading is associated to the surface 푆 Ă 푀. When using the language of marked closures, the closure p푌, 푅q corresponds to a marked closure 풟 “ p푌, 푅, 푚, 푟, 휂q, and we write the grading as

푆퐻퐺p풟, 푆, 푖q.

The grading on 푆퐻퐺p풟q also induces a grading on SHGp푀, 훾q, as stated in Theorem 4.1.4. We also say it is associated to 푆 and write

SHGp푀, 훾, 푆, 푖q.

Theorem 4.1.4. When B푆 is connected, the grading on SHGp푀, 훾q associated to 푆 is well-defined. That is, it is independent of all the choices made in the construction.

Proof. There are four types of choices we made in the construction of the grading:

157 I. The point 푝1 on B푆 X 훾.

II. The choice of the arcs 훼1, ..., 훼푛 on 푇 . III. The choice of the gluing diffeomorphism ℎ. IV. The genus of the closure. The proof of Theorem occupies the next two sections. In particular, the results are stated in Corollary 4.3.9, Corollary 4.1.7, Proposition 4.2.2, and Lemma 4.1.5.

In [8], Baldwin and Sivek have already dealt with the choices of type II, III, and IV. Among them, the idea for type IV can be adapted to the setting of the current paper verbatim, so we do not bother to write down the proof again.

Lemma 4.1.5 (Baldwin and Sivek [8]). The definition of the grading on SHGp푀, 훾q associated to the surface 푆 Ă 푀 is independent of choices of type IV.

To deal with the choices of type II, we have the following lemma.

Lemma 4.1.6. Suppose 푇 is a compact connected oriented surface-with-boundary and is of large enough genus. Suppose further that t훼1, ..., 훼푛u is a set of properly embedded simple arcs on 푇 so that the following is true.

(1) The arcs 훼1, ..., 훼푛 are pair-wise disjoint.

(2) The arcs represent linearly independent classes r훼1s, ..., r훼푛s in 퐻1p푇, B푇 q.

1 1 Suppose t훼1, ..., 훼푛u is another set of properly embedded simple arcs so that the following is true.

1 (3) For 푖 “ 1, ..., 푛, we have B훼푖 “B훼푖. 1 1 (4) The set of arcs t훼1, ..., 훼푛u also satisfies the above conditions (1) and (2). Then, there is an orientation preserving diffeomorphism ℎ : 푇 Ñ 푇 so that ℎ fixes the boundary of 푇 , and, for 푖 “ 1, ..., 푛, we have

1 ℎp훼푖q “ 훼푖.

158 Proof. Suppose 푁 is a product neighborhood of

훼1 Y ... Y 훼푛 Ă 푇.

Let 푇 “ 푇 zintp푁q. The boundary B푇 consists of the following:

r r 푛 B푇 “ pB푇 X 푇 q Y p 훼푖,` Y 훼푖,´q. 푖“1 ď r r Here, 훼푖,˘ are parallel copies of 훼푖, being part of the boundary of the product neighborhood 푁. From condition (2), we know that 푇 is connected. Also, by construction,

r 휒p푇 q “ 휒p푇 q ` 푛.

Similarly, we can pick 푁 1 to be a product neighborhood of

1 1 훼1 Y ... Y 훼푛 Ă 푇, and take 푛 1 1 1 1 1 1 푇 “ 푇 zintp푁 q, and B푇 “ pB푇 X 푇 q Y p 훼푖,` Y 훼푖,´q. 푖“1 ď r r r

By condition (3), we can assume that 푁 XB푇 “ 푁 1 XB푇 , so there is an orientation preserving diffeomorphism 푓 : B푇 ÑB푇 1 so that r r

1 푓|B푇 X푇 “ 푖푑, and 푓p훼푖,˘q “ 훼푖,˘

r 159 for all 푖 “ 1, ..., 푛. Since we have

휒p푇 1q “ 휒p푇 q ` 푛 “ 휒p푇 q,

r r the diffeomorphism 푓 extends to a diffeomorphism

푔 : 푇 Ñ 푇 1.

r r 1 1 Thus, we can glue 푇 and 푇 along 훼푖,˘ and 훼푖,˘, and 푔 is glued to become a diffeomorphism r r ℎ : 푇 Ñ 푇 that is the desired one.

As discussed in [8], Lemma 4.1.6 gives rise to the following corollary.

Corollary 4.1.7. The grading on SHGp푀, 훾q associated to the surface 푆 Ă 푀 is independent of choices of type II.

We deal with the choices of type III in Section 4.2 and the choices of type I in Section 4.3.

4.2 A reformulation of Canonical maps

In this section, we give an alternative description of the canonical maps Φ풟,풟1 , which was originally constructed by Baldwin and Sivek in [4] for two different marked closures of the same genus. For our convenience, we only study the special case as described in the following paragraph.

160 Suppose p푀, 훾q is a balanced sutured manifold and 푇 is a connected auxiliary surface. Let

푀 “ 푀 Y r´1, 1s ˆ 푇, B푀 “ 푅` Y 푅´.

Suppose ℎ1 and ℎ2 areĂ two different gluing diffeomorphisms, and there are corresponding marked closures 풟1 “ p푌1, 푅`, 푟1, 푚, 휂q and 풟2 “ p푌2, 푅`, 푟2, 푚, 휂q, respectively. Here, we choose the same non-separating simple closed curve 휂 on 푅` to support local coefficients.

´1 ℎ Let ℎ “ ℎ1 ˝ ℎ2, and 푌 be the mapping torus of ℎ, i.e., the manifold obtained from 푅` ˆr´1, 1s by identifying 푅` ˆt1u with 푅` ˆt´1u via ℎ. Then, we can obtain ℎ ℎ 푌2 from 푌1 and 푌 as follows. Cut 푌1 open along 푅` ˆt0u and cut 푌 along 푅` ˆt0u.

We can re-glue them via the identity map on 푅` to get a connected manifold. This resulting manifold is precisely 푌2. As in Theorem 2.5.1 and Theorem 2.5.2, there is ℎ a cobordism 푊 from 푌1 \ 푌 to 푌2, and 푊 induces an isomorphism:

ℎ 퐻퐺p푊 q : 퐻퐺p푌1 \ 푌 |푅` Y 푅`q Ñ 퐻퐺p푌2|푅`q.

Note, from Lemma 2.6.2, we know that

ℎ 퐻퐺p푌 |푅`q – ℛ.

ℎ Let 푎 be a generator of 퐻퐺p푌 |푅`q and let 휄 be the map

ℎ ℎ 휄 : 퐻퐺p푌1|푅`q Ñ 퐻퐺p푌1|푅`q b 퐻퐺p푌 |푅`q – 퐻퐺p푌1 \ 푌 |푅` Y 푅`q defined by 휄p푥q “ 푥 b 푎.

161 We have the following proposition.

Proposition 4.2.1. The canonical map Φ풟1,풟2 can be re-interpreted as

. Φ풟1,풟2 “ 퐻퐺p푊 q ˝ 휄.

Before proving the proposition, we first use it to prove the fact that the definition of the grading is independent of the choices of type III. Suppose p푀, 훾q is a balanced sutured manifold and 푆 Ă 푀 is a properly embedded surface with precisely one boundary component, so that B푆 intersects 훾 at 2푛 points for some odd 푛 “ 2푘 ` 1. Suppose further that, in the construction of the grading induced by 푆, the choices of type I, II, IV are fixed. This means that there is a connected auxiliary surface 푇 for p푀, 훾q and n arcs 훼1, ..., 훼푛 so that the following holds (1) We have

Bp훼1 Y ... Y 훼푛q “ B푆 X 훾.

(2) Let

Bp푀 Y r´1, 1s ˆ 푇 q “ 푅` Y 푅´, and 푆 “ 푆 pr´1, 1s ˆ 훼푖q, 푖“1푛 ď r then we have

B푆 X 푅˘ “ 퐶1,˘, ..., 퐶푘`1,˘.

Suppose there are two gluingr diffeomorphisms ℎ1 and ℎ2 so that, for 푖 “ 1, 2

ℎ푖p퐶1,` Y ... Y 퐶푘`1,`q “ 퐶1,´ Y ... Y 퐶푘`1,´.

Suppose further that there are marked closures 풟1 “ p푌1, 푅`, 푚, 푟1, 휂q and 풟2 “

162 p푌2, 푅`, 푚, 푟2, 휂q corresponding to ℎ1 and ℎ2, respectively. Here, we choose the same non-separating simple closed curve 휂 Ă 푅` to construct local coefficients. We have the following proposition.

Proposition 4.2.2. For any 푖 P Z, we have

– Φ풟1,풟2 : 푆퐻퐺p풟1, 푆, 푖q ÝÑ 푆퐻퐺p풟2, 푆, 푖q.

As a result, the definition of the grading on SHGp푀, 훾q is independent of the choices of type III.

Proof. We carry out this proof in the monopole settings. The proof in the instanton settings is exactly the same, with (first Chern classes of) spin푐 structures being

´1 ℎ replaced by eigenvalue functions. Let ℎ “ ℎ1 ˝ ℎ2, and form 푌 as in Proposition 푐 4.2.1. From Lemma 2.6.2, there is a unique spin structure s0 so that

ℎ ℎ 퐻푀p푌 |푅`q “ 퐻푀 ‚p푌 , s0;Γ휂q – ℛ.

~ ℎ There are tori inside 푌 : The cylinders 퐶푖,` ˆ r´1, 1s Ă 푅` ˆ r´1, 1s are glued via ℎ to become a union of tori 푇 . Lemma 2.6.12 tells us that

푐1ps0qr푇 s “ 0.

¯ ¯ Let 푆1 Ă 푌1 and 푆2 Ă 푌2 be the surfaces induced by 푆 Ă 푀 as in the construction of the grading. We know that there is a 3-dimensional cobordism from 푆1 \ 푇 to

푆2 inside the the cobordism 푊 . The construction of this (3-dimensional) cobordism is similar to that of the Floer’s excisions. If s is a spin푐 structure on 푊 , which contributes non-trivially to the cobordism map 퐻푀p푊 q, then s must restrict to s0

163 on 푌 ℎ. Hence, we know that

¯ ¯ ¯ ¯ 푐1psqpr푆2sq “ 푐1psqpr푆1s ` r푇 sq “ 푐1psqpr푆1sq ` 푐1ps0qpr푇 sq “ 푐1psqpr푆1sq.

Thus, 퐻푀 푊 preserves the grading and so does Φ푔 , by Proposition 4.2.1. p q 풟1,풟2

Now we proceed to prove proposition 4.2.1. There are a few preparations we need.

Lemma 4.2.3. Under the settings of Proposition 4.2.1, suppose we have a third

1 ´1 2 1 ´1 1 gluing diffeomorphism ℎ3, ℎ “ ℎ2 ˝ ℎ3, and ℎ “ ℎ ˝ ℎ “ ℎ1 ˝ ℎ3. Construct 푊 , 푊 2, 휄1, and 휄2 just in the same way as we construct 푊 and 휄. Then, we have an identity: . 퐻퐺p푊 2q ˝ 휄2 “ 퐻퐺p푊 1q ˝ 휄1 ˝ 퐻퐺p푊 q ˝ 휄. (4.1)

1 2 2 1 Proof. Let 푌ℎ1 and 푌ℎ2 be the mapping tori of ℎ and ℎ , respectively. Since ℎ “ ℎ˝ℎ ,

1 there is an excision cobordism from 푌ℎ \ 푌ℎ2 to 푌ℎ2 just as we construct 푊 , 푊 , and

2 _ 푊 . Call this cobordism ´푊푒 , and let 푊푒 be the cobordism from 푌ℎ2 to 푌ℎ \ 푌ℎ1 , _ obtained by putting ´푊푒 up side down and then reversing the orientation. By Theorem 2.5.1, Theorem 2.5.2, and Lemma 2.6.2, it is straightforward to see that

1 . 1 1 퐻퐺p푊 Y 푊 Y 푊푒q ˝ 휄3 “ 퐻퐺p푊 q ˝ 휄 ˝ 퐻퐺p푊 q ˝ 휄.

Hence, to prove (4.1), it is enough to show that

1 . 2 퐻퐺p푊 Y 푊 Y 푊푒q “ 퐻퐺p푊 q. (4.2)

1 1 1 However, we can cut 푊 Y푊 Y푊푒 open along the 3-manifold 푆 ˆ푅`, as depicted 2 in Figure 4-2 and glue back two copies of 퐷 ˆ푅`. The resulting 4-manifold is exactly

164 푊 2. Hence, from Proposition 2.5 in Kronheimer and Mrowka [53], (4.2) holds true and we conclude the proof of Lemma 4.2.3.

푌ℎ1 푌ℎ1

푌ℎ2 푊푒 1 푆 ˆ 푅` 1 푊 푌3 푌 ℎ

푊 푌2

푌1 푌1

1 Figure 4-2: The union 푊 Y 푊 Y 푊푒. The (blue) curve in the middle represents the 1 3-manifold 푆 ˆ 푅` to cut along.

Corollary 4.2.4. If ℎ1 “ ℎ2, then we have

. 퐻퐺p푊 q ˝ 휄 “ 푖푑.

Proof. From Theorem 2.5.1 and Theorem 2.5.2, we know that

퐻퐺p푊 q ˝ 휄 is an isomorphism. From Lemma 4.2.3, we know that

. 퐻퐺p푊 q ˝ 휄 ˝ 퐻퐺p푊 q ˝ 휄 “ 퐻퐺p푊 q ˝ 휄.

Hence, the corollary follows.

165 Proof of Proposition 4.2.1. Suppose ℎ is decomposed into Dehn twists:

푒1 푒푛 ℎ „ 퐷푎1 ˝ ... ˝ 퐷푎푛 , as in Baldwin and Sivek [4]. From Proposition 2.4.2 and Lemma 4.2.3, it is suffice to deal with the case when 푛 “ 1, i.e., there is only one Dehn twist involved.

When 푒 1, the Dehn twist is positive. In this case, the canonical map Φ푔 1 “ 풟1,풟2 is constructed using the cobordism 푊 , as in the hypothesis of Proposition 4.2.1, with the boundary component 푌 ℎ capped off by the total space of a relative minimal Lefschetz fibration, see Lemma 4.9 in Baldwin and Sivek [4]. Since such a Lefschetz fibration has relative monopole invariant being a unit in ℛ, as in Proposition B1 in [4], we conclude that . Φ푔 퐻퐺 푊 휄. 풟1,풟2 “ p q ˝

When 푒1 “ ´1, the Dehn twist is negative. We can instead look at the canonical map Φ푔 . It corresponds to ℎ´1 and is constructed using a positive Dehn twist. 풟2,풟1 Suppose we construct 푊 1 and 휄1 out of ℎ´1, just as we construct 푊 and 휄 out of ℎ. Then, from the previous case we know that

. Φ푔 퐻퐺 푊 1 휄1. 풟2,풟1 “ p q ˝

Then, the identity . Φ푔 퐻퐺 푊 휄. 풟1,풟2 “ p q ˝ follows from Theorem 2.5.1, Theorem 2.5.2, Lemma 4.2.3 and Corollary 4.2.4.

166 4.3 Pairing of the intersection points

In this section, we deal with type I choices, i.e., the choice of 푝1 among all intersection points in 푆 X 훾.

Let us first pick an arbitrary intersection point in B푆 X 훾 as 푝1. We need to relax the requirement in the construction of the grading that B훼푖 are chosen to be a special pair of points in 푆 X 훾. To record the data of the endpoints of 훼푖, we make the following definition.

Definition 4.3.1. Suppose we have a collection of 푛 pair of numbers

풫 “ tp푖1, 푗1q, ..., p푖푛, 푗푛qu so that

t푖1, 푗1, ..., 푖푛, 푗푛u “ t1, 2, ..., 2푛u, and, for all 푙 “ 1, ..., 푛, we have

푖푙 ı 푗푙 pmod 2q.

Then, we call such a collection 풫 a pairing of size 푛.

Suppose p푀, 훾q is a balanced sutured manifold and 푆 Ă 푀 is a properly embedded oriented surface. Suppose further that 푆 has a connected boundary, and it intersects

훾 at 2푛 “ 4푘 ` 2 points. Those points are labeled by 푝1, ..., 푝4푘`2, according to the orientation of B푆, with an arbitrary chosen starting point 푝1. Continuing, suppose 푛 풫 “ tp푖푙, 푗푙qu푙“1 is a pairing of size 푛, 푇 is an auxiliary surface of 푀, and 훼1, ..., 훼푛 are pair-wise disjoint simple arcs so that the following is true.

(1) The arcs 훼1,..., 훼푛 represent linearly independent classes in 퐻1p푇, B푇 q.

167 (2) For 푙 “ 1, ..., 푛, we have

B훼푙 “ t푝푖푙 , 푝푗푙 u.

Then, as in Definition 4.1.3, we can construct

푛

푀 “ 푀 Y 푇 ˆ r´1, 1s, 푆풫 “ 푆 Y p 훼푙 ˆ r´1, 1sq. 푙“1 ď Ă r We have

B푀 “ 푅` Y 푅´, B푆풫 X 푅˘ “ 퐶1,˘ Y 퐶푠˘,˘.

In general, the numbersĂ of intersectionr circles, 푠` and 푠´, are not necessarily equal to each other, so we make the following definition.

Definition 4.3.2. A pairing 풫 is called balanced if 푠´ “ 푠`.

Example 4.3.3. Here are some examples of the pairings. Assume 푛 “ 2푘 `1 is odd. (1) The simplest possible pairing

풫 “ tp1, 2q, p3, 4q, ..., p4푘 ` 1, 4푘 ` 2qu

has 푠´ “ 1 and 푠` “ 푛, or 푠´ “ 푛 and 푠` “ 1, depending on the choice of the starting point 푝1, so it is not a balanced paring for 푛 ą 1. (2) In Definition 4.1.3, we have a paring arising from the construction ofthe grading: 풫푔 “ tp1, 2q, p3, 6q, p4, 5q, ..., p4푘 ´ 1, 4푘 ` 2q, p4푘, 4푘 ` 1qu.

This is an example of a balanced pairing, with 푠` “ 푠´ “ 푘 ` 1.

168 (3) There is a very special balanced pairing with 푠` “ 푠´ “ 1:

풫푠 “ tp1, 2푘 ` 2q, p2, 2푘 ` 3q, ..., p2푘 ` 1, 4푘 ` 2qu.

If p푀, 훾q, 푆, and 푝1 are chosen as above, and we are equipped with a balanced pairing 풫, then we can repeat the construction in Definition 4.1.3 and define a grading on SHGp푀, 훾q. By Corollary 4.1.7, Proposition 4.2.2, and Lemma 4.1.5, the grading depends only on the choice of 푝1 and 풫. Since 푆 and 푝1 are fixed throughout this section, we omit them from the notation and write, in a moment, the grading as

SHGp푀, 훾, 풫, 푖q.

There is an operation we can perform on balanced pairings. Suppose 풫 is a balanced pairing, and we pick two indices 푙1 and 푙2 so that the following two conditions hold.

(i) The two arcs t1u ˆ 훼푙1 and t1u ˆ 훼푙2 are not contained in the same boundary component of 푆풫 . (ii) The two arcs t´1uˆ훼 and t´1uˆ훼 are not contained in the same boundary r 푙1 푙2 component of B푆. Then, we can perform the following operation on 풫: Suppose, in the two pairs r p푖푙1 , 푗푙1 q and p푖푙2 , 푗푙2 q, 푖푙1 and 푖푙2 are odd (and the two other numbers must be even), 1 then we can obtain a new pairing 풫 out of 풫 by removing the two pairs p푖푙1 , 푗푙1 q and p푖푙2 , 푗푙2 q from 풫 and add two new pairings p푖푙1 , 푗푙2 q and p푖푙2 , 푗푙1 q.

Definition 4.3.4. We call the above operation the cut and glue on parings. Two pairings are called equivalent if one is obtained from the other by a cut and glue operation.

169 Example 4.3.5. If 푛 “ 3, 풫 “ tp1, 2q, p3, 6q, p5, 4qu, 푙1 “ 1, and 푙2 “ 3 (푙1 “ 1 and

푙2 “ 2 do not meet the requirements of performing a cut and glue operation), then the resulting pairing 풫1 is

풫1 “ tp1, 4q, p3, 6q, p2, 5qu, and it is balanced. The equivalence introduced above is an equivalence relation. Also, the result of a cut and glue operation on a balanced pairing is still a balanced one.

Lemma 4.3.6. Suppose a cut and glue operation on a balanced pairing 풫 associated

1 to the two indices 푙1 and 푙2 gives rise to a new balanced pairing 풫 , then, for all 푖 P Z, we have SHGp푀, 훾, 풫, 푖q “ SHGp푀, 훾, 풫1, 푖q.

Proof. At this point, we have shown that the choices of type II, III, and IV do not make difference on the definition of the grading. So,once 풫 is chosen, we can freely choose other auxiliary data to construct the grading. Let 푇 and 훼1, ..., 훼푛 be chosen, and the pre-closure 푀 as well as the properly embedded surface 푆풫 have been constructed. We can assume that they are chosen so that there is a Ă r curve 푐 intersecting both 훼푙1 and 훼푙2 transversely at one point. See Figure 4-3. The requirements (i) and (ii) make sure that t˘1uˆ훼푙1 and t˘1uˆ훼푙2 lie in four different boundary components of 푆풫 . So, there is an orientation preserving diffeomorphism ℎ : 푅 Ñ 푅 , where B푀 “ 푅 Y 푅 , so that ` ´ r ` ´ Ă ℎpB푆 X 푅`q “ B푆 X 푅´, ℎp푐 ˆ t1uq “ 푐 ˆ t´1u,

r r ℎp훼푙1 ˆ t1uq “ 훼푙1 ˆ t´1u, and ℎp훼푙2 ˆ t1uq “ 훼푙2 ˆ t´1u.

