On the Structure of Automorphisms of Manifolds

Abstract

Thurston [16] proved that the group Diff$^{\infty}(M)$ of a smooth manifold $M$ is perfect, which implies the first homology group is trivial. If $M$ has a geometric structure, then the first homology of the group of automorphisms of $M$ preserving the geometric structure is not necessarily trivial. There are many results concerning this field. In this paper, we shall summarize the results of the first homology groups of automorphisms of manifolds with geometric structure.