The Structure of Classical Diffeomorphism Groups 9780792344759, 0792344758

Table of contents :
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