Harvard University Department of Mathematics Lefschetz Fibrations on 4-Manifolds Author: Advisor: Gregory J. Parker Prof. Clifford Taubes [email protected] This thesis is submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors in Mathematics and Physics. March 20, 2017 Contents 1 Lefschetz Fibrations on Complex Algebraic Varieties 4 1.1 Lefschetz Pencils and Fibrations . 4 1.2 Monodromy and the Picard-Lefschetz Theorem . 10 1.3 The Topology of Lefschetz Fibrations . 21 2 Lefschetz Bifibrations and Vanishing Cycles 30 2.1 Matching Cycles . 33 2.2 Lefschetz Bifibrations . 34 2.3 Vanishing Cycles as Matching Cycles . 36 2.4 Bases of Matching Cycles . 42 2.5 Expressions for Vanishing Cycles . 48 3 Symplectic Lefschetz Pencils 52 3.1 Outline of the Proof . 53 3.2 Symplectic Submanifolds . 54 3.3 Asymptotically Holomorphic Sections . 57 3.4 Quantitative Transversality . 60 3.5 Constructing Pencils . 63 3.6 The Converse Result . 64 3.7 Asymptotic Uniqueness . 66 4 Broken Fibrations and Morse 2-functions 69 4.1 The Near-Symplectic Case . 69 4.2 Broken Lefschetz Fibrations . 70 4.3 Morse 2-functions . 77 4.4 Uniqueness of Broken Lefschetz Fibrations . 83 4.5 Trisections . 86 Acknowledgements: First and foremost, I would like to thank my advisor, Clifford Taubes, for his guidance throughout the process of learning this material and writing this thesis. His insights and suggestions have helped me grow as a writer and as a mathematician. Second, I would like to thank Roger Casals for inspiring me to learn these topics and for sharing his expertise on them. In particular, he taught me the material that appears in Chapter 2. I would also like to thank Matt Hedden and Selman Akbulut for helpful conversations regarding the material in Chapter 4. Finally, I would like to thank my friends and family for their support and encouragement over the course of this project. Recommended Background: Throughout this thesis, it is assumed that the reader is familiar with smooth manifolds, say at the level of [1], and Morse theory both in the language of handle attachments, as in [2], and in the language of gradient flows, as in [3]. The language of homology and cohomology at the level of [4] is also used throughout without exposition. For Chapter 3 in particular, some familiarity with connections and curvature is also helpful [5]. 1 Introduction Morse theory studies the topology of smooth manifolds by analyzing the critical points of generic smooth functions on them. It is one of the most fruitful tools in differential topology, and since its invention, it has been applied to almost every area of geometry and topology from the classification of compact surfaces to the study of infinite-dimensional manifolds in Floer theory. The basic premise of Morse theory is to consider a Morse function on a manifold, i.e. a function with non-degenerate critical points. A fundamental lemma is: Lemma 0.1. Let X be a smooth manifold and f : X Ñ R a Morse function . If the interval pa; bq contains no critical values, then the level sets f ´1pcq are smooth submanifolds and are diffeomorphic for all c P pa; bq. This is proved by flowing along a normalized negative gradient. This lemma implies that the topology of the level sets can only change when crossing a critical value. At the critical value itself, the level set is a not a submanifold, and is instead a singular level set. singular level sets singular fiber critical points f 1 CP R Figure 1: (Left) a smooth manifold with a Morse function f. Critical points and critical values are indicated by crosses (red). The level sets of f (top) are smooth submanifolds except at level sets containing a critical point, which are singular level sets. (Right) a Lefschetz fibration above a region of the Riemann sphere. There can be non-trivial monodromy around loops encircling critical points, such as the one indicated (blue). 2 Lefschetz fibrations, originally due to Solomon Lefschetz, are the holomorphic analogue of Morse functions and have proved equally fruitful in their application. Here, one takes a holomorphic function to the complex numbers or the Riemann sphere CP 1 with non-degenerate critical points. Similar to the real case, the fibers away from the critical values are all diffeomorphic submanifolds, and there are singular fibers above the critical values. In the holomorphic case, however, one can ask the additional question of what happens when considering the fibers above a path that encircles a critical value. Looping around a path that encircles a critical value will result in a non-trivial twisting of the fiber above the base point of the loop. This action is called the monodromy and understanding it is a central question in the study of Lefschetz fibrations. The goal of this thesis is to give an introduction to the theory and applications of Lefschetz fibrations in smooth topology. Lefschetz fibrations were originally developed in the context of complex