Suppose that $\{U_i\}_{i\in \N}$ is a pairwise disjoint, locally finite family of open sets of $M^o$. Put $U=\bigcup_iU_i$. Let $\H_{[U]}(M)$ (resp. $\H_{[U]}^{\p}(M)$) denote the group of all homeomorphisms from $\hm$ (resp. $\hdm$) supported in $U$ such that for the decomposition $h=h_1h_2\ldots$ resulting from the partition $U=\bigcup_iU_i$ one has $h_i\in\H_{U_i}(M)$ for all $i$.

Let $\overline{B_i}\s U_i$, $i\in \N$, and let the pair $(B_i,U_i)$ be such as in Lemma 3.1 or Corol. 3.2. Then any element $h\in\H_{[B]}(M)$ (where $B=\bigcup_iB_i$) is expressed as $h=\tilde h\bar h$, where $\tilde h,\bar h\in\H_{[U]}(M)$. Moreover, we can arrange so that if $h\in\H_{[B]}^{\p}(M)$ then $\tilde h,\bar h\in\H_{[U]}^{\p}(M)$.

In fact, we can glue together $S(h_i)$ and $\phi_i$ obtained for particular $U_i$.

*Proof of Theorem 1.1 for $n>1$ or $\pm=\emptyset$*. For $M$ compact it follows from Corol. 2.4, Lemma 3.1, Corol. 3.2(1) and, for $\pm \neq\emptyset$ and $\dim M=1$, from the fact that $\H([0,1])$ is perfect. The proof of the latter fact will follow from the proof of Theorem 1.2 below. Suppose now that $M$ admits a compact exhaustion. If $h\in\H_c(M)$ then there are $j\in\N$ and an isotopy $h_t$ such that $h_0=\operatorname{id}$, $h_1=h$, and $\operatorname{supp}(h_t)\s M_j$ for all $t$. In view of Corol. 2.2 it follows that $h|_{M_j}$ can be 