A surjective diffeomorphism is a global one?

Question

My question is really simple. I know that if a local diffeomorphism is injective, then it's a global one. I would like to know if the same implication is true in the case of surjective functions, i.e., is it true that every surjective local diffeomorphism is a global one? If not, could someone give me some counterexamples?

Answer 1

It's not really true that an injective local diffeomorphism is a global diffeomorphism. That is, if $f:M\to N$ is an injective local diffeomorphism between smooth manifolds $M$ and $N$, then it is not true that $f$ is a diffeomorphism $M\to N$. What is true is that if you change $f$'s codomain so that it is a bijection, then it becomes a diffeomorphism. That is, $f$ is a diffeomorphism when considered as a map $M\to f(M)$.

What's really going on here is that any bijective local diffeomorphism is a global diffeomorphism. Since an injective map can always be made bijective in a canonical way by restricting its codomain, you can thus think of any injective local diffeomorphism as a global diffeomorphism onto its image.

On the other hand, given a surjective map, there is not any canonical way to restrict it to be a bijection. So if $f:M\to N$ is a surjective local homeomorphism that is not injective, there is not going to be any natural way to modify its domain or codomain so that it is a bijection, and so there is not any reasonable sense in which you can say "$f$ is a global diffeomorphism".

For an explicit example, you could take $M=\{0,1\}$, $N=\{0\}$, and $f:M\to N$ defined by $f(0)=f(1)=0$. Then $f$ is a surjective local diffeomorphism between two $0$-dimensional manifolds $M$ and $N$, but is not a global diffeomorphism because it is not injective.

Or if you want an example with connected manifolds, take $M=\mathbb{R}$, $N=S^1\subset\mathbb{C}$, and $f:M\to N$ defined by $f(t)=e^{it}$. Then $f$ is surjective and is a local diffeomorphism between the $1$-dimensional manifolds $M$ and $N$, but it is not injective (and in fact every point has infinitely many preimages).