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.

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Banyaga, Augustin

Springer Science & Business Media

Mathematics and its applications ;, v. 400, Mathematics and its applications (Kluwer Academic Publishers) ;, v. 400., Dordrecht, Boston, Netherlands, 1997

Mathematics and its applications (Kluwer Academic Publishers), Dordrecht, 2010

Mathematics and its applications (Dordrecht), Dordrecht, cop. 1997

Mathematics and Its Applications, 400, Boston, MA, 1997

United States, United States of America

Springer Nature, New York, NY, 2013

1997, PT, 1997

Dec 08, 2010

1, 2010

Bookmarks added

lg2838232

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Source title: The Structure of Classical Diffeomorphism Groups (Mathematics and Its Applications)

Includes bibliographical references (p. 184-195) and index.

Table of Contents
Preface
1. Diffeomorphism Groups: A First Glance
1.1. The group Diff^r(M)
1.2. The smooth structure of Diff^r(M)
1.3. Basic examples and classical diffeomorphism groups
1.3.5. Actions of Lie groups
1.3.6. Flows
1.3.7. Classical diffeomorphism groups and corresponding Lie algebras
1.4. Some properties of Lie algebras of vector fields
1.5. Splitting of the de Rham complex
2. The Simplicity of Diffeomorphism Groups
2.1. From perfectness to simplicity
Some definitions from group theory [174], [79]
Thurston's tricks
The simplicial set Bar{G}
A Kan complex
2.2. Epstein's theorem
End of proof of Epsteins' theorem
2.3. Herman's theorem (74]
Sketch of the proof of Herman-Sergeraert theorem
2.4. Foliations and diffeomorphism groups
Haefliger structures and their classifying spaces
2.5. Equivariant diffeomorphisms
Automorphisms Group of G-principal Bundles
Automorphisms of a trivial bundle
3. The Geometry of the Flux
3.1. The flux homomorphism
The subgroup Γ_w
Fathi-Visetti and Ismagilov constructions
Prequantization construction of the flux
Question
Interpretation of the Subgroup
3.2. The transgression of the flux
An extension of the group Diff_c^∞(M)_0
The generalized Liouville form
Cohomology and homology of diffeomorphism groups
The transgression homomorphism
3.3 The flux and gauge groups
The class of a gauge transformation
Examples
3.4. More cohomology classes related to the flux
Ismagilov construction
The Action Functional Construction
A More General Setting for the Action
Proof of proposition 3.4.5.
4. Symplectic Diffeomorphism
4.1. The Weinstein chart
Remark 4.1.3
4.2. A first glance at the kernel of the flux and new invariants
Remark 4.2.3
The geometry of the Hofer metric
The homomorphisms R and μ
Remark 4.2.8
Computation of R on commutators
An different construction of R in case Ω is exact
The homomorphism μ
4.3. Statement of the main results
4.4. Symplectic of the torus T^{2n}
Proof of theorem 4.3.1 for M = T^{2n}
4.5. The symplectic deformation lemma
Isotopies of symplectic embedding
End of proofs of the main results of this chapter
5. Volume-Preserving Diffeomorphism Groups
5.1. Statement of the main results
Volume-preserving deformation lemma 5.1.4.
5.2. Proof of Thurston's fragmentation lemma
Complements to the proof of the fragmentation lemma
5.3. Proof of the volume-preserving deformation lemma
Remark 5.3.2
Continuation of the proof of the volume preserving deformation lemma
PROOF OF LEMMA 5.3.3.
6. Contact Diffeomorphisms
6.1 Contact Geometry Preliminaries [105]
DARBOUX' THEOREM.
A contact invariant built-in the definition of contact diffeomorphisms
6.2. The Lychagin chart
Legendre distributions.
6.3. Epstein's Axioms hold for contact diffcomorphisms
Remark 6.3.3
Proof of lemma 6.3.1
CONTACT FRAGMENTATION LEMMA.
6.4. The transverse flux
The structure of the kernels of S and s
6.5. The group of strictly contact diffeomorphisms
7. Isomorphisms between Diffeomorphism Groups
7.1. Statement of the main results
7.2. Pursell-Shanks and Omori's theorems
7.3. Takeus' theorem aud its generalizations
7.4. The General Theory
Existence of proper F-invariant closed subset
7.5. The Contact Case
7.6. The symplectic and volume preserving cases
7.7. The Measure preserving homeomorphisms
7.8. Miscellaneous problems
Bibliography
1-11
12-28
29-44
45-61
62-78
79-96
97-113
114-130
131-147
148-165
166-184
185-191
Index

<p><p>the Book Introduces And Explains Most Of The Main Techniques And Ideas In The Study Of The Structure Of Diffeomorphism Groups. A Quite Complete Proof Of Thurston's Theorem On The Simplicity Of Some Diffeomorphism Groups Is Given. The Method Of The Proof Is Generalized To Symplectic And Volume-preserving Diffeomorphisms. The Mather-thurston Theory Relating Foliations With Diffeomorphism Groups Is Outlined. A Central Role Is Played By The Flux Homomorphism. Various Cohomology Classes Connected With The Flux Are Defined On The Group Of Diffeomorphisms. The Main Results On The Structure Of Diffeomorphism Groups Are Applied To Showing That Classical Structures Are Determined By Their Automorphism Groups, A Contribution To The Erlanger Program Of Klein. <br> Audience&#58; Graduate Students And Researchers In Mathematics And Physics.</p> <h3>booknews</h3> <p>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.</p>

2020-11-12