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Displaceability in Symplectic Geometry DISPLACEABILITY IN SYMPLECTIC GEOMETRY A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Frederik De Keersmaeker August 2020 c 2020 Frederik De Keersmaeker ALL RIGHTS RESERVED DISPLACEABILITY IN SYMPLECTIC GEOMETRY Frederik De Keersmaeker, Ph.D. Cornell University 2020 We use a topological condition by Albers that is sufficient for nondisplaceabil- ity to describe a large class of nondisplaceable Lagrangians, namely the anti- diagonals of closed monotone symplectic manifolds. We provide some exam- ples of closed monotone symplectic manifolds of Euler characteristic zero. Furthermore, we study the displaceability of fibers of the bending flow sys- tem on equilateral pentagon space. Besides torus fibers over points in the in- terior of moment map image, this completely integrable system has two La- grangian sphere fibers over boundary points of the moment polytope. They are nondisplaceable for topological reasons. Most of the regular torus fibers are displaceable using McDuff’s probes technique. We prove that the central torus fiber is nondisplaceable by showing that we can lift a Hamiltonian diffeo- morphism displacing it to a Hamiltonian diffeomorphism displacing the cen- tral torus fiber of the Gelfand-Cetlin system on the complex Grassmannian of 2-planes. This fiber is known to be nondisplaceable. BIOGRAPHICAL SKETCH Frederik De Keersmaeker was born in Belgium in 1989. He started studying at Ghent University in 2007, where he got an M.S. in Engineering Physics in 2012 and an M.S. in Mathematics in 2013. After working as a teaching assistant for a year at KULeuven KULAK, he enrolled in the Ph.D. program in Mathematics at Cornell University. iii This document is dedicated to all Cornell graduate students. iv ACKNOWLEDGEMENTS I cannot imagine what my experience in graduate school would have been like without my adviser, Tara Holm. She is the most supportive and caring per- son. Thank you for guiding me through my research over the past six years, for encouraging me to be a better teacher and to do outreach, for suggesting I go to as many interesting workshops and conferences as possible, and for providing a safe space to talk about the ups and downs. The Cornell Math Department is a wonderful place, and I have a lot of peo- ple to thank there. Let me start with my minor committee members: Reyer Sjamaar and Allen Knutson. Thank you for reading this dissertation and for the many useful comments. I have loved teaching at Cornell; Kelly Delp and Ja- son Manning played a big role is making it a positive experience. Crossing the border to get into the U.S. would have been much harder without those letter Melissa Totman wrote for me, saying I was a valued member of the department. The heart of the department is its graduate students. There are a lot of names to list here, and I hope I have conveyed my appreciation for you all during our time at Cornell. Thank you Ian, My, Thomas, Balazs, Daoji, Emily, David, Lila, Smaranda, Trevor, Ryan, Ellie, Nicki, Fiona, Portia, Joe, Amin, Sasha, Jeff, Kelsey, Valente, and Voula. My symplectic buddy, Benjamin Hoffman, deserves a special shout-out. So does Hannah Keese for teaching in prison with me for two semesters. I learned a lot during the research visits I went on. I want to thank Nguyen Tien Zung at Universite´ Paul Sabatier in Toulouse and Katrin Wehrheim at UC Berkeley for talking to me about my research. Ithaca has been a wonderful place to live with my friends. Thank you Lau- ren, Michelle, Naomi, Drew, Gennie, Jaron, Emily, Smaranda, Patrick, Margaux, v Trevor, Benjamin, Phoebe, Jan, Patsy, and Valeria! Whenever I was back in Belgium, Emilie, Annabelle, Margo, Emma, and Margot made me feel like I had never left! Thank you for all the support! Finally, I want to thank my parents and my family for cheering me on and believing in me. vi TABLE OF CONTENTS Biographical Sketch . iii Dedication . iv Acknowledgements . .v Table of Contents . vii List of Tables . ix List of Figures . .x 1 Introduction 1 2 Preliminary Material 3 2.1 A Brief Overview of Symplectic Geometry . .3 2.1.1 Symplectic Manifolds: Definition and Examples . .3 2.1.2 Local Normal Forms . .6 2.1.3 Hamiltonian Vector Fields . .7 2.1.4 Hamiltonian Diffeomorphisms . .9 2.1.5 Hamiltonian Group Actions and Symplectic Reduction . 13 2.1.6 Compatible Almost Complex Structures . 15 2.2 Nondisplaceable Lagrangians . 16 2.2.1 Definition and Properties . 16 2.2.2 Examples . 17 2.3 Monotone and Semipositive Symplectic Manifolds . 18 2.4 The Maslov Class and Monotone Lagrangians . 21 3 Floer Homology and the PSS Isomorphism 29 3.1 Morse Theory . 29 3.1.1 Morse Functions . 29 3.1.2 The Morse Complex . 30 3.1.3 Continuation Maps . 33 3.1.4 Functoriality and Induced Maps . 