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

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