The Structure of Classical Diffeomorphism Groups / Edition 1

by Augustin Banyaga

Overview

In the 60's, the work of Anderson, Chernavski, Kirby and Edwards showed that the group of homeomorphisms of a smooth manifold which are isotopic to the identity is a simple group. This led Smale to conjecture that the group Diff'" (M)o of cr diffeomorphisms, r ~ 1, of a smooth manifold M, with compact supports, and isotopic to the identity through compactly supported isotopies, is a simple group as well. In this monograph, we give a fairly detailed proof that DifF(M)o is a simple group. This theorem was proved by Herman in the case M is the torus rn in 1971, as a consequence of the Nash-Moser-Sergeraert implicit function theorem. Thurston showed in 1974 how Herman's result on rn implies the general theorem for any smooth manifold M. The key idea was to vision an isotopy in Diff'"(M) as a foliation on M x [0, 1]. In fact he discovered a deep connection between the local homology of the group of diffeomorphisms and the homology of the Haefliger classifying space for foliations. Thurston's paper [180] contains just a brief sketch of the proof. The details have been worked out by Mather [120], [124], [125], and the author [12]. This circle of ideas that we call the "Thurston tricks" is discussed in chapter 2. It explains how in certain groups of diffeomorphisms, perfectness leads to simplicity. In connection with these ideas, we discuss Epstein's theory [52], which we apply to contact diffeomorphisms in chapter 6.

Product Details

ISBN-13:	9780792344759
Publisher:	Springer US
Publication date:	03/31/1997
Series:	Mathematics and Its Applications , #400
Edition description:	1997
Pages:	202
Product dimensions:	6.10(w) x 9.45(h) x 0.02(d)

Table of Contents

1. Diffeomorphism Groups: A First Glance.- 2. The Simplicity of Diffeomorphism Groups.- 3. The Geometry of the Flux.- 4. Symplectic Diffeomorphisms.- 5. Volume Preserving Diffeomorphisms.- 6. Contact Diffeomorphisms.- 7. Isomorphisms Between Diffeomorphism Groups.

Editorial Reviews

An introduction of the main techniques and ideas in the study of the structure of diffeomorphism groups, giving a quite complete proof of Thurston's theorem on the simplicity of some diffeormorphism. Banyaga (mathematics, Pennsylvania State U.) supplies the method of the proof generalized to symplectic and volume-preserving diffeomorphisms, outlines the Mather-Thurston theory relating foliations with diffeomorphism groups, defines various cohomology classes connected with the flux, and contributes to the Erlanger Program of Klein by applying the main results on the structure of diffeomorphism groups to showing that classical structures are determined by their automorphism groups. Annotation c. by Book News, Inc., Portland, Or.

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