On the perfectness of C^{∞,s}-diffeomorphism groups on a foliated manifold

Abstract

The notion of \(C^{r,s}\) and \(C^{\infty,s}\)-diffeomorphisms is introduced. It is shown that the identity component of the group of leaf preserving \(C^{\infty,s}\)-diffeomorphisms with compact supports is perfect. This result is a modification of the Mather and Epstein perfectness theorem.

Keywords

• group of \(C^{\infty}\)-diffeomorphisms
• perfectness
• commutator
• foliation