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IAS/PARK CITY SERIES Volume 15

Low Dimensional

Tomasz S. Mrowka Peter S. Ozsváth Editors

American Mathematical Society Institute for Advanced Study

Low Dimensional Topology

https://doi.org/10.1090//pcms/015

IAS/PARK CITY MATHEMATICS SERIES Volume 15

Low Dimensional Topology

Tomasz S. Mrowka Peter S. Ozsváth Editors

American Mathematical Society Institute for Advanced Study John C. Polking, Series Editor Ezra Miller, Volume Editor Victor Reiner, Volume Editor Bernd Sturmfels, Volume Editor

IAS/Park City Mathematics Institute runs mathematics education programs that bring together high school mathematics teachers, researchers in mathematics and mathematics education, undergraduate mathematics faculty, graduate students, and undergraduates to participate in distinct but overlapping programs of research and education. This volume contains the lecture notes from the Graduate Summer School program on Low Dimensional Topology held in Park City, Utah in the summer of 2006. 2000 Mathematics Subject Classification.Primary53–XX,57–XX,58–XX.

Library of Congress Cataloging-in-Publication Data Low dimensional topology / Tomasz S. Mrowka, Peter S. Ozsv´ath, editors. p. cm. — (IAS/Park City mathematicsseries,ISSN1079-5634;v.15) Includes bibliographical references. ISBN 978-0-8218-4766-4 (alk. paper) 1. theory (Topology) I. Mrowka, Tomasz. II. Ozsv´ath, Peter Steven, 1967–

QA611.3.L677 2009 514—dc22 2009000616

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprintshouldbeaddresseddirectlytotheauthor(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2009 by the American Mathematical Society. All rights reserved. ⃝ The American Mathematical Society retains all rights except those granted to the Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines ∞ ⃝ established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 09 Contents

Preface xi

Peter S. Ozsv´ath and Tomasz S. Mrowka Int roduction 1

1. The subject 3

2. The PCMI Graduate Summer School 3

3. Acknowledgments 5

John Milnor Fifty Years Ago: Topology of in the 50’s and 60’s 7

1. 3-dimensional manifolds 9

2. Higher 10

3. Why are higher dimensions sometimes easier? 14

4. Questions from the audience 15

5. Bibliography 18

Cameron Gordon and 3-Manifolds 21

Introduction 23

Lecture 1. 3-manifolds and knots 25 1.1. 3-manifolds 25 1.2. Knots 27 1.3. Exercises 29

Lecture 2. Dehn surgery 31 2.1. Overview 31 2.2. Framed surgery on knots on surfaces 34 2.3. Exercises 35

Lecture 3. Exceptional Dehn surgeries 37 3.1. Exceptional surgeries 37 3.2. Lens space surgeries 37 3.3. Seifert fiber space surgeries 38 3.4. Toroidal surgeries 39 3.5. Knots in solid tori 39 3.6. Exercises 41

v vi CONTENTS

Lecture 4. Rational tangle filling 43 4.1. Dehn filling 43 4.2. Tangles 43 4.3. Tangles with non-simple double branched covers 47 4.4. Example: the Whitehead link 47 4.5. Exercises 50

Lecture 5. Examples of exceptional Dehn fillings 51 5.1. Some examples 51 5.2. Chain links 54 5.3. Simplicity of the double branched cover 56 5.4. Exercises 57

Lecture 6. Classification of some exceptional fillings 59 6.1. Some classification theorems 59 6.2. Seifert fiber spaces 65 6.3. Methods of proof; non-integral toroidal surgeries 65

Bibliography 69

David Gabai Hyperbolic and 3- Topology 73

Introduction 75

1. Topological tameness; examples and foundations 76

2. Background material for hyperbolic 3-manifolds 79

3. Shrinkwrapping 82

4. Proof of Canary’s theorem 86

5.Thetamenesscriterion 88

6. Proof of the 96

Bibliography 101

John W. Morgan (notes by Max Lipyanskiy) Ricci Flow and Thurston’s Geometrization 105

Introduction 107

Lecture 1. Statement of Thurston’s geometrization conjecture 109 1. Prime decomposition 109 2. Examples of geometric manifolds 109 3. Thurston geometrization conjecture 111

