# Equivalent definitions of winding number

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.

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