Singularity Theory I

252 Pages·1998·19.39 MB·English

by  V. I. Arnold, V. V. Goryunov, O. V. Lyashko, V. A. Vasil’ev (auth.)

SingularityTheory I
Springer-Verlag Berlin Heidelberg GmbH
V. 1. Arnold V. V. Goryunov
O. V. Lyashko V. A.Vasil'ev
Singularity Theory 1
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Springer
Consulting Editors of the Series:
A.A. Agrachev, A.A.Gonchar, E.F. Mishchenko,
N.M. Ostianu, V. P. Sakharova, A.B. Zhishchenko
Title of the Russian edition:
Itogi nauki i tekhniki, Sovremennye problemy matematiki,
Fundamental'nye napravleniya, VoI. 6, Dinamicheskie sistemy 6
Publisher VINITI, Moscow 1988
Second Printing 1998 ofthe First Edition 1993, which was original1y
published as Dynamical Systems VI,
Volume 6 of the Encydopaedia of Mathematical SCÎences.
Die Dcut5che Bibliothek -Clp·EinbeitsaufDabme
Sinplarity lbro'1/ by v. 1. Amal'd. ... -Berlin ; Heid.elberg ; New
York; Barcelonl ; BudapeSl ; HODpollg ; London ; Mailand. ; Paris;
SaDla Oara; Siogapur ; To~o : Springer
1. -1. ed.., 2. prinling. -1998
(Eocyclopacd.ia of DlIlbc:lIlllicll SCkDCeI ; \tiI. 6)
ISBN 978-3-540-63711-0 ISBN 978-3-642-58009-3 (eBook)
DOI 10.1007/978-3-642-58009-3
Mathematics Subject Classification (1991):
Primary 58C27, Secondary 14805, 14E15, 32S05, 32SXX, 58C28
ISBN 978-3-540-63711-0
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List of Editors,Authors and Translators
Editor-in-Chief
R.V.Gamkrelidze,RussianAcademyofSciences,SteklovMathematical Institute,
ul.Gubkina8,117966Moscow,Institutefor Scientific Information(VINITI),
ul.Usievicha 20 a,125219Moscow,Russia;e-mail:[email protected]
ConsultingEditor
V.I.Arnold,SteklovMathematicalInstitute,ul.Gubkina8,117966Moscow,
Russia,e-mail:[email protected];CEREMADE,Universite Paris9
Dauphine,PlaceduMarechalde Lattrede Tassigny,75775 ParisCedex16-e,
France,e-mail:[email protected]
Authors
V.I.Arnold,SteklovMathematical Institute,ul.Gubkina8,117966Moscow,
Russia,e-mail:[email protected];CEREMADE,UniversiteParis9
Dauphine,Placedu MarechaldeLattrede Tassigny,75775 ParisCedex16-e,
France,e-mail:[email protected]
V.V.Goryunov,MoscowAviation Institute,Volokolamskoesh.4,125871Moscow,
Russia;DivisionofPureMathematics,The UniversityofLiverpool,
Liverpool L693BX,UK,e-mail:[email protected]
O.V. Lyashko,AirforceEngineeringAcademy,Leningradskij pro40,
125167Moscow,Russia;e-mail:[email protected]
V.A.Vasil'ev,SteklovMathematicalInstitute,ul.Gubkina8,117966Moscow,
Russia;e-mail:[email protected]
Translator
A. Iacob,MathematicalReviews,416 FourthStreet,P.O.Box8604,AnnArbor,
MI 48107,USA
Singularities
Local and Global Theory
V.I. Arnol'd, V.A. Vasil'ev,
V.V. Goryunov, G.V. Lyashko
Translatedfrom the Russian
by A. Iacob
Contents
Foreword 7
Chapter 1. Critical PointsofFunctions 10
§1. Invariants ofCritical Points 10
1.1. Degenerateand NondegenerateCritical Points 10
1.2. EquivalenceofCritical Points ................... 11
1.3. Stable Equivalence 12
1.4. The LocalAlgebraand the Multiplicity ofa Singularity 13
1.5. FiniteDeterminacy ofan Isolated Singularity 14
1.6. LieGroupActions on Manifolds 15
1.7. Versal DeformationsofaCritical Point 16
1.8. Infinitesimal Versality ................................. 17
1.9. The ModalityofaCritical Point 18
1.10. The Level BifurcationSet 19
1.11. Truncated Versal Deformationsand the Function
BifurcationSet .................................. ..... 20
§2. TheClassification ofCritical Points .......................... 22
2.1. Normal Forms ....................................... 22
2.2. ClassesofLow Modality 23
2.3. SingularitiesofModality ~ 2 24
2.4. SimpleSingularitiesand Klein Singularities 25
2.5. Resolution ofSimpleSingularities 26
2.6. Unimodal and BimodalSingularities ................ 28
2.7. Adjacency ofSingularities 30
2.8. Real Singularities 32
2 Contents
§3. Reduction to Normal Forms ................................ 34
