# 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 $$\nu S$$.

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 $$a,b\in(0,\epsilon)$$.