Singularity Theory I

ISBN-13: 9783540637110 / Angielski / Miękka / 1998 / 245 str.

Vladimir I. Arnol'd; V. A. Vasil'ev; O. V. Lyashko

Opis

From the reviews: ..". My general impression is of a particularly nice book, with a well-balanced bibliography, recommended "
Mededelingen van Het Wiskundig Genootschap, 1995
..". The authors offer here an up to date guide to the topic and its main applications, including a number of new results. It is very convenient for the reader, a carefully prepared and extensive bibliography ... makes it easy to find the necessary details when needed. The books (EMS 6 and EMS 39) describe a lot of interesting topics. ... Both volumes are a very valuable addition to the library of any mathematician or physicist interested in modern mathematical analysis."
European Mathematical Society Newsletter, 1994

Spis treści

1. Critical Points of Functions.- 1. Invariants of Critical Points.- 1.1. Degenerate and Nondegenerate Critical Points.- 1.2. Equivalence of Critical Points.- 1.3. Stable Equivalence.- 1.4. The Local Algebra and the Multiplicity of a Singularity.- 1.5. Finite Determinacy of an Isolated Singularity.- 1.6. Lie Group Actions on Manifolds.- 1.7. Versal Deformations of a Critical Point.- 1.8. Infinitesimal Versality.- 1.9. The Modality of a Critical Point.- 1.10. The Level Bifurcation Set.- 1.11. Truncated Versal Deformations and the Function Bifurcation Set.- 2. The Classification of Critical Points.- 2.1. Normal Forms.- 2.2. Classes of Low Modality.- 2.3. Singularities of Modality ? 2.- 2.4. Simple Singularities and Klein Singularities.- 2.5. Resolution of Simple Singularities.- 2.6. Unimodal and Bimodal Singularities.- 2.7. Adjacency of Singularities.- 2.8. Real Singularities.- 3. Reduction to Normal Forms.- 3.1. The Newton Diagram.- 3.2. Quasihomogeneous Functions and Filtrations.- 3.3. The Multiplicity and the Generators of the Local Algebra of a Semi-Quasihomogeneous Function.- 3.4. Quasihomogeneous Maps.- 3.5. Quasihomogeneous Diffeomorphisms and Vector Fields.- 3.6. The Normal Form of a Semi-Quasihomogeneous Function.- 3.7. The Normal Form of a Quasihomogeneous Function.- 3.8. The Newton Filtration.- 3.9. The Spectral Sequence.- 3.10. Theorems on Normal Forms for the Spectral Sequence.- 2. Monodromy Groups of Critical Points.- 1. The Picard-Lefschetz Theory.- 1.1. Topology of the Nonsingular Level Manifold.- 1.2. The Classical Monodromy and the Variation Operator.- 1.3. The Monodromy of a Morse Singularity.- 1.4. The Monodromy Group of an Isolated Singularity.- 1.5. Vanishing Cycles and Distinguished Bases.- 1.6. The Intersection Matrix of a Singularity.- 1.7. Stabilization of Singularities.- 1.8. Dynkin Diagrams.- 1.9. Transformations of a Basis and of its Dynkin Diagram.- 1.10. The Milnor Fibration over the Complement of the Level Bifurcation Set.- 1.11. The Topological Type of a Singularity Along the ?-Constant Stratum.- 2. Dynkin Diagrams and Monodromy Groups.- 2.1. Intersection Matrices of Singularities of Functions of Two Variables.- 2.2. The Intersection Matrix of a Direct Sum of Singularities.- 2.3. Pham Singularities.- 2.4. The Polar Curve and the Intersection Matrix.- 2.5. Modality and Quadratic Forms of Singularities.- 2.6. The Monodromy Group and the Intersection Form.- 2.7. The Monodromy Group in the Skew-Symmetric Case.- 3. Complex Monodromy and Period Maps.- 3.1. The Cohomology Bundle and the Gauss-Manin Connection.- 3.2. Sections of the Cohomology Bundle.