algebraic varieties, and have been a standard tool in algebraic geometry for decades. In 1995, however, a startling theorem of Simon Donaldson proved the existence of Lefschetz fibrations on compact symplectic manifolds, which need not have a holomorphic structure. Since then, Lefschetz fibrations and slight generalizations of them called broken Lefschetz fibrations have been defined on arbitrary smooth 4-manifolds. In all these cases, Lefschetz fibrations have a chance to reduce the study of the geometry and topology of compact manifolds to the study of the monodromy around critical points, which is, in principle, a question only about the self-diffeomorphisms of fibers. For the majority of this exposition, the focus will be on manifolds whose real dimension is 4. There are two main reasons for this. First, Lefschetz fibrations are an especially powerful tool in 4 dimensions. A Lefschetz fibration will, up to the critical fibers, decompose a 4-manifold into a fiber bundle of surfaces over another surface. Explicit understanding of (real) surfaces and their diffeomorphisms can provide significant insight into 4-dimensional topology that is lost in higher dimensions. Second, 4-manifolds are the most interesting case from the perspective of smooth topology. In many ways, 4-manifolds are more complicated than either lower or higher dimensional manifolds. In low dimensions, there are simply not many possibilities for complexity. The Geometrization conjecture and the 3-dimensional Poincar´econjecture were resolved by G. Perelman in 2003, yielding a relatively complete understanding of 3-manifolds [6]. In dimensions strictly greater than 4, there is more fluidity than in 4-dimensions. For example, the \Whitney Trick" allows one to \untangle" submanifolds by moving them past each other in the extra dimensions. S. Smale used this trick to resolve the Poincar´econjecture in dimension ¥ 5 in 1961 [7], showing that certain aspects of high dimensional topology are quite tractable. The smooth Poincar´eConjecture in 4 dimensions remains open, and symbolizes the lack of understanding in this boundary case between the simplicity of low dimensions and the fluidity of high dimensions. The discovery of exotic smooth structures on 4-manifolds [8] added further complexity to the study of 4-manifolds. Seiberg-Witten theory, developed in the 1980s, gave a fleeting hope that a classification of 4- manifolds would be possible. Since then, however, many new examples of strange and exotic 4-manifolds have been constructed, and Seiberg-Witten theory has thus far been unable to parse the complexity of 4-manifolds. As of 2017, the study of 4-manifold topology remains a quagmire that defies understanding. Lefschetz fibrations have been on both sides of the study of 4-manifolds, at times being used to aid understanding of 4-manifolds, while at others being used in bizarre and counter-intuitive constructions. This thesis is divided into four chapters. Chapter 1 discusses Lefschetz fibrations in the classical setting of complex algebraic varieties. This includes the basic definitions, the Picard-Lefschetz Theorem, which is the essential result for understanding the monodromy of Lefschetz fibrations, and results relating the monodromy to the topology of the manifold. Chapter 2 introduces a computational process that allows the results of Chapter 1 to be realized concretely on specific varieties. Chapter 3 gives a survey of Donaldson's theorem on the existence and uniqueness of Lefschetz fibrations on compact symplectic manifolds. The majority of this chapter is devoted to the development of \approximately-holomorphic" theory and has a rather more geometric flavor than the other chapters. Chapter 4 discusses more recent results about generalizations of Lefschetz fibrations on arbitrary smooth 4-manifolds, many of which are due to Robion Kirby and David Gay. 3 Chapter 1 Lefschetz Fibrations on Complex Algebraic Varieties 1.1 Lefschetz Pencils and Fibrations This section gives the basic definitions and constructions of Lefschetz fibrations on complex algebraic varieties. Lefschetz fibrations in this case arise from preliminary structures called Lefschetz pencils. The construction of Lefschetz pencils is a holomorphic analogue of a height function on a compact smooth manifold embedded in Euclidean space. Given such a manifold M Ď RN a \height" function can be constructed on it by choosing a vector v and taking the projection to the span Rv. It can be shown that this projection is a Morse function for almost every v P RN . Viewed another way, choosing v specifies a smoothly N varying family of hyperplanes Ht for t P R | the planes perpendicular to v | such that t Ht “ R and the height function is f : M Ñ R is defined by fpxq being the unique t0 such that x P Ht0 . The intersections Ů M X Xt are the level sets of f and are submanifolds for all but finitely many values of t, at which there are singular level sets.