170 훼푙1

훼푙2 휂 푐 푐1 훿 휂

훼푙1 훽

훼푙2 푇 Σ2

훼1 푙1 훼1 푙2 휂1 휂1 훼1 푙2 훼1 푙1

Figure 4-3: The auxiliary surface 푇 and the surface Σ2

Let

푌 “ 푀 Y r´1, 1s ˆ 푅`, and 푅 “ t0u ˆ 푅 푖푑Yℎ ¯ be a closure of p푀, 훾q. TheĂ surface 푆풫 becomes a closed surface 푆풫 Ă 푌 . We can also choose a simple closed curve 휂 on 푅 “ t0u ˆ 푅 so that 휂 is disjoint from 푆 r ` 풫 and 휂 intersects 푐ˆt0u transversely at one point. Hence, we obtain a marked closure r 풟 “ p푌, 푅, 푚, 푟, 휂q, where 푚 and 푟 are both inclusions.

As in Definition 4.1.3, we have

푆퐻푀p푀, 훾, 푆, 푖q “ 퐻푀 ‚p푌, s;Γ휂q, sPSp푌 |푅q ¯ 푐1psàqr푆s“2푖 ~ 171 and

휔Y휂 푆퐻퐼p푀, 훾, 푆, 푖q “ 퐼˚ p푌, 휆q. 휆PHp푌 |푅q 휆rà푆¯s“2푖

1 Let Σ2 be a closed connected oriented surface of genus 2. Let 푐 , 훿 and 훽 be three simple closed curves on Σ2, as depicted in Figure 4-3.

1 1 Let 푌Σ be the 3-manifold 푆 ˆ Σ2. There is a torus Σ “ 푆 ˆ 푐 Ă 푌 and a torus

1 1 1 1 Σ “ 푆 ˆ 푐 Ă 푆 ˆ Σ2. We can choose an orientation preserving diffeomorphism ℎ1 :Σ Ñ Σ1 so that, for all 푡 P 푆1, we have ℎ1pt푡u ˆ 푐q “ t푡u ˆ 푐1 as well as

1 1 ℎ pt푡u ˆ pp훼푙1 X 푐q Y p훼푙2 X 푐qqq “ t푡u ˆ p훽 X 푐 q.

1 1 We can use Σ, Σ , and ℎ to perform a Floer’s excision on 푌 \ 푌Σ. The result is

1 1 a 3-manifold 푌 , with a distinguishing surface 푅 , obtained from 푅 \ Σ2 by cutting 1 ¯ and re-gluing along the two curves 푐 and 푐 . The surface 푆풫 Ă 푌 also becomes a new ¯ 1 ¯ 1 closed surface 푆풫1 Ă 푌 , obtained from 푆\p푆 ˆ훽q by cutting and re-gluing along four

1 1 1 1 curves 푆 ˆp훼푙1 X푐q, 푆 ˆp훼푙2 X푐q, and 푆 ˆp훽 X푐 q (there are two intersection points 1 of 훽 with 푐 ). The curve 휂 together with 훿 Ă Σ2 gives rise to a simple closed curve 휂1 Ă 푅1. See Figure 4-3. Hence, we get a new marked closure 풟1 “ p푌 1, 푅1, 푚1, 푟1, 휂1q.

1 Then, Theorem 2.5.1 and Theorem 2.5.2 result in a cobordism 푊 from 푌 \ 푌Σ to 푌 and a map

1 1 퐻퐺p푊 q : 퐻퐺p푌 \ 푌Σ|푅 Y Σ2q Ñ 퐻퐺p푌 |푅 q.

Let 푎 P 퐻퐺p푌Σ|Σ2q – ℛ be a generator, as in Lemma 2.6.2. Then, we can define

1 1 휄 : 퐻퐺p푌 |푅;Γ휂q Ñ 퐻퐺p푌 |푅 ;Γ휂1 q

172 as 휄p푥q “ 푥 b 푎 and we know that

Φ풟,풟1 “ 퐻퐺p푊 q ˝ 휄, by the definition of Canonical maps in Baldwin and Sivek [4]. ¯ 1 1 The surface 푆풫1 Ă 푌 can also be obtained from the balanced pairing 풫 , which is obtained by performing a cut and glue operation on 풫 associated to the two indices

푙1 and 푙2. Just as we did in the proof of Proposition 4.2.2, we conclude that, for all 푖,

1 1 Φ풟,풟1 p푆퐻푀p풟, 풫, 푖qq “ 푆퐻푀p풟 , 풫 , 푖q.

This concludes the proof of Lemma 4.3.6.

Definition 4.3.7. Two balanced pairings 풫, 풫1 are called connected if there is a sequence of balanced pairings

1 풫0 “ 풫, 풫1, ..., 풫푛 “ 풫 ,

so that, for all 푖 “ 0, 1, ..., 푛 ´ 1, 풫푖 and 풫푖`1 are equivalent.

Lemma 4.3.8. For any odd 푛, the two special balanced pairings 풫푔 and 풫푠 in Ex- ample 4.3.3 are connected to each other.

Proof. In Example 4.3.5, we have shown that

tp1, 2q, p3, 6q, p4, 5qu and tp1, 4q, p2, 5q, p3, 6qu are equivalent. In a similar way, we can also show that

tp1, 6q, p2, 4q, p3, 5qu and tp1, 4q, p2, 5q, p3, 6qu

173 are equivalent. So,

tp1, 2q, p3, 6q, p4, 5qu and tp1, 6q, p2, 4q, p3, 5qu are connected. The later one can be thought of being obtained from the former one by sliding the arc 훼1, which originally joined the points 푝1 and 푝2, over the two arcs

훼2 and 훼3.

If we ignore the pairs p2, 4q and p3, 5q and look at tp1, 6q, p7, 10q, p8, 9qu, then the above argument applies again and we can connect it to tp1, 10q, p6, 9q, p7, 8qu, and this can be thought of further sliding 훼1 over 훼4 and 훼5. We can repeat this step for many times.

Case 1. If 푛 is of the form 4푘 ` 1. In this case, we can slide 훼1 over to join 푝1 푔 with 푝4푘`2. Hence, 풫 is connected to a new balanced pairing

풫1 “tp1, 푛 ` 1 “ 4푘 ` 2q, p2, 5q, p3, 4q, ..., p4푘 ´ 2, 4푘 ` 1q, p4푘 ´ 1, 4푘q,

p4푘 ` 3, 4푘 ` 6q, p4푘 ` 4, 4푘 ` 5q, ..., p8푘 ´ 1, 8푘 ` 2q, p8푘, 8푘 ` 1qu.

Then, we can perform cut and glue operations on pairs p4푙 ´ 2, 4푙 ` 1q and p4푙 ´ 2 ` 푛, 4푙 ` 1 ` 푛q as well as on pairs p4푙 ´ 1, 4푙q and p4푙 ´ 1 ` 푛, 4푙 ` 푛q, for all 1 ď 푙 ď 푘. The result of these operations is nothing but the special balanced paring 풫푠 introduced in Example 4.3.3. Hence, we are done.

Case 2. If 푛 is of the form 4푘 ` 3. In this case, we can slide 훼1 to join 푝1 with 푔 푝4푘`2, so the balanced pairing 풫 is connected to

풫1 “tp1, 4푘 ` 2q, p2, 5q, p3, 4q, ..., p4푘 ´ 2, 4푘 ` 1q, p4푘 ´ 1, 4푘q,

p4푘 ` 3, 4푘 ` 6q, p4푘 ` 4, 4푘 ` 5q, ..., p8푘 ` 3, 8푘 ` 6q, p8푘 ` 4, 8푘 ` 5qu.

174 Perform another cut and glue operation on pairs p1, 4푘 ` 2q and p4푘 ` 4, 4푘 ` 5q, then we get a new balanced pairing

풫1 “tp1, 푛 ` 1 “ 4푘 ` 4q, p2, 5q, p3, 4q, ..., p4푘 ´ 2, 4푘 ` 1q, p4푘 ´ 1, 4푘q,

p4푘 ` 2, 4푘 ` 5q, p4푘 ` 3, 4푘 ` 6q, ..., p8푘 ` 3, 8푘 ` 6q, p8푘 ` 4, 8푘 ` 5qu.

There is, then, an arc joining 푝4푘`2 and 푝4푘`5, and we can slide it over to join 푝4푘`5 and 푝2. Similarly, there is an arc joining 푝4푘`3 with 푝4푘`6, and we can slide it over 푔 to join 푝4푘`3 with 푝8푘`6. Then, 풫 is connected to a new balanced pairing

풫2 “tp1, 푛 ` 1 “ 4푘 ` 4q, p2, 푛 ` 2 “ 4푘 ` 5q, p푛 “ 4푘 ` 3, 2푛 “ 8푘 ` 6q,

p3, 6q, p4, 5q...p4푘 ´ 1, 4푘 ` 2q, p4푘, 4푘 ` 1q

p4푘 ` 6, 4푘 ` 9q, p4푘 ` 7, 4푘 ` 8q, ..., p8푘 ` 2, 8푘 ` 5q, p8푘 ` 3, 8푘 ` 4qu.

Finally, we can perform cut and glue operations on pairs p4푙 ´ 1, 4푙 ` 2q and p4푙 ´ 1 ` 푛, 4푙 ` 2 ` 푛q as well as on p4푙, 4푙 ` 1q and p4푙 ` 푛, 4푙 ` 1 ` 푛q, for all 1 ď 푙 ď 푘, then the final result is 풫푠, and we conclude the proof of Lemma 4.3.8.

Corollary 4.3.9. The definition of the grading on SHGp푀, 훾q is independent of choices of type I.

Proof. It is straightforward to check that if we use the special balanced pairing 풫푠, then the surface 푆풫푠 is the same for all possible choices of the starting point 푝1. Hence the corollary follows from Lemma 4.3.6 and Lemma 4.3.8. r

Remark 4.3.10. We want to use 풫푔 in the definition of grading because it is more convenient to use this construction to discuss the positive and negative stabilizations (see Definition 4.1.1), as we will see in the following two sections.

175 4.4 A naive version of grading shifting property

Suppose p푀, 훾q is a balanced sutured manifold and suppose 푆 is a properly embedded surface in 푀 with a connected boundary. In Definition 4.1.3, we constructed a grading on SHGp푀, 훾q associated to 푆, when |B푆 X 훾| “ 2푛 with 푛 being odd. If 푛 is even, then we introduce, in Definition 4.1.1, positive and negative stabilizations 푆˘ that both increase 푛 by 1. It is a natural question to ask how the gradings associated to 푆` and 푆´ are related to each other. The following proposition is the first answer to this question.

Proposition 4.4.1. Suppose p푀, 훾q is a balanced sutured manifold, 푆 Ă 푀 is a properly embedded surface with a connected boundary, and that B푆 intersects 훾 transversely at 2푛 points with 푛 “ 2푘 ą 0 odd. Suppose further that the balanced sutured manifold obtained by decomposing p푀, 훾q along 푆 is taut. Let 푆` and 푆´ are the positive and negative stabilizations of 푆, respectively. Suppose 푆 is of genus 푔 and let

푔푐 “ 푔 ` 푘.

Then, we have

` ´ SHGp푀, 훾, 푆 , 푔푐q Ă SHGp´푀, ´훾, 푆 , 푔푐 ´ 1q.

` ` Proof. If we have two different negative stabilizations 푆1 and 푆2 , then we know from Lemma 4.1.2 and Theorem 2.6.20 that

` 1 1 ` SHGp푀, 훾, 푆1 , 푔푐q – SHGp푀 , 훾 q – SHGp푀, 훾, 푆2 , 푔푐q, where p푀 1, 훾1q is obtained from p푀, 훾q by performing a sutured manifold decompo-

176 sition along 푆. Hence, we can choose a special negative stabilization to deal with.

Suppose the intersection points of B푆 X 훾 are labeled as 푝1, ..., 푝2푛 according to the orientation of B푆. When labeling the points, we need to pick a suitable 푝1 so that the new pair of intersection points created by the positive or negative stabilization lie

1 1 1 between 푝3 and 푝4. Let 훽 ĂB푆 be part of B푆 so that B훽 “ t푝3, 푝4u and 훽 contains no other intersection points 푝푗 for 푗 ‰ 3, 4. Let 훽 Ă 푆 be a properly embedded arc

1 so that B훽 “ t푝3, 푝4u, 훽 and 훽 co-bound a disk on 퐷, and when performing positive and negative stabilizations, the isotopy on 푆 can be fixed outside the disk 퐷. Now if

˘ we use the same starting point 푝1 to label B푆 X 훾, then the new pair of intersection points are both 푝4 and 푝5 in the two cases. See Figure 4-4.

B푀 B푀 훾 훾 푝 4 푝6

퐷 푝3 푝3 푊 훽 푆 푆

Figure 4-4: A negative stabilization of 푆. Positive stabilizations are similar.

Suppose 푇 is an auxiliary surface for p푀, 훾q of large enough genus. When constructing the grading associated to 푆˘, we need to choose linearly independent arcs

˘ 푔 훼1, 훼2, 훼3 , 훼4..., 훼푛`1 Ă 푇 and the special pairing 풫 , which is defined in Example ˘ 4.3.3, to make it clear what are the endpoints of the arcs 훼푖. Here, 훼3 correspond ˘ to the different surfaces 푆 , while 푇 and all other arcs 훼푖, for 푖 ‰ 3, can be chosen

177 to be the same for both 푆` and 푆´. In the pre-closure 푀 “ 푀 Y r´1, 1s ˆ 푇 , we have two surfaces 푆` and 푆´. After picking suitable gluing diffeomorphisms ℎ˘, we Ă get two marked closures r r

풟` “ p푌 `, 푅`, 푟`, 푚`, 휂`q and 풟´ “ p푌 ´, 푅´, 푟´, 푚´, 휂´q so that there are closed surfaces 푆¯` and 푆¯` inside 푌 ` and 푌 ´, respectively, and the gradings associated to 푆` and 푆´ are defined by looking at the pairings between the first Chern classes of the spin푐 structures on 푌 ` and 푌 ´ with the fundamental classes ¯` ¯´ ¯` ¯´ of 푆 and 푆 , respectively. Note the genus of 푆 and 푆 are both 푔푐 `1 “ 푔 `푘 `1.

From Proposition 4.2.1, we know that the canonical map Φ풟`,풟´ can be interpreted in terms of a Floer’s excision cobordism 푊 from 푌 ` \ 푌 ℎ, where 푌 ℎ is the mapping torus of ℎ “ pℎ`q´1 ˝ ℎ´, to 푌 ´. We can construct a special closed surface of genus 2 as follows. Recall we have an arc 훽 Ă 푆, and since the isotopies for positive or negative stabilizations are supported ¯˘ ¯˘ in the interior of the disk 퐷, 훽 also lies in 푆 . Let 훿 “ 훽 Y p훼2 ˆ t0uq Ă 푆 be a closed curve. Then, the curve 훿 cuts each of 푆¯˘ into two parts. One part contains 푆zintp퐷q and the other part is a connected oriented surface 푇 ˘ Ă 푆¯˘ of genus 1 and with boundary 훿. Inside 푊 , we can define

` ´ Σ2 “ 푇 Y r0, 1s ˆ 훿 Y ´푇 Ă 푊.

It is straightforward to see that, in 퐻2p푊 q,

¯` ¯´ r푆 s “ r푆 s ` rΣ2s.

Hence, by the adjunction inequality in dimension 4, which is a 4-dimensional analogue

178 of Lemma 2.6.12 (In the monopole settings, see Theorem 40.2.3 in Kronheimer and Mrowka [52]; in the instanton settings, see Proposition 2.8 in Baldwin and Sivek [8]), we have

` ´ ´ Φ풟`,풟´ p푆퐻푀p풟, 푆 , 푔푐qq Ă 푆퐻퐺p풟 , 푆 , 푔푐 ` 1q

´ ` ‘ 푆퐻퐺p풟 , 푆 , 푔푐q

´ ` ‘ 푆퐻퐺p풟 , 푆 , 푔푐 ´ 1q.

´ ´ The adjunction inequality also implies that 푆퐻퐺p풟 , 푆 , 푔푐 ` 1q “ 0. If we decompose p푀, 훾q along 푆´, and suppose p푀 2, 훾2q is the resulting balanced sutured

2 manifold, then, by Lemma 4.1.2, 푅˘p훾 q is compressible and so

´ ´ 2 2 푆퐻퐺p´풟 , 푆 , 푔푐q – 푆퐻퐺p푀 , 훾 q “ 0.

The first isomorphism follows from Theorem 2.6.20 and the second equality follows again from the adjunction inequality in Lemma 2.6.12 and Proposition 2.3.4. Hence, the only possibility left is

´ ` ´ ´ Φ풟`,풟´ p푆퐻퐺p풟 , 푆 , 푔푐qq Ă 푆퐻퐺p´풟 , 푆 , 푔푐 ´ 1q and we we conclude the proof of Proposition 4.4.1.

4.5 Supporting spin푐 structures and eigenvalue functions

Let 푌 be a connected closed oriented 3-manifold. Suppose 퐾 Ă 푌 is a null- homologous knot, and 푆 Ă 푌 is a Seifert surface of 퐾. The surface 푆 then induces

179 a framing on the knot complement 푌 p퐾q “ 푌 z푁p퐾q. Let 휇 be the meridian, and 휆 “ 푆 XB푌 p퐾q be the longitude. We can regard 푆 as a properly embedded surface inside 푌 p퐾q. Pick 훾 ĂB푌 p퐾q be a suture having two components. We further assume the following: if we write the slope of 훾 as 푝{푞 (meridian/longitude), we require that |푝| ą 1. The surface 푆 induces gradings on SHMp푀, 훾q, after perturbations if necessary. In this and next subsection, we will study how different gradings associated to perturbations of 푆 are related to each other. Suppose p푌 , 푅q is a closure of p푌 p퐾q, 훾q, as in Definition 2.2.2. In this subsection, we want to study the set of supporting spin푐 structures and eigenvalue functions on s s 푌 which is introduced in Definition 2.2.7 and Definition 2.3.7. In particular, weprove the following proposition. s

Proposition 4.5.1. Suppose, after possible perturbation, 푆 intersects 훾 transversely at 2푛 points with 푛 even. Suppose p푌 , 푅q is a closure of p푌 p퐾q, 훾q so that 푆 extends to a closed surface 푆 Ă 푌 as in the construction of gradings in Section 4.1. Let s s s , s P S˚p푌 |푅q be two supporting spin푐 structures on 푌 . Then, there is a 1-cycle 푥 1 2 s s inside 푌 p퐾q, so that s s s

푃.퐷.푐1ps1q ´ 푃.퐷.푐1ps2q “ r푥s P 퐻1p푌 ; Qq.

s Note the cycle is contained in 푌 p퐾q but the identity is on the whole 푌 . Similarly, if 휆 and 휆 are two supporting eigenvalue functions on 푌 , then there 1 2 s exists a 1-cycle 푥 inside 푌 p퐾q so that s

푃.퐷.휆1 ´ 푃.퐷.휆2 “ r푥s P 퐻1p푌 ; Qq.

s Remark 4.5.2. Here, we only need Q coefficients since our aim is to study the grading

180 that arises from the pairing of the first Chern classes of supporting spin푐 structures with the fundamental classes of some surfaces. So Q is enough for this purpose.

To prove Proposition 4.5.1, we need to understand the homology of the closures of p푌 p퐾q, 훾q as well as the homology of the excision cobordisms which induces the canonical map as in Subsection 4.2 better. Let us start with an alternative description of the closures of p푌 p퐾q, 훾q.

Let Σ푔 be a closed oriented connected surface of large enough genus 푔. Its first homology is generated by the classes r푎1s, r푏1s, ..., r푎푔s, r푏푔s, as depicted in Figure 4-5.

... 푏1 푏푔

푎1 푎푔

Figure 4-5: The surface Σ푔.

Let 푇 “ Σ푔zintp푁p푎1qq be a surface obtained from Σ푔 by cutting Σ푔 open along

푎1, then 푇 can be viewed as an auxiliary surface for p푌 p퐾q, 훾q. Let

푌 “ 푌 p퐾q Y r´1, 1s ˆ 푇

r be a pre-closure of p푀, 훾q, and let

B푌 “ 푅` Y 푅´.

r 0 If we choose a special gluing diffeomorphism ℎ : 푅` Ñ 푅´ so that ℎ|푇 ˆt1u “ 푖푑,

181 then we get a special marked closure

0 0 0 0 풟 “ p푌 , Σ푔, 푟 , 푚 , 휂q.

Similar to the closures described in Section 5.1 in [53], the closure p푌 0, 푅q can be 1 achieved as follows: Let Σ푔 be the surface as in Figure 4-5, and let 푌Σ “ 푆 ˆ Σ푔.

By abusing the notations, use 푎1 to also denote the curve t1u ˆ 훼1 Ă 푌Σ. Let 푁p푎1q be a tubular neighborhood of 푎1 Ă 푌Σ. Note 푎1 Ă t1u ˆ Σ푔, so there is a framing on B푁p푎1q induced by t1u ˆ Σ푔. Let 휆푎 and 휇푎 be the longitude and meridian, respectively.

Then, we have

0 푌 “ 푌 p퐾q Yp푌Σzintp푁p푎1qqq, 휑 where the map

휑 : B푁p푎1q Ñ B푌 p퐾q sends two copies of 휆푎 to the suture 훾. Note there are canonical ways to identify 푅˘ with Σ푔. So, in the marked closure 풟0, we have 푅 “ Σ푔.

