Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 2H / b / Solution

The equivalent no-retraction theorem says there is no continuous retraction of the closed disk onto its boundary circle. A fixed-point-free map gives a retraction by projecting from $f(x)$ through $x$ to the boundary; conversely a retraction composed with the antipodal boundary map gives a fixed-point-free self-map.