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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?

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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.

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  • $\begingroup$ what is $ \partial f$ in this case? $\endgroup$ – Idonotknow Nov 26 '18 at 22:06
  • $\begingroup$ $\partial f$ is $f$ restricted to $\partial X$, which is the image of boundary points in $U$ under $\phi^{-1}$. $\endgroup$ – Prototank Nov 26 '18 at 23:24

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