is supported by some $U_{i}$. Moreover, if $h_t|_{\pm}=\operatorname{id}$ for all $t$, then $h_{j,t}|_{\pm}=\operatorname{id}$ for all $j$ and $t$. The same is true for homeomorphisms instead of isotopies.

Another important consequence of Theorem 2.2 is the following Isotopy Extension Theorem.

[@ed-ki] Let $f_t$ be an isotopy in $\H(M)$ and let $C\s M$ be a compact set. Then for any open neighborhood $U$ of the track of $C$ by $f_t$ given by $\bigcup_{t\in [0,1]}f_t(C)$ there is an isotopy $g_t$ in $\H_c(M)$ such that $g_t=f_t$ on $C$ and $\operatorname{supp}(g_t)\s
U$.

Perfectness of $\H_c(M)$ and $\H^{\p}_c(M)$
===========================================

The goal of this section is to give the proof of Theorem 1.1. We begin with the following fact, with a straightforward proof, which plays a basic role in studies on homeomorphism groups.

[@Mat71](Basic lemma) Let $B\s M$ be a ball and $U\s M$ be an open subset such that $\overline{B}\s U$. 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 $h\in\H_B(M)$.

First choose a larger ball $B'$ such that $\overline{B}\s
B'\s\overline{B'}\s U$. Next, fix $p\in\p_{B'}$ and set $B_0=B$. There exists a sequence of balls $(B_k)_{k=1}^{\infty}$ such that $\operatorname{cl}(B_k)\s B'$ for all $k$, where the family $(B_k)_{k=0}^{\infty}$ is pairwise disjoint, locally finite 