BOLYAI SOCIETY MATHEMATICAL STUDIES BOLYAI SOCIETY MATHEMATICAL STUDIES Series Editor: Gabor Fejes Toth Publication Board: Laszlo Babai Istvan Juhasz Gyula 0. H. Katona Laszlo Lovasz Domokos Szasz Vilmos Totik Managing Editor: Dezs6 Miklos 1. Combinatorics, Paul Erdos is Eighty, Vol. 1 D. Miklos, V.T. Sos, T. Sz6nyi (Eds.) 2. Combinatorics, Paul Erdos is Eighty, Vol. 2 D. Miklos, V.T. Sos, T. Sz6nyi (Eds.) 3· Extremal Problems for Finite Sets P. Frankl, Z. Fiiredi, G. Katona, D. Miklos (Eds.) 4· Topology with Applications A. Csaszar (Ed.) 5· Approximation Theory and Function Series P. Ve rtesi, L. Leindler, Sz. Revesz, J. Szabados, V. Totik (Eds.) 6. Intuitive Geometry I. Barany, K. Boroczky (Eds.) 7· Graph Theory and Combinatorial Biology L. Lovasz,A. Gyarfas, G. Katona, A. Recski (Eds.) 8. Low Dimensional Topology K. Bori:iczky, Jr., W. Neumann, A. Stipsicz (Eds.) 9· Random Walks P. Revesz, B. Toth (Eds.) 10. Contemporary Combinatorics B. Bollobas (Ed.) n. Paul Erdos and His Mathematics I+ II G. Halasz, L. Lovasz, M. Simonovits, V. T. Sos (Eds.) 12. Higher Dimensional Varieties and Rational Points K. Bi:iri:iczky, Jr., J. Kollar, T. Szamuely (Eds.) 13. Surgery on Contact 3-Manifolds and Stein Surfaces B. Ozbagci, A. I. Stipsicz Burak Ozbagci Andras I. Stipsicz Surgery on Contact 3-Manifolds and Stein Surfaces ~Springer JANOS BOLYAI MATHEMATICAL SOCIETY Burak Ozbagci Koc University Rumelifeneri Yolu, Sariyer 34450 Istanbul, Turkey e-mall: [email protected] Andnis 1. Stipsicz Hungarian Academy of Sciences, Alfred Renyi Institute of Mathematics ReaItanoda U.13-15 1053 Budapest, Hungary e-mail: [email protected] Mathematics Subject Classification (2000): 57Rxx, 53Dxx, 58KlO, 32Q28 Library of Congress Control Number: 2004110893 ISSN 1217-4696 ISBN 978-3-642-06184-4 ISBN 978-3-662-10167-4 (eBook) DOI 10.1007/978-3-662-10167-4 This work is subject to copyright. 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Violations are liable for prosecution under the German Copyright Law. springeronline.com © 2004 Springer-Verlag Berlin Heidelberg Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Softcover reprint of the hardcover 1s t edition 2004 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro tective laws and regulations and therefore free for general use. Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper 41/3142/XT - 5 43 2 1 o CONTENTS CONTENTS 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 PREFACE 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1. INTRODUCTION 00 00 0 0 0 0 11 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1.1. Why symplectic and contact? 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 1.20 Results concerning Stein surfaces 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 1.30 Some contact results 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 20 TOPOLOGICAL SURGERIES 00 0 0 0 00 0 0 0 0 0 25 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 2°1. Surgeries and handlebodies 00 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 o o 2020 Dehn surgery 00 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 31 o o 2030 Kirby calculus 00 0 0 00 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 38 o o o o o 30 SYMPLECTIC 4-MANIFOLDS 00 0 0 0 0 0 49 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 301. Generalities about symplectic manifolds 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 49 3020 Moser's method and neighborhood theorems 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 55 3030 Appendix: The complex classification scheme for symplectic 4-lnanifolds 0 00 0 00 00 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 58 o o o o o o o 40 CONTACT 3-MANIFOLDS 63 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 401. Generalities on contact 3-manifolds 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 63 4020 Legendrian knots 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 72 4030 Tight versus overtwisted structures 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 76 50 CONVEX SURFACES IN CONTACT 3-MANIFOLDS 85 o o o o o o o o o o o o o o o o o o o o 501. Convex surfaces and dividing sets 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 85 5020 Contact structures and Heegaard decompositions 00 0 0 0 0 0 0 0 0 0 0 96 60 Spine STRUCTURES ON 3- AND 4-MANIFOLDS 99 o o o o o o o o o o o o o o o o o o o o o o 601. Generalities on spin and spine structures 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 99 4 Contents 6.2. Spine structures and oriented 2-plane fields .................. 102 6.3. Spine structures and almost-complex structures .............. 