Monopoles and Three-Manifolds Peter Kronheimer and Tomasz Mrowka Frontmatter More Information

Cambridge University Press 978-0-521-88022-0 - Monopoles and Three-Manifolds Peter Kronheimer and Tomasz Mrowka Frontmatter More information

Monopoles and Three-Manifolds

Originating with Andreas Floer in the 1980s, Floer homology has proved to be an effective tool in tackling many important problems in 3- and 4-dimensional geometry and topology. This book provides a comprehensive treatment of Floer homology, based on the Seiberg–Witten monopole equations. After first providing an overview of the results, the authors develop the analytic properties of the Seiberg–Witten monopole equations, assuming only a basic grounding in differential geometry and analysis. The Floer groups of a general 3-manifold are then defined, and their properties studied in detail. Two final chapters are devoted to the calculation of Floer groups, and to applications of the theory in topology. Suitable for beginning graduate students and researchers in the field, this book provides the first full discussion of a central part of the study of the topology of manifolds since the mid 1990s.

New Mathematical Monographs

Editorial Board Béla Bollobás, University of Memphis William Fulton, University of Michigan Frances Kirwan, Mathematical Institute, University of Oxford Peter Sarnak, Princeton University Barry Simon, California Institute of Technology Burt Totaro, University of Cambridge For information about Cambridge University Press mathematics publications visit http://www.cambridge.org/mathematics

Monopoles and Three-Manifolds

PETER KRONHEIMER Harvard University TOMASZ MROWKA Massachusetts Institute of Technology

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To Jenny and Gigliola

Contents

Preface page xi

I Outlines 1 1 Monopole invariants of four-manifolds 2 2 Morse theory 14 3 Monopole Floer homology for three-manifolds 49 Notes and references for Chapter I 82

II The Seiberg–Witten equations and compactness 84 4Basic terms 85 5 Compactness and properness 99 6 The blown-up conﬁguration space 112 7 Unique continuation 120 8 Compactness in the blown-up conﬁguration space 130 Notes and references for Chapter II 132

III Hilbert manifolds and perturbations 134 9 Completions and Hilbert manifolds 134 10 Abstract perturbations 152 11 Constructing tame perturbations 171 Notes and references for Chapter III 194

IV Moduli spaces and transversality 195 12 Transversality for the three-dimensional equations 196 13 Moduli spaces of trajectories 217 14Local structure of moduli spaces 239 15 Transversality for moduli spaces of trajectories 265 Notes and references for Chapter IV 272

vii

viii Contents

V Compactness and gluing 274 16 Compactness of trajectory spaces 275 17 The moduli space on a ﬁnite cylinder 294 18 Stable manifolds and gluing near critical points 317 19 Gluing trajectories 343 Notes and references for Chapter V 374

VI Floer homology 375 20 Orienting moduli spaces 375 21 A version of Stokes’ theorem 405 22 Floer homology 410 Notes and references for Chapter VI 448

VII Cobordisms and invariance 449 23 Summary of results 449 24The moduli space on a manifold with boundary 461 25 Maps from cobordisms 508 26 Composing cobordisms 535 27 Closed four-manifolds 551 28 Canonical gradings 581 Notes and references for Chapter VII 589

VIII Non-exact perturbations 590 29 Closed two-forms as perturbations 590 30 Floer groups and non-exact perturbations 597 31 Some isomorphisms 605 32 Applications to gluing 622 Notes and references for Chapter VIII 633

IX Calculations 634 33 Coupled Morse theory 634 34Calculation of coupled homology 658 35 Application to the Floer groups HM 678 36 The manifold S1 × S2 695 37 The three-torus 699 38 Elliptic surfaces 711 Notes and references for Chapter IX 719

Contents ix

X Further developments 721 39 Homology spheres and negative-deﬁnite cobordisms 722 40 Genus bounds and scalar curvature 733 41 Foliations and non-vanishing theorems 741 42 Surgery and exact triangles 757 Notes and references for Chapter X 778 References 779 Glossary of notation 785 Index 792

Preface

Gauge theory and related areas of geometry have been an important tool for the study of 4-dimensional manifolds since the early 1980s, when Donaldson introduced ideas from Yang–Mills theory to solve long-standing problems in topology. In dimension 3, the same techniques formed the basis of Floer’s construction of his “instanton homology” groups of 3-manifolds [32]. Today, Floer homology is an active area, and there are several varieties of Floer homology theory, all with closely related structures. While Floer’s construction used the anti-self-dual Yang–Mills (or instanton) equations, the theory presented in this book is based instead on the Seiberg–Witten equations (or monopole equations). We have aimed to lay a secure foundation for the study of the Seiberg–Witten equations on a general 3-manifold, and for the construction of the associated Floer groups. Our goal has been to write a book that is complete in its coverage of several aspects of the theory that are hard to find in the existing literature, providing at the same time an introduction to the techniques from analysis and geometry that are used. We have omitted some background topics that are now well treated in several good sources: in particular, the Seiberg–Witten invariants of closed 4-manifolds and related topics in gauge theory are given only a brief exposition here. The main results of this book – the formal properties of the Floer groups that we construct – can be summarized without delving too far into the techniques which lie beneath; so we present such a summary in Chapter I. The final chapter of the book touches on some further topics and describes how the theory has been applied to questions in topology. The definition of the Floer groups that we present here is new in some aspects. We believe that our approach to the Morse homology of a manifold with circle

action has not appeared before. It is described in Section 2, along with a closely related approach to Morse theory on a manifold with boundary. Our deﬁnition of the groups that we call HM(Y ) and HM(Y ) has roots in lectures given by Donaldson in Oxford in 1993. For the case that the ﬁrst Betti number of Y is

xii Preface

zero, a similar construction is described in [22] for the case of the instanton Floer theory, and there is related material due to Frøyshov in [40]. Another approach to the Seiberg–Witten version of Floer homology is presented by Marcolli and Wang in [71]. During the course of this work, a completely different approach to Floer homology was introduced by Ozsváth and Szabó in [93]. The construction of their “Heegaard homology” of 3-manifolds is not based on gauge theory, but appears to be entirely equivalent to the Seiberg–Witten version. Ozsváth and Szabó’s theory has influenced the development of this book, most particularly because of the way in which it has clarified the formal structure of Floer homology. We have sometimes tailored our account to emphasize the similar- ities between the two versions. Heegaard homology has spurred tremendous activity in the topological applications of Floer theory. Chapter X provides a small sample of results from this rapidly moving field. Acknowledgements. Gauge theory is now a mature subject, and the analysis on which it rests has deep roots. Much of the material that we present is therefore not original. When a particular argument is taken directly from a unique source, we have tried to cite the source at the relevant point in the text. More often, however, pointers to the earlier literature are to be found in the remarks collected at the end of each chapter. Among the many mathematicians who have contributed to this field, we would like to acknowledge particularly our debt to Simon Donaldson, Kim Frøyshov, Peter Ozsváth, Zoltán Szabó and Cliff Taubes. This work was supported in part by the Institute for Advanced Study and the National Science Foundation through grants DMS-9531964, DMS-9803166, DMS-0100771, DMS-011129, DMS-0244663, DMS-0206485 and DMS- 0405271. The authors would also like to thank Ron Fintushel, Larry Guth, Yi-Jen Lee, Yanki Lekili, Max Lipyanski, Tim Perutz, Yann Rollin, Jake Rasmussen, Peter Ozsváth, Zoltán Szabó and Fangyun Yang for many com- ments and corrections.