# 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.

As is often the case, it seems helpful to introduce a little terminology.  A finite collection of horizontal and vertical lines in $$\mathbb C$$ will be called a grid.  The grid lines of a given grid $$\mathfrak G$$ subdivide one another into line segments (some finite and some not), and a cycle which is a finite linear combination of finite line segments of this type will be called a  grid cycle  for $$\mathfrak G$$.  The complement of the grid $$\mathfrak G$$  is a disjoint union of open rectangles (some finite and some not), and a given grid cycle will have a well-defined winding number around each of these complementary rectangles; let us call the collection of these numbers the grid winding numbers  of the grid cycle in question.  For the book I drew the figure below which illustrates these ideas:

Now we just need to prove two simple lemmas.

Lemma 1: Every cycle in an open set $$\Omega$$ is homologous to a grid cycle. (An easy approximation argument.)

Lemma 2: Any cycle all of whose grid winding numbers are zero is (homologous to) the zero cycle.  (Indeed, the coefficient with which any edge appears in a given cycle is equal to the difference of its grid winding numbers around the rectangles on the two sides of that edge.)

Given these, let $$\Gamma$$ be a grid cycle and define another cycle $$\Gamma’$$ by

$\Gamma’ = \sum_R n(\Gamma;R) [\partial R].$

Here the sum is extended over rectangles $$R$$, and $$n(\Gamma;R)$$ denotes the winding number of $$\Gamma$$ about rectangle $$R$$.   The hypothesis of Artin’s criterion ensures that the only rectangles $$R$$ that appear with nonzero coefficients in the above sum are completely contained in $$\Omega$$.  Thus $$\Gamma’$$ is a boundary in $$\Omega$$.  On the other hand, $$\Gamma$$ and $$\Gamma’$$ have the same grid winding numbers by construction, so they are homologous by Lemma 2.  It suffices to consider grid cycles $$\Gamma$$ by Lemma 1, so Artin’s criterion is proved.