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.

A simplicial homeomorphism is the homeomorphism of geometric realizations induced by an isomorphism of simplicial complexes: a bijection of their vertex sets such that a finite vertex set is a simplex exactly when its image is a simplex. The map acts linearly in each barycentric coordinate system. It and its inverse are continuous by the realization topology. Between the boundaries of two triangles, any bijection of the three vertices extends linearly on edges to a simplicial homeomorphism. This ensures that the edge identifications agree at common endpoints.

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.