Stable Mappings and Their SingularitiesThis 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. |
Contents
Preliminaries on Manifolds | 1 |
Transversality | 30 |
Stable Mappings | 72 |
Copyright | |
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1:1 immersion a₁ assume B₁ canonical chart choose coordinates codim codimension compact compute coordinate functions coordinates x1 Corollary countable critical point curve defined Definition deformation denote dense Diff(X diffeomorphism differentiable dimension dimensional equations Exercise F₁ fiber finite Frechet given H₁ homeomorphism immersed submanifold induced infinitesimally stable isomorphism J¹(X k-deformation k-jet Lemma Let f Let G Lie group linear map local ring locally Malgrange Preparation Theorem map f measure zero metric module normal crossings Note open nbhd open set open subset parameter group polynomial prove ring S₁ S₁(f S₁(ƒ satisfying singularities smooth function smooth manifolds smooth mapping stable mappings submersion subspace Suppose T₁ tangent Thom Transversality Theorem topology Transversality Theorem trivial tubular nbhd U₁ V₁ vector bundle vector field vector space W₁ Whitney x₁ y₁