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The concept of winding number of a closed curve relatively to a point not on the curve has several possible definitions.

One can define the concept with the path integral $$Wn(\gamma,a) = \frac{1}{2i\pi}\int_{\gamma} \frac{1}{z-a} dz$$ or simply count the number of turns around the point.

One can also use more abstracted tools like Alexander duality or the degree of an application $S^1 \to S^1.$

My question is the following : Is there any good reference which proves the equivalence between all these definitions ? Thank you.

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  • $\begingroup$ "Number of turns around a point $a$". How do you define this if not by the integral $W_n(\gamma, a)$? $\endgroup$ – Git Gud Feb 23 '16 at 20:01
  • $\begingroup$ Of course, but the link with homology is more complicated. $\endgroup$ – C. Dubussy Feb 23 '16 at 20:07
  • $\begingroup$ @GitGud: By its homotopy class in $\pi_1(\Bbb C \backslash a) \simeq \Bbb Z$, for instance. $\endgroup$ – Shalop Feb 23 '16 at 20:07
  • $\begingroup$ @GitGud Maybe if we think of the winding number as the index of curve at a point $a$ ? "The index of an isolated critical point of a vector field is the winding number of a small counterclockwise oriented circle with center at that point". Source $\endgroup$ – GaussTheBauss Feb 23 '16 at 20:09
  • $\begingroup$ I think there should be a division by $2\pi i$ up there. At least if you want an integer winding number... $\endgroup$ – Shalop Feb 23 '16 at 20:13

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