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?


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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.