Singularity Theory I Springer-Verlag Berlin Heidelberg GmbH V. 1. Arnold V. V. Goryunov O. V. Lyashko V. A.Vasil'ev

Singularity Theory 1

.~. ~ 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.

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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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Editor-in-Chief

R.V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute, ul. Gubkina 8,117966 Moscow, Institute for Scientific Information (VINITI), ul. Usievicha 20 a, 125219 Moscow, Russia; e-mail: [email protected]

Consulting Editor

V. I. Arnold, Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Russia, e-mail: [email protected]; CEREMADE, Universite Paris 9 • Dauphine, Place du Marechal de Lattre de Tassigny, 75775 Paris Cedex 16-e, France, e-mail: [email protected]

Authors

V. I. Arnold, Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Russia, e-mail: [email protected]; CEREMADE, Universite Paris 9 • Dauphine, Place du Marechal de Lattre de Tassigny, 75775 Paris Cedex 16-e, France, e-mail: [email protected] V. V. Goryunov, Moscow Aviation Institute, Volokolamskoe sh. 4, 125871 Moscow, Russia; Division of Pure , The University of Liverpool, Liverpool L69 3BX,UK, e-mail: [email protected] O. V. Lyashko, Airforce Engineering Academy, Leningradskij pro 40, 125167 Moscow, Russia; e-mail: [email protected] V. A. Vasil'ev, Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Russia; e-mail: [email protected]

Translator

A. Iacob, Mathematical Reviews, 416 Fourth Street, P.O. Box 8604, Ann Arbor, MI 48107, USA Singularities Local and Global Theory V.I. Arnol'd, V.A. Vasil'ev, V.V. Goryunov, G.V. Lyashko

Translated from the Russian by A. Iacob

Contents

Foreword 7

Chapter 1. Critical Points of Functions 10 § 1. Invariants of Critical Points 10 1.1. Degenerate and Nondegenerate Critical Points 10 1.2. Equivalence of Critical Points ...... 11 1.3. Stable Equivalence 12 1.4. The Local Algebra and the Multiplicity of a Singularity 13 1.5. Finite Determinacy of an Isolated Singularity 14 1.6. Actions on 15 1.7. Versal Deformations ofa Critical Point 16 1.8. Infinitesimal Versality ...... 17 1.9. The Modality ofa Critical Point 18 1.10. The Level Bifurcation Set 19 1.11. Truncated Versal Deformations and the Bifurcation Set ...... 20 §2. The Classification of Critical Points ...... 22 2.1. Normal Forms ...... 22 2.2. Classes of Low Modality 23 2.3. Singularities of Modality ~ 2 24 2.4. Simple Singularities and Klein Singularities 25 2.5. Resolution of Simple Singularities 26 2.6. Unimodal and Bimodal Singularities ...... 28 2.7. Adjacency of Singularities 30 2.8. Real Singularities 32 2 Contents

§ 3. Reduction to Normal Forms ...... 34 3.1. The Newton Diagram 35 3.2. Quasihomogeneous Functions and Filtrations ...... 36 3.3. The Multiplicity and the Generators of the Local Algebra of a Semi-Quasihomogeneous Function 38 3.4. Quasihomogeneous Maps 38 3.5. Quasihomogeneous DifTeomorphisms and Vector Fields 40 3.6. The Normal Form of a Semi-Quasihomogeneous Function. 42 3.7. The Normal Form of a Quasihomogeneous Function 43 3.8. The Newton Filtration 45 3.9. The Spectral Sequence 47 3.10. Theorems on Normal Forms for the Spectral Sequence 49 Chapter 2. Groups of Critical Points 50 § 1. The Picard-Lefschetz Theory 51 1.1. of the Nonsingular Level 51 1.2. The Classical Monodromy and the Variation Operator 53 1.3. The Monodromy of a Morse Singularity 54 1.4. The Monodromy Group of an Isolated Singularity ...... 56 1.5. Vanishing Cycles and Distinguished Bases 58 1.6. The Intersection Matrix of a Singularity 61 1.7. Stabilization of Singularities 63 1.8. Dynkin Diagrams 64 1.9. Transformations of a Basis and of its Dynkin Diagram 64 1.10. The Milnor Fibration over the Complement of the Level Bifurcation Set ...... 68 1.11. The Topological Type of a Singularity Along the jl-Constant Stratum 70 §2. Dynkin Diagrams and Monodromy Groups 72 2.1. Intersection Matrices of Singularities of Functions ofTwo Variables 72 2.2. The Intersection Matrix of a Direct Sum of Singularities 75 2.3. Pham Singularities 77 2.4. The Polar Curve and the Intersection Matrix 77 2.5. Modality and Quadratic Forms of Singularities 82 2.6. The Monodromy Group and the Intersection Form 84 2.7. The Monodromy Group in the Skew-Symmetric Case 87 §3. Complex Monodromy and Period Maps 88 3.1. The Cohomology Bundle and the Gauss-Manin Connection ...... 88 3.2. Sections ofthe Cohomology Bundle 89 3.3. The Vanishing Cohomology Bundle 90 3.4. The Period Map 91 3.5. The Residue Form 91 3.6. Trivializations of the Cohomology Bundle 92 Contents 3

