Maps to a sphere above the dimension of a compact smooth manifold are null-homotopic ($\dim X<q,q\ge 1⟹[X,S^q]=\{\ast \}$)

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