the remaining ones being analogous. First we have to shrink the cover $(U_i)_{i=1}^d$ $d$ times, that is we choose an open $U_{i,j}$ for every $i=1\ld d$ and $j=0\ld d$ with $U_{i,0}=U_i$ such that $\bigcup_{i=1}^dU_{i,j}=M$ for all $j$ and such that $\operatorname{cl}(U_{i,j+1})\s U_{i,j}$ for all $i,j$. We make use of Theorem 2.2 $d$ times with $q=1$. Namely, for $i=1\ld d$ we have a neighborhood $\P_i$ of the identity in $I(M,\bigcup_{\alpha=1}^{i-1}U_{\alpha,i-1}; M)$ and a deformation $\phi_i:\P_i\t I\r \H_c(M)$ which is modulo $M\setminus U_{i,0}$ and which takes its values in $I(M,\bigcup_{\alpha=1}^{i}\operatorname{cl}(U_{\alpha,i}); M)$ and such that $\phi_i(\operatorname{id}, t)=\operatorname{id}$ for all $t$. Here we apply Theorem 2.2 with $C=\operatorname{cl}(U_{i,i})$, $U=U_{i,0}$, $D_1=\bigcup_{\alpha=1}^{i-1}\operatorname{cl}(U_{\alpha,i})$ and $V_1=\bigcup_{\alpha=1}^{i-1}U_{\alpha,i-1}$. Taking a neighborhood $\P$ of id small enough, we have that $\phi_d\star\cdots\star\phi_1$ restricted to $\P\t I$ is well defined. For every $h\in\P$ we set $h_0=h$ and $h_i=\phi_i\star\cdots\star\phi_1(h,1)$, $i=1\ld d$. It follows that $h_d=\operatorname{id}$ and $h=\prod_{i=1}^dh_ih_{i-1}^{-1}$. It suffices to define $\sigma_i:\P\r\H_c(M)$ by $\sigma_i(h)=h_ih_{i-1}^{-1}$ for all $i$.

\[isot\] Let $h_{t}:M \rightarrow M$, $t \in I$, be an isotopy of a compact manifold $M$ with $h_0=\operatorname{id}$, and let $(U_{i})_{i=1}^d$ be an open cover of $M$. Then $h_{t} $ can be written as a composition of isotopies $h_{t}=h_{k,t}h_{k-1,t}\ldots h_{1,t}$, where each isotopy $h_{j,t}: M \rightarrow M$ 