if $f: X \rightarrow Y$ is a diffeomorphism of manifolds with boundary, then $\partial f$ maps $\partial X$ diffeomorphically onto $\partial Y$.

Question

Prove that if $f: X \rightarrow Y$ is a diffeomorphism of manifolds with boundary, then $\partial f$ maps $\partial X$ diffeomorphically onto $\partial Y$.

Could anyone give me a hint for the proof of this please?

Answer 1

If you look at page 57 you'll find the definition of $\partial X$. Then, as usual, get the local diagram for $f$ - here $U$ is an open set in $H^k$ (p.14).

\begin{array}{lcl} X & \overset{f}{\longrightarrow} & Y \\ \downarrow\varphi & & \downarrow\psi \\ U & \overset{id}{\longrightarrow} & U \end{array}

Given commutativity of the diagram and the definition of $\partial X$, you should be able to prove it from here.