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")](http://www.math.brown.edu/~mann/papers/RealizationProblems.pdf), 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. 



Restriction of diffeomorphisms homotopic to identity to the boundary
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. 


