is bounded, where $\operatorname{frag}_M$ is the fragmentation norm on $M$ with respect to homeomorphisms. In particular, $\H_c(\R^n)$ is bounded.

Also we have the following boundedness theorem.

Let $\pm$ be compact. If the group $\H_c(M)$ is bounded then $\H(\pm)$ is bounded also. Moreover, if the group $\H_c(M^o)$ is bounded then so is the group $\hdm$.

Observe that $\H_c(M^o)$ is bounded if $M^o$ is portable (sect. 6). In [@ry6] the second-named author proved that $\H(M^o)$ is bounded provided so is $\H_c(M^o)$.

The last part of the paper is devoted to the structure of the universal coverings of some homeomorphism groups. Let $\H_c(M)^{\sim}$ denote the universal covering group of $\H_c(M)$.

Let $n=\dim M\geq 2$ or $\pm=\emptyset$. The group $\H_c(M)^{\sim}$ is perfect. Moreover, the groups $\H_c(\R^n)^{\sim}$ and $\H_c(\R^n_+)^{\sim}$ are acyclic, where $\R^n_+=\{x\in\R^n: x_n\geq 0\}$ is the half-space.

The proof will be given in section 7. We will also study the problem of boundedness.

Let $\operatorname{frag}^{iso}_M$ be the isotopy fragmentation norm on the universal covering group $\H_c(M)^{\sim}$ (c.f. sect.7). Suppose that $\dim M\geq 2$ or $\pm=\emptyset$. Then $\H_c(M)^{\sim}$ is bounded if and only if $\operatorname{frag}^{iso}_{M}$ is bounded.

We emphasize that many facts presented in this paper are specific for the topological category, that is there 