Symplectic topology and Floer homology:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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Cambridge
Cambridge University Press
2015
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| Series: | New mathematical monographs
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| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| ISBN: | 9781107535688 |
Staff View
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| adam_text | new mathematical monographs
Published in two volumes, this is the first book to provide a thorough and systematic explanation of symplectic topology, and the analytical details and techniques used in applying the machinery arising from Floor theory as a whole.
Volume 1 covers the basic materials of Hamiltonian dynamics and symplectic geometry and the analytic foundations of Gromov’s pseudoholomorphic curve theory. One novel aspect of this treatment is the uniform treatment of both closed and open cases and a complete proof of the boundary regularity theorem of weak solutions of pseudoholomorphic curves with totally real boundary conditions. Volume 2 provides a comprehensive introduction to both Hamiltonian Floer theory and Lagrangian Floer theory, including many examples of their applications to various problems in symplectic topology.
Symplectic Topology and Floer Homology is a comprehensive resource suitable for experts and newcomers alike.
Yong-Geun Oh is Director of the IBS Center for Geometry and Physics and is Professor in the Department of Mathematics at POSTECH (Pohang University of Science and Technology) in Korea. He was also Professor in the Department of Mathematics at the University of Wisconsin-Madison. He is a member of the KMS, the AMS, the Korean National Academy of Sciences, and the inaugural class of AMS Fellows. In 2012 he received the Kyung-Ahm Prize of Science in Korea.
1
The New Mathematical Monographs are dedicated to books containing an in-depth discussion of a substantial area of mathematics. They bring the reader to the forefront of research by presenting a synthesis of the key results, while also acknowledging the wider mathematical context. As well as being detailed, they are readable and contain the motivational material necessary for those entering a field. For established researchers they are a valuable resource. Books are edited and typeset to a high standard and published in hardback.
Contents of Volume 1
ft
Contents of Volume 2 page ix
Preface xiii
PART 1 HAMILTONIAN DYNAMICS AND SYMPLECTIC GEOMETRY
1 The least action principle and Hamiltonian mechanics 3
1.1 The Lagrangian action functional and its first variation 3
1.2 Hamilton’s action principle 7
1.3 The Legendre transform 8
L4 Classical Poisson brackets 18
2 Symplectic manifolds and Hamilton’s equation 21
2.1 The cotangent bundle 21
2.2 Symplectic forms and Darboux’ theorem 24
2.3 The Hamiltonian diffeomorphism group 37
2.4 Banyaga’s theorem and the flux homomorphism 45
2.5 Calabi homomorphisms on open manifolds 52
3 Lagrangian submanifolds 60
3.1 The conormal bundles 60
3.2 Symplectic linear algebra 62
3.3 The Darboux-Weinstein theorem 71
3.4 Exact Lagrangian submanifolds 74
3.5 Classical deformations of Lagrangian submanifolds 77
3.6 Exact Lagrangian isotopy = Hamiltonian isotopy 82
3.7 Construction of Lagrangian submanifolds 87
3.8 The canonical relation and the natural boundary condition 96
3.9 Generating functions and Viterbo invariants 99
v
vi Contents of Volume 1
4 Symplectic fibrations 106
4.1 Symplectic fibrations and symplectic connections 106
4.2 Hamiltonian fibration 110
4.3 Hamiltonian fibrations, connections and holonomies 120
5 Hofer’s geometry of Ham(M, a ) 130
5.1 Normalization of Hamiltonians 130
5.2 Invariant norms on and the Hofer length 135
5.3 The Hofer topology of Ham(Af, oj) 137
5.4 Nondegeneracy and symplectic displacement energy 139
5.5 Hofer’s geodesics on Ham(Af, oj) 143
