Author | Golubitsky, Martin. author |
---|---|

Title | Stable Mappings and Their Singularities [electronic resource] / by Martin Golubitsky, Victor Guillemin |

Imprint | New York, NY : Springer US, 1973 |

Connect to | http://dx.doi.org/10.1007/978-1-4615-7904-5 |

Descript | 209 p. online resource |

SUMMARY

This book aims to present to first and second year graduate students a beautiful and relatively accessible field of mathematics-the theory of singuยญ 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

CONTENT

I: Preliminaries on Manifolds -- ยง1. Manifolds -- ยง2. Differentiable Mappings and Submanifolds -- ยง3. Tangent Spaces -- ยง4. Partitions of Unity -- ยง5. Vector Bundles -- ยง6. Integration of Vector Fields -- II: Transversality -- ยง1. Sardโ{128}{153}s Theorem -- ยง2. Jet Bundles -- ยง3. The Whitney C? Topology -- ยง4. Transversality -- ยง5. The Whitney Embedding Theorem -- ยง6. Morse Theory -- ยง7. The Tubular Neighborhood Theorem -- III: Stable Mappings -- ยง1. Stable and Infinitesimally Stable Mappings -- ยง2. Examples -- ยง3. Immersions with Normal Crossings -- ยง4. Submersions with Folds -- IV: The Malgrange Preparation Theorem -- ยง1. The Weierstrass Preparation Theorem -- ยง2. The Malgrange Preparation Theorem -- ยง3. The Generalized Malgrange Preparation Theorem -- V: Various Equivalent Notions of Stability -- ยง1. Another Formulation of Infinitesimal Stability -- ยง2. Stability Under Deformations -- ยง3. A Characterization of Trivial Deformations -- ยง4. Infinitesimal Stability => Stability -- ยง5. Local Transverse Stability -- ยง6. Transverse Stability -- ยง7. Summary -- VI: Classification of Singularities, Part I: The Thom-Boardman Invariants -- ยง1. The Sr Classification -- ยง2. The Whitney Theory for Generic Mappings between 2-Manifolds -- ยง3. The Intrinsic Derivative -- ยง4. The Sr,s Singularities -- ยง5. The Thom-Boardman Stratification -- ยง6. Stable Maps Are Not Dense -- VII: Classification of Singularities, Part II: The Local Ring of a Singularity -- ยง1. Introduction -- ยง2. Finite Mappings -- ยง3. Contact Classes and Morin Singularities -- ยง4. Canonical Forms for Morin Singularities -- ยง5. Umbilics -- ยง6. Stable Mappings in Low Dimensions -- ยงA. Lie Groups -- Symbol Index

Mathematics
Global analysis (Mathematics)
Manifolds (Mathematics)
Differential geometry
Complex manifolds
Mathematics
Differential Geometry
Global Analysis and Analysis on Manifolds
Manifolds and Cell Complexes (incl. Diff.Topology)