Lemma 4.5.3. The conclusion of proposition 4.5.1 is true for 푌 0, despite of the fact that 푆 may not extend to 푌 0.

Proof. We only prove the lemma in the monopole settings, and the proof in the instanton settings is the same.

From the Mayer-Vietoris sequece, we know that there is an exact sequence

2 0 퐻1p푇 ; Qq Ñ 퐻1p푌 p퐾q; Qq ‘ 퐻1p푌Σzintp푁p푎1qq; Qq Ñ 퐻1p푌 ; Qq Ñ 0,

182 2 where 푇 “B푀 “ Bp푌Σzintp푁p푎1qqq. Hence, we conclude that

0 퐻1p푌 ; Qq “ 퐻1p푌 p퐾q; Qq ‘ 퐻1p푌Σzintp푁p푎1qq; Qq{ „, where „ is the relation induced by the gluing map 휑 :

r휆푎s „ 휑˚pr휆푎sq, r휇푎s „ 휑˚pr휇푎sq.

A direct calculation shows that

0 퐻1p푌Σzintp푁p푎1qq; Qq “ xr푎1s, r푏1s, ..., r푎푔s, r푏푔s, r푠 sy,

0 1 1 where 푠 corresponds to the 푆 direction in 푌Σ “ 푆 ˆ Σ푔, and

r휇푎s “ 0 P 퐻1p푌Σzintp푁p푎1qq; Qq. (4.3)

Hence, we can write

0 0 퐻1p푌 ; Qq “ 퐻1p푌 p퐾q; Qq ‘ xr푏1s, r푎2s, r푏2s, ..., r푎푔s, r푏푔s, r푠 sy. (4.4)

This is because r푎1s is absorbed into 퐻1p푌 p퐾q; Qq.

˚ 0 Suppose s P S p푌 |Σ푔q, then we can express 푃.퐷.푐1psq in terms of the above basis. The coefficient of r푠s can be fixed by the evaluation

푐1psqrΣ푔s “ 2푔 ´ 2.

There are no r푏1s, r푎2s, r푏2s...r푎푔s, r푏푔s terms, since we can apply the adjunction in- 1 1 1 0 equality in Lemma 2.6.12 to tori 푆 ˆ 푎1, 푆 ˆ 푏2..., 푆 ˆ 푎푔 Ă 푌 to rule out those

183 classes. The rest of the terms must then lie in 퐻1p푌 p퐾q; Qq. So, if we look at the difference (of the Poincaré dual of their first Chern class) of two supporting spin푐 structures, it must lie in 푌 p퐾q.

Next, we deal with general closures of p푌 p퐾q, 훾q. As above, we have the pre- closure 푌 “ 푌 p퐾q Y r´1, 1s ˆ 푇, where 푇 “ Σ푔z푁p푎1q. Also, recallr

B푀 “ 푅` Y 푅´.

Ă Note, as in the above discussion, there are canonical ways to identify 푅` and 푅´ with Σ푔. We can pick any orientation preserving diffeomorphism ℎ : 푅` Ñ 푅´ to get a closure p푌 , Σ푔q of p푌 p퐾q, 훾q, or a marked closure

s 풟 “ p푌 , Σ푔, 푟, 푚, 휂q.

s In particular, the special marked closure 풟0 in Lemma 4.5.3 corresponds to taking ℎ “ ℎ0 “ 푖푑.

ℎ Let 푌 be the mapping torus of the diffeomorphism ℎ :Σ푔 Ñ Σ푔, then we can reinterpret 푌 as

ℎ 푌 “ 푌 p퐾q Yp푌 zintp푁p푎1qqq. s 휑

From Proposition 4.2.1, we knows that the canonical map Φ풟0,풟 can be obtained from a cobordism 푊 from 푌 0 \ 푌 ℎ to 푌 . The cobordism 푊 arises from the Floer excision as in Section 4.2. s Next, we deal with the class r휇푎s coming from the meridian of 푎1. In general, we

184 ℎ don’t know if the class r푎1s is trivial inside 퐻1p푌 q or not, so we also don’t know if r휇푎s is trivial or not, either. However, when 푌 satisfies the hypothesis of Proposition 4.5.1, we do. In the rest of this subsection, we always assume that 푌 satisfies the s hypothesis of Proposition 4.5.1. s

Lemma 4.5.4. We know that

r휇푎s ‰ 0 P 퐻1p푌 ; Qq, and r푎1s “ 0 P 퐻1p푌 ; Qq.

s s Proof. Note we have

ℎ 푌 “ 푌 p퐾q Yp푌 zintp푁p푎1qqq, 휑 where the gluing map 휑 mapss 푎1 to a component of the suture 훾. Recall we have assumed that the slope of the suture is 푝{푞 with |푝| ą 1, so 휑˚pr푎1sq “ 푝r휇s ` 푞r휆s. 1 1 1 Hence, we know that 휑˚pr휇푎sq “ 푝 r휇s ` 푞 r휆s with |푝 | ą 1. Note inside 푌 , we have r휇s ¨ r푆s “ 1, and r휆s “ 0, this means that r휇 s ‰ 0. If r푎 s ‰ 0, then there 푎 1 s is a closed oriented surface Σ Ă 푌 so that r푎 s ¨ rΣs ‰ 0. Then, the boundary of s 1 ΣX푌 ℎzintp푁p푎 qq r휇 s represent a class that equals a non-zero multiple of r휇 s. This 1 푎 s 푎 implies that r휇푎s “ 0 P 퐻1p푌 ; Qq, which is a contradiction.

s With Lemma 4.5.4, the computation of the first homologies of 푌 , 푌 ℎ and 푊 are straightforward, and we can describe them as follows. s

퐻1p푌 q “ 퐻1p푌 p퐾q; Qq ‘ xr휇푎s, r푎1s, r푏1s, ..., r푎푔s, r푏푔s, r푠sy{ „휑,ℎ (4.5)

s ℎ ℎ 퐻1p푌 ; Qq “ xr푎1s, r푏1s..., r푎푔s, r푏푔s, r푠 sy{ „ℎ (4.6)

0 ℎ 퐻1p푊 ; Qq “ 퐻1p푌 p퐾q; Qq ‘ xr푎1s, r푏1s, ..., r푎푔s, r푏푔s, r푠 s, r푠 sy{ „휑,ℎ . (4.7)

185 Here, 푠 is a circle inside 푌 which intersects Σ푔 once. We can isotope ℎ so that ℎ has 1 ℎ a fixed point 푝 P Σ푔, then, inside 푌 , there is a circle 푠 “ t푝u ˆ 푆 . The class 푠 is similar. The relations „휑,ℎ are

r푎1s „ 휑˚pr푎1sq, r휇푎s „ 휑˚pr휇푎sq, r푎푖s „ ℎpr푎푖sq, r푏푖s „ ℎpr푏푖sq.

The relations „ℎ are

r푎푖s „ ℎpr푎푖sq, r푏푖s „ ℎpr푏푖sq.

0 Note r휇푎s “ 0 P 퐻1p푊 ; Qq, since r휇푎s “ 0 P 퐻1p푌 ; Qq. From (4.5) and (4.7), we know that

Lemma 4.5.5. Suppose 푖 : 푌 ãÑ 푊 is the inclusion, and let

s 푖˚ : 퐻1p푌 q Ñ 퐻1p푊 ; Qq

s be the induced map on first homology. Then, we have

푘푒푟p푖˚q “ xr휇푎sy Ă 퐻1p푌 q

s Proof of Proposition 4.5.1. Again, we only prove the proposition in the monopole settings, and the proof in the instanton settings is the same.

Suppose 푌 is a closure satisfying the hypothesis of the proposition. Let 푌 0, 푌 ℎ, and 푊 be chosen as above. From Lemma 2.6.2 we know that there is a unique s 푐 ℎ ℎ supporting spin structure on 푌 , which we denote by s . Suppose s1 and s2 are two supporting spin푐 structures on 푌 . Then, from the fact that 푊 induces an isomorphism and the nature of the cobordism map in monopole Floer homology, we s 186 푐 0 0 0 know that there are two supporting spin structures s1 and s2 on 푌 , so that

0 ℎ 푃.퐷.푐1ps1q “ 푃.퐷.푐1ps1q ` 푃.퐷.푐1ps q P 퐻1p푊 ; Qq, and

0 ℎ 푃.퐷.푐1ps2q “ 푃.퐷.푐1ps2q ` 푃.퐷.푐1ps q P 퐻1p푊 ; Qq.

Thus, we know that

0 0 푃.퐷.푐1ps1q ´ 푃.퐷.푐1ps2q “ 푃.퐷.푐1ps1q ´ 푃.퐷.푐1ps2q P 퐻1p푊 ; Qq.

From Lemma 4.5.3, we know that there is a 1-cycle 푥 inside 푌 p퐾q so that

0 0 0 푃.퐷.푐1ps1q ´ 푃.퐷.푐1ps2q “ r푥s P 퐻1p푌 ; Qq.

We conclude that

푃.퐷.푐1ps1q ´ 푃.퐷.푐1ps2q P r푥s ` 푘푒푟p푖˚q Ă 퐻1p푌 ; Qq.

s Then, the proposition follows from Lemma 4.5.5.

4.6 The grading shifting property

In this section, we prove the following proposition.

Proposition 4.6.1. Suppose p푌 p퐾q, 훾q is the balanced sutured manifold and 푆 is a Seifert surface of the knot 퐾, both as described in Subsection 4.5. Suppose further that 푆 has minimal genus among its homology class and has minimal intersection

187 with 훾 so that |B푆 X 훾| “ 2푛 ą 0. Suppose further that decomposing p푌 p퐾q, 훾q along 푆 and ´푆 are both taut. Then, for any 푝, 푘, 푙 P Z such that 푛 ` 푝 is odd, we have

SHGp푌 p퐾q, 훾, 푆푝, 푙q “ SHGp푌 p퐾q, 훾, 푆푝`2푘, 푙 ` 푘q.

Note 푆푝 is a stabilization of 푆 as introduced in Definition 4.1.1, and, in particular, 푆0 “ 푆.

Proof. Again we prove the proposition in the monopole settings, and the proof for the instanton settings is exactly the same. Suppose we have chosen 푆푝 and 푆푝`2푘 as two stabilizations of 푆.

Claim 1. There is a fixed integer 푙0 so that, for any 푙 P Z, we have

푝 푝`2푘 SHMp푌 p퐾q, 훾, 푆 , 푙q “ SHMp푌 p퐾q, 훾, 푆 , 푙 ` 푙0q.

To prove this claim, we start with two disjoint copies of 푆. Since the perturbation can be made in an arbitrary neighborhood of 푆, we can perform 푝 perturbations on one copy and 푝 ` 2푘 perturbations on the other. The result is two surfaces 푆푝 and 푆푝`2푘 embedded disjointly into 푀.

Next, we can carry out the construction of gradings in Section 4.1. We can apply the construction in that section on both 푆푝 and 푆푝`2푘 simultaneously and obtain a closure 푌 with two surfaces 푆¯푝 and 푆¯푝`2푘, extending 푆푝 and 푆푝`2푘, respectively.

푐 Supposes s1 and s2 are two supporting spin structures on 푌 , then, by Proposition 4.5.1, there is a 1-cycle 푥 inside 푌 p퐾q so that s

푃.퐷.푐1ps1q ´ 푃.퐷.푐1ps2q “ r푥s P 퐻1p푌 ; Qq.

188 s Since 푆푝 and 푆푝`2푘 are the same inside 푌 p퐾q, we conclude that

s s r푥s ¨ pr푆s ´ r푆2푘sq “ 0,

s s and thus

2푘 2푘 푐1ps1qpr푆sq ´ .푐1ps1qpr푆 sq “ 푐1ps2qpr푆sq ´ 푐1ps2qpr푆 sq.

Note the above equalitys is equivalents to the existences of 푙0. s

Claim 2. We have 푙0 “ 푘.

Case 1. We have 푝 ą 0 and 푝 ` 2푘 ą 0. From Lemma 4.1.2, we know that if we decompose p푀, 훾q along both 푆푝 and 푆푝`2푘, we obtain the same taut balanced sutured manifold p푀 1, 훾1q. From Theorem 2.6.20, we conclude that

푝 푝 1 1 푝`2푘 푝`2푘 SHMp푌 p퐾q, 훾, 푆 , 푔푐p푆 qq – SHMp푀 , 훾 q – SHMp푌 p퐾q, 훾, 푆 q, 푔푐p푆 q, where 푝 ´ 휒p푆q 푝 ` 2푘 ´ 휒p푆q 푔 p푆푝q “ and 푔 p푆푝q “ . 푐 2 푐 2

푝 푝`2푘 Theorem 2.6.20 also states that 푔푐p푆 q and 푔푐p푆 q are the top non-vanishing grading with respect to the grading induced by each surface. Hence, from Claim 1 we conclude that 푙0 “ 푘.

Case 2. We have 푝 ă 0 and 푝 ` 2푘 ă 0. This follows exactly the same line as in Case 1, though working with ´푆 instead of 푆.

Case 3. We have t푝, 푝 ` 2푘u “ t´1, 1u. We can apply Claim 1 and Proposition 4.4.1.

Case 4. We have 푝 and 푝 ` 2푘 to be of different sign. This is a combination of Case 1, 2, and 3 and Claim 1.

189 This concludes the proof of Proposition 5.1.9.

4.7 Floer homologies on a sutured solid torus

As a first application of the grading shifting property, we compute the sutured monopole Floer homology of any sutured solid tori. The same result in sutured Heegaard Floer theory can be found in Juhász [40]. Suppose 푉 “ 푆1 ˆ 퐷2 is a solid torus. Let 휆 denote a longitude 푆1 ˆ t푡u where 푡 PB퐷2 and let 휇 denote a meridian t푠u ˆ B퐷2 where 푠 P 푆1. Suppose further 훾 is a suture on 푉 so that p푉, 훾q is a balanced sutured manifold. Then, 훾 is parametrized by two quantities, 푛 and 푠, where 2푛 is the number of components of 훾 and 푠 is

푛 the slope of the suture. In this section, we write the suture 훾 as 훾p푞,´푝q. We write the slope 푠 as p푞, ´푝q, and this is to keep our notations consistent with the ones in Honda [33]. Note p푞, ´푝q means going around longitude ´푝 times and meridian 푞 times. We always assume that 푝 ě 0.

2 Proposition 4.7.1. Suppose p푉, 훾p푞,´푝qq is defined as in the above paragraph. Then, we have

2 푝 SHGp´푉, ´훾p푞,´푝qq “ ℛ .

2 Proof. If 푝 “ |푞|, then 푝 “ ˘푞 “ 1, since they are co-prime. Then, p푉, 훾p1,˘1qq is diffeomorphic to a product sutured manifold p퐴 ˆ r´1, 1s, B퐴 ˆ t0uq, where 퐴 is an annulus. Thus, we know

2 SHGp´푉, ´훾p1,´1qq – ℛ.

From now on, we assume that 푝 ą 푞 ą 0. If not, we can achieve this assumption by applying diffeomorphisms of the solid torus 푉 . We want to re-interpret the by- pass exact triangle in Theorem 2.7.21 as follows: We have a basic by-pass exact

190 triangles:

2 SHGp´푉, ´훾p1,´1qq (4.8) 휓´,1 5 휓´,2

) SHGp´푉, ´훾2 q o SHGp´푉, ´훾2 q p1,0q 휓´,0 p0,´1q

Recall, from Section 2.7, that the map 휓´,1 (as well as the other two) is interpreted 2 2 1 2 as a gluing map: Suppose we have p´푉, ´훾p1,0qq and an identification 푇 “ 푆 ˆB퐷 , then we can glue r0, 1sˆ푇 2 to 푉 via the identification 푖푑 : B푉 “ 푆1 ˆB퐷2 ÝÑt“ 0uˆ푇 2. 2 2 2 Suppose t0uˆ푇 is equipped with the suture 훾p1,0q, and 푇 ˆt1u is equipped with the 2 2 2 2 suture 훾p1,´1q, then we can identify p푉, 훾p1,´1qq with p푉 Y r0, 1s ˆ 푇 , 훾p1,´1qq. There 2 2 2 exists a compatible contact structure 휉´,1 on pr0, 1s ˆ 푇 , 훾p1,0q Y 훾p1,´1qq so that we have

2 2 휓´,0 “ Φ휉´,0 : SHGp´푉, ´훾p1,0qq Ñ SHGp´푉, ´훾p1,´1qq.

2 2 2 When dealing with other sutures, we can also glue pr0, 1s ˆ 푇 , 훾p1,0q Y 훾p1,´1qq to 푉 , but along a diffeomorphism

푔 : t0u ˆ 푇 2 ÑB푉, instead of the identity map. Such a map needs to be orientation preserving and, hence, is parametrized by an element in 푆퐿2pZq. We can pick the map 푔 corresponding to the matrix 푞 ´ 푞1 ´푞1 퐴 “ P 푆퐿2pZq, ¨ 푝1 ´ 푝 푝1 ˛ ˝ ‚ where 푝1푞 ´ 푝푞1 “ 1, 푝1 ď 푝, 푞1 ď 푞, 푞2 “ 푝 ´ 푝1, and 푝2 “ 푝 ´ 푝1. (Such 푝1, 푞1, 푝2, 푞2

191 are unique.)

2 2 2 2 Then, the suture 훾p1,0q on 푇 ˆ t0u is glued to 훾p푞,´푝q on B푉 and the suture 훾p1,´1q 2 2 on 푇 ˆ t1u now becomes the suture 훾p푞1,´푝1q. As in Formula (4.8), they still fit into an exact triangle

2 SHGp´푉, ´훾p푞,´푝qq (4.9) 휓´,1 5 휓´,2

) 2 2 SHGp´푉, ´훾 2 2 q o SHGp´푉, ´훾 1 1 q p푞 ,´푝 q 휓´,0 p푞 ,´푝 q

2 We claim that 휓´,0 “ 0. Let 퐷푝 be a meridian disk of 푉 which intersects 훾p푞,´푝q at 2푝 points, then, from a similar argument as in Proposition 5.1.9 (which we will prove later), we have

2 ´p푝´푝1q 2 `p푝´푝2q 휓´,0pSHGp´푉, ´훾p푞1,´푝1q, 퐷푝1 , 푖qq Ă SHGp´푉, ´훾p푞2,´푝2q, 퐷푝2 , 푖q for any 푖 P Z.

We only deal with the case when 푝1 is odd and 푝2 is even. Other cases are similar. From the construction of the grading in Definition 4.1.3, we know that there isa ¯ suitable marked closure 풟푝1 “ p푌푝1 , 푅, 푟, 푚, 휂q and a closed surface 퐷푝1 Ă 푌푝1 so that the grading is defined via the evaluations of the first Chern classes ofspin푐 structures ¯ on the fundamental class of 퐷푝1 . From the construction, we know that

¯ 1 1 휒p퐷푝1 q “ 휒p퐷푝1 q ´ 푝 “ 1 ´ 푝 .

192 Hence, the adjunction inequality in Lemma 2.6.12 implies that

2 SHGp´푉, ´훾p푞1,´푝1q, 퐷푝1 , 푖q “ 0

1´푝1 if 푖 ă 2 . Then, from the grading shifting property in Proposition 4.6.1, we know that 2 2 ´푝2 2 푝 SHGp´푉, ´훾 1 1 , 퐷 1 , 푖q “ SHGp´푉, ´훾 1 1 , 퐷 1 , 푖 ` p qq. p푞 ,´푝 q 푝 p푞 ,´푝 q 푝 2 Thus, we know

2 ´푝2 SHGp´푉, ´훾p푞1,´푝1q, 퐷푝1 , 푖q “ 0 (4.10)

1´푝1`푝2 2 1 if 푖 ă 2 . Note, by definition, 푝 “ 푝 ´ 푝 .

` 2 The above argument for 퐷푝1 applies to 퐷푝2 as well. Note 푝 is assumed to be even, so we need to perform a positive stabilization on 퐷푝2 to construct the grading. The adjunction inequality in Lemma 2.6.12 and Proposition 2.3.4 again implies that

2 ` SHGp´푉, ´훾p푞2,´푝2q, 퐷푝2 , 푖q “ 0 (4.11)

푝2 if 푖 ą 2 . However, from Theorem 2.6.20, we know that

2 2 ` 푝 1 1 SHGp´푉, ´훾 2 2 , 퐷 2 , q – SHGp푀 , 훾 q, p푞 ,´푝 q 푝 2

1 1 2 where p푀 , 훾 q is the result of doing a sutured manifold decomposition on p´푉, ´훾p푞2,´푝2qq ` along the surface 퐷푝2 . From Lemma 4.1.2, we know that

2 2 ` 푝 1 1 SHGp´푉, ´훾 2 2 , 퐷 2 , q – SHGp푀 , 훾 q “ 0. (4.12) p푞 ,´푝 q 푝 2 193 The grading shifting property in Proposition 4.6.1, then, implies

1 2 `푝1 2 ` 푝 ´ 1 SHGp´푉, ´훾 2 2 , 퐷 2 , 푖q “ SHGp´푉, ´훾 2 2 , 퐷 2 , 푖 ´ q. p푞 ,´푝 q 푝 p푞 ,´푝 q 푝 2

The above equality, together with (4.11) and (4.12), implies that

2 `푝1 SHGp´푉, ´훾p푞2,´푝2q, 퐷푝2 , 푖q “ 0

1´푝1`푝2 if 푖 ě 2 . Compare this with (4.10), we can see that 휓´,0 “ 0. 2 Once we conclude that 휓´,0 “ 0, we can compute SHGp´푉, ´훾p푞,´푝qq by the induction, and Proposition 4.7.1 follows.

