On the structure of the homeomorphism and diffeomorphism groups fixing a point

Abstract

Let M be a manifold, p ∈ M and let H(M, p) be the identity component of the group of all compactly supported homeomorphisms of M fixing p. It is shown that H(M, p) is a perfect group. Next, we prove that the group H(R n , 0) is bounded. In contrast, in the C ∞ category the diffeomorphism group D ∞ (R n , 0), analogous to H(R n , 0), is neither perfect nor bounded. Finally, the boundedness and uniform perfectness of H(M, p) is studied.

Diffeomorphisms preserving Morse–Bott functions

“…However for singular foliations their groups of diffeomorphisms are less studied, e.g. [14], [15], [16], [17].…”