105 7. SYMPLECTIC SURGERY ............................................ 111 7.1. Symplectic cut-and-paste .................................... 111 7.2. Weinstein handles ........................................... 115 7.3. Another handle attachment .................................. 119 8. STEIN MANIFOLDS ................................................. 121 8.1. Recollections and definitions ................................. 121 8.2. Handle attachment to Stein manifolds ....................... 125 8.3. Stein neighborhoods of surfaces .............................. 127 9. OPEN BOOKS AND CONTACT STRUCTURES ........................ 131 9.1. Open book decompositions of 3-manifolds ................... 131 9.2. Compatible contact structures ............................... 138 9.3. Branched covers and contact structures ...................... 150 10. LEFSCHETZ FIBRATIONS ON 4-MANIFOLDS ........................ 155 10.1. Lefschetz pencils and fibrations ............................. 155 10.2. Lefschetz fibrations on Stein domains ....................... 162 10.3. Some applications .......................................... 173 11. CONTACT DEHN SURGERY ....................................... 179 11.1. Contact structures on S1 x D2 ••..•....••.•......•.....•... 179 11.2. Contact Dehn surgery ...................................... 185 11.3. Invariants of contact structures given by surgery diagrams .. 191 12. FILLINGS OF CONTACT 3-MANIFOLDS ............................ 201 12.1. Fillings ..................................................... 201 12.2. Nonfillable contact 3-manifolds ............................. 206 12.3. Topology of Stein fillings ................................... 215 13. APPENDIX: SElBERG-WITTEN INVARIANTS ...................... 223 13.1. Seiberg-Witten invariants of closed 4-manifolds ............ 223 13.2. Seiberg-Witten invariants of 4-manifolds with contact boun- dary ......................................................... 229 13.3. The adjunction inequality .................................. 231 14. APPENDIX: HEEGAARD FLOER THEORY .......................... 235 14.1. Topological preliminaries ................................... 235 Contents 5 14.2. Heegaard Floer theory for 3- and 4-manifolds ............... 239 14.3. Surgery triangles ........................................... 244 14.4. Contact Ozsvath-Szab6 invariants .......................... 249 15. APPENDIX: MAPPING CLASS GROUPS ............................ 255 15.1. Short introduction ......................................... 255 15.2. Mapping class groups and geometric structures ............. 263 15.3. Some proofs ................................................ 265 BIBLIOGRAPHY ...................................................... 269 INDEX ............................................................... 279 PREFACE The groundbreaking results of the near past - Donaldson's result on Lef schetz pencils on symplectic manifolds and Giroux's correspondence be tween contact structures and open book decompositions - brought a top ological flavor to global symplectic and contact geometry. This topological aspect is strengthened by the existing results of Weinstein and Eliashberg (and Gompf in dimension 4) on handle attachment in the symplectic and Stein category, and by Giroux's theory of convex surfaces, enabling us to perform surgeries on contact 3-manifolds. The main objective of these notes is to provide a self-contained introduction to the theory of surgeries one can perform on contact 3-manifolds and Stein surfaces. We will adopt a very topological point of view based on handlebody theory, in particular, on Kirby calculus for 3- and 4-dimensionalmanifolds. Surgery is a constructive method by its very nature. Applying it in an intricate way one can see what can be done. These results are nicely com plemented by the results relying on gauge theory - a theory designed to prove that certain things cannot be done. We will freely apply recent results of gauge theory without a detailed introduction to these topics; we will be content with a short introduction to some forms of Seiberg-Witten theory and some discussions regarding Heegaard Floer theory in two Appendices. As work of Taubes in the closed, and Kronheimer-Mrowka in the manifold with-boundary case shows, the analytic approach towards symplectic and contact topology can be very fruitfully capitalized when coupled with some form of Seiberg-Witten theory. On the other hand, Lefschetz pencils on symplectic, and open book decompositions on contact manifolds are well suited for the newly invented contact Ozsvath-Szab6 invariants. Under some fortunate