Lecture 2. Basics of Ricci flow 113 1. on the geometric structure of 3-manifolds 113 2. Review of the Riemann curvature tensor 114 3. The Ricci flow equation 114 CONTENTS vii

4. Examples: Einstein manifolds, Ricci solitons, and gradient shrinking solitons 115 5. Maximum principle 116 6. Consequences of the maximum principle 117

Lecture 3. Blow-up limits and κ-non-collapsed solutions 119 1. Definition of blow-up limit 119 2. Criteria for existence of blow-up limits 119 3. L-geodesics and reduced volume 120 4. Non-collapsing 122 5. Curvature bounds 122

Lecture 4. Structure of κ-solutions and canonical neighborhoods 123 1. Gradient shrinking solitons 123 2. Classification of gradient shrinking solitons 124 3. κ-non-collapsed solutions 124 4.Canonicalneighborhoods 124 5. Existence of canonical neighborhoods for κ-solutions 125 6. Canonical neighborhoods for Ricci flows 125

Lecture 5. Structure of regions of large curvature and surgery 127 1. Canonical neighborhoods for regions of high curvature 127 2. Global versions of tubes and caps 127 3. Global structure of the regions of large curvature 129 4. The structure at the singular time 129 5.Thesurgeryprocess 129 6. Auxiliary Ricci flow to glue in 130

Lecture 6. Ricci flow with surgery and finite-time extinction 131 1.Ricciflowwithsurgery 131 2. Geometrization 132

Bibliography 137

Marta Asaeda and Mikhail Khovanov Notes on Link 139

Introduction 141

Lecture 1. A braid action on a category of complexes 143 1. rings 143 2. Zigzag rings An 144 3. A realization of the Temperley-Lieb algebra 145 4. The category of complexes 147 5. representation 148

Lecture 2. More on braid group actions 151 1. Invertibility of Ri 151 2. Braid group action on complexes of projective modules and topology of curves 151 3. Reduced Burau representation 154 viii CONTENTS

Lecture 3. A Categorification of the Jones polynomial 157 1. The Jones polynomial 157 2. Categorification and a bigraded link homology theory 158 3. Properties and examples 164

Lecture 4. Flat tangles and bimodules 169 1. Two-dimensional TQFTs and Frobenius algebras 169 2. Algebras Hn 171 3. Flat tangles and their cobordisms 173

Lecture 5. A homological of tangles and tangle cobordisms 177 1. An invariant of tangles 177 2. Tangle cobordisms 179 3. Equivariant versions and applications 181

Lecture 6. Categorifications of the HOMFLY-PT polynomial 185 1. The HOMFLY-PT polynomial and its generalizations 185 2. Hochschild homology 186 3. A categorification of the HOMFLY-PT polynomial 189

Bibliography 193

Zolt´an Szab´o Lecture Notes on Heegaard Floer Homology 197

Introduction. 199

Acknowledgments. 199

1. Three-manifolds and Heegaard decompositions. 199 1.1. Basic examples. 199 1.2. Heegaard decompositions. 200 1.3. Exercises for Section 1. 202

2. Heegaard diagrams. 202 2.1. Handle decompositions. 202 2.2. Attaching circles. 203 2.3. Examples. 203 2.4. Heegaard moves. 204 2.5. Exercises for Section 2. 205

3. Morse functions. 205 3.1. Exercises for Section 3. 207

4. Symmetric products and generators. 207 4.1. Heegaard generators. 208 4.2. Symmetric products. 208 4.3. Whitney disks. 209 4.4. An obstruction for Whitney disks. 210 4.5. More on homotopy classes and shadows. 210 4.6. Exercises for Section 4. 211 CONTENTS ix

5. Pointed diagrams. 212 5.1. Admissible diagrams. 213 5.2. Two-pointed Heegaard diagrams. 213 5.3. Chain-complexes for pointed Heegaard diagrams. 213 5.4. Chain complexes for knots. 214 5.5. Exercises for Section 5. 214

