= Maps to a sphere above the dimension of a compact smooth manifold are null-homotopic
{title2=$\dim X<q,\quad q\geq1\quad\Longrightarrow\quad[X,S^q]=\{*\}$}
Approximate a continuous sphere-valued map by smooth real coordinate functions, then normalize. A sufficiently close approximation is homotopic to the original map by normalized straight interpolation. <Sard theorem> implies the smooth map misses a point when its source dimension is smaller than the sphere dimension. The punctured sphere is contractible, so the map is null-homotopic, even if the source is disconnected.
// Target: geometry-and-topology.bigb
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