On the structure of automorphisms of manifolds. (English) Zbl 0978.58003

Mladenov, I. M. (ed.) et al., Proceedings of the international conference on geometry, integrability and quantization, Varna, Bulgaria, September 1-10, 1999. Sofia: Coral Press Scientific Publishing. 7-16 (2000).
Let \(M\) be a connected closed smooth manifold. Let \(\text{Diff}^{\infty}(M)\) denote the group of \(C^{\infty}\)-diffeomorphisms of \(M\) which are isotopic to the identity. W. Thurston [Bull. Am. Math. Soc. 80, 304-307 (1974; Zbl 0295.57014)] proved that \(\text{Diff}^{\infty}(M)\) is perfect which implies the first homology group is trivial. There are many analogous results on the group of automorphisms of a manifold \(M\) which preserve a geometric structure on \(M\) such as volume structure, symplectic structure, submanifold structure, foliated structure, \(G\)-manifold structure. In those cases the first homology groups are not necessarily trivial.
The purpose of this paper is to try to summarize the results of the first homology groups of the automorphisms stated above. Many interesting results concerning those fields are found in [A. Banyaga, Comment. Math. Helv. 53, 174-227 (1978; Zbl 0393.58007)].
For the entire collection see [Zbl 0940.00039].


58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)