3.1. The Newton Diagram 35
3.2. Quasihomogeneous Functionsand Filtrations ........ 36
3.3. The Multiplicityand theGeneratorsofthe Local Algebra
ofa Semi-Quasihomogeneous Function 38
3.4. Quasihomogeneous Maps 38
3.5. QuasihomogeneousDifTeomorphismsand Vector Fields 40
3.6. The Normal Formofa Semi-Quasihomogeneous Function. 42
3.7. The Normal Formofa Quasihomogeneous Function 43
3.8. The Newton Filtration 45
3.9. The SpectralSequence 47
3.10. Theoremson Normal Formsfor the SpectralSequence 49
Chapter 2. Monodromy Groups ofCritical Points 50
§1. The Picard-LefschetzTheory 51
1.1. Topology ofthe Nonsingular Level Manifold 51
1.2. TheClassical Monodromyand the Variation Operator 53
1.3. The Monodromy ofa MorseSingularity 54
1.4. The MonodromyGroupofan Isolated Singularity ........ 56
1.5. VanishingCyclesand Distinguished Bases 58
1.6. The Intersection Matrix ofa Singularity 61
1.7. StabilizationofSingularities 63
1.8. Dynkin Diagrams 64
1.9. Transformationsofa Basisand ofits Dynkin Diagram 64
1.10. The Milnor Fibration overtheComplementofthe Level
Bifurcation Set .............................. 68
1.11. TheTopologicalTypeofa Singularity Along the jl-Constant
Stratum 70
§2. Dynkin Diagramsand MonodromyGroups 72
2.1. Intersection MatricesofSingularitiesofFunctionsofTwo
Variables 72
2.2. TheIntersection Matrix ofa Direct Sum ofSingularities 75
2.3. Pham Singularities 77
2.4. The PolarCurveand the Intersection Matrix 77
2.5. Modalityand Quadratic FormsofSingularities 82
2.6. The MonodromyGroupand the Intersection Form 84
2.7. The MonodromyGroupin theSkew-SymmetricCase 87
§3. Complex Monodromyand Period Maps 88
3.1. TheCohomology Bundleand theGauss-Manin
Connection .......................................... 88
3.2. SectionsoftheCohomology Bundle 89
3.3. TheVanishingCohomology Bundle 90
3.4. The Period Map 91
3.5. The Residue Form 91
3.6. TrivializationsoftheCohomology Bundle 92
Contents 3
3.7. The ClassicalComplex Monodromy 94
3.8. Differential Equations and Asymptotics ofIntegrals 95
3.9. Nondegenerate Period Maps 98
3.10. Stability ofPeriod Maps 100
3.11. Period Maps and Intersection Forms 101
3.12. TheCharacteristic Polynomial and the Zeta Function ofthe
Monodromy Operator 102
§4. The Mixed Hodge Structure in the Vanishing Cohomology 105
4.1. The Pure Hodge Structure '" .......................... 105
4.2. The Mixed Hodge Structure 106
4.3. The AsymptoticHodge Filtration in the Fibres ofthe
Cohomology Bundle 108
4.4. The Weight Filtration 108
4.5. The Asymptotic Mixed HodgeStructure 110
4.6. The Hodge Numbers and the Spectrum ofa Singularity III
4.7. Computing the Spectrum 112
4.8. Semicontinuity ofthe Spectrum 114
4.9. The Spectrum and the GeometricGenus 115
4.1O. The Mixed Hodge Structureand the Intersection Form 116
4.11. The Number ofSingular Points ofa Complex Projective
Hypersurface 116
4.12. The Generalized Petrovskii-Oleinik Inequalities 118
§5. SimpleSingularities 119
5.1. Reflection Groups 119
5.2. The Swallowtail ofa Reflection Group 122
5.3. The Artin-Brieskorn Braid Group 124
5.4. Convolution ofInvariants ofa CoxeterGroup 125
5.5. Root Systems and WeylGroups 127
5.6. SimpleSingularities and Weyl Groups 129
5.7. Vector FieldsTangent to the Level Bifurcation Set 130
5.8. The Complement ofthe Function Bifurcation Set 132
5.9. Adjacency and Decomposition ofSimpleSingularities 132
5.10. FiniteSubgroups ofSU2, Simple Singularities, and Weyl
Groups 133
5.11. Parabolic Singularities 134
§6. Topology ofComplementsofDiscriminants ofSingularities 138
6.1. Complements ofDiscriminants and Braid Groups 138
6.2. The mod-2 Cohomology ofBraid Groups 138
6.3. An Application: Superposition ofAlgebraic Functions 140
6.4. The IntegerCohomology ofBraid Groups 140
6.5. The Cohomology ofBraid Groups withTwisted
Coefficients 141
6.6. Genus ofCoverings Associated with an Algebraic Function,
