Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 21H / e / Solution

Triangulate $S^n$ and $S^m$. By the Simplicial approximation theorem, after subdividing the domain, a continuous map $f:S^n\to S^m$ is homotopic to the realization of a simplicial map $g$. Since $n<m$, the image of $|g|$ lies in the $n$-skeleton of the target triangulation and therefore misses an interior point of every $m$-simplex. A sphere with one point removed is homeomorphic to the Euclidean space ${\mathbb{R}}^m$, which is contractible. Hence $|g|$, and therefore $f$, is homotopic to a constant map. Thus