Simplicial approximation

A simplicial map $g:K\to L$ is a simplicial approximation to a continuous map $f:|K|\to |L|$ whenfor every vertex $v$ of $K$.

If $g$ simplicially approximates $f$, then $f(x)$ and $|g|(x)$ lie in a common simplex for every $x$. The affine formulatherefore remains in the realization and gives a homotopy from $f$ to $|g|$.

For a finite simplicial complex $K$, a simplicial complex $L$, and a continuous map $f:|K|\to |L|$, some iterated barycentric subdivision $K^{(r)}$ admits a simplicial approximation to $f$. The proof applies the Lebesgue number lemma to the inverse images of target open stars and uses that the mesh of a simplicial complex tends to zero under repeated barycentric subdivision.

If $n<m$, every continuous map $S^n\to S^m$ is 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.