circumstances these dual viewpoints provide interesting re sults in the subject. As a preview, Chapter 1 is devoted to the description of problems where the above discussed techniques can be applied. For setting up the topological background of surgeries on contact 3- manifolds and Stein surfaces we will first examine the smooth surgery con struction, with a special emphasis on 2-handle attachments to 4-manifolds 8 Preface and Dehn surgeries on 3-manifolds. This is done in Chapter 2. Then we turn to the symplectic cut-and-paste operation, which enables us to glue sym plectic 4-manifolds along contact type boundaries. To put this operation in the right perspective, in Chapters 3 and 4 we first briefly review some parts of symplectic and contact topology in dimensions 4 and 3, respec tively. We pay special attention to convex surfaces in contact 3-manifolds (Chapter 5), with an eye on its later applicability in contact surgery. Be fore giving the general scheme of symplectic surgery in Chapter 7, we make a little digression and discuss spine structures from a point of view suit able for our later purposes. As a special case of the general gluing scheme, we will meet Weinstein's construction for attaching symplectic 2-handles to w-convex boundaries along Legendrian knots. After having these prepara tions, we can turn to the discussion of the famous result of Eliashberg that shows how to attach a Stein 2-handle to the pseudoconvex boundary of a Stein domain along a Legendrian knot. For the convenience of the more topologically minded reader, in Chapter 8 a short recollection of rudiments of the theory of Stein manifolds is included. Once the gluing construction given, we can turn to its applications, including the search for Lefschetz fi bration structures on Stein domains, embeddability of Stein domains into closed surfaces with extra (symplectic or complex) structures, or the study of Stein fillings of contact 3-manifolds (Chapters 10 and 12). In the contact setting, the most important technique for being able to do surgery is the convex surface theory developed by Giroux. After recalling relevant parts of this beautiful theory, and proving the neighborhood theorems we need in this subject, in Chapter 11 we will be able to do contact surgeries. With this construction at our disposal, now we can seek for applications: we will be able to draw explicit diagrams of many contact 3-manifolds, show ways to distinguish them and to determine the homotopy type of contact struc tures given by various constructions. These results - together with various versions of gauge theories, including Seiberg-Witten theory and Heegaard Floer theory- provide ways to examine tightness and fillability properties of numerous contact structures, which are given in Chapter 12. To make the presentation more complete, we include Chapter 9 on open book de compositions and their relation to contact structures. The appearance of mapping class groups in these theories, together with some nice applications allows us to conclude the discussion with a short recollection of definitions and results in that field. To guide the interested reader, we close this preface by listing some monographs discussing topics we only outline here. Handlebody theory and Preface 9 Kirby calculus, which is only sketched in Chapter 2, is discussed more thor oughly in [66]. A more complete introduction to symplectic geometry and topology is provided by [111]. For additional reading on contact topology, the reader is advised to turn to [1, 2, 57]. Seiberg-Witten theory is covered by many volumes, including for example [119, 126, 149]. These notes are based on two lecture series given by the second author at the Banach Center (Warsaw, Poland) and at the University ofLille (France). He wants to thank these institutions for their hospitality. The final form of the notes were shaped while the authors visited KIAS (Seoul, Korea); they wish to thank KIAS for its hospitality. The authors would like to thank Selman Akbulut, John Etnyre, Sergey Finashin, David Gay, Paolo Lisca, Gordana Matic and Robert Szoke for many enlightening conversations. Special thanks go to Hansjorg Geiges for suggesting numerous corrections and improvements of an earlier version of the text. The second author also wants to express his thanks to his family - without their support this volume would not have come into existence. The first author acknowledges support from the Turkish Academy of Sciences and from Koc; University. The second author acknowledges partial support by OTKA T034885 and T037735. Istanbul and Budapest, 2004. Burak Ozbagci and Andras Stipsicz