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Mathematics and Its Applications

The Structure of Classical Diffeomorphism Groups

Authors: Banyaga, Augustin

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About this book

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.

Table of contents (7 chapters)

Table of contents (7 chapters)

Buy this book

eBook $149.00
price for USA in USD (gross)
  • ISBN 978-1-4757-6800-8
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $199.99
price for USA in USD
  • ISBN 978-0-7923-4475-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $199.99
price for USA in USD
  • ISBN 978-1-4419-4774-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
The Structure of Classical Diffeomorphism Groups
Authors
Series Title
Mathematics and Its Applications
Series Volume
400
Copyright
1997
Publisher
Springer US
Copyright Holder
Springer Science+Business Media Dordrecht
eBook ISBN
978-1-4757-6800-8
DOI
10.1007/978-1-4757-6800-8
Hardcover ISBN
978-0-7923-4475-9
Softcover ISBN
978-1-4419-4774-1
Edition Number
1
Number of Pages
XII, 202
Topics