6 C°-Sympleetic topology and Hamiltonian dynamics 146
6.1 C° symplectic rigidity theorem 146
6.2 Topological Hamiltonian flows and Hamiltonians 158
6.3 Uniqueness of the topological Hamiltonian and its flow 163
6.4 The hameomorphism group 171
PART 2 RUDIMENTS OF PSEUDOHOLOMORPHIC CURVES
7 Geometric calculations 177
7.1 Natural connection on almost-Kahler manifolds 177
7.2 Global properties of y-holomorphic curves 185
7.3 Calculations of Ae(u) on shell 189
7.4 Boundary conditions 193
8 Local study of J-holomorphie curves 197
8.1 Interior a-priori estimates 197
8.2 Off-shell elliptic estimates 202
8.3 Removing boundary contributions 209
8.4 Proof of 6-regularity and density estimates 212
8.5 Boundary regularity of weakly J-holomorphic maps 221
8.6 The removable singularity theorem 227
8.7 Isoperimetric inequality and the monotonicity formula 235
8.8 The similarity principle and the local structure of the image 239
9 Gromov compactification and stable maps 247
9.1 The moduli space of pseudoholomorphic curves 247
9.2 Sachs—Uhlenbeck rescaling and bubbling 251
9.3 Definition of stable curves 257
9.4 Deformations of stable curves 267
9.5 Stable map and stable map topology 291
Contents of Volume 1
vii
10 Fredholm theory 323
10.1 A quick review of Banach manifolds 323
10.2 Off-shell description of the moduli space 328
10.3 Linearizations of 332
10.4 Mapping transversality and linearization of d 335
10.5 Evaluation transversality 345
10.6 The problem of negative multiple covers 353
11 Applications to symplectic topology 355
11.1 Gromov’s non-squeezing theorem 356
11.2 Nondegeneracy of the Hofer norm 364
References 377
index 392
Contents of Volume 2
Preface xiii
PART 3 LAG RANG I AN INTERSECTION FLOER HOMOLOGY
12 Floer homology on cotangent bundles 3
12.1 The action functional as a generating function 4
12.2 L2-gradient flow of the action functional 7
12.3 C° bounds of Floer trajectories 11
12.4 Floer-regular parameters 15
12.5 Floer homology of submanifold S c N 16
12.6 Lagrangian spectral invariants 23
12.7 Deformation of Floer equations 33
12.8 The wave front and the basic phase function 36
13 The off-shell framework of a Floer complex with bubbles 41
13.1 Lagrangian subspaces versus totally real subspaces 41
13.2 The bundle pair and its Maslov index 43
13.3 Maslov indices of polygonal maps 47
13.4 Novikov covering and Novikov ring 50
13.5 Action functional 53
13.6 The Maslov—Morse index 55
13.7 Anchored Lagrangian submanifolds 57
13.8 Abstract Floer complex and its homology 60
13.9 Floer chain modules 64
14 On-shell analysis of Floer moduli spaces 73
14.1 Exponential decay 73
14.2 Splitting ends of M(x,y; B) 82
IX
X
Contents of Volume 2
14.3 Broken cusp-trajectory moduli spaces 93
14.4 Chain map moduli space and energy estimates 97
15 Off-shell analysis of the Floer moduli space 109
15.1 Off-shell framework of smooth Floer moduli spaces 109
15.2 Off-shell description of the cusp-trajectory spaces 115
15.3 Index calculation 119
15.4 Orientation of the moduli space of disc instantons 122
15.5 Gluing of Floer moduli spaces 127
15.6 Coherent orientations of Floer moduli spaces 140
16 Floer homology of monotone Lagrangian submanifolds 150
16.1 Primary obstruction and holomorphic discs 150
16.2 Examples of monotone Lagrangian submanifolds 156
16.3 The one-point open Gromov—Witten invariant 161
16.4 The anomaly of the Floer boundary operator 166
16.5 Product structure; triangle product 176
17 Applications to symplectic topology 182
17.1 Nearby Lagrangian pairs: thick-thin dichotomy 183
17.2 Local Floer homology 188
17.3 Construction of the spectral sequence 194
17.4 Biran and Cieliebak’s theorem 202
17.5 Audin’s question for monotone Lagrangian submanifolds 208
17.6 Polterovich’s theorem on Ham(S2) 211
PART 4 HAMILTONIAN FIXED-POINT FLOER HOMOLOGY
18 The action functional and the Conley-Zehnder index 219
18.1 Free loop space and its 5 1 action 220
18.2 The free loop space of a symplectic manifold 221
