Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 2I / Solution

The planar Brouwer fixed-point theorem says every continuous $F:D^2\to D^2$ has a fixed point. The equivalent no-retraction theorem says no continuous $r:D^2\to S^1$ restricts to the identity on $S^1$. A fixed-point-free map gives a retraction by extending the ray from $F(x)$ through $x$ to the boundary; a retraction followed by the antipodal map gives a fixed-point-free self-map.

Suppose the stated sphere map is not onto and omits $p$. Compactness gives $\epsilon >0$ such that its image lies in $S^2\setminus B_{\epsilon }(p)$. That set is homeomorphic to a closed disc. Restricting $f$ to it gives a continuous self-map of a disc, so Brouwer supplies a fixed point, contrary to the hypothesis. Hence $f$ is surjective.