Symplectic Topology and Floer Homology, Volume 2: Floer Homology and its Applications

471 Pages·2015·5.413 MB·English

by  Yong-Geun Oh

SymplecticTopologyandFloerHomology
Volume2
Publishedintwovolumes,thisisthefirstbooktoprovideathoroughandsystematic
explanationofsymplectictopology,andtheanalyticaldetailsandtechniquesusedin
applyingthemachineryarisingfromFloertheoryasawhole.
Volume1coversthebasicmaterialsofHamiltoniandynamicsandsymplecticgeom-
etryandtheanalyticfoundationsofGromov’spseudoholomorphiccurvetheory.One
novelaspectofthistreatmentistheuniformtreatmentofbothclosedandopencases
and a complete proof of the boundary regularity theorem of weak solutions of pseu-
doholomorphic curves with totally real boundary conditions. Volume 2 provides a
comprehensive introduction to both Hamiltonian Floer theory and Lagrangian Floer
theory,includingmanyexamplesoftheirapplicationstovariousproblemsinsymplectic
topology.
SymplecticTopologyandFloerHomologyisacomprehensiveresourcesuitablefor
expertsandnewcomersalike.
Yong-GeunOhisDirectoroftheIBSCenterforGeometryandPhysicsandisProfes-
sorintheDepartmentofMathematicsatPOSTECH(PohangUniversityofScienceand
Technology)inKorea.HewasalsoProfessorintheDepartmentofMathematicsatthe
UniversityofWisconsin–Madison.HeisamemberoftheKMS,theAMS,theKorean
NationalAcademyofSciences,andtheinauguralclassofAMSFellows.In2012he
receivedtheKyung-AhmPrizeofScienceinKorea.
NEWMATHEMATICALMONOGRAPHS
EditorialBoard
Béla Bollobás, William Fulton, Anatole Katok, Frances Kirwan, Peter Sarnak, Barry
Simon,BurtTotaro
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisitwww.cambridge.org/mathematics.
1. M.CabanesandM.EnguehardRepresentationTheoryofFiniteReductive
Groups
2. J.B.GarnettandD.E.MarshallHarmonicMeasure
3. P.CohnFreeIdealRingsandLocalizationinGeneralRings
4. E.BombieriandW.GublerHeightsinDiophantineGeometry
5. Y.J.IoninandM.S.ShrikhandeCombinatoricsofSymmetricDesigns
6. S.Berhanu,P.D.CordaroandJ.HounieAnIntroductiontoInvolutive
Structures
7. A.ShlapentokhHilbert’sTenthProblem
8. G.MichlerTheoryofFiniteSimpleGroupsI
9. A.BakerandG.WüstholzLogarithmicFormsandDiophantineGeometry
10. P.KronheimerandT.MrowkaMonopolesandThree-Manifolds
11. B.Bekka,P.delaHarpeandA.ValetteKazhdan’sProperty(T)
12. J.NeisendorferAlgebraicMethodsinUnstableHomotopyTheory
13. M.GrandisDirectedAlgebraicTopology
14. G.MichlerTheoryofFiniteSimpleGroupsII
15. R.SchertzComplexMultiplication
16. S.BlochLecturesonAlgebraicCycles(2ndEdition)
17. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups
18. T.DownarowiczEntropyinDynamicalSystems
19. C.SimpsonHomotopyTheoryofHigherCategories
20. E.FricainandJ.MashreghiTheTheoryofH(b)SpacesI
21. E.FricainandJ.MashreghiTheTheoryofH(b)SpacesII
22. J.Goubault-LarrecqNon-HausdorffTopologyandDomainTheory
23. J.S´niatyckiDifferentialGeometryofSingularSpacesandReductionof
Symmetry
24. E.RiehlCategoricalHomotopyTheory
25. B.MunsonandI.Volic´CubicalHomotopyTheory
26. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups(2ndEdition)
27. J.Heinonen,P.Koskela,N.ShanmugalingamandJ.T.TysonSobolevSpaces
onMetricMeasureSpaces
28. Y.-G.OhSymplecticTopologyandFloerHomologyI
29. Y.-G.OhSymplecticTopologyandFloerHomologyII
Symplectic Topology and Floer Homology
Volume 2: Floer Homology and its Applications
YONG-GEUN OH
IBSCenterforGeometryandPhysics,PohangUniversityofScience
andTechnology,RepublicofKorea
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107109674
(cid:2)c Yong-GeunOh2015
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2015
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
ISBN–2VolumeSet978-1-107-53568-8Hardback
ISBN–Volume1978-1-107-07245-9Hardback
ISBN–Volume2978-1-107-10967-4Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
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Contents of Volume 2
ContentsofVolume1 pageix
Preface xiii
PART 3 LAGRANGIAN INTERSECTION FLOER
HOMOLOGY
12 Floerhomologyoncotangentbundles 3
12.1 Theactionfunctionalasageneratingfunction 4
12.2 L2-gradientflowoftheactionfunctional 7
12.3 C0boundsofFloertrajectories 11
12.4 Floer-regularparameters 15
12.5 FloerhomologyofsubmanifoldS ⊂ N 16
12.6 Lagrangianspectralinvariants 23
12.7 DeformationofFloerequations 33
12.8 Thewavefrontandthebasicphasefunction 36
13 Theoff-shellframeworkofaFloercomplexwithbubbles 41
13.1 Lagrangiansubspacesversustotallyrealsubspaces 41
13.2 ThebundlepairanditsMaslovindex 43
13.3 Maslovindicesofpolygonalmaps 47
13.4 NovikovcoveringandNovikovring 50
13.5 Actionfunctional 53
13.6 TheMaslov–Morseindex 55
13.7 AnchoredLagrangiansubmanifolds 57
13.8 AbstractFloercomplexanditshomology 60
13.9 Floerchainmodules 64
14 On-shellanalysisofFloermodulispaces 73
14.1 Exponentialdecay 73
14.2 SplittingendsofM(x,y;B) 82
v
vi ContentsofVolume2
14.3 Brokencusp-trajectorymodulispaces 93
14.4 Chainmapmodulispaceandenergyestimates 97
15 Off-shellanalysisoftheFloermodulispace 109
15.1 Off-shellframeworkofsmoothFloermodulispaces 109
15.2 Off-shelldescriptionofthecusp-trajectoryspaces 115
15.3 Indexcalculation 119
15.4 Orientationofthemodulispaceofdiscinstantons 122
15.5 GluingofFloermodulispaces 127
15.6 CoherentorientationsofFloermodulispaces 140
16 FloerhomologyofmonotoneLagrangiansubmanifolds 150
16.1 Primaryobstructionandholomorphicdiscs 150
16.2 ExamplesofmonotoneLagrangiansubmanifolds 156
16.3 Theone-pointopenGromov–Witteninvariant 161
16.4 TheanomalyoftheFloerboundaryoperator 166
16.5 Productstructure;triangleproduct 176
17 Applicationstosymplectictopology 182
17.1 NearbyLagrangianpairs:thick–thindichotomy 183
17.2 LocalFloerhomology 188
17.3 Constructionofthespectralsequence 194
17.4 BiranandCieliebak’stheorem 202
17.5 Audin’squestionformonotoneLagrangiansubmanifolds 208
17.6 Polterovich’stheoremonHam(S2) 211
PART 4 HAMILTONIAN FIXED-POINT FLOER
HOMOLOGY
18 TheactionfunctionalandtheConley–Zehnderindex 219
18.1 FreeloopspaceanditsS1action 220
18.2 Thefreeloopspaceofasymplecticmanifold 221
18.3 Perturbedactionfunctionalsandtheiractionspectrum 228
18.4 TheConley–Zehnderindexof[z,w] 233
18.5 TheHamiltonian-perturbedCauchy–Riemannequation 240
19 HamiltonianFloerhomology 244
19.1 NovikovFloerchainsandtheNovikovring 244
19.2 DefinitionoftheFloerboundarymap 248
19.3 DefinitionofaFloerchainmap 254
19.4 Constructionofachainhomotopymap 256
19.5 ThecompositionlawofFloerchainmaps 258
ContentsofVolume2 vii
19.6 Transversality 261
19.7 Time-reversalflowandduality 268
19.8 TheFloercomplexofasmallMorsefunction 278
20 Thepantsproductandquantumcohomology 281
20.1 Thestructureofaquantumcohomologyring 282
20.2 Hamiltonianfibrationswithprescribedmonodromy 289
20.3 ThePSSmapanditsisomorphismproperty 299
20.4 Frobeniuspairingandduality 312
21 Spectralinvariants:construction 314
21.1 EnergyestimatesandHofer’sgeometry 315
21.2 TheboundarydepthoftheHamiltonianH 322
21.3 Definitionofspectralinvariantsandtheiraxioms 324
21.4 Proofofthetriangleinequality 332
21.5 Thespectralityaxiom 337
21.6 Homotopyinvariance 342
22 Spectralinvariants:applications 348
22.1 ThespectralnormofHamiltoniandiffeomorphisms 349
22.2 Hofer’sgeodesicsandperiodicorbits 352
22.3 Spectralcapacitiesandsharpenergy–capacityinequality 366
22.4 EntovandPolterovich’spartialsymplecticquasi-states 372
22.5 EntovandPolterovich’sCalabiquasimorphism 381
22.6 BacktotopologicalHamiltoniandynamics 397
22.7 Wildarea-preservinghomeomorphismsonD2 403
AppendixA TheWeitzenböckformulaforvector-valuedforms 408
AppendixB Thethree-intervalmethodofexponentialestimates 412
AppendixC TheMaslovindex,theConley–Zehnderindex
andtheindexformula 417
References 429
Index 444