36 3.1.5 Poincare´ Duality . 37 3.2 Periodic Orbits of Hamiltonian Vector Fields . 39 3.2.1 Definition and Properties . 40 3.2.2 The Arnold Conjecture . 41 3.3 Floer Homology . 42 3.3.1 The Novikov Ring . 43 3.3.2 The Loop Space and the Action Functional . 45 3.3.3 The Conley-Zehnder Index . 47 3.3.4 The Floer Chain Groups . 51 3.3.5 The Boundary Operator . 52 3.4 Moduli Spaces of Floer Cylinders . 59 3.4.1 Fredholm Operators and the Implicit Function Theorem . 59 3.4.2 Smoothness, Linearization, and Transversality . 61 vii 3.4.3 Compactness . 62 3.5 The Piunikhin-Salamon-Schwarz (PSS) Isomorphism . 69 3.6 Representing Submanifolds Using PSS . 75 3.7 An Application to Displaceability . 82 3.8 Nondisplaceable Anti-Diagonals . 84 4 Polygon Spaces 89 4.1 Definition and Elementary Properties . 89 4.2 The Bending Flow System . 90 4.3 McDuff’s Probes Technique . 94 4.4 Displacing Fibers and Symplectic Reduction . 98 4.5 Polygon Spaces and Grassmannians . 100 4.6 The Gelfand-Cetlin System . 104 4.7 Displacing the Central Fiber . 107 Bibliography 110 viii LIST OF TABLES 2.1 Semipositivity of products of two complex projective spaces in low dimension. 21 2.2 Semipositivity of products of three complex projective spaces in low dimension. 21 3.1 Different moduli spaces of Floer cylinders . 62 ix LIST OF FIGURES 3.1 Floer cylinder with disk cappings connecting two contractible periodic solutions (in blue). 55 3.2 Broken configuration with two Floer cylinders, with periodic or- bits in blue. 57 3.3 A plumber’s helper solution. 70 3.4 The two possible cases for a broken plumber’s helper solution with one break: either the Morse flow half-line breaks (top) or the cylindrical part breaks (bottom). 73 3.5 A broken configuration: periodic orbits in red connected by Floer cylinders, and Lagrangian boundary condition for the rightmost cylinder, with sphere bubbles in green and disk bub- bles in blue. 78 4.1 A pentagon with its two diagonals as red dashed line segments. 91 4.2 The image of the bending flow system for r = (2; 3; 3; 2; 3)..... 92 4.3 The image of the bending flow system for r = (1; 1; 1; 1; 1). The polytope is not smooth at the two blue vertices. 93 4.4 A equilateral pentagon with a vanishing diagonal consists of an equilateral triangle and a line segment. 93 4.5 Bending flow fibers displaced by vertical (left) and horizontal (right) probes in equilateral pentagon space. 96 4.6 Bending flow fibers displaced by probes in equilateral pentagon space. 97 x CHAPTER 1 INTRODUCTION The main subject of this dissertation is nondispaceable Lagrangians. They are symplectic analogues of n-dimensional submanifolds of an 2n-dimensional smooth manifold with nonzero self-intersection number. We start Chapter 2 with a brief overview of symplectic geometry. We define displaceable and nondisplaceable Lagrangians and introduce monotonicity of symplectic manifolds and their Lagrangians. In Chapter 3 we present a topological condition (Theorem 15) by Peter Al- bers that guarantees nondisplaceability. We describe a large class of nondis- placeable Lagrangians, namely the anti-diagonals of closed monotone symplec- tic manifolds (Theorem 16). If their Euler characteristic is nonzero, they are not displaceable by maps isotopic to the identity (a class that contains Hamiltonian diffeomorphisms). We provide some examples of closed monotone symplectic manifolds of Euler characteristic zero (Examples 9 and 10). In Chapter 4 we study the displaceability of fibers of the bending flow sys- tem on equilateral pentagon space. Besides torus fibers over points in the in- terior of moment map image, this completely integrable system has two La- grangian sphere fibers over boundary points of the moment polytope. They are nondisplaceable for topological reasons (Proposition 25). Most of the regular torus fibers are displaceable using McDuff’s probes technique (Theorem 17). We prove that the central torus fiber is nondisplaceable (Theorem 23) by showing that we can lift a Hamiltonian diffeomorphism displacing it to a Hamiltonian diffeomorphism displacing the central torus fiber of the Gelfand-Cetlin system 1 on the complex Grassmannian of 2-planes. This fiber is known to be nondis- placeable. 2 CHAPTER 2 PRELIMINARY MATERIAL After a brief overview of symplectic geometry, we define what it means for a Lagrangian submanifold in a symplectic manifold to be displaceable (by Hamil- tonian diffeomorphisms). We introduce monotonicity for symplectic manifolds and Lagrangians. 2.1 A Brief Overview of Symplectic Geometry We review the basic concepts from symplectic geometry that we will need this dissertation. We use this opportunity to introduce notation and conventions. Unless stated otherwise, the material presented in this section can be found in [MS17] or [CdS01].
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