Stoilow Theorem

Question

I want to see the precise statement and a proof for a theorem of Stoilow on "inner" functions (I do not know what this exactly means, I suppose it is an open map with other natural properties). A vague statement is this:

An inner function is a continuous deformation of a holomorphic function.

I am looking for a simple version rather than the most general one. Any accessible reference is welcome (I understand Romanian).

Thanks in advance.

Answer 1

First of all, it seems that every author has his own favourite definition of "inner mapping". The best thing would probably be to adopt the one used by Martin Jurchescu: a continuous, open, zero-dimensional mapping.

Stoilow's theorem says, then, that if $\phi : R \to S^2$ is a map from a Hausdorff, connected topological surface $R$ to the Riemann sphere $S^2$, then the following are equivalent:

• $\phi$ is an interior mapping
• $\phi$ behaves locally like the mapping $\{ |z| \le 1 \} \ni z \mapsto z^n \in \{ |z| \le 1 \}$
• there exist a unique Weyl-Rado Riemann surface on $R$ such that $\phi$ becomes a holomorphic mapping.

To make things a bit confusing, the article about "inner mapping" on the Springer encyclopedia of mathematics gives a slightly different statement of the theorem.

For more details, check "Analysis and Topology: A Volume Dedicated to the Memory of S Stoilow" (World Scientific, 1998), pages 420-421.

Answer 2

I'm a bit late to the party but I thought I'd offer my two cents both to Stoilow's theorem and on some related terminology.

Stoilow's theorem is one of the more classical results in my field, and even though I rarely use it, it gets mentioned a lot. ("In dimension two problem X is completely understood due to Stoilow's theorem.") A bit over a year ago me and my then adviser were talking that it might be beneficial to study the proof of the theorem to see if it could be applied in a more general setting. What we learned that there were very few sources from where the proof could be found. Long story short - we decided it would be beneficiary to write up an exposition containing a self-contained proof of the theorem. This can now be found from arXiv: 1701.05726. (Note that there are several different theorems that people call Stoilow's theorem. Our exposition focuses on results found in Stoilow's 1928 paper and what Whyburn calls Stoilow's theorem in his book Topological Analysis.)

The mappings Stoilow is interested in the 1928 paper are mappings $f \colon \Sigma \to \Sigma'$ between topological surfaces which are

• continuous,
• open (images of open sets in $\Sigma$ are open in $\Sigma'$), and
• light (pre-image of any point $y \in \Sigma'$ is totally disconnected).

As Alex.M mentioned in their answer, some authors use the term "inner map" to mean this class of mappings. Some other authors use the term "inner mapping" to describe mappings for which the image of any open set in the domain is open in the image of the mapping. (Another term for this I've seen is quasi-open.) For example the plane map $\mathbb{R}^2 \to \mathbb{R}^2$, $(x,y) \mapsto (|x|,y)$ is, with this terminology, a quasi-open map but not an open one.

One of the first things Stoilow proves is that continuous, open and light mappings are in fact discrete, i.e. the pre-image of any point $y \in \Sigma'$ is discrete. Such continuous, open and discrete mappings are called branched covers by many authors, but this term is used also for different, but related, types of mappings; see e.g. Aaltonen or Bonk-Meyer. I would advice to be careful when reading papers about either inner maps or branched covers since the definitions vary slightly. (This makes it sometimes hard to read older papers since the authors' might not explicitly state which definition they are using.)

In modern times there seems to be, from my subjective point of view, more interest in continuous, open and discrete mappings than in continuous, open and light mappings. Some reasons for this shift of interest seem to be:

• Stoilow's theorem. All planar continuous, open and light maps are discrete).
• Results of Church-Hemmingsen stating that between Euclidean domains of equal dimension, a topologically orientable continuous, light and open map is, in fact, discrete.
• Reshetnyak's theorem stating that a quasiregular mapping between Riemannian $n$-manifolds is a continuous, open and discrete map.

For more details I suggest looking at previously mentioned exposition by me and Pekka Pankka, we tried to make it as reader friendly as possible.