$\psi\in
\H(\R_+)$ with $\psi(a_k)=\tilde a_k$ and $\psi(b_k)=\tilde b_k$ for $k=1,2,\ldots$.

In order to prove the first assertion it suffices to choose $f_1$ (and similarly $f_2$) of the form $\phi=\bigcup_{k=1}^{\infty}\phi_k$ with $\phi_k([a_{2k-1},b_{2k-1}]\cup[ a_{2k}, b_{2k}])\s ( a_{2k},
b_{2k})$ for all $k$ and with $\operatorname{supp}(\phi_k)$ mutually disjoint. The $\psi$ in the second assertion is obtained by gluing together linear homeomorphisms on the consecutive intervals $[a_1,1]$, $[b_1,a_1]$, and so on.

*Proof of Theorems 1.4 (for $n=1$ and $\pm\neq\emptyset$) and 1.5.* Let $P=\p\t[0,1]$ be a collar neighborhood embedded in $M$ such that $\p$ identifies with $\p\t\{0\}$. Since $\p=\pm$ is compact, in view of Theorem 2.5 the restriction mapping $$\hm\ni f\mapsto f|_{\p}\in\H(\pm)$$ is an epimorphism. It follows from Lemma 1.10 in [@BIP] that $\H_c(\pm)$ is bounded. Thus it suffices to show the second assertion of Theorem 1.5. Let $g\in \H_c(\p\t\R_+)$. Arguing as in the proof of Theorem 1.2, there is a sequence, converging to 0, of the form $$1>b_1>\bar b_1>\bar a_1>a_1>b_2>\ldots>b_k>\bar b_k>\bar
a_k>a_k>\ldots>0$$ and homeomorphisms $h_1,h_2\in \H_c(\p\t\R_+)$ such that $$h_1=g\;\hbox{ on }\;\bigcup_{k=1}^{\infty} \p\t[\bar a_{2k-1},\bar
b_{2k-1}],\quad \operatorname{supp}(h_1)\s U_1:=\bigcup_{k=1}^{\infty} \p\t(
a_{2k-1}, b_{2k-1}),$$ $$h_2=g\;\hbox{
on} \;\bigcup_{k=1}^{\infty} \p\t[\bar a_{2k},\bar b_{2k}],\quad
\operatorname{supp}(h_2)\s U_2:=\bigcup_{k=1}^{\infty} \p\t( a_{2k},
b_{2k}).$$ Continuing the reasoning from the proof of Theorem 1.2 for $h'=h^{-1}g$, it can be checked 