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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.

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I haven't looked at Ahlfors in awhile. Does the superscript $c$ mean closure (why doesn't he use a bar)? Also, do you mean $H_1(\Omega, \mathbb{Z})$ rather than the cohomology group? –  Matt May 22 '13 at 3:08
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@Matt It means complement. (Is there a less confusing notation I could use?) And yes, it should be a subscript. Let me fix that. –  Potato May 22 '13 at 3:10
    
Sorry. That's what I originally thought it meant, but for some reason became convinced it had to mean something else. I was being dumb and what I thought was a counterexample really wasn't. –  Matt May 22 '13 at 3:30

2 Answers 2

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.

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It tells you whether each subrectangle of your grid is "inside" the curve or not (and with multiplicities, too). We need a blackboard to draw pictures :) Draw a few sample curves and try it. Note that Ahlfors's definition says that the region can't have any holes inside the curve. –  Ted Shifrin May 22 '13 at 3:27
    
P.S. And to get the other direction in the equivalence, use the integral formula for winding number and Stokes's Theorem. –  Ted Shifrin May 22 '13 at 3:43
    
The integral formula for winding number won't work if the curve is merely continuous and not piecewise differentiable. –  Potato May 22 '13 at 3:44
    
(Although I suppose we could attempt an approximation by a piecewise differentiable curve? I will need to think more.) –  Potato May 22 '13 at 3:45
    
Well, actually, it'll work for rectifiable curves, but, personally, I'm happy to stay in the piecewise-$C^1$ setting. Yes, you should be able to make various approximation arguments (e.g., with mollifiers). –  Ted Shifrin May 22 '13 at 4:04
up vote 2 down vote accepted

This result appears as proposition 1.9.13 in Berenstein and Gay's book on complex analysis.

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