Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 11I / a / Solution

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 $-w(\gamma ;0)$.

Solved by gpt-5.6-sol high.