The presentation of the homology version of Cauchy's theorem in Ahlfors is slick, but sweeps a lot of the topology under the rug using clever arguments. This question is an attempt to reconcile Ahlfors' analytic notion of a curve being homologous to zero (presented in his book Complex Analysis and originally due to E. Artin, I believe) with the standard definition in homology as found in Hatcher.
We work in the complex plane and fix $a\in \mathbb C$. Let $\gamma$ be a continuous map $[0,1]\rightarrow C\backslash \{ a\}$. Following Munkres in his book Topology, we define the winding number of $\gamma$ with respect to the point $a$ by considering
$$g(t)=\frac{\gamma(t)-a}{|\gamma(t)-a|}.$$
This is clearly a loop in $S^1$ and corresponds to some multiple of of the generator of the fundamental group of $S^1$. If the generator is $\tau$ and $g(t)$ corresponds to $m\tau$, $m\in\mathbb Z$, we define the winding number $n(\gamma, a)$ to be $m$. This is the definition of winding number I will use in this question, but in case $\gamma$ is piecewise differentiable, it corresponds to the analytic definition by integration given in Ahlfors.
Ahlfors calls a curve contained in an open region $\Omega$ homologous to zero if $n(\gamma,a)=0$ for all $a\in \Omega^c$.
In homology theory, as I understand it, we would call a curve homologous to zero if it represents the zero element in $H_1(\Omega, \mathbb Z)$. That is, $\gamma$ is the boundary of some singular $2$-chain.
My question: Why do all of these notions agree? Why is Ahlfors' definition of being homologous to zero (using Munkres' definition of winding number) agree with the usual homological one?
I would like to use Munkres' definition because it works for continuous curves, not just piecewise differentiable ones, and it seems to me the equivalence should hold in this generality.
Edit: This result appears as proposition 1.9.13 in Berenstein and Gay's book on complex analysis.
Best Answer
Imagine the curve as made of segments parallel to the coordinate axes. Create a grid on a large rectangle containing the curve in its boundary and interior so that every segment of our curve is an edge of some subrectangle. Now define a singular $2$-chain by using each subrectangle with the coefficient given by the winding number of our curve around the center of the subrectangle. Check that the boundary of this $2$-chain is our original curve.