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Contributions to Sutured Monopole and Sutured Instanton Floer 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 spinc 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 휏G....................... 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 (math- ematical) 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 un- derlying 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 re- lation between dimensions 3 and 4: For a closed oriented 3-manifold Y , the negative gradient flow equation of the Chern-Simons functional is exactly the anti-self-dual equation on the infinite cylinder R ˆ Y . 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 Y 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 Y , 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 Y . 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 descrip- tion 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 man- ifold 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.
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