Note that $p:\mathbb{C}\rightarrow \mathbb{C}\setminus\{0\}:z\mapsto e^z$ is a covering map.

Let $\alpha:[0,1]\rightarrow \mathbb{C}\setminus\{0\}$ be a closed curve.

Let $\gamma$ be any lift of $\alpha$ such that $\alpha=p\circ \gamma$.

Call $\frac{\gamma(1)-\gamma(0)}{2\pi i}$ the winding number of $\alpha$.

Is it okay to define "winding number" in this way? That is, does this definition exactly mean the winding number?

  • $\begingroup$ What is your definition of winding number? $\endgroup$ – Seirios Dec 1 '14 at 9:35
  • $\begingroup$ @Seirios I don't have one. This is the first time I'm studying somewhat related to that. $\endgroup$ – Rubertos Dec 1 '14 at 10:05
  • $\begingroup$ @Rubertos Sorry, I deleted my initial comment, realised that I hadn't read your question properly (my fault for trying to be clever before my morning coffee). $\endgroup$ – Benjamin Alderson Dec 1 '14 at 11:58
  • $\begingroup$ @BenjaminAlderson No problem and I deleted the comment :) $\endgroup$ – Rubertos Dec 1 '14 at 12:09
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    $\begingroup$ @student Exactly, but now I prefer the definition here: math.stackexchange.com/questions/186512/… $\endgroup$ – Rubertos Dec 2 '14 at 3:50

This is correct, yes. If you have a closed curve $\alpha$, then from beginning to end it only differs by a phase $e^{2\pi i n}$. Using the exponential map as you have, you see that this gives you exactly the number of times that your curve has wound around the origin.


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