Remark 4.7.2. As in Honda [33], the two slopes p푞1, ´푝1q and p푞2, ´푝2q can be written out explicitly in terms of the continued fraction of p푞, ´푝q. Note we have assumed 푝 ą 푞. Suppose 푝 1 ´ “ 푟1 ´ 1 , 푞 푟2 ´ 푟3´... where it is a finite continued fraction, and 푟푗 ă ´1 for all 푗. We can write

푝 ´ “ r푟 , 푟 , ..., 푟 s. (4.13) 푞 1 2 푘

Under this notation, we have

푝1 푝2 ´ “ r푟 , 푟 ..., 푟 s, ´ “ r푟 , 푟 ..., 푟 ` 1s, 푞1 1 2 푘´1 푞2 1 2 푘´1

and in the above notation, we identify r푟1, ..., 푟푗´1, 푟푗, ´1s with r푟1, ..., 푟푗´1, 푟푗 ` 1s.

푛 Now we deal with the general sutures 훾p푞,´푝q for 푛 ą 1. There are two types 2푛`2 2푛 by-passes relating p푉, 훾p푞,´푝qq and p푉, 훾p푞,´푝qq. We call them positive and negative

194 by-passes according to Figure 4-6. They give rise to by-pass exact triangles as in Theorem 2.7.21:

2푛`2 SHGp´푉, ´훾p푞,´푝qq (4.14) 푛 푛`1 휓˘,푛`1 5 휓˘,푛

) 2푛 2푛 SHGp´푉, ´훾p푞,´푝qq o 푛 SHGp´푉, ´훾p푞,´푝qq 휓˘,푛

Positive by-passes Negative by-passes Figure 4-6: The positive and negative by-passes.

Remark 4.7.3. Unlike the case of two sutures where there are exactly two different possibilities of by-passes, in the case where 훾 has more than two components, positive and negative by-passes are not unique. Here, we just pick two specific by-passes so that they are ’adjacent’ to each other. This is crucial to the proof of Lemma 4.7.4.

Question 27. What can we say about the by-pass maps associated to non-adjacent by-pass attachments?

195 Lemma 4.7.4. For any 푛 P Z and slope p푞, ´푝q, we have

푛`1 푛 푛`1 푛 2푛 2푛 휓´,푛 ˝ 휓`,푛`1 “ 휓`,푛 ˝ 휓´,푛`1 “ 푖푑 : SHGp´푉, ´훾p푞,´푝qq Ñ SHGp´푉, ´훾p푞,´푝qq.

푛`1 푛 Proof. We will only prove that 휓´,푛 ˝ 휓`,푛`1 “ 푖푑. The other is the same. From [5] or [72] we know that a by-pass attached along an arc 훼 can be thought of as attaching a pair of contact 1-handle and 2-handle. The contact one handle is attached along the two endpoints B훼 while the contact 2-handle is attached along a Legendrian curve 훽 “ 훼 Y 훼1, where 훼1 is an arc on the contact 1-handle intersecting the dividing set once.

푛`1 푛 Now 휓´,푛 ˝ 휓`,푛`1 corresponds to first attaching a by-pass along 훼` and then attaching another one along 훼´, as in Figure 4-7. However, in terms of contact handle attachments, the two pairs of handles are disjoint from each other, so we can reverse the order of attachments: Instead, we can first attach a by-pass along 훼´ and then along 훼`. If we attach a by-pass along 훼´ first, we can see from Figure 4-7 that this is a trivial by-pass as discussed in Honda [34]. In that paper, it is proved that a trivial by-pass does not change the contact structure. From Theorem 2.7.16, we conclude that a trivial by-pass induces the identity map. Then, the second by-pass attached along 훼` is also trivial and, hence, again induces the identity map. Thus, 푛`1 푛 we conclude that 휓´,푛 ˝ 휓`,푛`1 “ 푖푑.

Corollary 4.7.5. We know that

2푛 p2푛´1¨푝q SHGp´푉, ´훾p푞,´푝qq – ℛ .

196 훼`

훼´ 훼` Above: first attach along 훼´ then 훼`

Below: first attach along 훼` then 훼´

훼´ 훼´

Figure 4-7: Reversing the order of by-pass attachments. Bottom right picture: we can isotope 훼´ to this new position, where we can see directly that the by-pass is trivial.

푛`1 푛 Proof. From Lemma 4.7.4, we know that 휓˘,푛 is surjective while 휓˘,푛`1 is injective. Hence, we can conclude the statement by using the by-pass exact triangles and the induction.

Corollary 4.7.6. We have

2푛 푛´1 |휋0pTightp푉, 훾p푞,´푝qqq| ě 2 ¨ |푟1 ` 1| ¨ ... ¨ |푟푘´1 ` 1| ¨ |푟푘|.

Proof. First assume 푛 “ 1. In [33], Honda explained how to construct any compatible tight contact structures on a sutured solid torus: First we start with the standard

2 tight contact structure on p푉, 훾p1,´1qq. Then, we can glue 푘 different layers r푖 ´ 1, 푖s ˆ 푇 2, for 1 ď 푖 ď 푘, to 푉 , so that, on r푖 ´ 1, 푖s ˆ 푇 2, t푖 ´ 1u ˆ 푇 2 has the dividing

197 set 훾2 , while 푖 푇 2 has the dividing set 훾2 . We glue 0 푇 2 to 푉 via p1,´1q t u ˆ p1,1´푟푖q t u ˆ B identity, while glue t푖u ˆ 푇 2 Ă r푖, 푖 ` 1s ˆ 푇 2 to t푖u ˆ 푇 2 Ă r푖 ´ 1, 푖s ˆ 푇 2 so that the dividing sets on these two surfaces are identified.

2 Each layer 푇 ˆr푖´1, 푖s is further decomposed into the composition of ´1´푟푖 (or

´푟푘 for the last layer) many by-passes. There are two by-passes: One corresponds to the map 휓´,1 in formula (4.9), and the other corresponds to some 휓`,1 in a similar by-pass exact triangle. Use the inductive step as introduced in [33], which Honda used to construct tight contact structures on a sutured solid torus, we see that all the contact structures that Honda constructed have distinct contact elements. Hence, there are at least |푟1 ` 1| ¨ ... ¨ |푟푘´1 ` 1| ¨ |푟푘| many different contact structures. When 푛 is bigger than 1, we proceed by induction. Suppose, for 푛 “ 푙, there

푙´1 are at least 푚푙 “ 2 ¨ |푟1 ` 1| ¨ ... ¨ |푟푘´1 ` 1| ¨ |푟푘| many different non-zero contact elements 휓 , ..., 휓 P SHGp´푉, 훾2푙 q. From Lemma 4.7.4, we know that 휓푙 휉1 휉푚푙 p푞,´푝q `,푙`1 푙 and 휓´,푙`1 are both injective,

푙`1 푙 푙`1 푙 휓˘,푙 ˝ 휓˘,푙`1 “ 0, 푎푛푑 휓¯,푙 ˝ 휓˘,푙`1 “ 푖푑.

The first equality is due to the exactness of the by-pass triangle, and the secondis

2푙`2 again Lemma 4.7.4. Hence, we know that, inside SHMp´푉, 훾p푞,´푝qq, there are at least 푙 푚푙`1 “ 2 ¨ |푟1 ` 1| ¨ ... ¨ |푟푘´1 ` 1| ¨ |푟푘| many different contact elements

휓푙 p휑 q, ..., 휓푙 p휑 q. ˘,푙`1 휉1 ˘,푙`1 휉푚푙

Hence, we are done.

Remark 4.7.7. When 푛 “ 1, the above argument gives an alternative way to provide 2 a tight lower bound of |휋0pTightp푉, 훾p푞,´푝qqq|, which is originally done by Honda [33].

198 When 푛 ą 1, as mentioned in Remark 4.7.3, there are not just two by-passes, so this lower bound, a priori, need not be tight. However, one could try to study the impact of all other by-pass attachments to see if we could improve the lower bound.

2푛 Question 28. What is the exact value of |휋0pTightp푉, 훾p푞,´푝qqq|?

Remark 4.7.8. We can use a meridian disk of the solid torus to define a grading

2푛 on SHGp´푉, ´훾p푞,´푝qq. The above method is also capable of computing the graded homology.

4.8 The connected sum formula

In this section, we exploit the computational result in the previous section and derive the connected sum formula for sutured monopole Floer homology. First, we prove the following proposition.

Proposition 4.8.1. We use Z2 coefficients. Suppose three oriented links 퐿0, 퐿1 and 3 3 퐿2 are the same outside a 3-ball 퐵 , and, inside 퐵 , they are depicted as in Figure 4-8. We have the following.

(1) If 퐿2 has one more component than 퐿0 and 퐿1, then there is an exact triangle:

3 3 퐾퐻푀p푆 , 퐿0q / 퐾퐻푀p푆 , 퐿1q h

v 3 퐾퐻푀p푆 , 퐿2q

(2) If 퐿2 has one less component than 퐿0 and 퐿1, then there is an exact triangle:

199 3 3 퐾퐻푀p푆 , 퐿0q / 퐾퐻푀p푆 , 퐿1q j

t 3 4 퐾퐻푀p푆 , 퐿2q b pZ2q

퐿0 퐿1 퐿2

Figure 4-8: The oriented skein relation.

Proof. It follows from an analogous argument in sutured instanton Floer theory in Kronheimer and Mrowka [51]. We sketch the proof as follows: The monopole

3 3 knot Floer homology 퐾퐻푀p푆 , 퐿0q for a link 퐿0 Ă 푆 is defined by taking the 3 sutured monopole Floer homology of the balanced sutured manifold p푆 p퐿0q, Γ휇q, 3 where 푆 p퐿0q is the link complement, and Γ휇 consists of a pair of meridians on each 3 3 boundary component of 푆 p퐿0q. Let p푌0, 푅q be a closure of p푆 p퐿1q, Γ휇q. To obtain an exact triangle for the oriented skein relation, we pick a small circle 훼 linking around the crossing, on which we perform the crossing change and oriented-smoothing. The curve 훼 naturally embeds into 푌0 and is disjoint from the surface 푅 Ă 푌0. Then, from Kronheimer, Mrowka, Ozsváth, and Szabó [44], there is a surgery exact triangle relating the monopole Floer homologies of the 3-manifolds obtained by performing p´1q, 0, and 8 surgery along the curve 훼 Ă 푌0. There is a canonical framing for the curve 훼 Ă 푆3, and the surgeries slopes are the ones with respect to the canonical framing.

200 When performing the 8 surgery, we obtain 푌0. When performing the p´1q 3 3 surgery, we obtain a closure of p푆 p퐿2q, Γ휇q, which gives rise to 퐾퐻푀p푆 , 퐿2q. When performing the 0 surgery, we get a closure of the balanced sutured manifold p푀, 훾q, 3 which is obtained from p푆 p퐿0q, Γ휇q by performing a 0-surgery along 훼. To further relate p푀, 훾q to 퐿0, we need to perform a sutured manifold decomposition of p푀, 훾q, along an annulus 퐴 arising from 훼 and the 0-surgery, as explained in Kronheimer and Mrowka [51]. Suppose that p푀 1, 훾1q is obtained from p푀, 훾q by such a decomposition along 퐴. There are two cases.

1 1 Case 1. When 퐿2 has one more component than 퐿0 and 퐿1, then p푀 , 훾 q is 3 diffeomorphic to p푆 p퐿2q, Γ휇q. Hence, we are done.

1 1 Case 2. When 퐿2 has one less component than 퐿0 and 퐿1, then p푀 , 훾 q is 3 3 p푆 p퐿2q, Γ휇q except that on one component of B푆 p퐿2q, there are six meridians as 3 the suture rather than two. So, to obtain p푆 p퐿2q, Γ휇q, we need to further decompose p푀 1, 훾1q by another annulus, just as we did in the proof of Corollary 4.7.5, where we obtained two copies of p푉, 훾4q from p푉, 훾6q. The result of this second sutured 3 6 manifold decomposition is a disjoint union of p푆 p퐿2q, Γ휇q with p푉, 훾 q. Hence, we are done.

As a corollary to Proposition 4.8.1, we derive the following corollary, independent of the works by [20] or [63].

Corollary 4.8.2. With Z2 coefficients and the canonical Z2 grading of monopole Floer homology, the Euler characteristics of KHMp푆3, 퐾, 푖q (for definition, see [53]) corresponds to the coefficients of a suitable version of Alexander polynomial ofthe knot 퐾 Ă 푆3.

Proof. It follows from an analogous argument in sutured instanton Floer theory in [51].

201 Suppose that 푌 is a closed oriented 3-manifold. Let 푌 p푛q denote the manifold obtained by removing 푛 disjoint 3-balls from 푌 . We can make 푌 p푛q to be a balanced sutured manifold p푌 p푛q, 훿푛q, where 훿푛 consists of one simple closed curve on each boundary sphere of 푌 p푛q. The following two lemmas are straightforward.

Lemma 4.8.3. Suppose 푌 is a closed oriented 3-manifold and 푛 P Z is no less than 2, then 푌 p푛q – p푌 p푛 ´ 1q \ 푆3p2q, 훿푛´1 Y 훿2q Y ℎ, where ℎ “ p휑, 푆, 퐷3, 훿q is a contact 1-handle so that 휑 sends one component of 푆 to B푌 p푛 ´ 1q and the other component to B푆3p2q.

Lemma 4.8.4. Suppose p푀1, 훾1q and p푀2, 훾2q are two balanced sutured manifolds. Also, suppose that p푆3p2q, 훿2q is defined as above, and its two boundary components are

3 2 2 B푆 p2q “ 푆1 Y 푆2 .

Then, we have

3 2 p푀17푀2, 훾 Y 훾2q – p푀1 \ 푀2 \ 푆 p2q, 훾1 Y 훾2 Y 훿 q Y ℎ1 Y ℎ2.

3 Here, for 푖 “ 1, 2, ℎ푖 “ p휑푖, 푆푖, 퐷푖 , 훿푖q is a contact 1-handle so that 휑푖 maps one 2 component of 푆푖 to B푀푖 and the other component of 푆푖 to 푆푖 .

Remark 4.8.5. In Lemmas 4.8.3 and 4.8.4, we do not require a sutured manifold p푀, 훾q to have a global compatible contact structure. However, we can identify a collar of the boundary of 푀 with r0, 1sˆB푀, and assume that there is an 퐼-invariant contact structure in the collar so that, under this contact structure, B푀 is a convex

202 surface with 훾 being the dividing set. Thus, the contact handle attachment makes sense.

From Lemmas 4.8.3 and 4.8.4, we can see the significant role played by p푆3p2q, 훿2q. So, we proceed to compute its sutured monopole Floer homology.

Lemma 4.8.6. For any closed 3-manifold 푌 and every positive integer 푛, there is an injective map

푆퐻푀p푌 p푛q, 훿푛q Ñ 푆퐻푀p푌 p푛 ` 1q, 훿푛`1q.

Proof. We can obtain p푌 p푛 ` 1q, 훿푛`1q from p푌 p푛q, 훿푛q by attaching a contact 2- handle. If we further attach a contact 3-handle to it, the result will be p푌 p푛q, 훿푛q again. The pair of handles forms a 2-3 cancelation pair, as in Lemma 3.3.3. Thus, the composition is the identity, and the 2-handle attachment induces the desired injective map.

3 2 2 Corollary 4.8.7. We have 푆퐻푀p푆 p2q, 훿 ; Z2q – pZ2q .

Proof. It follows from the proof of an analogous statement in sutured instanton Floer theory in Baldwin and Sivek [6]. A sketch of the proof is as follows: First, as in [5], there is an isomorphism

3 2 3 푆퐻푀p푆 p2q, 훿 q – 퐾퐻푀p푆 , 푈2q,

where 푈2 is the unlink with two components. Note that there is an oriented skein relation, which relates two copies of the unknot 푈1 and one copy of 푈2. Thus, from

203 Proposition 4.8.1 we have

3 3 퐾퐻푀p푆 , 푈1q / 퐾퐻푀p푆 , 푈1q h

v 3 퐾퐻푀p푆 , 푈2q

3 Since 퐾퐻푀p푆 , 푈1q – Z2, we know that

3 2 퐾퐻푀p푆 , 푈2q – pZ2q or 0.

The second possibility is then ruled out by Lemma 4.8.6 since

3 1 푆퐻푀p푆 p1q, 훿 q – Z2.

So, we conclude that

3 2 3 2 푆퐻푀p푆 p2q, 훿 q – 퐾퐻푀p푆 , 푈2q – pZ2q .

Corollary 4.8.8. Suppose p푀1, 훾1q and p푀2, 훾2q are balanced sutured manifolds. Then, we have

2 푆퐻푀p푀17푀2, 훾1 Y 훾2q – 푆퐻푀p푀1 \ 푀2, 훾1 Y 훾2q b pZ2q .

Proof. This follows directly from Lemma 4.8.4 and Corollary 4.8.7.

Corollary 4.8.9. Suppose 퐿 is a link in 푆3. Then, with any coefficients, KHMp푆3, 퐿q ‰ 0.

204 Proof. Lemma 4.8.6 makes sure that 푆퐻푀p푆3p2q, 훿2q has a rank of at least one with 3 any coefficients. If 퐿 is non-splitting, then the balanced sutured manifold p푆 p퐿q, Γ휇q is taut, and the non-vanishing statement follows from Kronheimer and Mrowka [53]. If 퐿 has separable components, we can apply Lemma 4.8.4.

The discussion in the instanton settings would be completely analogous. We use the field of complex numbers C as coefficients and have the following proposition.

Proposition 4.8.10. Suppose p푀1, 훾1q and p푀2, 훾2q are two balanced sutured manifolds, then

2 푆퐻퐼p푀17푀2, 훾1 Y 훾2q – 푆퐻퐼p푀1, 훾1q b 푆퐻퐼p푀2, 훾2q b C .

This formula can also be applied to the framed instanton Floer homology of closed 3-manifolds. Suppose 푌 is a closed oriented 3-manifold, then we can connected sum

3 3 푌 with 푇 and pick 휔 to be a circle that represents a generator of 퐻1p푇 q. The pair p푌 7푇 3, 휔q is then admissible and we can form the framed instanton Floer homology of 푌 : 퐼7p푌 q “ 퐼휔p푌 7푇 3q.

In [53], Kronheimer and Mrowka discuss the relation between the framed instanton Floer homology of a closed 3-manifold and the sutured instanton Floer homology of p푌 p1q, 훿1q. As a corollary to the connected sum formula for sutured instanton Floer theory, we have the following:

Corollary 4.8.11. Suppose 푌1 and 푌2 are two closed oriented 3-manifolds. Then, as vector spaces over complex numbers, we have

7 7 7 퐼 p푌1q b 퐼 p푌2q – 퐼 p푌17푌2q.

205 206 Chapter 5

Applications to knot theory

In this chapter, we present how the constructions in the previous two sections could be applied to the study of knot theory. In particular, we construct minus versions of knot Floer homology, as well as a concordance invariant, which we call the tau invariant, in monopole and instanton theories. In the last section, we also present the computations for the twisted knots.

5.1 The construction

Suppose 푌 is a closed oriented 3-manifold, and 퐾 Ă 푌 is an oriented knot with a Seifert surface 푆 Ă 푌 . Suppose further that 푝 P 퐾 is a fixed base point and 휙 : 푆1 ˆ 퐷2 ãÑ 푌 is an embedding as in Section 2.4, i.e., we require that

휙p푆1 ˆ t0uq “ 푘, and 휙pt1u ˆ t0uq “ 푝.

Then, we have a 3-manifold with boundary 푌휙 “ 푌 zintpimp휙qq. The Seifert surface

푆 induces a framing on B푌휙. We call the meridian 휇휙 and the longitude 휆휙. Let Γ푛,휙

207 be a collection of two disjoin parallel oppositely oriented simple closed curves on B푌휙, each of class ˘p휆휙 ´ 푛휇휙q. Then, we have a balanced sutured manifold p푌휙, Γ푛,휙q.

1 Suppose 휙 is another embedding, then we also have pp푌휙1 , Γ푛,휙1 qq. Suppose 푓푡 is the ambient isotopy defined as in Section 2.4, relating 휙 and 휙1. We have the following lemma.

Lemma 5.1.1. The diffeomorphism 푓1 is a diffeomorphism from p푌휙, Γ푛,휙q to p푌휙1 , Γ푛,휙1 q.

Proof. It is enough to show that 푓1 sends the framing p휇휙, 휆휙q on B푌휙 to the framing p휇휙1 , 휆휙1 q on B푌휙1 .

By construction, 푓1 sends 휇휙 to 휇휙1 . 푓1 must also preserve 휆휙, since 푓푡 is an isotopy, and 휆휙 can be characterized by the fact that it represents a generator of the map

푖˚ : 퐻1pB푌휙q Ñ 퐻1p푌휙q, where 푖 : B푌휙 Ñ 푌휙 is the inclusion.

Corollary 5.1.2. There is a transitive system (of projective transitive systems)

tSHMp푌휙, Γ푛,휙qu and tΨ휙,휙1 “ SHMp푓1qu.

So, we obtain a canonical module SHMp푌, 퐾, 푝, 푛q associated to the quadruple p푌, 퐾, 푝, 푛q.

Once set up Lemma 5.1.1, we can fix a knot complement to study with. Suppose 푌 p퐾q “ 푌 zintp푁p퐾qq be a knot complement and let 휆 and 휇 be the longitude and meridian, respectively, with respect to the framing on B푌 p퐾q induced by the Seifert surface 푆. For any 푛 P Z`, use Γ푛 to denote the suture on B푌 p퐾q consisting of a pair of simple closed curves of class ˘p휆 ´ 푛휇q, and use Γ8 to denote the suture on B푌 p퐾q consisting of a pair of meridians.

