Any self-diffeomorphism of a compact manifold with boundary rel boundary can be isotoped to fix a collar of boundary Let $M$ be a compact manifold with boundary. Is it true that any self-diffeomorphism $f:M\to M$ fixing $\partial M$ is isotopic to a self-diffeomorphism that fixes a collar neighborhood $\partial M \times [0,\epsilon)\subset M$ of $\partial M$?
If this is true, I want to use this in the following case: suppose $S$ is a properly embedded submanifold (without boundary) in a compact manifold $X$ (without boundary),  and there is a self-diffeomorphism $f:S\to S$ that extends to a diffeomorphism $\tilde{f}: \nu S\to \nu S$ rel $\partial \nu S$, where $\nu  S$ is a closed tubular neighborhood of $S$ in $X$. Then applying the above situation with $M=\nu S$, we may assume $\tilde{f}$ fixes a collar neighborhood of $\partial \nu S$ in $\nu S$. Then we can extend $\tilde{f}$ to a global diffeomorphism of $X$ by defining $\tilde{f}$ to be identity outside $\nu S$.
 A: Here's a sketch of one possible construction.
Choose an extendible collar $C:\partial M\times[0,L)\to M$, and let $\pi_1,\pi_2$ be the projection onto the first and second factors of $\partial M\times[0,1)$. These are only defined on the image of $C$.
Note that $\max_{x\in\partial M}\pi_2(f(C(t,x)))$ is a well defined and continuous function of $t$ for sufficiently small $t$, so there is an $\epsilon\in(0,l)$ such that $f$ maps $C(\partial M\times[0,\epsilon))$ into the image of $C$.
One can view the restriction of $f$ to $C(\partial M\times[0,\epsilon))$ as a family of embeddings $f_t:\partial M\to\partial M\times[0,l)$, and, by projecting onto factors, a familty of smooth maps $\varphi_t:=\pi_1\circ f_t:\partial M\to\partial M$ and $\psi_t:=\pi_2\circ f_t:\partial M\to[0,l)$. Since $\varphi_0$ is the identity (and diffeomorphisms are stable for compact manifolds without boundary), we may assume by shrinking $\epsilon$ as needed that $\varphi_t$ is a diffeomorphism for $t\in[0,\epsilon)$.
From here, we split $f$ ito a "horizontal" and "vertical" part, both with domain $\partial M\times[0,\epsilon)$. Let $h(x,t)=(\varphi_t(x),t)$ and $v(x,t)=(x,(\psi_t\circ\varphi_t^{-1})(x))$. Note that $f|_{\partial M\times[0,\epsilon)}=v\circ h$. From here, one can show that $h$ can be isotopically deformed into a function $\hat{h}(x,t)=(\varphi_{\lambda(t)}(x),t)$ where $\lambda:[0,\epsilon)\to[0,\epsilon)$ vanishes on a neighborhood of $0$ and is equal to the identity on a neighborhood of $\epsilon_2$, likewise, one can deform $v$ to a funtion $\hat{v}$ which is equal to the identity on $\partial M\times=[0,a)$ and equal to $v$ on $\partial M\times(b,\epsilon)$ for some $a,b\in(0,\epsilon)$.