6. Holomorphic disks. 214 6.1. A mod 2 invariant. 215 6.2. Properties of the Maslov index. 216 6.3. Exercises for Section 6. 217

7. A local formula for the Maslov index. 217 7.1. Exercises for Section 7. 218

8. The . 218 8.1. Exercises for Section 8. 220

9. Generators and spin-c structures. 221 9.1. Exercises for Section 9. 222

10. Further constructions. 222 10.1. The construction of HF+(Y,s). 222 10.2. Orientations. 223 10.3. Floer homology. 223 10.4. Computations. 225

11. Problems. 225

Bibliography 226

John Etnyre Contact Geometry in Low Dimensional Topology 229

1. Introduction 231

2. Contact structures and foliations 233

3. From foliations to contact structures 242 3.1. Part2oftheproofofTheorem3.1 244 3.2. Part1oftheproofofTheorem3.1 246

4. Taut foliations and symplectic fillings 253

5. Symplectic handle attachment and Legendrian surgery 255

6. Open book decompositions and symplectic caps 258

Bibliography 262 xCONTENTS

Ronald Fintushel and Ronald J. Stern Six Lectures on Four 4-Manifolds 265 Introduction 267 Lecture 1. How to construct 4-manifolds 269 1. 270 2. Techniques used for the construction of simply connected smooth and symplectic 4-manifolds 270 3. Some examples: Horikawa surfaces 271 4. Complex surfaces 273 5. More symplectic manifolds 273 Lecture 2. A user’s guide to Seiberg-Witten theory 275 1. The set-up 275 2. The equations 277 3. Adjunction Inequality 278 4. K¨ahler manifolds 279 5. Blowup formula 279 6. Gluing formula 279 7. Seiberg-Witten invariants of elliptic surfaces 280 8. Nullhomologous tori 280 9. Seiberg-Witten invariants for log transforms 281 Lecture 3. Knot surgery 283 1.Theknotsurgerytheorem 283 2. Proof of knot surgery theorem: the role of nullhomologous tori 284 Lecture 4. Rational blowdowns 291 1. Configurations of spheres and associated rational balls 291 2. Effect on Seiberg-Witten invariants 293 3. Taut configurations 294

Lecture 5. Manifolds with b+ =1 297 1. Seiberg-Witten invariants 297 2. Smooth structures on blow-ups of CP2 299 Lecture 6. Putting it all together: The geography and botany of 4-manifolds 305 1. Existence: the geography problem 305 2. Uniqueness: the botany problem 307 3. Horikawa surfaces: how to go from one deformation type to another 308

4. Manifolds with c =9χh :afakeprojectiveplane 309 5. Small 4-manifolds 310 6. What were the four 4-manifolds? 311 Bibliography 313 Preface

The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as part of the “Regional Geometry Institute” initiative of the National Science Foundation. In mid 1993 the program found an institutional home at the Institute for Advanced Study (IAS) in Princeton, New Jersey. The IAS/Park City Mathematics Institute encourages both research and ed- ucation in mathematics and fosters interaction between the two. The three-week summer institute offers programs for researchers and postdoctoral scholars, gradu- ate students, undergraduate students, high school teachers, undergraduate faculty, and researchers in mathematics education. One of PCMI’s main goals is to make all of the participants aware of the total of activities that occur in math- ematics education and research: we wish to involve professional mathematicians in education and to bring modern concepts in mathematics to the attention of educators. To that the summer institute features general sessions designed to encourage interaction among the various groups. In-year activities at the sites around the country form an integral part of the High School Teachers Program. Each summer a different topic is chosen as the focus of the Research Program and Graduate Summer School. Activities in the Undergraduate Summer School deal with this topic as well. Lecture notes from the Graduate Summer School are being published each year in this series. The first fifteen volumes are: • Volume 1: Geometry and Quantum Field Theory (1991) • Volume 2: Nonlinear Partial Differential Equations in Differential Geom- etry (1992) • Volume 3: Complex (1993) • Volume 4: and the Topology of Four-Manifolds (1994) • Volume 5: Hyperbolic Equations and Frequency Interactions (1995) • Volume 6: Probability Theory and Applications (1996) • Volume 7: and Topology (1997) • Volume 8: Representation Theory of Lie Groups (1998) • Volume 9: Arithmetic Algebraic Geometry (1999) • Volume 10: Computational Complexity Theory (2000) • Volume 11: Quantum Field Theory, Supersymmetry, and Enumerative Geometry (2001) • Volume 12: Automorphic Forms and their Applications (2002) • Volume 13: Geometric Combinatorics (2004) • Volume 14: Mathematical Biology (2005) • Volume 15: Low Dimensional Topology (2006) Volumes are in preparation for subsequent years. Some material from the Undergraduate Summer School is published as part of the Student Mathematical Library series of the American Mathematical Soci- ety. We hope to publish material from other parts of the IAS/PCMI in the future. This will include material from the High School Teachers Program and publications documenting the interactive activities which are a primary focus of the PCMI. At the summer institute late afternoons are devoted to seminars of common interest