208 Definition 5.1.3 (Kronheimer and Mrowka [53], or Baldwin and Sivek [4]). Define

KHMp푌, 퐾, 푝q “ SHMp푌 p퐾q, Γ8q.

3 On the balanced sutured manifold p푆 p퐾q, Γ푛q, there are two different by-passes we can attach. The following theorem is essentially due to Honda [33].

Theorem 5.1.4. There are two tight and minimal-twisting contact structures on

2 2 r0, 1s ˆ 푇 so that, for 푖 “ 1, 2, t푖u ˆ 푇 is convex with dividing set being Γ푛`푖. These two contact structures correspond to two different by-pass attachments on p푌 p퐾q, Γ푛q.

Definition 5.1.5. We denote the two contact structures in Theorem 5.1.4 by 휉`,푛 and

휉´,푛, respectively. The corresponding two by-passes are called positive and negative, respectively. The two by-passes can be distinguished by Figure 5-1.

훼` 훼´

B푆 B푆

휆

´휇 positive by-pass negative by-pass

Figure 5-1: The positive and negative by-pass attachments for p푌 p퐾q, Γ3qq. The squares represent the toroidal boundary of 푌 p퐾q. Note the contact structures 휉˘,2 correspond to the by-passes from the bottom one to the top left one in each by-pass triangle.

209 There are by-pass exact triangles associated to the positive and negative by- passes, as in Theorem 2.7.21.

푛`1 휓˘,8 SHGp´푌 p퐾q, ´Γ푛`1q / SHGp´푌 p퐾q, ´Γ8q (5.1) j

휓푛`1 휓8 ˘,푛 t ˘,푛 SHGp´푌 p퐾q, ´Γ푛q

푛`1 Note we have 휓˘,푛 “ Φ휉˘,푛 . We also need the following facts.

2 Proposition 5.1.6 (Honda [33]). On r0, 2s ˆ 푇 , the two contact structures 휉´,푛 Y

휉`,푛`1 and 휉`,푛 Y 휉´,푛`1 are the same.

Corollary 5.1.7. We have a commutative diagram

푛 휓´,푛`1 SHGp푌 p퐾q, Γ푛q / SHGp푌 p퐾q, Γ푛`1q

휓푛 푛`1 `,푛`1 휓`,푛`2

푛`1 휓´,푛`2 SHGp푌 p퐾q, Γ푛`1q / SHGp푌 p퐾q, Γ푛`2q

Proof. The corollary follows from Proposition 5.1.6 and Theorem 2.7.16.

Definition 5.1.8. Define the minus version of monopole knot Floer homology of a based knot 퐾 Ă ´푌 , which is denoted by KHG´p´푌, 퐾, 푝q, to be the direct limit of the direct system

푛 휓´,푛`1 ... Ñ SHGp´푌 p퐾q, Γ푛q ÝÝÝÝÑ SHGp´푌 p퐾q, Γ푛`1q Ñ ...

푛 Here, the maps 휓´,푛`1 are defined in the exact triangle (5.1). By Corollary 5.1.7,

210 푛 ´ the maps t휓`,푛`1u푛PZ` induce a map on KHG , which we call 푈:

푈 : KHG´p´푌, 퐾, 푝q Ñ KHG´p´푌, 퐾, 푝q.

´ Next, we construct a grading on the direct limit KHG p´푌, 퐾, 푝q. Suppose 푆푛 is the Seifert surface of 퐾 so that 푆푛 intersects Γ푛 at 2푛 points. Then, we have the following proposition.

Proposition 5.1.9. Suppose 푛 is even, then, for any 푖 P Z, we have

푛 ˘ 휓˘,푛`1pSHGp´푌 p퐾q, ´Γ푛, 푆푛 , 푖qq Ă SHGp´푌 p퐾q, ´Γ푛`1, 푆푛`1, 푖q.

Suppose 푛 is odd, then we have for any 푖 P Z

푛 ˘2 ˘ 휓˘,푛`1pSHGp´푌 p퐾q, ´Γ푛, 푆푛 , 푖qq Ă SHGp´푌 p퐾q, ´Γ푛`1, 푆푛`1, 푖q.

푛 Proof. We only prove the proposition for 휑´,푛`1 with 푛 even. Other cases are similar.

In Figure 5-2, it is clear that the surface 푆푛`1 Ă p푌 p퐾q, Γ푛q can also be obtained from the surface 푆푛 by a negative stabilization:

´ 푆푛`1 “ 푆푛 .

Thus, for any 푖 P Z, we have

´ SHGp´푌 p퐾q, ´Γ푛, 푆푛 , 푖q “ SHGp´푌 p퐾q, ´Γ푛, 푆푛`1, 푖q.

´ For 푆푛 “ 푆푛`1 Ă p푌 p퐾q, Γ푛q, we can choose some auxiliary data to construct a

211 휓2 ÝÝÑ´,3 푆3 푆3

휆 푆2

´휇 푆3

´ Figure 5-2: The solid vertical arc represents the surface 푆3 “ 푆2 and the dashed arc represents 푆2. marked closure

´ ´ 풟푛 “ p푌푛 , 푅, 푟푛, 푚푛, 휂q,

´ ¯´ ´ so that 푆푛 extends to a closed surface 푆푛 Ă 푌푛 and it induces a grading on ´ SHGp´푌 p퐾q, ´Γ푛q that is exactly the one associated to 푆푛 . (See Definition 4.1.3.)

´ We can obtain p푌 p퐾q, Γ푛`1q by attaching a by-pass disjoint from 푆푛`1 “ 푆푛 . 푛 From Baldwin and Sivek [5], we know the map 휑´,푛`1 associated to the by-pass can ´ be described as follows: There is a curve 훽 Ă p푚푛p푌 p퐾qqq Ă 푌푛 so that a 0-framed Dehn surgery on 훽, with respect to the B푌 p퐾q framing, will result in a 3-manifold

212 푌푛`1. Since 훽 is disjoint from imp푟푛q, the data 푅, 푟푛 and 휂 survive and we get a marked closure

풟푛`1 “ p푌푛`1, 푅, 푟푛`1, 푚푛`1, 휂q which is a marked closure of p푌 p퐾q, Γ푛`1q. The surgery description gives rise to a ´ cobordism 푊 from 푌푛 to 푌푛`1 and the cobordism map associated to this cobordism 푛 induces the by-pass attaching map 휑´,푛`1.

´ It is a key observation that the surface 푆푛 “ 푆푛`1 is disjoint from the region we attach the by-pass and, hence, is disjoint from the curve 훽 along which we perform ¯´ ¯ the Dehn surgery. As a result, the surface 푆푛 remains as a closed surface 푆푛`1 Ă 푌푛`1 and induces a grading on SHGp푌 p퐾q, Γ푛`1q. It is clear that the grading induced by ¯ 푆푛`1 is nothing but the one associated to the surface 푆푛`1 Ă p푌 p퐾q, Γ푛`1q as in Definition 4.1.3.

¯´ ¯´ ´ ¯ There is a product cobordism r0, 1s ˆ 푆푛 Ă 푊 , from 푆푛 Ă 푌푛 to 푆푛`1 Ă 푌푛`1, and, thus, we conclude that

푛 ´ 휑´,푛`1pSHGp푌 p퐾q, Γ푛, 푆푛 , 푖qq Ă SHGp푌 p퐾q, Γ푛`1, 푆푛`1, 푖q.

This concludes the proof of Proposition 5.1.9.

푛 The following Figures 5-3 and 5-4 might be helpful for figuring out how do 휓`,푛`1 푛 1 and 휓´,푛`1 change the gradings. In the figures, 푘 “ 푘 ` 푔p푆q. Now, we perform a grading shifting as follows:

휏p푛q 휏p푛q SHGp´푌 p퐾q, ´Γ푛, 푆푛 , 푖qr휎p푛qs “ SHGp´푌 p퐾q, ´Γ푛, 푆푛 , 푖 ` 휎p푛qq.

213 ´ ´ ` 퐷2푘 퐷2푘`1 퐷2푘 퐷2푘 퐷2푘`1 푘1 ○- ○ ○ ○

푘1 ´ 1 ○- ○ ○ ○- ○

푘1 ´ 2 ○- ○ ○ ○- ○ ......

2 ´ 푘1 ○- ○ ○ ○- ○

1 ´ 푘1 ○- ○ ○ ○- ○

´푘1 ○ ○- ○

Figure 5-3: The maps 휑˘ from SHMp´푌 p퐾q, ´Γ2푘q to SHMp´푌 p퐾q, ´Γ2푘`1q. The 2푘 2푘 map 휑´,2푘`1 is depicted on the left and 휑`,2푘`1 on the right. They are represented by the solid arrows. The circles ○ denote the graded homologies. The dashed lines represent the grading shifting when using different surfaces to construct the grading.

Here, 휏p푛q “ ´1 if 푛 is even and 휏p푛q “ 0 if 푛 is odd, and

푛 ´ 1 ` 휏p푛q 휎p푛q “ . 2

We will simply write

휏 SHGp´푌 p퐾q, ´Γ푛, 푆푛qr휎s, and the direct system becomes

휑푛 휏 ´,푛`1 휏 ... Ñ SHGp´푌 p퐾q, ´Γ푛, 푆푛qr휎s ÝÝÝÝÑ SHGp´푌 p퐾q, ´Γ푛`1, 푆푛`1qr휎s Ñ ...

푛 It is straightforward to check that, after the shifting, 휑´,푛`1 is grading preserving 푛 and 휑`,푛`1 shifts the grading down by 1. Thus, we conclude the following.

Proposition 5.1.10. If 푆 is a Seifert surface of 퐾 Ă 푌 , then 푆 induces a grading

214 ´2 ´ `2 ` ´ 퐷2푘´1 퐷2푘´1 퐷2푘 퐷2푘´1 퐷2푘´1 퐷2푘 퐷2푘 푘1 ○- ○ ○

푘1 ´ 1 ○ ○- ○ ○ ○ ○

푘1 ´ 2 ○ ○- ○ ○ ○- ○ ○ ......

2 ´ 푘1 ○ ○- ○ ○ ○- ○ ○

1 ´ 푘1 ○ ○ ○ ○- ○ ○

´푘1 ○- ○

Figure 5-4: The maps 휑˘ from SHMp´푌 p퐾q, ´Γ2푘´1q to SHMp´푌 p퐾q, ´Γ2푘q.

on KHM´p´푌, 퐾, 푝q, which we write as

KHG´p´푌, 퐾, 푝, 푆, 푖q.

Under this grading, the map 푈 is of degree 1.

Definition 5.1.11. Suppose 퐾 Ă 푌 is an oriented knot and 푆 is a Seifert surface of 퐾. We can define the tau invariant 휏p푌, 퐾, 푆q of 퐾 Ă 푌 with respect to 푆 as follows:

휏p푌, 퐾, 푆q “ ´ maxt푖|D푥 P KHM´p푌, 퐾, 푝, 푆, 푖q, 푈 푗푥 ‰ 0 for any 푗 ě 0.u

Here the base point can be fixed arbitrarily.

215 5.2 Basic properties of the minus version

Proposition 5.2.1. Suppose 푌 is a closed oriented 3-manifold and 퐾 Ă 푌 is a knot so that there exists an embedded disk 푆 “ 퐷2 with B푆 “ 퐾. Then

´ KHG p´푌, 퐾, 푝q – SHGp´푌 p1q, ´훿q bℛ ℛr푈s.

Here, 푝 P 퐾 is any choice of the base point. p푌 p1q, 훿q is the balanced sutured manifold obtained from 푌 by removing a 3-ball and picking one simple closed curve on the spherical boundary as the suture.

Proof. First assume that 푌 “ 푆3, then p푌 p1q, 훿q is a product sutured manifold 2 2 and p푌 p퐾q, Γ푛q “ p푉, 훾p1,´푛qq, where p푉, 훾p1,´푛qq is the balanced sutured manifold as defined in Section 4.7. From Proposition 4.7.1, we know that

2 푛 SHGp´푉, ´훾p1,´푛qq – ℛ .

2 Suppose 푆푛 is a Seifert surface of 퐾 that intersects Γ푛 “ 훾p1,´푛q at 2푛 points, then the argument in the proof of Proposition 4.7.1 can be applied to calculate the graded homology, and we conclude that: (Note 푆푛 are disks when 퐾 is the unknot.)

2 휏 SHGp´푉, ´훾p1,´푛q, 푆푛, 푖qr휎s – ℛ for all 푖 such that 1 ´ 푛 ď 푖 ď 0. Moreover, the map

푛 2 휏 2 휏 휓`,푛`1 : SHGp´푉, ´훾p1,´푛q, 푆푛qr휎s Ñ SHGp´푉, ´훾p1,´푛´1q, 푆푛`1qr휎s is of degree ´1 and is an isomorphism for all 푖 such that 1 ´ 푛 ď 푖 ď 0. Thus, we

216 conclude that KHG´p´푆3, 퐾, 푝q – ℛr푈s.

When 푌 is an arbitrary 3-manifold, we know that

3 2 p푌 p퐾q, Γ푛q “ pp푌 p1q, 훿q \ p푆 p퐾q, 훾p1,´푛qqq Y ℎ, where ℎ is a contact 1-handle, as introduced in Baldwin and Sivek [3], which connects

3 2 the two disjoint balanced sutured manifolds pp푌 p1q, 훿q and p´푆 p퐾q, ´훾p1,´푛qq. Thus, we know that

3 2 SHGp´푌 p퐾q, ´Γ푛q – SHGp´푌 p1q, ´훿q b p´푆 p퐾q, ´훾p1,´푛qq.

푛 Moreover, the the above isomorphism intertwines with the maps 휓˘,푛`1 on SHGp´푌 p퐾q, ´Γ푛q 푛 3 2 and the maps 푖푑 b 휓˘,푛`1 on SHGp´푌 p1q, ´훿q b p´푆 p퐾q, ´훾p1,´푛qq, since the corresponding contact handle attachments are clearly disjoint from each other. Thus, we conclude that KHG´p´푌, 퐾, 푝q – SHGp´푌 p1q, ´훿q b ℛr푈s.

Proposition 5.2.2. Suppose 퐾 Ă 푌 is a null-homologous knot and 푆 is a minimal genus Seifert surface of 퐿. Then, the direct system stabilizes: For any 푖 P Z, if 푛 ą 푔p푆q ´ 푖, then we have an isomorphism

푛 휏 휏 휑´,푛`1 : SHGp´푌 p퐾q, ´Γ푛, 푆푛, 푖qr휎s–SHGp´푌 p퐾q, ´Γ푛`1, 푆푛`1, 푖qr휎s.

217 Proof. We have the following exact triangle:

푛`1 휓´,8 SHGp´푌 p퐾q, ´Γ푛`1q / SHGp´푌 p퐾q, ´Γ8q j

휓푛 휓8 ´,푛`1 t ´,푛 SHGp´푌 p퐾q, ´Γ푛q

We prove the proposition under the assumption that 푛 “ 2푘 is even. The other case, when 푛 is odd, is similar. When 푛 is even, we know from Proposition 5.1.9 that

푛 ´ 휑´,푛`1pSHGp´푌 p퐾q, ´Γ푛, 푆푛 , 푗qq Ă SHGp´푌 p퐾q, ´Γ푛`1, 푆푛`1, 푗q.

By a similar argument, we have

푛`1 `푛 휑´,8pSHGp´푌 p퐾q, ´Γ푛`1, 푆푛`1, 푗qq Ă SHGp´푌 p퐾q, ´Γ8, 푆8 , 푗q

where 푆8 is a Seifert surface of 퐾 that intersects the suture Γ8 twice. Proposition 4.6.1 then implies that (recall 푛 “ 2푘)

`2푘 SHGp´푌 p퐾q, ´Γ8, 푆8 , 푗q “ SHGp´푌 p퐾q, ´Γ8, 푆8, 푗 ` 푘q.

However, the adjunction inequality in Lemma 2.6.12 and Proposition 2.3.4 imply that if 푗 ` 푘 ą 푔p푆q, then

SHGp´푌 p퐾q, ´Γ8, 푆8, 푗 ` 푘q “ 0.

Thus, for 푗 P Z so that 푗 ` 푘 ą 푔p푆q, we have

푛 ´ 휑´,푛`1 : SHGp´푌 p퐾q, ´Γ푛, 푆푛 , 푗q Ñ SHGp푌 p퐾q, Γ푛`1, 푆푛`1, 푗q

218 is an isomorphism. From the way we perform the grading shifting in Proposition

5.1.10, we know that, for any 푗 P Z,

휏 휏 SHGp´푌 p퐾q, ´Γ푛, 푆푛, 푗qr휎s “ SHGp´푌 p퐾q, ´Γ푛, 푆푛, 푗 ` 푘q.

Thus, for the fixed grading 푖 P Z as in the hypothesis of the proposition, when 푛 ą 푔p푆q ´ 푖, we have p푖 ` 푘q ` 푘 ą 푔p푆q and this implies that the map

푛 푛 휏 휏 휑´,푛`1|SHGp´푌 p퐾q,´Γ푛,푆푛,푖qr휎s “ 휑´,푛`1|SHGp´푌 p퐾q,´Γ푛,푆푛,푖`푘q is an isomorphism.

Corollary 5.2.3. Under the above conditions, there exists an integer 푁0 so that, for any 푖 ă 푁0, the 푈 map induces an isomorphism:

KHG´p´푌, 퐾, 푝, 푆, 푖q – KHG´p´푌, 퐾, 푝, 푆, 푖 ´ 1q,

Proof. The proof is exactly the same as the above proposition.

Corollary 5.2.4. For a knot 퐾 Ă 푌 , a Seifert surface 푆 of 퐾, and a fixed point 푝 P 퐾, we have KHG´p´푌, 퐾, 푝, 푆, 푖q “ 0 for 푖 ą 푔, and KHG´p´푌, 퐾, 푝, 푆, 푔q – KHGp´푌, 퐾, 푝, 푆, 푔q.

Here, 푔 is the genus of the Seifert surface 푆, and KHGp´푌, 퐾, 푝, 푆, 푔q is defined in Definition 5.1.3.

219 Proof. The first statement that

KHG´p´푌, 퐾, 푝, 푆, 푖q “ 0 for 푖 ą 푔 follows from the adjunction inequality in Lemma 2.6.12 and Prop 2.3.4. For the second part of the statement, we prove the case where 푛 “ 2푘 ` 1 is odd and the other case is exactly the same. Suppose p푀 1, 훾1q is obtained from p푌 p퐾q, Γ푛q by a sutured manifold decomposition of 푆푛 Ă 푌 p퐾q. It is straight forward to check that if we decompose p푌 p퐾q, Γ8q along 푆8, then we will get exactly the same balanced sutured manifold p푀 1, 훾1q. Hence, from Theorem 2.6.20 in [53], we know that

1 1 SHGp´푌 p퐾q, ´Γ푛, 푆푛`1, 푔p푆q ` 푘 ` 1q – SHGp푀 , 훾 q – KHGp´푌, 퐾, 푝, 푆8, 푔p푆qq.

Then, the corollary follows from Proposition 5.2.2 and the grading shifting we performed in Proposition 5.1.10.

Suppose 퐾 Ă 푌 is a fibred knot with fibre 푆 of genus 푔. Suppose p푆, ℎq is an open book corresponding to the fibration of 퐾 Ă 푌 . It supports a contact structure 휉 on 푌 . We call ℎ not right-veering if there is an arc 훼 Ă 푆 and one end point 푝 PB훼 so that near 푝 Ă 푆, ℎp훼q is to the left of 훼. See Figure 5-5. See Baldwin and Sivek [8] for more details.

Corollary 5.2.5. Under the above setting, if ℎ is not right-veering, we have

KHG´p´푌, 퐾, 푝, 푆, 푔q – ℛ, and the generator is in the kernel of the 푈 map.

220 ℎp훼q 훼

푝

Figure 5-5: Not right-veering

Proof. This result is the main result in Baldwin and Sivek [8]. The only difference is that we translate it into our language involving KHG´.

Proposition 5.2.6. We have an exact triangle:

푈 KHG´p´푌, 퐾, 푝q / KHG´p´푌, 퐾, 푝q i

휓1 휓 u KHMp´푌, 퐾, 푝q

Proof. We will use the by-pass exact triangle

푛`1 휓`,8 SHGp´푌 p퐾q, ´Γ푛`1q / SHGp´푌 p퐾q, ´Γ8q (5.2) j

휓푛`1 휓8 `,푛 t `,푛 SHGp´푌 p퐾q, ´Γ푛q

푛 푛`1 The maps t휑`,푛`1u푛PZ` induce the 푈 map. By a similar argument, the maps t휑`,8uu푛PZ` 8 1 and t휑`,푛u푛PZ` induce the maps 휓 and 휓 in the statement of the proposition. Then, it is formal to check that the by-pass exact triangles (5.2) for all 푛 P Z` induce the desired one as stated in the proposition.

