Commutators of diffeomorphisms of a manifold with boundary
Annales Polonici Mathematici
A well known theorem of Herman-Thurston states that the identity component of the group of diffeomorphisms of a boundaryless manifold is perfect and simple. We generalize this result to manifolds with boundary. Remarks on C r -diffeomorphisms are included. 1. Introduction. The aim of this paper is to extend a well known theorem of M. Herman and W. Thurston to manifolds with boundary. Let us fix the notation. Let M be an n-dimensional smooth manifold, and Diff r (M ) 0 denote the totality of C r
... the totality of C r -diffeomorphisms of M which are isotopic to the identity through a compactly supported isotopy. It is clear that (as a result of local contractibility) Diff r (M ) 0 is the identity component in the C r topology iff M is compact. Theorem 1 (Herman, Thurston, Mather). If M is a boundaryless manifold , and 1 ≤ r ≤ ∞, r = n + 1, then Diff r (M ) 0 is a simple group. D. B. A. Epstein  demonstrated for a large class of transitive groups of homeomorphisms that the perfectness yields the simplicity (the converse statement is trivial). By appealing to a difficult K.A.M. theory Herman  proved that Diff ∞ (T n ) 0 is perfect, T n being the n-dimensional torus. Next, Thurston announced in  (for the proof, see  ) that the result of Herman can be extended to an arbitrary manifold by making use of Kan simplices. Finally, J. N. Mather in  showed the assertion for any positive integer r not equal to n + 1 by a completely different argument. The case of manifolds with boundary has been considered by A. Masson  who extended the results of F. Sergeraert  . By making use of a different method than the two above they proved 1991 Mathematics Subject Classification: Primary 57R50; Secondary 58D05.