3.7. The Classical Complex Monodromy 94 3.8. Differential Equations and Asymptotics of Integrals 95 3.9. Nondegenerate Period Maps 98 3.10. Stability of Period Maps 100 3.11. Period Maps and Intersection Forms 101 3.12. The Characteristic Polynomial and the Zeta Function of the 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 Asymptotic Hodge Filtration in the Fibres of the Cohomology Bundle 108 4.4. The Weight Filtration 108 4.5. The Asymptotic Mixed Hodge Structure 110 4.6. The Hodge Numbers and the Spectrum of a Singularity III 4.7. Computing the Spectrum 112 4.8. Semicontinuity of the Spectrum 114 4.9. The Spectrum and the Geometric Genus 115 4.1 O. The Mixed Hodge Structure and the Intersection Form 116 4.11. The Number of Singular Points of a Complex Projective Hypersurface 116 4.12. The Generalized Petrovskii-Oleinik Inequalities 118 §5. Simple Singularities 119 5.1. Reflection Groups 119 5.2. The Swallowtail of a Reflection Group 122 5.3. The Artin-Brieskorn 124 5.4. Convolution of Invariants of a Coxeter Group 125 5.5. Root Systems and Weyl Groups 127 5.6. Simple Singularities and Weyl Groups 129 5.7. Vector Fields to the Level Bifurcation Set 130 5.8. The Complement of the Function Bifurcation Set 132 5.9. Adjacency and Decomposition of Simple Singularities 132

5.10. Finite Subgroups of SU2 , Simple Singularities, and Weyl Groups 133 5.11. Parabolic Singularities 134 §6. Topology of Complements of Discriminants of Singularities 138 6.1. Complements of Discriminants and Braid Groups 138 6.2. The mod-2 Cohomology of Braid Groups 138 6.3. An Application: Superposition of Algebraic Functions 140 6.4. The Integer Cohomology of Braid Groups 140 6.5. The Cohomology of Braid Groups with Twisted Coefficients 141 6.6. Genus of Coverings Associated with an Algebraic Function, and Complexity of Algorithms for Computing Roots of Polynomials 142 4 Contents

6.7. The Cohomology of Brieskorn Braid Groups and Complements of the Discriminants of Singularities of the Series C and D 143 6.8. The Stable Cohomology of Complements of Level Bifurcation Sets 143 6.9. Characteristic Classes of Milnor Cohomology Bundles 146 6.10. Stable Irreducibility of Strata of Discriminants 147

Chapter 3. Basic Properties of Maps 147 § 1. Stable Maps and Maps of Finite Multiplicity ...... 148 1.1. The Left-Right Equivalence 148 1.2. Stability 149 1.3. Transversality 151 1.4. The Thorn-Boardman Classes 153 1.5. Infinitesimal Stability 155 1.6. The Groups ~ and j( 156 1.7. Normal Forms of Stable Genns 157 1.8. Examples 157 1.9. Nice and Semi-Nice Dimensions 160 1.10. Maps of Finite Multiplicity 161 1.11. The Number of Roots of a System of Equations 162 1.12. The Index of a Singular Point of a Real , and Polynomial Vector Fields 163 §2. Finite Determinacy of Map-Germs, and Their Versal Deformations 165 2.1. Tangent Spaces and Codimensions 166 2.2. Finite Determinacy 166 2.3. Versal Deformations ...... 167 2.4. Examples 168 2.5. Geometric Subgroups ...... 170 2.6. The Order of a Sufficient 174 2.7. Determinacy with Respect to Transformations of Finite 178 §3. The Topological Equivalence 179 3.1. The Topologically Stable Maps are Dense 179 3.2. Whitney Stratifications ...... 179 3.3. The Topological Classification of Smooth Map-Germs 181 3.4. Topological Invariants 182 3.5. Topological Triviality and Topological Versality of Deformations of Semi-Quasihomogeneous Maps 183