221 5.3 Knots representing torsion classes

In this section, we extend the definition of KHG´ to the case where 퐾 is not necessarily null-homologous but represents a torsion class in 퐻1p푌 q. Suppose 푌 is a closed connected oriented 3-manifold. Suppose further that 퐾 Ă 푌 is an oriented knot that represents a torsion class in 퐻1p푌 q. It is a basic fact that the map

푖˚ : 퐻1pB푌 p퐾q; Qq Ñ 퐻1p푌 p퐾q; Qq, which is induced by the inclusion map 푖 : B푌 p퐾q Ñ 푌 p퐾q, has a kernel of dimension one. Thus, we can find a curve 훼 ĂB푌 p퐾q so that 훼 bounds a properly embedded surface 푆 Ă 푌 p퐾q. We always give 푆 an orientation so that B푆 “ 훼 is oriented in a coherent way as 퐾. This surface is usually called a Rational Seifert surface of 퐾. For more details, readers are referred to Ni and Vafaee [69]. We still look at the knot complement 푌 p퐾q. On B푌 p퐾q – 푇 2, there is a preferred class 휇, which is the meridian of 퐾. There is no preferred longitude class, but we can pick any oriented non-separating simple closed curve 휆 on B푌 p퐾q so that r휇s and r휆s is an oriented basis of 퐻1pB푌 p퐾qq. Then, on 푌 p퐾q, we can still define the sutures Γ푛 and Γ8, and there are by-pass exact triangles as in (5.1). Note the same formula as in (5.1) holds with our new definitions of Γ푛 and Γ8. Furthermore, Corollary 5.1.7 continues to hold for exactly the same reason, so we can make the following definition.

Definition 5.3.1. Suppose 퐾 Ă 푌 is a knot representing a torsion class in 퐻1p푌 q and 푝 P 퐾 is a base point. Then, define the minus version of monopole knot Floer homology, which is denoted by KHM´p´푌, 퐾, 푝q, to be the direct limit of the direct system 푛 휓´,푛`1 ... Ñ SHGp´푌 p퐾q, Γ푛q ÝÝÝÝÑ SHGp´푌 p퐾q, Γ푛`1q Ñ ...

222 푛 Here, the maps 휓´,푛`1 are defined in the exact triangle (5.1). By Corollary 5.1.7, 푛 ´ the maps t휓`,푛`1u푛PZ` induce a map on KHM , which we call 푈:

푈 : KHG´p´푌, 퐾, 푝q Ñ KHG´p´푌, 퐾, 푝q.

It is clear that Definition 5.3.1 is independent of the choice of the longitude 휆 on B푌 p퐾q. Next, we want to use the rational Seifert surface 푆 of 퐾 Ă 푌 to construct a grading on KHG´p´푌, 퐾, 푝q. As in Proposition 5.1.10, we need to perform a grading shifting. Instead of directly writing down the value of the shift, we define the shift in an indirect way. Suppose, for any 푛 P Z`, 푆푛 is a rational Seifert surface of 퐾, which 휏 has the minimal possible intersection with the suture Γ푛. Suppose 푆푛 is exactly the 휏 surface 푆푛 if |푆푛XΓ푛| is of the form 4푘`2, and 푆푛 is obtained from 푆푛 by performing a 휏 negative stabilization if else. We define a grading shifting, SHGp´푌 p퐾q, ´Γ푛, 푆푛qr휎s, 휏 of SHGp´푌 p퐾q, ´Γ푛, 푆푛q, so that

휏 휏 SHGp´푌 p퐾q, ´Γ푛, 푆푛, 푖qr휎s “ SHGp´푌 p퐾q, ´Γ푛, 푆푛, 푖 ` 휎p푛qq.

Here, the value 휎p푛q P Z is determined by the following property: The top non- 휏 vanishing grading of SHGp´푌 p퐾q, ´Γ푛, 푆푛qr휎s equals g(S), the genus of 푆.

Remark 5.3.2. Note the grading shifting we performed in Proposition 5.1.10 can also be described in the above way.

Proposition 5.3.3. If 푆 is a ration Seifert surface of 퐾 Ă 푌 , then 푆 induces a ´ Z-grading on KHG p´푌, 퐾, 푝q, which we write as

KHG´p´푌, 퐾, 푝, 푆, 푖q.

223 Under this grading, the map 푈 is of degree 푙, where 푙 is an integer depending on the knot 퐾 Ă 푌 .

As we did in Section 5.2, we can prove that the direct system in Definition 5.3.1 stabilizes.

Proposition 5.3.4. Suppose 퐾 Ă 푌 is a knot representing a torsion class in 퐻1p푌 q, and 푆 is a rational Seifert surface of 퐿. Then, the direct system stabilizes: For any

푖 P Z, there exists 푁 so that if 푛 ą 푁, then we have an isomorphism

푛 휏 휏 휑´,푛`1 : SHGp´푌 p퐾q, ´Γ푛, 푆푛, 푖qr휎s–SHGp´푌 p퐾q, ´Γ푛`1, 푆푛`1, 푖qr휎s.

The most common cases we might encounter a knot that represents a torsion first homology class is when performing Dehn surgeries. Suppose 퐾 Ă 푌 is a null- homologous knot, and 푆 is a Seifert surface of 퐾. Let 푌 p퐾q be the knot complement. Let 휆 and 휇 represent the longitude and meridian on B푌 p퐾q, respectively, according to the framing induced 푆. We can perform a Dehn surgery along the knot 퐾 and obtain a surgery manifold

1 2 푌휑 “ 푌 p퐾q Y 푆 ˆ 퐷 . 휑

2 1 Suppose 휇휑 “ 휑pt1u ˆ B퐷 q “ 푞0휆 ´ 푝0휇 and 휆휑 “ 휑p푆 ˆ t1uq “ 푟0휆 ´ 푠0휇. This

푝0 results in a surgery of slope ´ . Now 휆휑 and 휇휑 form another framing on B푌 p퐾q, 푞0 1 so that 휇휑 is the meridian of the knot 퐾휑 “ 푆 ˆ t0u Ă 푌휑. Note 푌 p퐾q is also a knot complement of 퐾휑 Ă 푌휑 and 퐾휑 is a knot inside 푌휑 which represents a torsion class in

퐻1p푌휑q. Hence, we can use the new framing to construct a minus version of monopole ´ knot Floer homology KHM p´푌휑, 퐾휑q of p푌휑, 퐾휑q. Here, we omit the choice of base points, since the discussion will be carried out on a fixed knot complement. We have the following property.

224 Proposition 5.3.5. For any fixed 푖0 P Z, there exists 푁 so that for any surgery slope ´ 푝0 ă ´푁, we have 푞0

´ ´ KHG p´푌, 퐾, 푆, 푖0q – KHG p´푌휑, 퐾휑, 푆, 푖0q.

Proof. We use the framing p휆, 휇q intricately and write both the curve 푞휆 ´ 푝휇 or the 푝 slope ´ 푞 as p푞, ´푝q. We use 훾p푞휆´푝휇q or 훾p푞,´푝q to denote the suture consisting of two curves of slope p푞, ´푝q. Note, 훾p1,´푛q “ Γ푛, for the notation Γ푛 as used in Section 5.1.

From the stabilization properties in Propositions 5.2.2 and 5.3.4, we know that there exists 푁1 ą 푔p푆q ´ 푖0 such that for any 푛 ą 푁1, we have

´ 휏 KHG p´푌, 퐾, 푆, 푖0q – SHGp´푌 p퐾q, ´훾p1,´푛q, 푆 , 푖0qr휎s, (5.3) and

´ 휏 KHG p´푌휑, 퐾휑, 푆, 푖0q – SHGp´푌 p퐾q, ´훾p휆휑´푛휇휑q, 푆 , 푖0qr휎s. (5.4)

Hence to prove the theorem, it is suffice to prove that for large enough 푛 and large enough surgery slope, we have

휏 휏 SHGp´푌 p퐾q, ´훾p1,´푛q, 푆 , 푖0qr휎s – SHGp´푌 p퐾q, ´훾p휆휑´푛휇휑q, 푆 , 푖0qr휎s. (5.5)

Fix an 푛2 ą 푁2, and write 휆휑 ´ 푛2휇휑 “ 푞휆 ´ 푝휇. From the proof of Proposition

1 1 2 2 4.7.1, we can construct two sequences of slopes tp푞푗, ´푝푗qu and tp푞푗 , ´푝푗 qu inductively 1 1 as follows: Let p푞0, ´푝0q “ p푞, ´푝q, and, for any 푗 ě 1, suppose we have the continued

225 1 1 fraction of p푞푗´1, ´푝푗´1q to be

1 1 p푞푗´1, ´푝푗´1q “ r푟1, ..., 푟푘´푗, 푟푘´푗`1s, then define

2 2 1 1 p푞푗 , ´푝푗 q “ r푟1, ..., 푟푘´푗, 푟푘´푗`1 ` 1s, p푞푗, ´푝푗q “ r푟1, ..., 푟푘´푗s.

Note we identify r푟1, ..., 푟푙, ´1s as r푟1, ..., 푟푙´1, 푟푙 ` 1s. We end the sequence when

1 1 p푞푘´1, ´푝푘´1q “ r푟1s “ p1, 푟1q. (5.6)

Here 푟1 ď ´2 is the first term in the continued fraction of p푞, ´푝q “ p휆휑 ´ 푛2휇휑q.

Remark 5.3.6. Note p푞0, ´푝0q is the slope of the surgery that gives rise to p푌휑, 퐾휑q,

1 1 while p푞0, ´푝0q “ p푞, ´푝q “ 휆휑 ´ 푛2휇휑. Also we can pick 푛2 as large as we want.

To proceed, we only carry out the proof in the case where 푛 is odd, and for any

1 2 푗, 푝푗 is odd and 푝푗 is even. Other cases are similar. Under this assumption, we can un-package the grading shifting we performed in Propositions 5.1.10 and 5.3.3, and to prove (5.5) is equivalent to proving (we omit the surface 푆 from the notation):

1 푛 ´ 1 푝0 ´ 1 SHGp´푌 p퐾q, ´훾 1, 푛 , 푖0 ` q – SHGp´푌 p퐾q, ´훾 푞1 , 푝1 , 푖0 ` q. (5.7) p ´ q 2 p 0 ´ 0q 2

For 푙 “ 0, ..., 푘 ´ 1, write 푝1 ´ 1 푖1 “ 푖 ` 푙 푙 0 2

Claim 1. There exists an 푁 ą 0 so that if the surgery slope ´ 푝0 ă ´푁, then, 푞0

226 for any 푙 P t0, ..., 푘 ´ 2u, there is an isomorphism:

1 1 SHGp´푌 p퐾q, ´훾 1 1 , 푆, 푖 q – SHGp´푌 p퐾q, ´훾 1 1 , 푆, 푖 q. p푞푙,´푝푙q 푙 p푞푙`1,´푝푙`1q 푙`1

Claim 2. There exists an 푁 ą 0 so that if the surgery slope ´ 푝0 ă ´푁, then 푞0 we have 푟1 ą 푔p푆q ´ 푖0.

Assuming Claim 1 and 2, we now prove the proposition. By Claim 1, Claim 2,

1 and Proposition 5.2.2, we have (note we have assumed that 푟1 “ ´푝푘´1 is odd)

휏 휏 SHGp´푌 p퐾q, ´훾p1,´푛q, 푆 , 푖0qr휎s – SHGp´푌 p퐾q, ´훾p1,푟1q, 푆 , 푖0qr휎s ´푟 ´ 1 “ SHGp´푌 p퐾q, ´훾 , 푆, 푖 ` 1 q p1,´푟1q 0 2 1 “ SHGp´푌 p퐾q, ´훾 1 1 , 푆, 푖 q p푞푘´1,´푝푘´1q 푘´1 1 SHG 푌 퐾 , 훾 1 1 , 푆, 푖 – p´ p q ´ p푞0,´푝0q 0q 1 푝0 ´ 1 “ SHGp´푌 p퐾q, ´훾 푞1 , 푝1 , 푖0 ` q. p 0 ´ 0q 2

Thus (5.7) is proved, and Proposition 5.3.5 follows.

To prove Claim 2, by definition, we have

푝 푝 푠0 ` 푛2푝0 푟1 “ ´pt u ` 1q and “ . (5.8) 푞 푞 푟0 ` 푛2푞0

If we choose large enough 푛2 (we can freely make 푛2 larger), then we know that

푝 푝 t u ě t 0 u ´ 1. (5.9) 푞 푞0

푝0 Hence, for any surgery slope ´ ă 푁 “ ´p푔p푆q ´ 푖0q, Claim 2 holds. 푞0

1 1 It remains to prove Claim 1. As in Section 4.7, the sutures of slopes p푞푙, ´푝푙q and

227 2 2 p푞푙 , ´푝푙 q fit into a by-pass exact triangle:

SHGp´푌 p퐾q, ´훾 1 1 q p푞푙´1,푝푙´1q 휓푙,1 4 휓푙,2

* SHGp´푌 p퐾q, ´훾p푞2,푝2qq o SHGp´푌 p퐾q, ´훾p푞1,푝1 qq 푙 푙 휓푙,0 푙 푙 (5.10)

3 If 푌 “ 푆 and 퐾 is the unknot, then 휓푗,푘 “ 휓´,푘 for 푘 “ 0, 1, 2 in the previous exact triangle (4.9). As in Section 4.7, for all 푙 P t1, ..., 푘 ´ 1u and 푗 P Z, we have

´푝2 `푝1 휓 : pSHGp´푌 p퐾q, ´훾 1 1 , 푆 푙 , 푗q Ñ SHGp´푌 p퐾q, ´훾 2 2 , 푆 푙 , 푗q, 푙,0 p푞푙,´푝푙q p푞푙 ,´푝푙 q

`푝1 휓 : SHGp´푌 p퐾q, ´훾 2 2 , 푆 푙 , 푗q Ñ SHGp´푌 p퐾q, ´훾 1 1 , 푆, 푗q, 푙,1 p푞푙 ,´푝푙 q p푞푙´1,´푝푙´1q

´푝2 휓 : SHGp´푌 p퐾q, ´훾 1 1 , 푆, 푗q Ñ pSHGp´푌 p퐾q, ´훾 1 1 , 푆 푙 , 푗q. 푙,2 p푞푙´1,´푝푙´1q p푞푙,´푝푙q

2 Note, in above formulae, we have assume that 푝푙´1 is odd for all 푙. From them, Claim

1 is equivalent to the fact that 휓푙2 is an isomorphism at the grading

푝1 ´ 1 푗 “ 푖1 “ 푖 ` 푙´1 , 푙´1 0 2 which is further equivalent to that

`푝1 1 SHGp´푌 p퐾q, ´훾 2 2 , 푆 푙 , 푖 q “ 0. (5.11) p푞푙 ,´푝푙 q 푙´1

2 1 1 Note, by assumption, 푝푙 “ 푝푙´1 ´ 푝푙 is even. From the grading shifting property, Proposition 4.6.1, we know that (5.11) is equivalent to

1 ` 1 푝푙 ´ 1 SHGp´푌 p퐾q, ´훾p푞2,´푝2q, 푆 , 푖 ` q “ 0. (5.12) 푙 푙 푙´1 2 228 ` 2 Note we have |B푆 X 훾 2 2 | “ 2푝 ` 2. From (the vanishing statement of) p푞푙 ,´푝푙 q 푙 Theorem 2.6.20, we know that (5.12) happens if

푝1 ´ 1 푝2 푖1 ` 푙 ą 푔p푆q ` 푙 . (5.13) 푙´1 2 2

Recall that 푝1 ´ 1 푖1 “ 푖 ` 푙´1 , 푙´1 0 2 so we know that 푝1 ´ 1 푝2 푖1 ` 푙 ą 푔p푆q ` 푙 푙´1 2 2 푝1 ´ 1 푝1 ´ 1 푝2 ô푖 ` 푙´1 ` 푙 ą 푔p푆q ` 푙 0 2 2 2 1 ô푝푙 ą 푔p푆q ´ 푖0 ` 1.

Since, by (5.8) and (5.9), we have

1 1 푝0 푝푙 ě 푝푘´1 “ ´푟1 ě t u ą 푁. 푞0

Thus, if we pick 푁 “ ´p푔p푆q ´ 푖0q, then (5.13) holds and Claim 1 follows. This concludes the proof of Proposition 5.3.5.

5.4 Concordance invariance of 휏퐺.

In Definition 5.1.11, we defined the tau invariant in monopole and instanton theories.

3 3 When 푌 “ 푆 , we simply write 휏퐺p퐾q for a knot 퐾 Ă 푆 . In this section, we will explore some basic properties of the tau invariant.

3 Throughout the section we have a knot 퐾 Ă 푆 and the suture Γ푛 and Γ휇 on 3 3 B푆 p퐾q which are described as in Section 5.1. Fix 푛 P Z`. Then, on B푆 p퐾q,

229 we can pick a meridional curve 훼 so that 훼 intersects the suture Γ푛 twice. Let r´1, 0s ˆ B푆3p퐾q Ă 푆3p퐾q be a collar of B푆3p퐾q inside the knot complement 푆3p퐾q, and we can give an r´1, 0s-invariant tight contact structure on r´1, 0s ˆ B푆3p퐾q so that each slice t푡u ˆ B푆3p퐾q for 푡 P r´1, 0s is convex and the dividing set is (isotopic to) Γ푛. By Legendrian realization principle, we can push 훼 into the interior of the collar r´1, 0s ˆ B푆3p퐾q and get a Legendrian curve 훽. With respect to the surface framing, the curve 훽 has 푡푏 “ ´1. When talking about framings of 훽, we will always refer to the surface framing with respect to B푆3p퐾q.

From Definition 2.7.5, since 훼 intersects the suture (or the dividing set) Γ푛 twice, 3 (after making 훼 Legendrian) we can glue a contact 2-handle to p푆 p퐾q, Γ푛q along 훼, and get a new balanced sutured manifold p푀, 훾q. Suppose p푌, 푅q is a closure 3 of p푆 p퐾q, Γ푛q, in the sense of Definition 2.2.2, so that 푔p푅q is large enough, then, by Proposition 2.7.8, we know that a closure p푌0, 푅q of p푀, 훾q can be obtained from p푌, 푅q by performing a 0-Dehn surgery along the curve 훽. Note that 훽 is disjoint from 푅, so the surgery can be made disjoint from 푅, and, hence, the surface 푅 survives

3 in 푌0. Let p푀´1, Γ푛q be the balanced sutured manifold obtained from p푆 p퐾q, Γ푛q by performing a p´1q-Dehn surgery along 훽. Since 훽 is contained in the interior of 푆3p퐾q, so the surgery does not influence the boundary as well as the suture.

Clearly, if we perform a p´1q-Dehn surgery along 훽 on 푌 , we will get a closure p푌´1, 푅q for the balanced sutured manifold p푀´1, Γ푛q. Applying the surgery exact triangle, we get the following.

3 SHGp´푀´1, ´Γ푛q / SHGp´푆 p퐾q, ´Γ푛q i

퐶 u ℎ,푛 SHGp´푀, ´훾q

230 Remark 5.4.1. Here, the surgery exact triangle seems to go in the wrong direction. However, the point is that the sutured manifolds have been reversed the orientations, so the maps in the surgery exact triangle shall also reverse the directions.

We need to understand the balanced manifolds p푀, 훾q and p푀´1, Γ푛q. First 3 p푀, 훾q is obtained from p푆 p퐾q, Γ푛q by attaching a contact 2-handle along a meridional curve 훼, so it is nothing but p퐷3, 훿q, where 훿 is a connected simple closed 3 curve on B퐷 . To figure out p푀´1, Γ푛q, note that we can view 훽 and 퐾 inside the 3-sphere 푆3 and 훽 is a meridional link around 퐾. So, a p´1q-Dehn surgery along 훽 on 푆3p퐾q results in the same 3-manifold 푆3p퐾q while the framings on it boundary is one larger than before performing the surgery (See Rolfsen [80]). Hence, we conclude

3 that p푀´1, Γ푛q – p푆 p퐾q, Γ푛´1q (Note the slope of Γ푛 is ´푛). Thus, the above exact triangle becomes

3 3 SHGp´푆 p퐾q, ´Γ푛´1q / SHGp´푆 p퐾q, ´Γ푛q (5.14) j

퐶 u ℎ,푛 SHGp´퐷3, ´훿q

Lemma 5.4.2. If 푛 ě ´푡푏p퐾q, then the map 퐶ℎ,푛 is surjective and hence

3 3 푟푘pSHGp´푆 p퐾q, ´Γ푛qq “ 푟푘pSHGp´푆 p퐾q, ´Γ푛´1q ` 1.

Here, 푡푏p퐾q is the maximal possible Thurston-Bennequin number of a Legendrian representative of the knot class of 퐾, with respect to the standard tight contact structure on 푆3. See Ng [66].

3 Proof. Suppose 휉푠푡 is the standard tight contact structure on 푆 . Since 푛 ą ´푡푏p퐾q, we can isotope 퐾 so that it is Legendrian with 푡푏 “ ´푛. We can dig out a standard

231 Legendrian neighborhood of 퐾, and then the dividing set on the boundary of the complement is the suture Γ푛. Hence, when we glue back a contact 2-handle, we get p퐷3, 훿q together with the standard tight contact structure on it. From Lemma 3.1.6 and Definition 2.7.12, we know that the corresponding contact element is a generator of SHGp´퐷3, ´훿q – ℛ.

Also we know that the contact 2-handle attaching map 퐶ℎ,푛 preserves the contact elements by Theorem 2.7.4 and Theorem 2.7.13, so 퐶ℎ,푛 is surjective and we are done.

Proposition 5.4.3. There is a unique infinite 푈-tower in KHG´p푆3, 퐾, 푝q for any knot 퐾 Ă 푆3.