- 3.3. The Vanishing Cohomology Bundle.- 3.4. The Period Map.- 3.5. The Residue Form.- 3.6. Trivializations of the Cohomology Bundle.- 3.7. The Classical Complex Monodromy.- 3.8. Differential Equations and Asymptotics of Integrals.- 3.9. Nondegenerate Period Maps.- 3.10. Stability of Period Maps.- 3.11. Period Maps and Intersection Forms.- 3.12. The Characteristic Polynomial and the Zeta Function of the Monodromy Operator.- 4. The Mixed Hodge Structure in the Vanishing Cohomology.- 4.1. The Pure Hodge Structure.- 4.2. The Mixed Hodge Structure.- 4.3. The Asymptotic Hodge Filtration in the Fibres of the Cohomology Bundle.- 4.4. The Weight Filtration.- 4.5. The Asymptotic Mixed Hodge Structure.- 4.6. The Hodge Numbers and the Spectrum of a Singularity.- 4.7. Computing the Spectrum.- 4.8. Semicontinuity of the Spectrum.- 4.9. The Spectrum and the Geometric Genus.- 4.10. The Mixed Hodge Structure and the Intersection Form.- 4.11. The Number of Singular Points of a Complex Projective Hypersurface.- 4.12. The Generalized Petrovski?-Ole?nik Inequalities.- 5. Simple Singularities.- 5.1. Reflection Groups.- 5.2. The Swallowtail of a Reflection Group.- 5.3. The Artin-Brieskorn Braid Group.- 5.4. Convolution of Invariants of a Coxeter Group.- 5.5. Root Systems and Weyl Groups.- 5.6. Simple Singularities and Weyl Groups.- 5.7. Vector Fields Tangent to the Level Bifurcation Set.- 5.8. The Complement of the Function Bifurcation Set.- 5.9. Adjacency and Decomposition of Simple Singularities.- 5.10. Finite Subgroups of SU2, Simple Singularities, and Weyl Groups.- 5.11. Parabolic Singularities.- 6. Topology of Complements of Discriminants of Singularities.- 6.1. Complements of Discriminants and Braid Groups.- 6.2. The mod-2 Cohomology of Braid Groups.- 6.3. An Application: Superposition of Algebraic Functions.- 6.4. The Integer Cohomology of Braid Groups.- 6.5. The Cohomology of Braid Groups with Twisted Coefficients.- 6.6. Genus of Coverings Associated with an Algebraic Function, and Complexity of Algorithms for Computing Roots of Polynomials.- 6.7. The Cohomology of Brieskorn Braid Groups and Complements of the Discriminants of Singularities of the Series C and D.- 6.8. The Stable Cohomology of Complements of Level Bifurcation Sets.- 6.9. Characteristic Classes of Milnor Cohomology Bundles.- 6.10. Stable Irreducibility of Strata of Discriminants.- 3. Basic Properties of Maps.- 1. Stable Maps and Maps of Finite Multiplicity.- 1.1. The Left-Right Equivalence.- 1.2. Stability.- 1.3. Transversality.- 1.4. The Thom-Boardman Classes.- 1.5. Infinitesimal Stability.- 1.6. The Groups l and K.- 1.7. Normal Forms of Stable Germs.- 1.8. Examples.- 1.9. Nice and Semi-Nice Dimensions.- 1.10. Maps of Finite Multiplicity.- 1.11. The Number of Roots of a System of Equations.- 1.12. The Index of a Singular Point of a Real Germ, and Polynomial Vector Fields.- 2. Finite Determinacy of Map-Germs, and Their Versal Deformations.- 2.1. Tangent Spaces and Codimensions.- 2.2. Finite Determinacy.- 2.3. Versal Deformations.- 2.4. Examples.- 2.5. Geometric Subgroups.- 2.6. The Order of a Sufficient Jet.- 2.7. Determinacy with Respect to Transformations of Finite Smoothness.- 3. The Topological Equivalence.- 3.1. The Topologically Stable Maps are Dense.- 3.2. Whitney Stratifications.- 3.3. The Topological Classification of Smooth Map-Germs.- 3.4. Topological Invariants.- 3.5. Topological Triviality and Topological Versality of Deformations of Semi-Quasihomogeneous Maps.