The Structure of Classical Diffeomorphism Groups

  • Augustin Banyaga

Part of the Mathematics and Its Applications book series (MAIA, volume 400)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Augustin Banyaga
    Pages 1-22
  3. Augustin Banyaga
    Pages 23-54
  4. Augustin Banyaga
    Pages 55-92
  5. Augustin Banyaga
    Pages 93-123
  6. Augustin Banyaga
    Pages 124-137
  7. Augustin Banyaga
    Pages 138-154
  8. Augustin Banyaga
    Pages 155-183
  9. Back Matter
    Pages 184-201

About this book

Introduction

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.

Keywords

DEX Erlang Morphism Volume automorphism boundary element method cohomology diffeomorphism foliation geometry group homology homomorphism mathematics techniques

Authors and affiliations

  • Augustin Banyaga
    • 1
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4757-6800-8
  • Copyright Information Springer Science+Business Media Dordrecht 1997
  • Publisher Name Springer, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4419-4774-1
  • Online ISBN 978-1-4757-6800-8
  • About this book