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Floer Homology, Gauge Theory, and Low-Dimensional Topology Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribu- tion, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Clay Mathematics Institute, One Bow Street, Cam- bridge, MA 02138, 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 reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2006 by the Clay Mathematics Institute. All rights reserved. Published by the American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA. Printed in the United States of America. The Clay Mathematics Institute retains all rights except those granted to the United States Government. ∞ 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/ Visit the Clay Mathematics Institute home page at http://www.claymath.org/ 10987654321 111009080706 Contents Introduction vii Heegaard Floer Homology and Knot Theory 1 An Introduction to Heegaard Floer Homology 3 Peter Ozsvath´ and Zoltan´ Szabo´ Lectures on Heegaard Floer Homology 29 Peter Ozsvath´ and Zoltan´ Szabo´ Circle Valued Morse Theory for Knots and Links 71 Hiroshi Goda Floer Homologies and Contact Structures 101 Lectures on Open Book Decompositions and Contact Structures 103 John B. Etnyre Contact Surgery and Heegaard Floer Theory 143 Andras´ I Stipsicz Ozsv´ath-Szab´o Invariants and Contact Surgery 171 Paolo Lisca and Andras´ I Stipsicz Double Points of Exact Lagrangian Immersions and Legendrian Contact Homology 181 Tobias Ekholm Symplectic 4–manifolds and Seiberg–Witten Invariants 193 Knot Surgery Revisited 195 Ronald Fintushel Will We Ever Classify Simply-Connected Smooth 4-manifolds? 225 Ronald J. Stern + 2 ≥ A Note on Symplectic 4-manifolds with b2 =1andK 0 241 Jongil Park The Kodaira Dimension of Symplectic 4-manifolds 249 Tian-Jun Li Symplectic 4-manifolds, Singular Plane Curves, and Isotopy Problems 263 Denis Auroux Monodromy, Vanishing Cycles, Knots and the Adjoint Quotient 277 Ivan Smith List of Participants 293 v Introduction The Clay Mathematical Institute hosted its 2004 Summer School on Floer ho- mology, gauge theory, and low–dimensional topology at the Alfr´ed R´enyi Institute of Mathematics in Budapest, Hungary. The aim of this school was to bring together students and researchers in the rapidly developing crossroads of gauge theory and low–dimensional topology. In part, the hope was to foster dialogue across closely related disciplines, some of which were developing in relative isolation until fairly recently. The lectures centered on several topics, including Heegaard Floer theory, knot theory, symplectic and contact topology, and Seiberg–Witten theory. This volume is based on lecture notes from the school, some of which were written in close collaboration with assigned teaching assistants. The lectures have revised the choice of material somewhat from that presented at the school, and the topics have been organized to fit together in logical categories. Each course consisted of two to five lectures, and some had associated problem sessions in the afternoons. Mathematical gauge theory studies connections on principal bundles, or, more precisely, the solution spaces of certain partial differential equations for such connec- tions. Historically, these equations have come from mathematical physics. Gauge theory as a tool for studying topological properties of four–manifolds was pioneered by the fundamental work of Simon Donaldson in the early 1980’s. Since the birth of the subject, it has retained its close connection with symplectic topology, a subject whose intricate structure was illuminated by Mikhail Gromov’s introduc- tion of pseudo–holomorphic curve techniques, also introduced in the early 1980’s. The analogy between these two fields of study was further underscored by Andreas Floer’s construction of an infinite–dimensional variant of Morse theory that applies in two aprioridifferent contexts: either to define symplectic invariants for pairs of Lagrangian submanifolds of a symplectic manifold (the so–called Lagrangian Floer homology), providing obstructions to disjoining the submanifolds through Hamil- tonian isotopies, or to give topological invariants for three–manifolds (the so–called instanton Floer homology), which fit into a framework for calculating Donaldson’s invariants for smooth four–manifolds. In the mid–1990’s, gauge–theoretic invariants for four–manifolds underwent a dramatic change with the introduction of a new set of partial differential equations introduced by Nathan Seiberg and Edward Witten in their study of string theory. Very closely connected with the underlying geometry of the four–manifolds over which they are defined, the Seiberg–Witten equations lead to four–manifold invari- ants which are in many ways much easier to work with than the anti–self–dual Yang–Mills equations which Donaldson had studied. The introduction of the new invariants led to a revolution in the field of smooth four–manifold topology. vii viii INTRODUCTION Highlights in four–manifold topology from this period include the deep theo- rems of Clifford Taubes about the differential topology of symplectic four–manifolds. These give an interpretation of some of Gromov’s invariants for symplectic man- ifolds in terms of the Seiberg–Witten invariants of the underlying smooth four– manifold. Another striking consequence of the new invariants was a quick, elegant proof by Kronheimer and Mrowka of a conjecture by Thom, stating that the alge- braic curves in the complex projective plane minimize genus in their homology class. The invariants were also used particularly effectively in work of Ron Fintushel and Ron Stern, who discovered several operations on smooth four–manifolds, for which the Seiberg–Witten invariants transform in a predictable manner. These operations include rational blow–downs, where the neighborhood of a certain chain of spheres is replaced by a space with vanishing second homology, and also knot surgery,for which the Alexander polynomial of a knot is reflected in the Seiberg–Witten invari- ants of a corresponding four–manifold. These operations can be used to construct a number of smooth four–manifolds with interesting properties. In an attempt to better understand the somewhat elusive gauge theoretic in- variants, a different construction was given by Peter Ozsv´ath and Zolt´an Szab´o. They formulated an invariant for three– and four–manifolds which takes as its starting point a Lagrangian Floer homology associated to Heegaard diagrams for three–manifolds. The resulting “Heegaard Floer homology” theory is conjecturally isomorphic to Seiberg–Witten theory, but more topological and combinatorial in its flavor and correspondingly easier to work with in certain contexts. Moreover, this theory has benefitted a great deal from an array of contemporary results rendering various analytical and geometric structures in a more topological and combinatorial form, such as Donaldson’s introduction of “Lefschetz pencils” in the symplectic cat- egory and Giroux’s correspondence between open book decompositions and contact structures. The two lecture series of Ozsv´ath and Szab´o in the first section of this volume provide a leisurely introduction to Heegaard Floer theory. The first
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