Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 11I (Topics in Analysis)

For a closed path avoiding zero, choose a continuous lift $\gamma (t)/|\gamma (t)|=e^{i\theta (t)}$ and set $w(\gamma ;0)=(\theta (1)-\theta (0))/(2\pi )$. Conjugation replaces $\theta$ by $-\theta$, as does reversing the path, so both winding numbers are $-w(\gamma ;0)$.

Homotopy invariance of winding number states that a homotopy $H(s,t)$ through closed paths in ${\mathbb{C}}^{\ast }$ preserves winding number. Uniform continuity divides the $s$ interval so successive paths differ pointwise by less than the minimum modulus of the earlier path; the supplied perturbation lemma makes their winding numbers equal. Chaining the subdivisions proves the result.

If no $z\in S^1$ obeyed $z^n=-g(z)$, the loops $z^n$ and $-g(z)$ could be joined without crossing zero by the straight-line homotopy, so they would have equal winding number. The first has winding $n$. Since $g{|}_{S^1}$ extends over the disc, it is null-homotopic and has winding $0$; multiplication by $-1$ does not change it. This contradicts $n\ne 0$.