[Grundlehren der mathematischen Wissenschaften] Singularities of Mappings Volume 357 (The Local Behaviour of Smooth and Complex Analytic Mappings) ||

"The first monograph on singularities of mappings for many years, this book provides an introduction to the subject and an account of recent developments concerning the local structure of complex analytic mappings. Part I of the book develops the now classical real C∞ and complex analytic theories jointly. Standard topics such as stability, deformation theory and finite determinacy, are covered in this part. In Part II of the book, the authors focus on the complex case. The treatment is centred around the idea of the "nearby stable object" associated to an unstable map-germ, which includes in particular the images and discriminants of stable perturbations of unstable singularities. This part includes recent research results, bringing the reader up to date on the topic. By focusing on singularities of mappings, rather than spaces, this book provides a necessary addition to the literature. Many examples and exercises, as well as appendices on background material, make it an invaluable guide for graduate students and a key reference for researchers. A number of graduate level courses on singularities of mappings could be based on the material it contains." -- Prové de l'editor Preface Contents 1 Introduction 1.1 Real or Complex? 1.2 Structure of the Book 1.3 The Nearby Stable Object 1.4 Exercises and Open Questions 1.5 Notation Part I Thom-Mather Theory: Right-Left Equivalence, Stability, Versal Unfoldings, Finite Determinacy 2 Manifolds and Smooth Mappings 2.1 Germs 2.2 Manifolds and Their Tangent Spaces Exercises for Sect.2.2 2.3 Inverse Mapping Theorem and Consequences Exercises for Sect.2.3 2.4 Submanifolds Exercises for Sect.2.4 2.5 Vector Fields and Flows Exercises for Sect.2.5 2.6 Transversality Exercises for Sect.2.6 2.7 Local Conical Structure 3 Left-Right Equivalence and Stability 3.1 Classification of Functions by Right Equivalence Exercises for Sect.3.1 3.2 Left-Right Equivalence and Stability 3.2.1 Right Equivalence and Left Equivalence Exercises for Sect.3.2 3.3 First Calculations Exercises for Sect.3.3 3.4 Multi-Germs 3.4.1 Notation Exercises for Sect.3.4 3.5 Infinitesimal Stability Implies Stability Exercises for Sect.3.5 3.6 Stability of Multi-Germs Exercises for Sect.3.6 4 Contact Equivalence 4.1 The Contact Tangent Space 4.2 Using TKef to Calculate TAef Exercises for Sect.4.2 4.3 Construction of Stable Germs as Unfoldings Exercises for Sect.4.3 4.4 Contact Equivalence Exercises for Sect.4.4 4.5 Geometric Criterion for Finite Ae-Codimension 4.5.1 Sheafification Exercises for Sect.4.5 4.6 Transversality 4.7 Thom–Boardman Singularities Exercises for Sect.4.7 5 Versal Unfoldings 5.1 Versality Exercises for Sect.5.1 5.2 Global Stability of C∞ Mappings 5.2.1 Stable Maps Are Not Always Dense 5.2.2 Mather's Nice Dimensions 5.3 Topological Stability 5.4 Bifurcation Sets Exercises for Sect.5.4 5.5 The Notion of Stable Perturbation of a Map-Germ 6 Finite Determinacy 6.1 Proof of the Finite Determinacy Theorem Exercises for Sect.6.1 6.2 Estimates for the Determinacy Degree 6.3 Determinacy and Unipotency 6.3.1 Unipotent Affine Algebraic Groups 6.3.2 Unipotent Groups of k-Jets of Diffeomorphisms 6.3.3 When Is a Closed Affine Space of Germs Contained in a G-Orbit? 6.3.4 Complexification and Determinacy Degrees 6.3.5 Notes 6.4 Complete Transversals Exercises for Sect.6.4 6.5 Notes and Further Developments 7 Classification of Stable Germs by Their Local Algebras 7.1 Stable Germs Are Classified by Their Local Algebras Exercises for Sect.7.1 7.2 Construction of Stable Germs as Unfoldings Exercises for Sect.7.2 7.3 The Isosingular Locus 7.3.1 Weighted Homogeneity and Local Quasihomogeneity 7.4 Quasihomogeneity and the Nice Dimensions 7.4.1 Multi-Germs 7.4.2 The Case n≥p 7.4.3 The Case n