the Alexander trick is no longer true in the smooth category.

Let us formulate the fragmentation property in the following stronger way.

Let $\U$ be an open covering of $M$. A subgroup $G\leq \H(M)$ is *locally continuously factorizable* if for any finite subcovering $(U_i)_{i=1}^d$ of $\B$, there exist a neighborhood $\P$ of $\operatorname{id}\in G$ and continuous mappings $\sigma_i:\P\r G$, $i=1\ld d$, such that for all $f\in
 \P$ one has $$f=\sigma_1(f)\ldots \sigma_d(f),\quad \operatorname{supp}(\sigma_i(f))\s U_i, \forall i.$$

Throughout for a topological group $G$ by $\mathcal{P}G$ we will denote the totality of paths $\gamma:I\r G$ with $\gamma(0)=e$ (where $I=[0,1]$). Observe that Def. 2.1 can also be formulated for $\P
G$ rather than $G$, where $G\leq\H(M)$.

From now on $M$ is a metrizable topological manifold. If $U$ is a subset of $M$, a *proper imbedding* of $U$ into $M$ is an imbedding $h: U
\rightarrow M$ such that $h^{-1}(\partial M)=U \cap \partial M$. An *isotopy* of $U$ into $M$ is a family of imbeddings $h_{t}: U \rightarrow M$, $t \in I$, such that the map $h: U
\times I \rightarrow M$ defined by $h(x,t)=h_{t}(x)$ is continuous. An isotopy is *proper* if each imbedding in it is proper. Now let $C$ and $U$ be subsets 