The structure of classical diffeomorphism groups. (English) Zbl 0874.58005

Mathematics and its Applications (Dordrecht). 400. Dordrecht: Kluwer Academic Publishers. xi, 197 p. (1997).
A detailed proof of Smale’s conjecture is given: Namely that the group of all diffeomorphisms, which have compact support and are isotopic to the identity through compactly supported isotopies, is a simple group. In the case of an \(n\)-torus this was shown by M. R. Herman [Publ. Math., Inst. Hautes Etud. Sci. 49, 5-233 (1979; Zbl 0448.58019)]. In W. P. Thurston [Bull. Am. Math. Soc. 80, 304-307 (1974; Zbl 0295.57014)] it was sketched how to deduce the general case. The details have been worked out by J. N. Mather [Bull. Am. Math. Soc. 77, 1111-1115 (1971; Zbl 0224.55022); Comment. Math. Helv. 49, 512-528 (1974; Zbl 0289.57014); ibid. Helv. 50, 33-40 (1975; Zbl 0299.58008)] and by A. Banyaga [Comment. Math. Helv. 53, 174-227 (1978; Zbl 0393.58007)]. Here a detailed account of “Thurston’s tricks” is given and they are applied to show that also for the following subgroups the commutator group is simple: That formed by the symplectomorphisms (on a symplectic manifold), that formed by the volume preserving diffeomorphisms (on a manifold with volume form), and that of contact diffeomorphisms (on a contact manifold). A special role in the proof is played by the flux homomorphisms, a homomorphism from the identity component of diffeomorphisms preserving a closed \(p\)-form into a certain quotient of the \((p-1)^{st}\) de-Rham cohomology of \(M\), and a hole chapter is devoted to it. Finally, it is shown that isomorphy of these automorphism groups implies isomorphy of the underlying structures.
This monograph is concerned only with the algebraic structure of diffeomorphism groups and not their “Lie-group” structure.
Reviewer: A.Kriegl (Wien)


58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
58-02 Research exposition (monographs, survey articles) pertaining to global analysis