of diffeomorphisms are also known ([@ban1], [@ban2], [@HR], [@ry6]).

In the case of a manifold with boundary $M$ we denote by $M^o$ the interior of $M$, and by $\pm$ the boundary of $M$. We will consider the following groups: $$\H_c(M^o)\leq\H_c^{\p}(M)\leq\H_c(M)\leq\H(M^o).$$ Here $h\in\H_c^{\p}(M)$ if there is a compactly supported isotopy $h_t$ connecting $h_0=\operatorname{id}$ with $h_1=h$ such that $h_t=\operatorname{id}$ on $\pm$ for all $t$. Moreover, $\hm$ identifies with a subgroup of $\H(M^o)$ by restricting elements of $\hm$ to $M^o$.

If the boundary $\pm$ is compact then $\hdm$ is a perfect group.

Concerning the simplicity of $\H_c(M)$ we have the following

Let $M$ be connected and satisfy the hypothesis of Theorem 1.1. Then $M$ is boundaryless (i.e. $\pm=\emptyset$) if and only if $\H_c(M)$ is simple.

The proof will be given in section 4 together with further comments on the simplicity by using some ideas of Ling [@li].

Conjugation-invariant norms related to homeomorphism groups on $M$ are considered in section 5. Recall that a group is *bounded* if every conjugation-invariant norm is bounded on it. Following an argument from [@BIP] we will prove in section 6 the following

Under the assumption of Theorem 1.1 on $M$, $\H_c(M)$ is bounded if and only if $\operatorname{frag}_{M}$ 