Artin’s Criterion, Part II

In this post I want to sketch the proof of “Artin’s criterion”, following Ahlfors’ book as referred to in my previous post (and presumably following Artin himself, though I still have not come up with any original reference to him).  The argument generalizes easily to $$(n-1)$$-dimensional cycles in an open subset $$\Omega\subseteq {\mathbb R}^n$$, but for simplicity I will stick to the planar case.

Artin’s Criterion

Emil Artin, picture from Wikipedia

There’s been a long lull in updates to this page.  I’ve posted elsewhere about the reason for that: in brief, I was found to have cancer, and subsequent treatments have kept me busy for months.  You can read about this on my personal website, but I’m not going to write more about it here.

Meanwhile though I have been slowly writing up a book-length version of my MASS 2013 course, “Winding Around”, whose central theme is “The Winding Number in Topology, Geometry and Analysis”.  As I was “winding around” myself and trying to complete Chapter 5 in a way that was satisfactory, I ran into an interesting “gap” in my own understanding.  This is related to the homology version of Cauchy’s theorem.  This is usually stated in the following way.

Theorem  Let $$f$$ be a function that is holomorphic on an open subset $$\Omega\subseteq\mathbb C$$, and let $$\Gamma$$ be a cycle in $$\Omega$$ that is nullhomologous, this being defined to mean that the winding number of $$\Gamma$$ about each point of $${\mathbb C}\setminus\Omega$$ is zero.   Then

$\int_\Gamma f(z)dz = 0.$