Let $f:\mathbb{D}=\{z\in\mathbb{C}\mid |z|<1\}\rightarrow\mathbb{C}$ be a local diffeomorphism (i.e. an immersion) from an open disk in the plane to the plane.

The only situation I can image where $f$ is not injective is that $f$ sends $\mathbb{D}$ to a "self-overlapping'' region, in which case $f$ can not have continuous injective boundary values. But it seems non-trivial to proof that such boundary values guarantee injectivity:

Question. Assume that $f$ extends to a continuous map $\overline{\mathbb{D}}\rightarrow\mathbb{C}$ such that $f|_{\partial\mathbb{D}}$ is injective and continuous, so that (by Jordan Curve Theorem) it maps $\partial\mathbb{D}$ homeomorphically to a Jordan curve which is the boundary of a simply connected domain $\Omega\subset\mathbb{C}$. Then it is true that $f$ is a homeomorphism from $\mathbb{D}$ to $\Omega$?

Essentially we can reduce the problem to the case where $f|_{\partial\mathbb{D}}$ maps $\partial\mathbb{D}$ identically to $\partial\mathbb{D}\subset\mathbb{C}$ and ask:

Question'. If $f:\overline{\mathbb{D}}\rightarrow\mathbb{C}$ is a continuous map such that $f|_{\partial\mathbb{D}}$ is the identity and $f|_{\mathbb{D}}$ is a local homeomorphism, then is $f$ always injective?


Yes, this sort of thing can be proved by degree theory. An outline of the argument is given here: https://math.stackexchange.com/questions/737358/locality-of-inverse-function-theorem


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

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