Artin’s Criterion

Picture of Artin

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

\[ \int_\Gamma f(z)dz = 0. \]

The homology language suggests a straightforward proof: \( f(z)dz \) is a closed 1-form (this is a restatement of the Cauchy-Riemann equations), and the integral of a closed form over a bounding cycle is 0 (homology version of Stokes’ theorem).  The trouble here is that the above definition of “nullhomologous” or bounding is not the standard one (instead of looking at \([\Gamma] \in H_1(\Omega)\), it looks at the pairing with \(H_0({\mathbb  C}\setminus\Omega) \) defined by the winding number).  While one can argue on the basis of Lefschetz duality that these definitions agree, that is not really an appropriate procedure for a course on plane topology – it is on the same level as using duality to prove the Jordan curve theorem.

Moreover, if one consults recent standard texts on complex analysis that give a proof of Cauchy’s theorem in the homology form, they don’t proceed as suggested above.  Instead, they tend to use some argument involving tricks with double integrals and special properties of analytic functions (the most elegant that I know is due to Dixon, see below).  These arguments do not seem to generalize to prove that \(\int_\Gamma \alpha = 0 \) for an arbitrary closed 1-form (anyone know if I am wrong?)

After a bit of digging around I found a proof of this statement in a classic source, Ahlfors’ complex analysis book, where it is Theorem 16 in the third edition.  The basic geometric construction in that proof shows that the “dual” definition of nullhomologous (as in the statement of the theorem above) implies the “usual” one (using singular homology).  Ahlfors attributes this argument to Emil Artin, and in my book I propose to call it Artin’s criterion.

One thing that I still need is a reference to Artin for “Artin’s criterion”.  I was excited to discover in another complex analysis book (which perhaps should be anonymous) such a reference, but disappointed when I turned to the back  to find that the reference was simply to Artin’s Collected Works without further elaboration.   Does anyone know of a more precise reference? Or was this a contribution by Artin to the “oral tradition” of the subject?

References

Ahlfors, Lars. 1979. Complex Analysis. 3 edition. New York: McGraw-Hill Science/Engineering/Math.

Dixon, John D. 1971. “Shorter Notes: A Brief Proof of Cauchy’s Integral Theorem.” Proceedings of the American Mathematical Society 29 (3): 625–26. doi:10.2307/2038614.

One thought on “Artin’s Criterion

  1. Paul Mitchener

    A few years back I inherited a complex analysis course in which it was shown that the integral stated is zero for any closed 1 form, using covering space type techniques. The old course was given by Holger Brenner; unfortunately I don’t have an original source for the argument. But in case they help, by version of the notes is here:

    http://www.mitchener.staff.shef.ac.uk/PMA444.pdf

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