Stable Mappings and Their Singularities
 Responsibility
 by Martin Golubitsky, Victor Guillemin.
 Digital
 text file
 Imprint
 New York, NY : Springer US, 1973.
 Physical description
 1 online resource (209 pages)
 Series
 Graduate texts in mathematics ; 14.
Online
More options
Description
Creators/Contributors
 Author/Creator
 Golubitsky, Martin.
 Contributor
 Guillemin, Victor.
Contents/Summary
 Contents

 I: Preliminaries on Manifolds.
 1. Manifolds.
 2. Differentiable Mappings and Submanifolds.
 3. Tangent Spaces.
 4. Partitions of Unity.
 5. Vector Bundles.
 6. Integration of Vector Fields. II: Transversality.
 1. Sard's Theorem.
 2. Jet Bundles.
 3. The Whitney C? Topology.
 4. Transversality.
 5. The Whitney Embedding Theorem.
 6. Morse Theory.
 7. The Tubular Neighborhood Theorem. III: Stable Mappings.
 1. Stable and Infinitesimally Stable Mappings.
 2. Examples.
 3. Immersions with Normal Crossings.
 4. Submersions with Folds. IV: The Malgrange Preparation Theorem.
 1. The Weierstrass Preparation Theorem.
 2. The Malgrange Preparation Theorem.
 3. The Generalized Malgrange Preparation Theorem. V: Various Equivalent Notions of Stability.
 1. Another Formulation of Infinitesimal Stability.
 2. Stability Under Deformations.
 3. A Characterization of Trivial Deformations.
 4. Infinitesimal Stability => Stability.
 5. Local Transverse Stability.
 6. Transverse Stability.
 7. Summary. VI: Classification of Singularities, Part I: The ThomBoardman Invariants.
 1. The Sr Classification.
 2. The Whitney Theory for Generic Mappings between 2Manifolds.
 3. The Intrinsic Derivative.
 4. The Sr, s Singularities.
 5. The ThomBoardman Stratification.
 6. Stable Maps Are Not Dense. VII: Classification of Singularities, Part II: The Local Ring of a Singularity.
 1. Introduction.
 2. Finite Mappings.
 3. Contact Classes and Morin Singularities.
 4. Canonical Forms for Morin Singularities.
 5. Umbilics.
 6. Stable Mappings in Low Dimensions. A. Lie Groups. Symbol Index.
 (source: Nielsen Book Data)
 Publisher's summary

This book aims to present to first and second year graduate students a beautiful and relatively accessible field of mathematicsthe theory of singu larities of stable differentiable mappings. The study of stable singularities is based on the now classical theories of Hassler Whitney, who determined the generic singularities (or lack of them) of Rn ~ Rm (m ~ 2n  1) and R2 ~ R2, and Marston Morse, for mappings who studied these singularities for Rn ~ R. It was Rene Thorn who noticed (in the late '50's) that all of these results could be incorporated into one theory. The 1960 Bonn notes of Thom and Harold Levine (reprinted in [42]) gave the first general exposition of this theory. However, these notes preceded the work of Bernard Malgrange [23] on what is now known as the Malgrange Preparation Theoremwhich allows the relatively easy computation of normal forms of stable singularities as well as the proof of the main theorem in the subjectand the definitive work of John Mather. More recently, two survey articles have appeared, by Arnold [4] and Wall [53], which have done much to codify the new material; still there is no totally accessible description of this subject for the beginning student. We hope that these notes will partially fill this gap. In writing this manuscript, we have repeatedly cribbed from the sources mentioned abovein particular, the ThomLevine notes and the six basic papers by Mather.
(source: Nielsen Book Data)
Subjects
 Subjects
 Mathematics.
Bibliographic information
 Publication date
 1973
 Series
 Graduate Texts in Mathematics, 00725285 ; 14
 ISBN
 9781461579045 (electronic bk.)
 146157904X (electronic bk.)
 9780387900735
 038790073X
 DOI
 10.1007/9781461579045.