18.3 Perturbed action functionals and their action spectrum 228
18.4 The Conley-Zehnder index of [z, w] 233
18.5 The Hamiltonian-perturbed Cauchy-Riemann equation 240
19 Hamiltonian Floer homology 244
19.1 Novikov Floer chains and the Novikov ring 244
19.2 Definition of the Floer boundary map 248
19.3 Definition of a Floer chain map 254
19.4 Construction of a chain homotopy map 256
19.5 The composition law of Floer chain maps 258
Contents of Volume 2 xi
19.6 Transversality 261
19.7 Time-reversal flow and duality 268
19.8 The Floer complex of a small Morse function 278
20 The pants product and quantum cohomology 281
20.1 The structure of a quantum cohomology ring 282
20.2 Hamiltonian fibrations with prescribed monodromy 289
20.3 The PSS map and its isomorphism property 299
20.4 Frobenius pairing and duality 312
21 Spectral invariants: construction 314
21.1 Energy estimates and Hofer’s geometry 315
21.2 The boundary depth of the Hamiltonian H 322
21.3 Definition of spectral invariants and their axioms 324
21.4 Proof of the triangle inequality 332
21.5 The spectrality axiom 337
21.6 Homotopy invariance 342
22 Spectral invariants: applications 348
22.1 The spectral norm of Hamiltonian diffeomorphisms 349
22.2 Hofer’s geodesics and periodic orbits 352
22.3 Spectral capacities and sharp energy-capacity inequality 366
22.4 Entov and Polterovich’s partial symplectic quasi-states 372
22.5 Entov and Polterovich’s Calabi quasimorphism 381
22.6 Back to topological Hamiltonian dynamics 397
22.7 Wild area-preserving homeomorphisms on D2 403
Appendix A The Weitzenbock formula for vector-valued forms 408
Appendix B The three-interval method of exponential estimates 412
Appendix C The Maslov index, the Conley-Zehnder index
and the index formula 417
References
Index
429
444
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| author | Oh, Yong-Geun 1961- |
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| discipline | Mathematik |
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| spelling | Oh, Yong-Geun 1961- Verfasser (DE-588)1079551433 aut Symplectic topology and Floer homology Yong-Geun Oh, IBS Center for Geometry and Physics, Pohang University of Science and Technology, Republic of Korea Cambridge Cambridge University Press 2015 txt rdacontent n rdamedia nc rdacarrier New mathematical monographs Floer-Homologie (DE-588)109388973X gnd rswk-swf Symplektische Geometrie (DE-588)4194232-2 gnd rswk-swf Floer-Homologie (DE-588)109388973X s Symplektische Geometrie (DE-588)4194232-2 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028791192&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028791192&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Oh, Yong-Geun 1961- Symplectic topology and Floer homology Floer-Homologie (DE-588)109388973X gnd Symplektische Geometrie (DE-588)4194232-2 gnd |
| subject_GND | (DE-588)109388973X (DE-588)4194232-2 |
| title | Symplectic topology and Floer homology |
| title_auth | Symplectic topology and Floer homology |
| title_exact_search | Symplectic topology and Floer homology |
| title_full | Symplectic topology and Floer homology Yong-Geun Oh, IBS Center for Geometry and Physics, Pohang University of Science and Technology, Republic of Korea |
| title_fullStr | Symplectic topology and Floer homology Yong-Geun Oh, IBS Center for Geometry and Physics, Pohang University of Science and Technology, Republic of Korea |
| title_full_unstemmed | Symplectic topology and Floer homology Yong-Geun Oh, IBS Center for Geometry and Physics, Pohang University of Science and Technology, Republic of Korea |
| title_short | Symplectic topology and Floer homology |
| title_sort | symplectic topology and floer homology |
| topic | Floer-Homologie (DE-588)109388973X gnd Symplektische Geometrie (DE-588)4194232-2 gnd |
| topic_facet | Floer-Homologie Symplektische Geometrie |
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