Stable Mappings and Their Singularities

Martin Golubitsky, Victor Guillemin (auth.)9780387900728, 0387900721, 038790073X

This book aims to present to first and second year graduate students a beautiful and relatively accessible field of mathematics-the 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 Theorem-which allows the relatively easy computation of normal forms of stable singularities as well as the proof of the main theorem in the subject-and 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 above-in particular, the Thom-Levine notes and the six basic papers by Mather.

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Table of contents :
Front Matter….Pages i-xi
Preliminaries on Manifolds….Pages 1-29
Transversality….Pages 30-71
Stable Mappings….Pages 72-90
The Malgrange Preparation Theorem….Pages 91-110
Various Equivalent Notions of Stability….Pages 111-142
Classification of Singularities. Part I: The Thom-Boardman Invariants….Pages 143-164
Classification of Singularities. Part II: The Local Ring of a Singularity….Pages 165-193
Back Matter….Pages 194-209