Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 2F / a / Solution

Normalize the nonvanishing loop to $\gamma (t)/|\gamma (t)|$. A continuous argument lift is a continuous function $\theta :[0,1]\to \mathbb{R}$ such that $\gamma (t)/|\gamma (t)|=e^{i\theta (t)}$. Such a lift exists on an interval; any two differ by a constant integer multiple of $2\pi$. The winding number isIts value is independent of the lift. This definition applies to continuous loops without assuming a differentiable contour.

The homotopy invariance of winding number states: if $H:[0,1{]}^2\to \mathbb{C}\setminus \{0\}$ is continuous and $H(s,0)=H(s,1)$ for every $s$, then $w(H(s,\cdot );0)$ is independent of $s$. Neither the loops nor their moving basepoints may pass through zero. One may justify invariance by uniformly subdividing the homotopy into nearby loops whose normalized values have a continuous argument difference of magnitude less than $\pi$; that argument difference has equal endpoint values, so their winding numbers agree.