and ComplexityofAlgorithmsfor ComputingRoots of
Polynomials 142
4 Contents
6.7. TheCohomology ofBrieskorn Braid Groupsand
Complements ofthe DiscriminantsofSingularitiesofthe
Series Cand D 143
6.8. The StableCohomology ofComplementsofLevel
BifurcationSets 143
6.9. CharacteristicClassesofMilnorCohomology Bundles 146
6.10. Stable Irreducibility ofStrata ofDiscriminants 147
Chapter 3. Basic PropertiesofMaps 147
§1. Stable Mapsand MapsofFinite Multiplicity ........... 148
1.1. The Left-Right Equivalence 148
1.2. Stability 149
1.3. Transversality 151
1.4. TheThorn-BoardmanClasses 153
1.5. Infinitesimal Stability 155
1.6. TheGroups~and j( 156
1.7. Normal Forms ofStableGenns 157
1.8. Examples 157
1.9. Niceand Semi-Nice Dimensions 160
1.10. MapsofFinite Multiplicity 161
1.11. The NumberofRoots ofa System ofEquations 162
1.12. The Index ofa Singular Pointofa RealGerm,and
Polynomial Vector Fields 163
§2. Finite Determinacy ofMap-Germs, and TheirVersal Deformations 165
2.1. Tangent Spacesand Codimensions 166
2.2. Finite Determinacy 166
2.3. Versal Deformations ............. 167
2.4. Examples 168
2.5. GeometricSubgroups ................................. 170
2.6. The Orderofa SufficientJet 174
2.7. Determinacy with Respect to TransformationsofFinite
Smoothness 178
§3. TheTopological Equivalence 179
3.1. TheTopologically Stable Mapsare Dense 179
3.2. Whitney Stratifications ................................ 179
3.3. TheTopological Classification ofSmooth Map-Germs 181
3.4. Topological Invariants 182
3.5. TopologicalTriviality and Topological Versality of
DeformationsofSemi-Quasihomogeneous Maps 183
Chapter4. TheGlobalTheory ofSingularities 185
§1. Thorn Polynomialsfor Maps ofSmooth Manifolds 186
1.1. Cycles ofSingularitiesand Topological InvariantsofMaps . 186
Contents 5
1.2. Thorn'sTheorem on the ExistenceofThorn Polynomials 187
1.3. Resolution ofthe Singularities oftheClosures ofthe
Thorn-Boardman Classes 188
1.4. Thorn Polynomialsfor Singularities ofFirst Order 189
1.5. Survey ofResults onThorn Polynomialsfor Singularities
ofHigherOrder 190
§2. IntegerCharacteristicClasses and Universal Complexesof
Singularities 191
2.1. Examples: the Maslov Indexand the First Pontryagin Class 192
2.2. The Universal Complex ofSingularities ofSmooth Functions 193
2.3. Cohomology oftheComplexesofRo-Invariant Singularities,
and Invariants ofFoliations 196
2.4. Computations in ComplexesofSingularitiesofFunctions.
GeometricConsequences 197
2.5. Universal Complexes ofLagrangianand Legendrian
Singularities 199
2.6. On Universal Complexes ofGeneral MapsofManifolds 201
§3. Multiple Pointsand Multisingularities 201
3.1. AFormula for Multiple Points ofImmersions, and
EmbeddingObstructionsfor Manifolds 201
3.2. Triple Points ofSingularSurfaces 202
3.3. Multiple Points ofComplex Maps 202
3.4. Self-Intersections ofLagrangian Manifolds 203
3.5. ComplexesofMultisingularities 203
3.6. Multisingularitiesand Multiplication in theCohomologyof
the Target Spaceofa Map 206
§4. Spaces ofFunctions with Critical Points ofMild Complexity 207
4.1. Functions with SingularitiesSimplerthan A3 •..•.•.....•.• 207
4.2. The Group ofCurves Without Horizontal Inflexional
Tangents 208
4.3. Homotopy Properties oftheComplementsofUnfurled
Swallowtails 211
§5. Elimination ofSingularitiesand Solution ofDifferential
Conditions 212
5.1. Cancellation ofWhitney Umbrellas and Cusps. The
Immersion Problem 212
5.2. The Smale-Hirsch Theorem 213
5.3. The w.h.e.- and h-Principles 213
5.4. The Gromov-LeesTheorem on Lagrangian Immersions 215
5.5. Elimination ofThorn-Boardman Singularities 215
5.6. The SpaceofFunctionswith no A Singularities 216
3
§6. Tangential Singularitiesand Vanishing Inflexions 216
6.1. TheCalculusofTangential Singularities 216
6.2. VanishingInflexions: TheCaseofPlaneCurves 217
6.3. Inflexions that Vanish at a Morse Singular Point 218