in $B'$, and $B_k\r p$ as $k\r\infty$. Choose a homeomorphism $\phi\in\H_U(M)$ such that $\varphi(B_{k-1})=B_k$ for $k=1,2,\ldots$. Here we use the fact that $\H_U(M)$ acts transitively on the family of balls in $B'$, c.f. [@hir].

Next we define a homomorphism $S:\H_B(M)\r\H_U(M)$ by the formula $$S(h)=\phi^kh\phi^{-k}\quad\hbox{on}\,B_k,\ k=0,1,\ldots$$ and $S(h)=\operatorname{id}$ outside $\bigcup_{k=0}^{\infty}B_k$. It is clear that $h=[S(h),\phi]$, as required.

The above reasoning appeared in Mather’s paper [@Mat71]. Actually Mather proved also the acyclicity of $\H(\R^n)$. It is easily seen that [@Mat71] and Lemma 3.1 are no longer true for $C^1$ homeomorphisms. However, Tsuboi gave an excellent improvement of this reasoning and adapted it for $C^r$-diffeomorphisms with small $r$, see [@tsu1].

Assume that either

1.  $\pm\neq\emptyset$ with $\dim M\geq 2$, and $B, U\s
    M$ are such that $B$ is a half-ball, and $U$ is open with $\overline{B}\s U$; or

2.  $M=N\t\R$, where $N$ is a manifold, and $B=N\t I$, $U=N\t J$ where $I, J\s\R$ are open intervals with the closure of $I$ contained in $J$.

Then there are $\phi\in\H_U(M)$ and a homomorphism $S:\H_B(M)\r \H_U(M)$ such that $h=[S(h),\phi]$ for all $h\in\H_B(M)$. Moreover, in the case (1), if $h\in\H_B(M)$ satisfies $h=\operatorname{id}$ on $\pm$ then $S(h)=\operatorname{id}$ on $\pm$.

The proof is analogous to that of Lemma 3.1.

