Restriction of diffeomorphisms homotopic to identity to the boundary

Question

Let $M$ be a smooth manifold with boundary $\partial M$. Let $Diff_0(M)$ be the group of all diffeomorphisms homotopic to identity. According to this article (Page 6, section " Beyond mapping class group"), the restriction of a diffeomorphism to the boundary gives a well defined surjactive homomorphism $$\phi: Diff_0(M)\rightarrow Diff_0(\partial M).$$

I could not find a reference for the above two results. So I have two questions.

1) Why is $\phi$ well defined, i.e., why the restriction of a difeomorphism homotopic to identity in $M$ is homotopic to identity in $\partial M$.

2) Why every diffeomorphism homotopic to identity in $\partial M$ is a restriction of a diffeomorphism of $M$ which is homotopic to identity.

As I am not quite familiar with diffeomorphism groups, any suggestion/reference/comment will be extremely helpful. Also I would request you to improve the tags if possible.

Thanks in advance.

Answer 1

As discussed in comments, $Diff_0$ stands for the subgroup of the diffeomorphism group, consisting of diffeomorphisms isotopic (rather than homotopic) to the identity. With this in mind, the fact that the restriction map $\phi: Diff(M)\to Diff(\partial M)$ sends $Diff_0(M)$ to $Diff_0(\partial M)$ is clear. Let's prove surjectivity. First of all, $\partial M$ admits a "collar" $C$ in $M$, a closed neighborhood of $\partial M$ in $M$, $C$ is diffeomorphic to $\partial M\times [0,1]$. Now, given $h\in Diff_0(\partial M)$, let $H(x, t), t\in [0,1]$, denote the isotopy of $h= H(\cdot, 0)$ to $id_{\partial M}= H(\cdot, 1)$. I leave it to you to prove that $H$ can be chosen so that $H(x,t)=x$ for all $t\in [1/4, 1]$. Then, using the diffeomorphism $C\cong \partial M\times [0,1]$, extend $h$ first to $C$ and then, by identity, to the rest of $M$. Call the extension $\hat{h}$. Clearly, $\phi(\hat{h})=h$. It remains to prove that $\hat{h}\in Diff_0(M)$. To prove this, play the same game as before: Given an isotopy $H(x,t)$ from $h$ to $id_{\partial M}$, extend it to $C\cong \partial M\times [0,1]$ by $$ (x,t,s)\mapsto H(x, t+s), $$ and then by identity to the rest of $M$. This will be an isotopy $\hat{H}$ from $\hat{h}$ to $id_M$.

Here is an example to ponder: Let $M$ be the annulus $S^1\times [0,1]$. Consider the diffeomorphism $f(s,t)=(s, 1-t)$; $f: M\to M$ is homotopic to the identity, but its restriction to $\partial M$ is not.