Proof. By Theorem 2.7.21, we know that there are exact triangles:

3 휓˘,푛 3 SHGp´푆 p퐾q, ´Γ푛q / SHGp´푆 p퐾q, ´Γ푛`1q j

휓 휓 ˘,휇 t ˘,푛`1 3 SHGp´푆 p퐾q, ´Γ휇q

Suppose 푆 is a minimal genus Seifert surface of 퐾, and choose 푛 “ 2푔p푆q. We have graded versions of the exact triangles as in the proof of Proposition 5.2.2 as follows:

휓 3 ` `,푛 3 SHGp´푆 p퐾q, ´Γ푛, 푆푛 , 푖q / SHGp´푆 p퐾q, ´Γ푛`1, 푆푛`1, 푖q (5.15) k 휓`,푛`1 휓 `,휇 3 ´푛 SHGp´푆 p퐾q, ´Γ휇, 푆휇 , 푖q

232 휓 3 ´ ´,푛 3 SHGp´푆 p퐾q, ´Γ푛, 푆푛 , 푖q / SHGp´푆 p퐾q, ´Γ푛`1, 푆푛`1, 푖q (5.16) k 휓´,푛`1 휓 ´,휇 3 `푛 SHGp´푆 p퐾q, ´Γ휇, 푆휇 , 푖q

Here, the notations 푆푘 and 푆휇 follow from the ones in the previous three sections, i.e. for all 푘, 푆푘 is an isotopy of 푆 so that B푆푘 intersects the suture Γ푘 transversely ˘푙 at 2푘 points. The supscripts in 푆푘 denote the positive or negative stabilizations as in Definition 4.1.1. From Theorem 2.6.20, we know thatif 푖 ą 푔p푆q or 푖 ă ´푔p푆q, then

3 SHGp´푆 p퐾q, ´Γ휇, 푆휇, 푖q “ 0.

From the grading shifting property, Proposition 4.6.1, we know that

3 ´푛 SHGp´푆 p퐾q, ´Γ휇, 푆휇 , 푖q “ 0 for 푛 푖 ă ´푔p푆q ` “ 0. 2

Hence, from (5.15), we conclude that (recall we have chosen 푛 “ 2푔p푆q)

3 3 ` SHGp´푆 p퐾q, Γ푛`1, 푆푛`1, 푖q – SHGp´푆 p퐾q, Γ푛`1, 푆푛 , 푖q (5.17) for all 푖 ă 0.

233 We can apply a similar argument and use (5.16) to show that

3 3 ´ SHGp´푆 p퐾q, Γ푛`1, 푆푛`1, 푖q – SHGp´푆 p퐾q, Γ푛`1, 푆푛 , 푖q (5.18) 3 ` – SHGp´푆 p퐾q, Γ푛`1, 푆푛 , 푖 ´ 1q for all 푖 ą 0. The last isomorphism also follows from the grading shifting property.

From the construction of the grading in Section 4.1, and the adjunction inequality in Lemma 2.6.12 and Proposition 2.3.4, we know that

3 3 ` SHGp´푆 p퐾q, Γ푛`1, 푆푛`1, 푖q “ 0, SHGp´푆 p퐾q, Γ푛`1, 푆푛 , 푖q “ 0 for 푖 ą 푔p푆q ` 푛 “ 2푔p푆q 표푟 푖 ă ´푔p푆q ´ 푛 “ ´2푔p푆q.

From Lemma 4.1.2 and Theorem 2.6.20, we know that

3 ` SHGp´푆 p퐾q, Γ푛`1, 푆푛 , 2푔p푆qq “ 0.

3 So, from (5.17) and (5.18), we can fix all the gradings of SHGp´푆 p퐾q, ´Γ푛`1q by 3 the corresponding term of SHGp´푆 p퐾q, ´Γ푛q, except at the grading 0. Suppose

3 푟푘pSHGp´푆 p퐾q, ´Γ푛`1, 푆푛`1, 0qq “ 푥, then we conclude that

3 3 푟푘pSHGp´푆 p퐾q, ´Γ푛`1q “ 푟푘pSHGp´푆 p퐾q, ´Γ푛q ` 푥.

Using the induction and the same argument as above, we can compute the graded

234 3 pSHGp´푆 p퐾q, ´Γ푛`푘q for all 푘 P Z` as follows. If 푘 is odd (recall 푛 “ 2푔p푆q), then we have

3 SHGp´푆 p퐾q, ´Γ푛`푘, 푆푛`푘, 푖q

푛`푘´1 0 푖 ą 푔p푆q ` 2 $ ’ ’ ’ SHGp´푆3, ´Γ , 푆`, 푖 ´ 푘`1 q 푘`1 ď 푖 ď 푔p푆q ` 푛`푘´1 ’ 푛 푛 2 2 2 ’ ’ ’ (5.19) ’ “ ’ 푥 1´푘 푘´1 ’ ℛ 2 ď 푖 ď 2 ’ &’

’ 3 ` 푘´1 푛`푘´1 푘`1 ’ SHGp´푆 , ´Γ푛, 푆푛 , 푖 ` 2 q ´푔p푆q ´ 2 ď 푖 ď ´ 2 ’ ’ ’ ’ ’ 푛`푘´1 ’ 0 푖 ă ´푔p푆q ´ ’ 2 ’ ’ If 푘 is even,% then

3 ` SHGp´푆 p퐾q, ´Γ푛`푘, 푆푛`푘, 푖q

푛`푘 0 푖 ě 푔p푆q ` 2 $ ’ ’ ’ SHGp´푆3, ´Γ , 푆`, 푖 ´ 푘 q 푘 ď 푖 ď 푔p푆q ` 푛`푘 ´ 1 ’ 푛 푛 2 2 2 ’ ’ ’ (5.20) ’ “ ’ 푥 푘 푘 ’ ℛ ´ 2 ď 푖 ď 2 ´ 1 ’ &’

’ SHGp´푆3, ´Γ , 푆`, 푖 ` 푘´1 q ´푔p푆q ´ 푛`푘 ď 푖 ď ´1 ´ 푘 ’ 푛 푛 2 2 2 ’ ’ ’ ’ ’ 0 푖 ă ´푔p푆q ´ 푛`푘 ’ 2 ’ ’ %’ 235 So, we conclude that

3 3 푟푘pSHGp´푆 p퐾q, ´Γ푛`푘q “ 푟푘pSHGp´푆 p퐾q, ´Γ푛q ` 푘 ¨ 푥.

Then, it follows from Lemma 5.4.2 that 푥 “ 1. From Proposition 5.2.2 and Corollary 5.2.3, we know that this rank 1 will implies that there is a unique infinite 푈 tower in KHG´.

Lemma 5.4.4. The map

3 3 퐶ℎ,푛 : SHGp´푆 p퐾q, Γ푛q Ñ SHGp´퐷 , ´훿q induces a surjective map

´ 3 3 퐶ℎ : KHG p´푆 , 퐾, 푝q Ñ SHGp´퐷 , ´훿q.

Furthermore, 퐶ℎ commutes with 푈.

Proof. The lemma follows from Corollary 5.4.2 and the following commutative diagrams.

3 휓´,푛 3 SHGp´푆 , Γ푛q / SHGp´푆 , Γ푛`1q 퐶ℎ,푛 퐶 ) u ℎ,푛`1 SHGp´퐷3, ´훿q

3 휓`,푛 3 SHGp´푆 , Γ푛q / SHGp´푆 , Γ푛`1q 퐶ℎ,푛 퐶 ) u ℎ,푛`1 SHGp´퐷3, ´훿q

To prove those commutative diagrams, Recall that the maps 휓˘,푛 are constructed via by-pass attachments and by-pass attachments can be interpreted as contact handle

236 attachments as in Section 2.7, and so is 퐶ℎ,푛. Then, the commutativity follows from the observation that the region to attach contact handles for 휓˘,푛 and 퐶푛,ℎ are disjoint from each other.

Corollary 5.4.5. We can give an alternative definition of 휏퐺p퐾q originally defined in Definition 5.1.11 as follows.

휏퐺p퐾q “ maxt푖 P Z 퐶ℎ|KHG´p´푆3,퐾,푝,푖q is surjectiveu ˇ ˇ Proof. This follows directly from Proposition 5.4.3 and Corollary 5.4.5.

Corollary 5.4.6. There is an exact triangle

KHG´p´푆3, 퐾, 푝q / KHG´p´푆3, 퐾, 푝q i

퐶 u ℎ SHGp´퐷3, ´훿q

Proof. The maps in the exact triangle (5.14) all commute with the maps 휓´,푛 in the construction of the direct system, so we can pass them to the direct limit and still get an exact triangle.

Convention 5.4.7. In the rest of the thesis, we will encounter many different maps. To index them, the subscripts might be a tuple. For example, we will have a map

3 3 휓0,´,푛 : SHGp´푆 p퐾0q, ´Γ푛q Ñ p´푆 p퐾0q, ´Γ푛`1q.

The subscripts will be ordered in the following way: the first will indicate which topological object it is associated to (in the above example, it is associated to 퐾0). The second will indicate how the map is constructed (in the above example, it comes

237 from a negative by-pass attachment as in Definition 5.1.5). The third will indicate the suture. The last will indicate the grading. Maybe some parts are omitted from the subscript, but the rest will still respect this order.

Proposition 5.4.8. The invariant 휏퐺p퐾q is a concordance invariant.

Proof. Suppose 퐾0 and 퐾1 are concordant. Then, there exists a properly embedded annulus 퐴 Ă r0, 1s ˆ 푆3 so that

3 3 3 3 3 3 pt0u ˆ 푆 , 퐴 X t0u ˆ 푆 q – p푆 , 퐾0q, pt1u ˆ 푆 , 퐴 X t1u ˆ 푆 q – p푆 , 퐾1q.

The pair pr0, 1s ˆ 푆3, 퐴q induces a map

3 3 퐹퐴,푛 : SHGp´푆 p퐾1q, ´Γ푛q Ñ SHGp´푆 p퐾2q, ´Γ푛q

3 as follows. The pair pr0, 1s ˆ 푆 , 퐴q induces a cobordism 푊푛 from 푌1,푛 to 푌2,푛, where 3 푌푖,푛 is a closure of p´푆 p퐾푖q, ´Γ푛q, and this cobordism induces the map 퐹퐴,푛. There are two ways to describe 푊 , which are both useful. Though both descriptions can be found in Section 3.2.1. For one description of 푊 , first we give a parametrization of 퐴 “ r0, 1s ˆ 푆1. Then, a tubular neighborhood of 퐴 Ă r0, 1s ˆ 푆3 can be identified with 퐴 ˆ 퐷2 “ r0, 1s ˆ 푆1 ˆ 퐷2, with

퐴 ˆ 퐷2 X t0, 1u ˆ 푆3 “ t0, 1u ˆ 푆1 ˆ 퐷2.

Thus, we know that

3 2 3 1 2 3 Bpr0, 1s ˆ 푆 z퐴 ˆ 퐷 q “ ´푆 p퐾0q Y pr0, 1s ˆ 푆 ˆ B퐷 q Y 푆 p퐾1q.

238 3 Suppose 푌0,푛 is a closure of p´푆 , ´Γ푛q and let

3 2 3 푊 “ ´pr0, 1s ˆ 푆 z퐴 ˆ 퐷 q Y r푌0,푛zr0, 1s ˆ 푆 p퐾1qs, via a natural identification

1 2 3 r0, 1s ˆ 푆 ˆ B퐷 “ r0, 1s ˆ B푆 p퐾0q.

A second description of 푊 is as follows. Recall that

3 2 3 1 2 3 Bpr0, 1s ˆ 푆 z퐴 ˆ 퐷 q “ ´푆 p퐾0q Y pr0, 1s ˆ 푆 ˆ B퐷 q Y 푆 p퐾1q and clearly

3 3 1 2 B푆 p퐾0q “ B푆 p퐾1q “ 푆 ˆ 퐷 .

3 2 3 As in Lemma 3.2.3, r0, 1s ˆ 푆 z퐴 ˆ 퐷 can be obtained from r0, 1s ˆ 푆 p퐾1q by 3 attaching a set of 4-dimensional handles ℋ to the interior of t1u ˆ 푆 p퐾0q. Thus, 3 as above, if we choose a closure 푌0,푛 of p푆 p퐾0q, Γ푛q, we can attach the same set of handles ℋ to t1u ˆ 푌0,푛 Ă r0, 1s ˆ 푌0,푛 and the result is just the cobordism 푊 .

Claim 1. 퐹퐴,푛 gives rise to a map

´ 3 ´ 3 퐹퐴 : KHG p´푆 , 퐾0, 푝0q Ñ KHG p´푆 , 퐾1, 푝1q,

1 where 푝0 and 푝1 are picked as follows. We pick a point 푝 P 푆 and let 푝푖 “ 푝 ˆ t푖u in the parametrization 퐴 “ r0, 1s ˆ 푆1.

239 To prove the claim, it is enough to show that we have a commutative diagram

퐹 3 퐴,푛 3 SHGp´푆 p퐾0q, ´Γ푛q / SHGp´푆 p퐾1q, ´Γ푛q

휓0,´,푛 휓1,´,푛 퐹 3 퐴,푛`1 3 SHGp´푆 p퐾0q, ´Γ푛`1q / SHGp´푆 p퐾1q, ´Γ푛`2q

This commutativity follows from the fact that when constructing 퐹퐴,푛, we attach 3 handles to r0, 1s ˆ 푌0,푛 to the region t1u ˆ rintp푆 p퐾0qqs (see above), while when con- 3 structing the map 휓푖,´,푛, we attach handles to r0, 1sˆ푌0,푛 to the region ˆrBp푆 p퐾0qqs, so the two set of handles are disjoint from each other and hence the corresponding maps commute.

´ Claim 2. 퐹퐴 commutes with the 푈 map on KHG . The proof of this claim is entirely analogous to one for Claim 1. Claim 3. There is a commutative diagram

´ 3 퐹퐴 ´ 3 KHG p´푆 , 퐾0, 푝0q / KHG p´푆 , 퐾1, 푝1q 퐶0,ℎ 퐶 * t 1,ℎ SHGp´푆3p1q, ´훿q

where 퐶ℎ is defined as in Lemma 5.4.4. To prove the claim, it is enough to prove that the following digram commutes for any 푛: 퐹 ´ 3 퐴,푛 ´ 3 KHG p´푆 p퐾0q, ´Γ푛q / KHG p´푆 p퐾1q, ´Γ푛q (5.21)

퐶0,ℎ 퐶1,ℎ 푖푑 SHGp´푆3p1q, ´훿q / SHGp´푆3p1q, ´훿q

3 As above, suppose we have a closure 푌0,푛 for p´푆 p퐾0q, Γ푛q. Let 푌1,푛 be the corre- 3 sponding closure for p´푆 p퐾1q, Γ푛q as in the construction of 푊 above. Recall, from

240 the construction of 퐶ℎ,푛, it is a 2-handle attaching map associated to a 2-handle 3 attached along a meridian curve 훼 ĂB푆 p퐾0q. So, we can slightly push it into the

1 3 interior and get a curve 훽. Then, we get a closure 푌0 for p´푆 p1q, ´훿q by performing 3 3 a 0-Dehn surgery on 푌0,푛 along 훽. Note the difference between 푆 p퐾0q and 푆 p퐾1q 3 are contained in the interior, so we also have the curve 훽 Ă 푆 p퐾1q Ă 푌1,푛. Thus, we

1 3 1 can obtain another closure 푌1 for p´푆 p1q, ´훿q. We can form a cobordism 푊 from 1 1 푌0 to 푌1 by attaching the set of 4-dimensional handles ℋ as in the proof of Claim 1 1 1 3 1 to t1u ˆ 푌0 Ă r0, 1s ˆ 푌0 , and the attaching region is contained in intp푆 p퐾0qq Ă 푌0 . Hence, there is a commutative diagram just as in the proof of Claim 1:

퐹 ´ 3 퐴,푛 ´ 3 KHG p´푆 p퐾0q, ´Γ푛q / KHG p´푆 p퐾1q, ´Γ푛q

퐶0,ℎ 퐶1,ℎ 1 퐹퐴 SHGp´푆3p1q, ´훿q / SHGp´푆3p1q, ´훿q

1 1 where 퐹퐴 is the map induced by the cobordism 푊 . So, to prove (5.21), it is suffice 1 1 1 to show that 푊 is actually a product r0, 1s ˆ 푌0 , and, hence, 퐹퐴 “ 푖푑. To do this, 1 1 recall that 푊 is obtained from 푌0 by attaching a set of handles ℋ, while the region 3 3 1 of attachment is contained in intp푆 p퐾0qq Ă intp푆 p1qq Ă t1u ˆ 푌0 . So this means that we can split 푊 1 into two parts

1 2 1 3 푊 “ 푊 Y r0, 1s ˆ p푌0 z푆 p1qq and 푊 2 is obtained from r0, 1s ˆ 푆3p1q by attaching the set of handles ℋ. Recall 3 3 that p푆 p1q, 훿q is obtained from p푆 p퐾0q, Γ푛q by attaching the contact 2-handle ℎ, so, topologically,

3 3 3 푆 p1q “ 푆 p퐾0q Y 퐵 .

241 3 3 Note the 3-ball 퐵 is attached to 푆 p퐾0q along part of the boundary, and the set of 3 3 3 handles ℋ is attached to r0, 1s ˆ 푆 p1q within the region intp푆 p퐾0qq Ă t1u ˆ 푆 p1q so the two attaching regions are disjoint. Thus, we have

푊 2 “ r0, 1s ˆ 푆3p1q Y ℋ

3 3 “ pr0, 1s ˆ p푆 p퐾0q Y 퐵 qq Y ℋ

3 3 “ pr0, 1s ˆ 푆 p퐾0qq Y ℋ Y r0, 1s ˆ 퐵

“ rpr0, 1s ˆ 푆3qzp퐴 ˆ 퐷2qs Y r0, 1s ˆ 퐵3.

3 3 Note 퐵 is glued to 푆 p퐾0q along an annulus on the boundary (the contact 2-handle attachment), so r0, 1s ˆ 퐵3 is glued to rpr0, 1s ˆ 푆3qzp퐴 ˆ 퐷2qs along annulus times r0, 1s. It is then straightforward to check that the resulting manifold (which is 푊 2) is just diffeomorphic to r0, 1s ˆ 푆3p1q. Hence, we are done.

Claim 4. The map

´ 3 ´ 3 퐹퐴 : KHG p´푆 , 퐾0, 푝0q Ñ KHG p´푆 , 퐾1, 푝1q is grading preserving.

Note from Proposition 5.2.2, we know that for any fixed 푘 P Z, we can pick a large enough odd 푛 so that, for 푖 “ 0, 1

푛 ´ 1 KHG´p´푆3, 퐾 , 푝 , 푘q “ SHGp´푆3p퐾 q, ´Γ , 푆 , 푘 ` q. 푖 푖 푖 푛 푛,푖 2

Hence, to show that 퐹퐴 is grading preserving, we only need to show that 퐹퐴,푛 is grading preserving. Recall, here, for 푖 “ 0 and 1, 푆푖,푛 is a minimal genus Seifert surface of 퐾푖, which intersects the suture Γ푛 exactly 푛 points. Note we can identify

242 the boundaries

3 3 B푆 p퐾0q “ B푆 p퐾1q via the parametrization 퐴 “ r0, 1s ˆ 푆1 and we can assume that under the above identification

3 3 푆0,푛 XB푆 p퐾0q “ 푆1,푛 XB푆 p퐾1q.

In Section 4.1, we know how to construct the grading on SHG based on a properly

3 embedded surface. Now let 푌0,푛 be a closure of p´푆 p퐾0q, ´Γ푛q so that the surface ¯ 푆푛,0 extends to a closed one 푆푛,0, as in the construction of grading. Then, we have a 3 corresponding closure 푌1,푛 for p´푆 p퐾1q, ´Γ푛q, inside which 푆1,푛 extends to a closed ¯ surface 푆1,푛. To describe this surface, recall that

3 3 푌1,푛 “ ´푆 p퐾1q Y r푌0,푛z푆 p퐾0qs 3 3 B푆 p퐾0q“B푆 p퐾1q as in the construction of 푊 at the beginning of the proof. Then we can take

¯ ¯ 3 푆1,푛 “ 푆1,푛 Y p푆0,푛z푆 p퐾0qq.

By using the Mayer-Vietoris sequence we know that

3 2 퐻2ppr0, 1s ˆ 푆 qzp퐴 ˆ 퐷 qq “ 0.

3 2 So the closed surface ´푆1 Y 퐴 Y 푆2 Ă pr0, 1s ˆ 푆 qzp퐴 ˆ 퐷 q bounds a 3-chain 푥 Ă pr0, 1s ˆ 푆3qzp퐴 ˆ 퐷2q. Recall we have

3 2 3 푊 “ ´pr0, 1s ˆ 푆 qzp퐴 ˆ 퐷 q Y r0, 1s ˆ r푌0,푛z푆 p퐾1qs

243 Now inside 푊 , let ¯ 3 푦 “ 푥 Y r0, 1s ˆ p푆0,푛z푆 p퐾0qq , ` ˘ where the two pieces are glued along

1 ¯ 3 퐴 “ r0, 1s ˆ 푆 “ Br0, 1s ˆ p푆0,푛z푆 p퐾0qq.

It is straightforward to check that

¯ ¯ B푦 “ ´푆0,푛 Y 푆1,푛.

Hence we conclude that ¯ ¯ r푆0,푛s “ r푆0,1s P 퐻1p푊 q.

Then, it follows that 퐹퐴,푛 preserves the grading. Finally, the four claims above, together with Corollary 5.4.5 and the fact that

퐾0 is concordant to 퐾1 if and only if 퐾1 is concordant to 퐾0 suffice to prove the proposition.

Corollary 5.4.9. If 퐴 is a Ribbon concordance from 퐾1 to 퐾2, then the map

´ 3 ´ 3 퐹퐴 : KHG p´푆 , 퐾0, 푝0q Ñ KHG p´푆 , 퐾1, 푝1q as in the Claim 1 in the proof of proposition 5.4.8 is injective.

Proof. From works by Daemi, Lidman, Vela-Vick, and Wong [11], for any 푛 P Z`, the map

3 3 퐹퐴,푛 : SHGp´푆 p퐾0q, ´Γ푛q Ñ SHGp´푆 p퐾1q, ´Γ푛q is injective. So, when passing to the direct limit, the 퐹퐴 is also injective.

