of $M$ with $C\subseteq
U$. By $I(U,C;M)$ we denote the space of proper imbeddings of $U$ into $M$ which equal the identity on $C$, endowed with the compact-open topology.

Suppose $X$ is a space with subsets $A$ and $B$. A *deformation of A into B* is a continuous mapping $\varphi : A \times I\rightarrow X$ such that $\varphi|_{A\times 0}=\operatorname{id}_{A}$ and $\varphi(A\times 1) \subseteq B$. If $\P$ is a subset of $I(U;M)$ and $\varphi:\P \times I\rightarrow I(U;M)$ is a deformation of $\P$, we may equivalently view $\varphi$ as a map $\varphi:\P\times I\times U\rightarrow M$ such that for each $h\in \P$ and $t\in I$, the map $\varphi(h,t):U\rightarrow M$ is a proper imbedding.

If $W\subseteq U$, a deformation $\varphi: \P\times I\rightarrow
I(U;M)$ is *modulo W* if $\varphi(h,t)|_{W}=h|_{W}$ for all $h\in \P$ and $t\in I$.

Suppose $\varphi: \P\times I \rightarrow
I(U;M)$ and $\psi: \Q\times I \rightarrow I(U;M)$ are deformations of subsets of $I(U;M)$ and suppose that $\varphi(\P\times
1)\subseteq \Q$. Then the *composition* of $\psi$ with $\varphi$, denoted by $\psi \star \varphi$, is the deformation $\psi \star \varphi :\P\times I\rightarrow I(U;M)$ defined by $$\psi \star\varphi(h,t)= \left\{
\begin{array}{lcl}
\varphi(h,2t)& for & t\in [0,1/2]\\
\psi(\varphi(h,1),2t-1)& for &t\in [1/2,1].
\end{array}
\right.$$

The main result of [@ed-ki] is the 