An introduction to contact topology

Hansjörg Geiges0521865859, 9780521865852, 9780511378850, 9780511377068

This text on contact topology is the first comprehensive introduction to the subject, including recent striking applications in geometric and differential topology: Eliashberg’s proof of Cerf’s theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds. Starting with the basic differential topology of contact manifolds, all aspects of 3-dimensional contact manifolds are treated in this book. One notable feature is a detailed exposition of Eliashberg’s classification of overtwisted contact structures. Later chapters also deal with higher-dimensional contact topology. Here the focus is on contact surgery, but other constructions of contact manifolds are described, such as open books or fibre connected sums. This book serves both as a self-contained introduction to the subject for advanced graduate students and as a reference for researchers.

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Table of contents :
Cover……Page 1
Series-title……Page 3
Title……Page 5
Copyright……Page 6
Dedication……Page 7
Contents……Page 9
Preface……Page 12
1 Facets of contact geometry……Page 19
1.1 Contact structures and Reeb vector fields……Page 20
1.2 The space of contact elements……Page 24
1.3 Interlude: symplectic linear algebra……Page 32
1.4 Classical mechanics……Page 37
1.5 The geodesic flow and Huygens’ principle……Page 43
1.6 Order of contact……Page 55
1.7.1 Cerf’s theorem……Page 58
1.7.2 Property P for knots……Page 61
2.1 Examples of contact manifolds……Page 69
2.2 Gray stability and the Moser trick……Page 77
2.3 Contact Hamiltonians……Page 80
2.4 Interlude: symplectic vector bundles……Page 82
2.5 Darboux’s theorem and neighbourhood theorems……Page 84
2.5.1 Darboux’s theorem……Page 85
2.5.2 Isotropic submanifolds……Page 86
2.5.3 Contact submanifolds……Page 93
2.5.4 Hypersurfaces……Page 95
2.6 Isotopy extension theorems……Page 99
2.6.1 Isotropic submanifolds……Page 100
2.6.2 The contact disc theorem……Page 103
2.6.3 Contact submanifolds……Page 107
2.6.4 Surfaces in 3–manifolds……Page 109
3 Knots in contact 3–manifolds……Page 111
3.1 Legendrian and transverse knots……Page 112
3.2 Front and Lagrangian projection……Page 113
3.2.1 Legendrian curves……Page 114
3.3 Approximation theorems……Page 118
3.3.1 Legendrian knots……Page 119
3.3.2 Transverse knots……Page 121
3.4.1 Hopf’s Umkehrhomomorphismus……Page 123
3.4.2 Representing homology classes by submanifolds……Page 125
3.4.3 Linking numbers……Page 128
3.5.1 Legendrian knots……Page 132
3.5.2 Transverse knots……Page 143
3.5.3 Transverse push-offs……Page 146
4 Contact structures on 3–manifolds……Page 148
4.1 Martinet’s construction……Page 150
4.2 2–plane fields on 3–manifolds……Page 152
4.2.1 Cobordism classes of links……Page 157
4.2.3 De.nition of the obstruction classes……Page 158
4.3 The Lutz twist……Page 160
4.4.1 Branched covers……Page 165
4.4.2 Open books……Page 167
4.5 Tight and overtwisted……Page 175
4.6 Surfaces in contact 3–manifolds……Page 180
4.6.1 The characteristic foliation……Page 181
4.6.2 Convex surfaces……Page 197
4.6.3 The elimination lemma……Page 202
4.6.4 Genus bounds……Page 211
4.6.5 The Bennequin inequality……Page 220
4.7.1 Statement of the classi.cation result……Page 222
4.7.2 Outline of the argument……Page 226
4.7.3 Characteristic foliations on spheres……Page 227
4.7.4 Construction near the 2–skeleton……Page 232
4.7.5 Proof of the classi.cation result……Page 243
4.8 Convex surface theory……Page 247
4.9 Tomography……Page 262
4.10 On the classi•cation of tight contact structures……Page 270
4.11 Proof of Cerf’s theorem……Page 272
4.12 Prime decomposition of tight contact manifolds……Page 275
5 Symplectic fillings and convexity……Page 286
5.1 Weak versus strong fillings……Page 287
5.2 Symplectic cobordisms……Page 291
5.3 Convexity and Levi pseudoconvexity……Page 294
5.4 Levi pseudoconvexity and………Page 299
6 Contact surgery……Page 304
6.1 Topological surgery……Page 305
6.2 Contact surgery and symplectic cobordisms……Page 311
6.3 Framings in contact surgery……Page 321
6.3.1 The h–principle for isotropic immersions……Page 324
6.3.2 Interlude: the Whitney–Graustein theorem……Page 327
6.3.3 Proof of the framing theorem……Page 332
6.4 Contact Dehn surgery……Page 338
6.5 Symplectic fillings……Page 342
7 Further constructions of contact manifolds……Page 350
7.1 Brieskorn manifolds……Page 351
7.2 The Boothby–Wang construction……Page 357
7.3 Open books……Page 362
7.4 Fibre connected sum……Page 368
7.5 Branched covers……Page 371
7.6 Plumbing……Page 373
7.7 Contact reduction……Page 378
8 Contact structures on 5–manifolds……Page 384
8.1 Almost contact structures……Page 385
8.2 On the structure of 5–manifolds……Page 400
8.3 Existence of contact structures……Page 416
Appendix A: The generalised Poincaré lemma……Page 419
Appendix B: Time-dependent vector fields……Page 422
References……Page 426
Notation index……Page 437
Author index……Page 444
Subject index……Page 447