244 5.5 Computations for twisted knots

´ In this section, we compute the KHG for the family of knots 퐾푚, as in Figure 5-6.

In particular, 퐾1 is the right-handed trefoil, 퐾0 is the unknot, and 퐾´1 is the figure eight.

훼

¯ 퐾푚 퐾푚 Figure 5-6: The one handle.

From the Seifert algorithm, we can easily construct a genus 1 Seifert surface for

퐾푚, which we denote by 푆푚. Hence, 푔p퐾푚q “ 1 and also from Rolfsen [80] we know that the (symmetrized) Alexander polynomial of 퐾푚 is

´1 ∆퐾푚 p푡q “ 푚푡 ` p1 ´ 2푚q ` 푚푡 . (5.22)

3 First, we will compute KHGp´푆 , 퐾푚q. Suppose p푆3p퐾푚q, Γ휇q is the balanced sutured manifold obtained by taking meridional sutures on knot complements. There

245 is a curve 훼 Ă intp푆3p퐾푚qq as in Figure 5-6 so that we have a surgery exact triangle:

3 3 SHGp´푆 p퐾푚q, ´Γ휇q / SHGp´푆 p퐾푚`1q, ´Γ휇q j

t SHGp´푀, ´Γ휇q

3 Here, 퐾푚 is described as above, and 푀 is obtained from 푆 p퐾푚q by performing a 0-Dehn surgery along 훼. We can use the surface 푆푚 which intersects the suture

Γ휇 twice to construct a grading on the sutured monopole and instanton Floer homologies. Since 훼 is disjoint from 푆푚, there is a graded version of the exact triangle

(since we will always use the surface 푆푚, we omit it from the notation):

3 3 SHGp´푆 p퐾푚q, ´Γ휇, 푖q / SHGp´푆 p퐾푚`1q, ´Γ휇, 푖q j

t SHGp´푀, ´Γ휇, 푖q (5.23)

Since 푆푚 has genus one and intersects the suture twice, all the graded sutured monopole and instanton Floer homologies in (5.23) could only possibly be non-trivial for ´1 ď 푖 ď 1. To understand what is SHGp´푀, ´Γ휇q, from Kronheimer and Mrowka [51] and Proposition 4.8.1, the surgery exact triangle (5.15) is just the same as the oriented Skein exact triangle and SHGp´푀, ´Γq is isomorphic to the monopole or instanton knot Floer homology of the oriented smoothing of 퐾푚, which is a Hopf link. Applying oriented Skein relation again on Hopf links, we can conclude that

rkpSHGp´푀, ´Γ휇qq ď 4. (5.24)

246 For the monopole and instanton knot Floer homologies of 퐾1 (trefoil), we could look at the surgery exact triangle along the curve 훽 in Figure 5-7 and argue in the same way as in Kronheimer and Mrowka [51] to conclude

3 rkpSHGp´푆 p퐾1q, ´Γ휇q ď 3.

From the knowledge of Alexander polynomial in (5.22) and Kronheimer and Mrowka [53, 51], we know that

3 SHGp´푆 p퐾1q, ´Γ휇, 푖q – ℛ (5.25) for 푖 “ ´1, 0, 1 and it vanishes in all other gradings.

훽

Figure 5-7: The trefoil and the circle 훽.

Now let 푚 “ 1 in (5.23). We know from (5.22) that

3 rkpSHGp´푆 p퐾2q, ´Γ휇q ě 7.

247 Then, from the exactness and inequalities (5.24) and (5.25), we know that

rkpSHGp´푀, ´Γ휇qq “ 4.

After further examining each gradings, we know that

ℛ 푖 “ 1, ´1 2 SHGp´푀, ´Γ휇, 푖q “ $ ℛ 푖 “ 0 (5.26) ’ ’ & 0 others ’ %’

Thus, by using the same argument and the induction, we can compute, for 푚 ą 0, that ℛ푚 푖 “ 1, ´1

3 2푚´1 SHGp´푆 p퐾푚q, ´Γ휇, 푖q “ $ ℛ 푖 “ 0 (5.27) ’ ’ & 0 others ’ %’

Since 퐾0 is the unknot, we can use the same technique to compute for, 푚 ď 0, that ℛ´푚 푖 “ 1, ´1

3 1´2푚 SHGp´푆 p퐾푚q, ´Γ휇, 푖q “ $ ℛ 푖 “ 0 (5.28) ’ ’ & 0 others ’ %’ Now we are ready to compute the minus version. Recall that the Seifert surface induces a framing on the boundary of the knot complements as well as 푀. Write Γ푛 the suture consists of two curves of slope ´푛. We have a graded version of by-pass

248 exact triangles (5.15) and (5.16) for even 푛 as well as their cousins

휓 3 `2 `,푛 3 ` SHGp´푆 p퐾푚q, ´Γ푛, 푆푚,푛, 푖q / SHGp´푆 p퐾푚q, ´Γ푛`1, 푆푚,푛`1, 푖q (5.29) k 휓`,푛`1 휓 `,휇 3 ´푛,` SHGp´푆 p퐾푚q, ´Γ휇, 푆푚 , 푖q and

휓 3 ´2 ´,푛 3 ´ SHGp´푆 p퐾푚q, ´Γ푛, 푆푚,푛, 푖q / SHGp´푆 p퐾푚q, ´Γ푛`1, 푆푚,푛`1, 푖q (5.30) k 휓´,푛`1 휓 ´,휇 3 `푛,´ SHGp´푆 p퐾푚q, ´Γ휇, 푆푚 , 푖q for odd 푛.

A simple case to analyze is when 푚 ă 0. For the knot 퐾푚 with 푚 ă 0, take 푛 “ 1 in (5.29), then we have Table 5.1.

3 Here, by Theorem 2.6.20, the top and bottom non-vanishing grading of SHGp´푆 p퐾푚q, ´Γ푛q can be computed via sutured manifold decomposition and coincides with the top and

3 bottom non-vanishing grading of SHGp´푆 p퐾푚q, ´Γ휇q.

From the graded exact triangles on the rows of the table and an extra exact triangle (5.14), we know that

푏 ě 1 ´ 푚, 푐 ě 푎 ` 푚, and 푏 ` 푐 ď 푎 ` 1.

Hence, the only possibility is 푏 “ 1 ´ 푚, 푐 “ 푎 ` 푚. Now take 푛 “ 2 in (5.15), we have Table 5.2.

3 Here, SHGp´푆 p퐾푚q, ´Γ3q can be computed by taking 푘 “ 1 in (5.19). We know

249 Γ휇 Γ1 Γ2

1 ´푚 ´푚

0 1 ´ 2푚 ´푚 푏

´1 ´푚 푎 푐

´2 ´푚 ´푚

Table 5.1: The map 휓`,1 for 퐾푚. There is a (graded) exact triangle between the three horizontal terms for each row. The leftmost column indicates the gradings. The numbers on other columns means the rank of corresponding (graded) homology. If it is a letter (rather than a formula in m), it means that, a priori, we don’t know what the rank is. from Proposition 5.2.2 that

´ 3 3 KHG p´푆 , 퐾푚, 푝푚, 푖q – SHGp´푆 p퐾푚q, ´Γ3, 푆푚,3, 푖 ` 1q

´ 3 for 푖 “ 1, 0, and ´1, and the 푈 maps on KHG p´푆 , 퐾푚, 푝푚, 푖q for 푖 “ 1 and 2 푖 coincide with the maps 휓`,2 as in Table 5.2. From the exactness, we know that 푈 map is actually zero at grading 1 and has a kernel of rank ´푚 at grading 0. Hence, we conclude that

Proposition 5.5.1. Suppose 푚 ď 0 and the knot 퐾푚 is described as above. Then

´ 3 ´푚 ´푚 KHG p´푆 , 퐾푚, 푝푚q – ℛr푈s0 ‘ pℛ1q ‘ pℛ0q .

Here, the subscripts means the grading of the element 1 P ℛ and the formal variable

250 Γ휇 Γ2 Γ3

2 ´푚 ´푚

1 휓`,2 1 1 ´ 2푚 ´푚 / 1 ´ 푚

0 휓`,2 0 ´푚 1 ´ 푚 / 1

´1 푎 ` 푚 푎 ` 푚

2 ´푚 ´푚

Table 5.2: The map 휓`,2 for 퐾푚. The sup-script means the map at a particular grading

푈 has degree ´1.

From the description it is clear that 휏퐺p퐾푚q “ 0.

´ To compute KHG of 퐾푚 for 푚 ą 0, we first deal with the case 푚 “ 1. Now 퐾1 is a right-handed trefoil. From Ng [66], we know that 푡푏p퐾1q “ 1, and, hence, from Corollary 5.4.2, we know that

3 3 rkpSHGp´푆 p퐾1q, ´Γ1qq “ rkpSHGp´푆 p퐾1q, ´Γ0qq ` 1.

3 Now let us compute SHGp´푆 p퐾1q, ´Γ0q. Pick 푆0 to be a genus 1 Seifert surface of ´ 퐾 so that 푆0 is disjoint from Γ0. We can use the surface 푆0 , a negative stabilization 3 of 푆0 as in Definition 4.1.1 to construct a grading on SHGp´푆 p퐾1q, ´Γ0q. From the construction of grading and the adjunction inequality, there could only be three non- vanishing grading ´1, 0, and 1. For the grading 1 part, we can apply Lemma 4.1.2

251 and Theorem 2.6.20 and look at the balanced sutured manifold p푀 1, 훾1q obtained 3 from p´푆 p퐾q, ´Γ0q by a (sutured manifold) decomposion along the surface 푆0:

3 ´ 1 1 SHGp´푆 p퐾1q, ´Γ0, 푆0 , 1q – SHGp푀 , 훾 q.

1 Since 퐾 is a fibred knot, the underlining manifold 푀 is just a product r´1, 1s ˆ 푆0. The suture 훾1 is not just t0u ˆ B푆 but is actually three parallel copies of t0u ˆ B푆 on r´1, 1s ˆ B푆. We can find an annulus 퐴 Ă r´1, 1s ˆ B푆 which contains the suture 훾1. Then, we can push the interior of 퐴 into the interior of r´1, 1sˆ푆 and get a properly embedded surface. If we further decompose p푀 1, 훾1q along (the pushed off of) 퐴, then we get a disjoint union of a product balanced sutured manifold pr´1, 1sˆ푆, t0uˆB푆q with a solid torus with four longitudes as the suture. The sutured monopole and instanton Floer homologies of the first are both of rank 1 and the second of rank2, as in Kronheimer and Mrowka [51]. Hence, we conclude

3 ´ 2 SHGp´푆 p퐾1q, ´Γ0, 푆0 , 1q – ℛ .

For the other two grading, note that from the grading shifting property, Proposition 5.1.9, we have

3 ´ 3 ` SHGp´푆 p퐾1q, ´Γ0, 푆0 , 푖q “ SHGp´푆 p퐾1q, ´Γ0, 푆0 , 푖 ´ 1q

3 ´ “ SHGp´푆 p퐾1q, ´Γ0, p´푆0q , 1 ´ 푖q.

The second equality follows from the basic observation that if we reverse the orien-

` ´ tation of the surface 푆0 , then we get p´푆0q . Hence,

3 ´ 3 ´ SHGp´푆 p퐾1q, ´Γ0, 푆0 , ´1q “ SHGp´푆 p퐾1q, ´Γ0, p´푆0q , 2q “ 0

252 by the adjunction inequality and

3 ´ 3 ´ 2 SHGp´푆 p퐾1q, ´Γ0, 푆0 , 0q “ SHGp´푆 p퐾1q, ´Γ0, p´푆0q , 1q – ℛ . by the same argument as above. Thus, as a conclusion,

3 5 SHGp´푆 p퐾1q, ´Γ1q – ℛ .

Similarly as above, there are only three possible non-vanishing gradings ´1, 0, 1. We have already known that the homology at top and bottom gradings are of rank 1 each, so the middle grading has rank 3. Let 푛 “ 1 in (5.29) and (5.30), we have Table 5.3.

Γ휇 Γ1 Γ2 | Γ휇 Γ1 Γ2

2 | 1 1

1 1 1 | 1 3 푏

0 1 1 푏 | 1 1 푐

´1 1 3 푐 | 1 1

´2 1 1 |

Table 5.3: The map 휓`,1 (on the left) and 휓´,1 (on the right) for 퐾1.

From the exactness, we know that 푏 “ 푐 “ 2. The rest of the computation is

253 straightforward and we conclude that

´ 3 KHG p´푆 , 퐾1, 푝1q – ℛr푈s1 ‘ ℛ0. (5.31)

Now we have the map

3 3 퐶1,ℎ,1 : SHGp´푆 p퐾1q, ´Γ1q Ñ SHGp´푆 p1q, 훿q

´ 3 and by the description of KHG p´푆 , 퐾1, 푝1q above, Lemma 5.4.5 and the fact that

퐶1,ℎ,푛 commutes with 휓´,푛 (Claim 1 in the proof of Proposition 5.4.8), we know that

3 3 퐶1,ℎ,1 : SHGp´푆 p퐾1q, ´Γ1, 1q Ñ SHGp´푆 p1q, ´훿q

3 is surjective, and, since SHGp´푆 p퐾1q, ´Γ1, 1q has rank 1 it is actually an isomorphism (for the monopole case, the argument is essentially the same as in the proof of Lemma 2.6.1). Now we go back to the surgery exact triangle in (5.23), which

3 corresponds to surgeries on the curve 훼 Ă intp푆 p퐾푚qq. Since 훼 is disjoint from the boundary, (and as above, disjoint from 푆푚), we have the following exact triangle for any 푚 and 푛.

3 3 SHGp´푆 p퐾푚q, ´Γ푛, 푖q / SHGp´푆 p퐾푚`1q, ´Γ푛, 푖q j

t SHGp´푀, ´Γ푛, 푖q (5.32)

There are contact 2-handle attaching maps

3 3 퐶푚,ℎ,푛 : SHGp´푆 p퐾푚q, ´Γ푛q Ñ SHGp´푆 p1q, ´훿q,

254 where the contact 2-handle is attached along a meridional curve on the knot complements. We can attach a contact 2-handle along the same curve on the boundary of M, and the handle attaching maps commute with the maps in the exact triangle (5.32). Thus, we have a diagram:

3 3 SHGp´푆 p퐾푚q, ´Γ푛, 푖q / SHGp´푆 p퐾푚`1q, ´Γ푛, 푖q j

휏푚,푛,푖 s 퐶푚,ℎ,푛 SHGp´푀, ´Γ푛, 푖q 퐶푚`1,ℎ,푛

3 휑8 3 SHGp´푆 p1q, ´훿q 퐶 / SHGp´푆 p1q, ´훿q j 푀,ℎ,푛

휑 휑 0 s 1 SHGp´푆2 ˆ 푆1p1q, ´훿q (5.33) Here, 푆2 ˆ 푆1 is obtained from 푆3 by performing a 0-surgery along the unknot 훼. The balanced sutured manifold p푆2 ˆ 푆1p1q, 훿q is obtained from 푆2 ˆ 푆1 by removing a 3-ball and assigning a connected simple closed curve on the spherical boundary as the suture. Its sutured monopole and instanton Floer homologies are computed in Baldwin and Sivek [6] or Corollary 4.8.7 and are both of rank 2. Thus, the exactness tells us that 휑8 “ 0, 휑1 is injective, and 휑0 is surjective.

Now take 푚 “ 0, 푛 “ 1, and 푖 “ 1, we know that

3 SHGp´푀, ´Γ1, 1q – SHGp´푆 p퐾1q, ´Γ1, 1q – ℛ,

and 퐶푀,ℎ,푛 is injective. Then, take 푚 to be an arbitrary non-negative integer and 푛 “ 1, 푖 “ 1 in (5.33). From (5.27), we know that

3 푚 SHGp´푆 p퐾푚q, ´Γ휇, 1q – ℛ .

255 By performing sutured manifold decompositions along 푆푚 and Applying Theorem 2.6.20, we know that

3 3 푚 SHGp´푆 p퐾푚q, ´Γ푛, 1q – SHGp´푆 p퐾푚q, ´Γ휇, 1q – ℛ .

Recall from above discussions we have

SHGp´푀, ´Γ1, 1q – ℛ,

so in the exact triangle (5.33), we know that 휏푚,1,1 is surjective. Then, we can use the commutativity part of (5.33) and conclude that

3 3 퐶푚`1,ℎ,푛 : SHGp´푆 p퐾푚`1q, ´Γ1, 1q Ñ SHGp´푆 p1q, ´훿q

is surjective. From the fact that 휓˘,푛 commutes with 퐶ℎ,푛 as in Claim 1 and 2 in the proof of Proposition 5.4.8, we know that this surjectivity means that the unique 푈

´ 3 tower in KHG p´푆 , 퐾푚, 푝푚q starts at grading 1:

휏퐺p퐾푚q “ 1 for 푚 ą 0.

Take 푛 “ 1 in (5.29), then we have Table 5.4.

0 0 The fact that 휏퐺p퐾푚q “ 1 means that 휓`,1 ‰ 0, as 휓`,1 corresponds to the 푈 ´ 3 map at grading 1 part of KHG p´푆 , 퐾푚, 푝푚q. Thus, from the exactness we know that 푏 ě 푚 ` 1, 푐 ě 푎 ´ 푚.

256 Γ휇 Γ1 Γ2

1 푚 푚

0 휓`,1 0 2푚 ´ 1 푚 / 푏

´1 푚 푎 푐

´2 푚 푚

Table 5.4: The map 휓`,1 for 퐾푚.

From the exact triangle (5.14) we know that

푏 ` 푐 ď 푎 ` 1 and hence 푏 “ 푚 ` 1, 푐 “ 푎 ´ 푚. Finally, we conclude the following.

Proposition 5.5.2. Suppose 푚 ą 0 and 퐾푚 is described as above. Then

´ 3 푚´1 푚 KHG p´푆 , 퐾푚, 푝푚q – ℛr푈s1 ‘ pℛ1q ‘ pℛ0q .

Furthermore, 휏퐺p퐾푚q “ 1.

´ We could also compute the KHG of the knots 퐾푚, the mirror image of 퐾푚. For 푚 ď 0, the computation is the same as before, and we conclude the following. s

Proposition 5.5.3. Suppose 푚 ď 0 and the knot 퐾푚 is described as above. Then

s ´ 3 ´푚 ´푚 KHG p´푆 , 퐾푚, 푝푚q – ℛr푈s0 ‘ pℛ1q ‘ pℛ0q .

s 257 From the description it is clear that 휏퐺p퐾푚q “ 0.

For 푚 ą 0, we have a similar diagram ass in (5.33) as follows.

3 3 SHGp´푆 p퐾푚`1q, ´Γ푛, 푖q / SHGp´푆 p퐾푚q, ´Γ푛, 푖q j

휏푚,푛,푖 s t s SHGp´푀, ´Γ푛, 푖q (5.34) ¯ Let us first compute the case 푚 “ 1, when 퐾푚 is the left-handed trefoil. In this case, take 푛 “ 1 in (5.29), then we get Table 5.5.

Γ휇 Γ1 Γ2

1 1 1

0 휓`,1 0 1 1 / 푏

´1 1 푎 푐

´2 1 1

Table 5.5: The map 휓`,1 for 퐾1.

s The left-handed trefoil is not right veering in the sense of Baldwin and Sivek [8], so from their discussion, we conclude that 휓`,1 “ 0. (This is how they prove that the second top grading of the instanton knot Floer homology of a not-right-veering knot is non-trivial. Though they only work in the instanton case, the monopole case is the same.) Thus we conclude that 푏 “ 0.

258 ` In (5.34), let 푚 “ 0, 푛 “ 2, 푖 “ 0. Note the grading is induced by 푆푚,2, i.e., a

Seifert surface of the knot 퐾푚 which intersects the suture Γ2 transversely at four points and with a positive stabilization. This is to incorporate with (5.15). Thus, s we know that

3 푏 3 SHGp´푆 p퐾1q, ´Γ2, 0q “ ℛ “ 0, SHGp´푆 p퐾0q, ´Γ2, 0q – ℛ.

s s Here, 퐾0 is the unknot and we have computed the SHG of such a sutured solid torus in 4.7.1. Thus, we conclude that s

SHGp´푀, ´Γ2, 0q – ℛ.

Use the exactness and the induction, then we have

3 푐푚 SHGp´푆 p퐾푚q, Γ2, 0q – ℛ , 푐푚 ď 푚 ´ 1.

s For the knot 퐾푚, take 푛 “ 2 in (5.15), then we have Table 5.6. Thus, we conclude from the exactness that 푐 “ 푚 ´ 1, 휓1 “ 0, and 휓0 “ 0. As above, the two s 푚 `,2 `,2 1 0 ´ 3 maps 휓`,2 and 휓`,2 correspond to the 푈 maps of KHG p´푆 , 퐾푚, 푝푚q at grading 1 and 0, respectively. Hence, we conclude s

Proposition 5.5.4. Suppose 푚 ą 0 and the knot 퐾푚 is described as above. Then

s ´ 3 푚 푚´1 KHG p´푆 , 퐾푚, 푝푚q – ℛr푈s´1 ‘ pℛ1q ‘ pℛ0q .

s From the description, it is clear that 휏퐺p퐾푚q “ 0.

259 Γ휇 Γ2 Γ3

2 푚 푚

1 휓`,2 1 2푚 ´ 1 푚 / 푐푚

0 휓`,2 0 푚 푐푚 / 1

´1 ? ?

2 푚 푚

Table 5.6: The map 휓`,2 for 퐾푚. The sup-script means the map at a particular grading

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