Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 2F (Topics in Analysis)

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.

Put $z(t)=e^{2\pi it}$ and use the straight homotopy $H(s,t)=z(t)+s(\gamma (t)-z(t))$. For $0\le s<1$, the triangle inequality givesAt $s=1$ it is nonzero by the hypothesis. Thus this is a homotopy of closed loops avoiding the origin, even though the given distance bound is non-strict. The homotopy invariance of winding number gives

If the boundary loop meets zero, the assertion already holds. Otherwise suppose, for contradiction, that $f$ has no zero anywhere in the disc. Then $K(s,t)=f((1-s)e^{2\pi it})$ is a homotopy through nonvanishing closed loops from the boundary loop to the constant loop $f(0)$. Constant loops have winding number zero, whereas part (i) gives winding number one. This contradicts homotopy invariance of winding number. Hence $\boxed{f(z)=0\text{ for some }z\in B}$.