On the homeomorphism groups of manifolds and their universal coverings

Abstract

Let $\mathcal H_c(M)$ stand for the path connected identity component of the group of all compactly supported homeomorphisms of a manifold $M$. It is shown that $\mathcal H_c(M)$ is perfect and simple under mild assumptions on $M$. Next, conjugation-invariant norms on $\H_c(M)$ are considered and the boundedness of $\mathcal H_c(M)$ is investigated. Finally, the structure of the universal covering group of $\mathcal H_c(M)$ is studied.