Lower-dimensional sphere maps are null-homotopic
= Lower-dimensional sphere maps are null-homotopic
If $n<m$, every continuous map $S^n\to S^m$ is <null-homotopic map>[null-homotopic]. After simplicial approximation, its image lies in the $n$-skeleton of a triangulation of $S^m$, hence omits a point. The punctured sphere is homeomorphic to $\mathbb R^m$ and is therefore contractible.