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.

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