- 4. The Global Theory of Singularities.- 1. Thom Polynomials for Maps of Smooth Manifolds.- 1.1. Cycles of Singularities and Topological Invariants of Maps.- 1.2. Thom’s Theorem on the Existence of Thom Polynomials.- 1.3. Resolution of the Singularities of the Closures of the Thom-Boardman Classes.- 1.4. Thorn Polynomials for Singularities of First Order.- 1.5. Survey of Results on Thom Polynomials for Singularities of Higher Order.- 2. Integer Characteristic Classes and Universal Complexes of Singularities.- 2.1. Examples: the Maslov Index and the First Pontryagin Class.- 2.2. The Universal Complex of Singularities of Smooth Functions.- 2.3. Cohomology of the Complexes of R0-Invariant Singularities, and Invariants of Foliations.- 2.4. Computations in Complexes of Singularities of Functions. Geometric Consequences.- 2.5. Universal Complexes of Lagrangian and Legendrian Singularities.- 2.6. On Universal Complexes of General Maps of Manifolds.- 3. Multiple Points and Multisingularities.- 3.1. A Formula for Multiple Points of Immersions, and Embedding Obstructions for Manifolds.- 3.2. Triple Points of Singular Surfaces.- 3.3. Multiple Points of Complex Maps.- 3.4. Self-Intersections of Lagrangian Manifolds.- 3.5. Complexes of Multisingularities.- 3.6. Multisingularities and Multiplication in the Cohomology of the Target Space of a Map.- 4. Spaces of Functions with Critical Points of Mild Complexity.- 4.1. Functions with Singularities Simpler than A3.- 4.2. The Group of Curves Without Horizontal Inflexional Tangents.- 4.3. Homotopy Properties of the Complements of Unfurled Swallowtails.- 5. Elimination of Singularities and Solution of Differential Conditions.- 5.1. Cancellation of Whitney Umbrellas and Cusps. The Immersion Problem.- 5.2. The Smale-Hirsch Theorem.- 5.3. The w.h.e.- and h-Principles.- 5.4. The Gromov-Lees Theorem on Lagrangian Immersions.- 5.5. Elimination of Thom-Boardman Singularities.- 5.6. The Space of Functions with no A3 Singularities.- 6. Tangential Singularities and Vanishing Inflexions.- 6.1. The Calculus of Tangential Singularities.- 6.2. Vanishing Inflexions: The Case of Plane Curves.- 6.3. Inflexions that Vanish at a Morse Singular Point.- 6.4. Integration with Respect to the Euler Characteristic, and its Applications.- References.- Author Index.

Blurb

From the reviews of the first printing of this book, published as volume 6 of the Encyclopaedia of Mathematical Sciences: "... My general impression is of a particularly nice book, with a well-balanced bibliography, recommended!"
Medelingen van Het Wiskundig Genootschap, 1995

"... The authors offer here an up to date guide to the topic and its main applications, including a number of new results. It is very convenient for the reader, a carefully prepared and extensive bibliography ... makes it easy to find the necessary details when needed. The books (EMS 6 and EMS 39) describe a lot of interesting topics. ... Both volumes are a very valuable addition to the library of any mathematician or physicist interested in modern mathematical analysis."
European Mathematical Society Newsletter, 1994
"...The authors are recognized experts in their fields and so are ideal choices to write such a survey. ...The text of the book is liberally sprinkled with illustrative examples and so the style is not heavy going or turgid... The bibliography is very good and extremely large ..."

IMS Bulletin, 1995