Complex Analysis/Integration over chains

Argument and winding numbers
Assume we are given a closed contour supported in $$\mathbb C$$, and suppose that we are an observer located at the origin. Suppose we want to measure how often a moving object rotates about us (ie. passes through a point which is chosen fixed and has a fixed angle with respect to us). The resulting number is called the winding number of the given closed contour. Note though that it is signed; that is, if we contour were to travel (regarding angular distance) first round the circle, and then again but in reverse direction, the winding number is supposed to be zero.

To make this precise,

argument definition to circle and lift to standard covering

homotopy invariance of the latter