Chapter 4. The Global Theory of Singularities 185 § 1. Thorn Polynomials for Maps of Smooth Manifolds 186 1.1. Cycles of Singularities and Topological Invariants of Maps . 186 Contents 5

1.2. Thorn's Theorem on the Existence of Thorn Polynomials 187 1.3. Resolution of the Singularities of the Closures of the Thorn-Boardman Classes 188 1.4. Thorn Polynomials for Singularities of First Order 189 1.5. Survey of Results on Thorn Polynomials for Singularities of Higher Order 190 §2. Integer Characteristic Classes and Universal Complexes of Singularities 191 2.1. Examples: the Maslov Index and the First Pontryagin Class 192 2.2. The Universal Complex of Singularities of Smooth Functions 193 2.3. Cohomology of the Complexes of Ro-Invariant Singularities, and Invariants of Foliations 196 2.4. Computations in Complexes of Singularities of Functions. Geometric Consequences 197 2.5. Universal Complexes of Lagrangian and Legendrian Singularities 199 2.6. On Universal Complexes of General Maps of Manifolds 201 § 3. Multiple Points and Multisingularities 201 3.1. A Formula for Multiple Points ofImmersions, and Obstructions for Manifolds 201 3.2. Triple Points of Singular Surfaces 202 3.3. Multiple Points of Complex Maps 202 3.4. Self-Intersections of Lagrangian Manifolds 203 3.5. Complexes of Multisingularities 203 3.6. Multisingularities and Multiplication in the Cohomology of the Target Space of a Map 206 §4. Spaces of Functions with Critical Points of Mild Complexity 207

4.1. Functions with Singularities Simpler than A 3 •..•.•.....•.• 207 4.2. The Group of Curves Without Horizontal Inflexional 208 4.3. Homotopy Properties of the Complements of Unfurled Swallowtails 211 § 5. Elimination of Singularities and Solution of Differential Conditions 212 5.1. Cancellation of Whitney Umbrellas and Cusps. The Problem 212 5.2. The Smale-Hirsch Theorem 213 5.3. The w.h.e.- and h-Principles 213 5.4. The Gromov-Lees Theorem on Lagrangian Immersions 215 5.5. Elimination of Thorn-Boardman Singularities 215

5.6. The Space of Functions with no A 3 Singularities 216 §6. Tangential Singularities and Vanishing Inflexions 216 6.1. The Calculus of Tangential Singularities 216 6.2. Vanishing Inflexions: The Case of Plane Curves 217 6.3. Inflexions that Vanish at a Morse Singular Point 218 6 Contents

6.4. Integration with Respect to the Euler Characteristic, and its Applications 219

References 221

Author Index 239

Subject Index 242 Foreword

Napoleon condemned Laplace for the attempt to "introduce the spirit of infinitesimals in government" and removed him from the post of minister. In this two-volume survey,l an exposition is given of the foundations of the part of the analysis of infinitesimals that is necessary for the deliberate control ofdynamical systems, for their optimization, and for understanding the behavior of complex systems that depend on several parameters. The theory of singularities of smooth maps is an apparatus for the study of abrupt, jump-like phenomena - bifurcations, perestroikas (restructurings), ca• tastrophes, metamorphoses - which occur in systems depending on parameters when the parameters vary in a smooth manner. Although the applications to the theory ofdynamical systems do not exhaust by far all the potential capabilities of the theory of singularities (it also has applications in geometric and physical optics, hydrodynamics, quantum me• chanics, crystallography, chemistry, acoustics, sinergetics, the theory of radio• wave propagation, cosmology, , differential topology, and so forth), the fundamental role that the theory ofsingularities plays in the investiga• tion of bifurcations of stationary and periodic regimes justifies the inclusion of this two-volume book in the series "Dynamical Systems". In writing the survey the authors had in mind a student-reader, mathemati• cian or physicist, who wishes to learn the modern mathematical apparatus of local as an instrument for applied studies, or a specialist in the respective applied domain seeking for the needed mathematical tools and reference information. Accordingly, we replaced proofs by references to the sources where they can be found, focussing on the methods, ideas, and results, rather than the technical details of the proofs. In doing so we counted on a reader prepared to accept many details on trust, or prefering to reconstruct the proofs by himself. A thorough exposition in any of the formalized mathematical languages (be it ""Ie 3l5", "Ext-Tor", or "GO TO") would have required a many times greater volume. The first two chapters ofthis volume are devoted to one ofthe most advanced parts of the theory of singularities - the study of degeneracies of critical points offunctions. In Chapter 1 we acquaint the reader with the basic notions of the theory of singularities of smooth maps, give the initial segment of the classification of smooth functions and present the technique of reducing functions to normal forms. In Chapter 2 we consider topological and algebro-geometric aspects of the theory of critical points of functions. Here we discuss the basic concepts of the local Picard-Lefschetz theory, that is, the discipline of branching of cycles and

