= Dimension reduction of sphere maps into CW skeleta
{title2=$[S^n,X^n]\longrightarrow[S^n,X]\text{ is surjective}$}
A map from the $n$-<sphere> to a <CW complex> can be homotoped into its $n$-skeleton. By <compact subsets of CW complexes lie in finite subcomplexes>, its image lies in a finite subcomplex. For a top-dimensional cell of dimension $d>n$, use <smooth approximation of maps into Euclidean space> in its interior, with a cutoff supported away from the cell boundary. The <Sard theorem> supplies a point in a smaller interior ball omitted by the smoothed map; the cutoff is chosen so the transition region also avoids that point. Radial retraction from the omitted point pushes the image into the cell boundary and fixes all other cells. Repeating over the finitely many higher-dimensional cells proves the assertion, without assuming the initial continuous image has measure zero.
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