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abstract: 'Let $\H_c(M)$ stand for the path connected identity component of the group of all compactly supported homeomorphisms of a manifold $M$. It is shown that $\H_c(M)$ is perfect and simple under mild assumptions on $M$. Next, conjugation-invariant norms on $\H_c(M)$ are considered and the boundedness of $\H_c(M)$ and its subgroups is investigated. Finally, the structure of the universal covering group of $\H_c(M)$ is studied.'
address: 'Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland'
author:
- 'Agnieszka Kowalik, Tomasz Rybicki'
date: 'March 18, 2011'
title: On the homeomorphism groups of manifolds and their universal coverings 
---

Introduction
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Let $M$ be a topological metrizable manifold of dimension $n\geq
1$, possibly with boundary, and let $\H(M)$ (resp. $\H_c(M)$) be the path connected identity component of the group of all (resp. compactly supported) homeomorphisms of a manifold $M$ endowed with the compact-open topology. In this paper we will deal with algebraic properties of the group $\H_c(M)$ and of its universal covering.

Recall that a group $G$ is called *perfect* if it is equal to its own commutator subgroup $[G,G]$. That is, $H_1(G)=0$. The following basic fact is probably well-known but we have not 