I The second part of the survey, "Singularity Theory II. Applications", will appear as Vol. 39 of the present series (Dynamical Systems VIII). 8 Foreword integrals depending on parameters. A detailed study is made of the main object of this theory - the bundle of vanishing cohomology (i.e., of branching integra• tion contours) connected with a critical point and, in particular, the base over which this bundle is defined, the complement of the discriminant of the singu• larity. We also consider the connection between simple singularities offunctions and the classification of simple Lie groups, reflection groups, and braid groups. Among the original results of this chapter let us mention the calculation of the cohomology groups with nonconstant coefficients ofthe complements ofdis• criminants of one-dimensional singularities and their application to the theory of algorithms, the description of the stable cohomology of the complements of discriminants of arbitrary singularities, theorems on stable irreducibility of the strata of a discriminant, the noncoincidence of the dimensions of the complex and real J1. = const strata of real singularities. In Chapter 3 we give an exposition of the general theory of equivalence of maps. Mathematical and physical problems arising in real situations lead to the investigation of the properties of maps with respect to a variety of equivalence relations. In analyzing a concrete equivalence relation one has to deal with a number ofstandard questions: Is the given map stable? Can one regard the map, even locally, as a polynomial, which would considerably simplify calculations? Does the map admit a versal deformation, i.e., can it be included in a family with a finite number ofparameters, which contains all small deformations ofthe map? How much simpler does the classification become when one passes from the rigid differentiable equivalence to the less demanding topological one? For many equivalence relations the answers to these questions look the same. The statements of the corresponding theorems and of sufficient conditions for their applicability constitute the main content of the third chapter. In the last, fourth chapter, we describe topological characteristics of singular sets of smooth maps: the cohomology classes dual to the sets of critical points and nonregular values; invariants of maps defined by these classes; the structure ofthe spaces ofmaps not having singularities ofone kind or another. Apparently for the first timc in the literature, we carry out the construction ofcharacteristic classes of foliations with the help of universal complexes of singularities and multisingularities, and also the computation of the of the space of functions with singularities no more complicated than x3 and of the topology of complements of open swallowtails. For the first time in monograph form, we discuss in this survey the results of S.V. Chmutov on the monodromy group of an isolated singularity in the skew• symmetric case, the theorems of a.v. Lyashko and P. Jaworski on the decom• position of simple and parabolic singularities, the estimates of the index of a polynomial vector field obtained by A.G. Khovanskii, and the results of E.I. Shustin and V.1. Arnol'd on the number of the flattening points that vanish under various degeneracies of algebraic hypersurfaces. The references within the volume are organized as follows. Ifthe reference lies within the same chapter, we indicate the number of the corresponding section Foreword 9 or subsection, as in the table of contents. Ifthe reference is to a different chapter, then we precede the number of the section or subsection by the number of the chapter. Chapters 1 and 2, with the exception of §2.6 and subsections 2.1.11, 2.5.11, were written by O.V. Lyashko, Chapter 3 and subsections 2.1.11, 2.5.11 by V.V. Goryunov, Chapter 4 (except for §4.4) and §2.6 by VA Vasil'ev, and §4.4 by V.l. Arnol'd. For the second volume of the survey V.I. Arnol'd and Y.A. Vasil'ev each contributed two chapters, and one chapter was written by V.V. Goryunov. The second volume gives a representation of a wide circle of problems that are currently being solved, and contains many new results.