Let $M$ a compact manifold, $N$ a connected manifold and $f\colon M\longrightarrow N$ an injective étale function. Why $f$ is a diffeomorphism?

I know that since $M$ is compact, $f(M)$ will be compact too, but I don't know why if $N$ is connected, I will get the diffeomorphism.

  • $\begingroup$ What is an etale function in the smooth category? $\endgroup$ – levap Nov 30 '16 at 23:32
  • $\begingroup$ From the book where I'm reading, an étale function is one that is between manifolds of the same dimension, and is an immersion if and only if is a submersion. $\endgroup$ – MonsieurGalois Nov 30 '16 at 23:33

An etale function is a local diffeomorphism. If $N$ is connected, $f$ is surjective: to see this remark that the image of $f$ is a closed subspace since $M$ is compact, $f(M)$ is compact. It is also open since $f$ is a local diffeomorphism. So it is a union of connected components of $N$, so it is $N$ since $N$ is connected. A local diffeomorphism which is bijective is a diffeomorphism.

  • $\begingroup$ So if the etale function can be understood as a local diffeomorphism given the definition in the comment I added, it is not enough to say that it is a diffeomorphism? $\endgroup$ – MonsieurGalois Dec 1 '16 at 0:06
  • 2
    $\begingroup$ No, the universal cover is an etale cover, if the fundamental group is finite but it is not always a diffeomorphism $\endgroup$ – Tsemo Aristide Dec 1 '16 at 0:08
  • $\begingroup$ Thanks for the explanation. $\endgroup$ – MonsieurGalois Dec 1 '16 at 0:11

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.