Structure of Classical Diffeomorphism Groups

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...