Contents of Volume 1
Preface xiii
PART 1 HAMILTONIAN DYNAMICS AND
SYMPLECTIC GEOMETRY
1 TheleastactionprincipleandHamiltonianmechanics 3
1.1 TheLagrangianactionfunctionalanditsfirstvariation 3
1.2 Hamilton’sactionprinciple 7
1.3 TheLegendretransform 8
1.4 ClassicalPoissonbrackets 18
2 SymplecticmanifoldsandHamilton’sequation 21
2.1 Thecotangentbundle 21
2.2 SymplecticformsandDarboux’theorem 24
2.3 TheHamiltoniandiffeomorphismgroup 37
2.4 Banyaga’stheoremandthefluxhomomorphism 45
2.5 Calabihomomorphismsonopenmanifolds 52
3 Lagrangiansubmanifolds 60
3.1 Theconormalbundles 60
3.2 Symplecticlinearalgebra 62
3.3 TheDarboux–Weinsteintheorem 71
3.4 ExactLagrangiansubmanifolds 74
3.5 ClassicaldeformationsofLagrangiansubmanifolds 77
3.6 ExactLagrangianisotopy=Hamiltonianisotopy 82
3.7 ConstructionofLagrangiansubmanifolds 87
3.8 Thecanonicalrelationandthenaturalboundarycondition 96
3.9 GeneratingfunctionsandViterboinvariants 99
ix