are no longer true in the smooth (or even Lipschitz) category. See, e.g., Remark 3.4 and Prop. 6.3.

Fragmentation property and isotopy extension theorem
====================================================

The results of this paper depend essentially on the deformation properties for the spaces of imbeddings obtained by Edwards and Kirby in [@ed-ki]. See also Siebenmann [@sie]. Let us recall basic notions and facts from [@ed-ki].

Given a subset $S\s M$, by $\H_S(M)$ we denote the path connected identity component of the subgroup of all elements of $\H(M)$ with compact support contained in $S$. By a *ball* (resp. *half-ball*) $B$ we mean rel. compact open ball (resp. half-ball with $\p_B=B\cap\p_M$) embedded in $M$ with its closure. By $\B$ we denote the family of all balls and half-balls in $M$.

Using the Alexander trick, we have that $\H(\R^n)$ coincides with the group of all compactly supported homeomorphisms of $\R^n$. In fact, if $\operatorname{supp}(g)$ is compact, we define an isotopy $g_{t} :
\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$, $t \in I$, from the identity to $g$, by

$$g_{t}(x)= \left\{
\begin{array}{lcl}
tg\left( \frac{1}{t}x \right)& for & t>0\\
x&for&t=0.
\end{array} \right.$$

In particular, for every ball $B$ in $M$ the group $\H_B(M)$ consists of all homeomorphisms compactly supported in $B$. Observe that 