found it explicitly proven in the literature.

Assume that either $M$ is compact (possibly with boundary), or $M$ admits a compact exhaustion, i.e. there is a sequence of compact submanifolds with boundary $(M_i)_{i=1}^{\infty}$ with $\dim M_i=\dim M=n$ such that $M_1\s
M^o_2\s M_2\s\ldots$ and $M=\bigcup_{i=1}^{\infty}M_i$. Then the group $\H_c(M)$ is perfect.

The proof of the perfectness is a consequence of Mather’s paper [@Mat71] combined with Edwards and Kirby [@ed-ki], Corollary 1.3. In the case $n=1$ and $M$ with boundary the proof requires an additional argument. See section 3. A special case of Theorem 1.1 was already proved by Fisher [@fis] (see also Anderson [@An]). Observe that McDuff in [@MD] proved that $\H(M)$ is perfect provided $M$ is the interior of a compact manifold with boundary. There exist some generalizations of Theorem 1.1 (see, e.g., Fukui and Imanishi [@fu-im], and Rybicki [@ry3]).

If $M$ is a smooth manifold then Theorem 1.1 has its smooth analogue. Let $\D(M)$ be the identity component of the group of all compactly supported $C^{\infty}$-diffeomorphisms of $M$. Thurston proved that $\D(M)$ is perfect and simple (see [@Thu74], [@ban2]). Also Mather in [@Mat74] proved the same in the class of $C^r$-diffeomorphisms unless $r=\dim M+1$. Analogous results for classical groups 