xi xii PREFACE to all participants. Many deal with current issues in education: others treat math- ematical topics at a level which encourages broad participation. The PCMI has also spawned interactions between universities and high schools at a local level. We hope to share these activities with a wider audience in future volumes.

John C. Polking Series Editor December 2008

Titles in This Series

16 Scott Sheffield and Thomas Spencer, Editors, Statistical Mechanics, 2009 15 Tomasz S. Mrowka and Peter S. Ozsv´ath, Editors, Low Dimensional Topology, 2009 14 Mark A. Lewis, Mark A. J. Chaplain, James P. Keener, and Philip K. Maini, Editors, Mathematical Biology, 2009 13 Ezra Miller, Victor Reiner, and Bernd Sturmfels, Editors, Geometric Combinatorics, 2007 12 Peter Sarnak and Freydoon Shahidi, Editors, Automorphic Forms and Applications, 2007 11 Daniel S. Freed, David R. Morrison, and Isadore Singer, Editors, Quantum Field Theory, Supersymmetry, and Enumerative Geometry, 2006 10 Steven Rudich and , Editors, Computation Complexity Theory, 2004 9 Brian Conrad and Karl Rubin, Editors, Arithmetic Algebraic Geometry, 2001 8 Jeffrey AdamsandDavidVogan,Editors, Representation Theory of Lie Groups, 2000 7 and Lisa Traynor, Editors, Symplectic , 1999 6 Elton P. Hsu and S. R. S. Varadhan, Editors, Probability Theory and Applications, 1999 5 Luis Caffarelli and Weinan E, Editors, Hyperbolic Equations and Frequency Interactions, 1999 4 Robert Friedman and John W. Morgan, Editors, Gauge Theory and the Topology of Four-Manifolds, 1998 3 J´anos Koll´ar, Editor, Complex Algebraic Geometry, 1997 2 Robert Hardt and Michael Wolf, Editors, Nonlinear Partial Differential Equations in Differential Geometry, 1996 1 Daniel S. Freed and Karen K. Uhlenbeck, Editors, Geometry and Quantum Field Theory, 1995 Low-dimensional topology has long been a fertile area for the interaction of many different disciplines of mathematics, including , , combinatorics, representation theory, global analysis, classical mechanics, and theoretical . The Park City Mathematics Institute summer school in 2006 explored in depth the most exciting recent aspects of this interaction, aimed at a broad audience of both graduate students and researchers. The present volume is based on lectures presented at the summer school on low- dimensional topology. These notes give fresh, concise, and high-level introductions to these developments, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field of low-dimensional topology and to senior researchers wishing to keep up with current developments. The volume begins with notes based on a special lecture by about the history of the topology of manifolds. It also contains notes from lectures by Cameron Gordon on the basics of three-manifold topology and surgery problems, Mikhail Khovanov on his homological invariants for knots, John Etnyre on contact geometry, Ron Fintushel and Ron Stern on constructions of exotic four-manifolds, David Gabai on the hyper- bolic geometry and the ending lamination theorem, Zoltán Szabó on Heegaard Floer homology for knots and three manifolds, and John Morgan on Hamilton’s and Perelman’s work on Ricci flow and geometrization.

PCMS/15 AMS on the Web www.ams.org