written as $h|_{M_j}=h_d\ldots h_1$ such that $h_i\in\H_{B_i}(M_j)$, where $B_i$ is a ball or half-ball of $M_i$ for $i=1\ld d$. Moreover, we have $h_i=\operatorname{id}$ on $\p_{M_j}$ for all $i$. Then due to Corol. 3.2(1) we have $h_i=[S_i(h_i),\phi_i]$ and $S_i(h_i)=\operatorname{id}$ on $\p_{M_j}$ for each $i$. Is is easily seen that $\phi_i$ may be defined as an element of $\H_c(M)$ supported in the interior of $M_{j+1}$. Thus extending each $h_i$ to $M$ by putting $h_i=\operatorname{id}$ off $M_j$, the perfectness of $\H_c(M)$ follows. $\square$

*Proof of Theorem 1.1 for $n=1$ and $\pm\neq\emptyset$, and of Theorem 1.2*. Let $M$ be a manifold with boundary $\p$. By a *collar* neighborhood of $\p$ we mean a set $P=\p\t[0,1]$ embedded in $M$, where $\p\t\{0\}$ identifies with $\p$. It is well-known that such a neighborhood exists.

In the case of 1.1 we have $\p=\{0\}$. In view of Theorem 2.3 it suffices to consider $\hr$, where $\R_+=[0,\infty)$. For any $f\in \hr$ there is a sequence of reals from (0,1) $$1>b_1>\bar b_1>\bar a_1>a_1>b_2>\ldots>b_k>\bar
b_k>\bar a_k>a_k>\ldots>0,$$ tending to 0, and $h\in
\hr$ such that $$h=f\quad\hbox{ on}\quad\p\t\quad\bigcup_{k=1}^{\infty} [\bar a_k,\bar b_k].$$ Moreover, setting $A_k:=\p\t(a_k,b_k)$ and $A:=\bigcup_{k=1}^{\infty}A_k$, we may also have that $$\operatorname{supp}(h)\s A,$$ and that for the decomposition $h=h_1h_2\ldots$ resulting from the partition $A=\bigcup_{k=1}^{\infty}A_k$ 