Simplicial map

A simplicial map sends vertices to vertices and sends the vertex set of every simplex to the vertex set of a simplex. It induces a continuous map between geometric realizations and a chain map between simplicial chain complexes.

For a simplicial map to the standard simplicial decomposition of a sphere, a $k$-simplex is positive alternating when its image has the displayed distinct labels. Reversing all signs makes it negative alternating. A collapsed or nonalternating simplex is neutral. Counting positive facets modulo two is the local ingredient in the antipodal alternating-simplex parity lemma.

An antipodal simplicial map from a regular antipodal triangulation of a sphere $S^k$ to $F^n$ has an odd number of positive alternating $k$-simplices. For each $k$-simplex, the number of positive alternating facets is odd exactly when the simplex is positive or negative alternating; otherwise it is even, including collapsed images. Count these incidences in one hemisphere: interior facets contribute twice, boundary facets once. Antipodal pairing then gives $p_k\equiv p_{k-1}$, starting with one positive vertex on $S^0$. In particular, no such map exists when $k>n$.

A map satisfying this condition is the object whose existence after subdivision is guaranteed by the Simplicial approximation theorem.

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.

Simplicial maps $f,g:K\to L$ are contiguous when $f(\sigma )\cup g(\sigma )$ lies in one simplex of $L$ for every simplex $\sigma$ of $K$. Contiguous maps have homotopic geometric realizations.