Simplicial map

A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of a simplex always span a simplex.^[1] Simplicial maps can be used to approximate continuous functions between topological spaces that can be triangulated; this is formalized by the simplicial approximation theorem.

A simplicial isomorphism is a bijective simplicial map such that both it and its inverse are simplicial.

A simplicial map is defined in slightly different ways in different contexts.

Let K and L be two abstract simplicial complexes (ASC). A simplicial map of K into L is a function from the vertices of K to the vertices of L, ${\displaystyle f:V(K)\to V(L)}$, that maps every simplex in K to a simplex in L. That is, for any ${\displaystyle \sigma \in K}$, ${\displaystyle f(\sigma )\in L}$.^{[2]: 14, Def.1.5.2} As an example, let K be the ASC containing the sets {1,2},{2,3},{3,1} and their subsets, and let L be the ASC containing the set {4,5,6} and its subsets. Define a mapping f by: f(1)=f(2)=4, f(3)=5. Then f is a simplicial mapping, since f({1,2})={4} which is a simplex in L, f({2,3})=f({3,1})={4,5} which is also a simplex in L, etc.

If ${\displaystyle f}$ is not bijective, it may map k-dimensional simplices in K to l-dimensional simplices in L, for any l ≤ k. In the above example, f maps the one-dimensional simplex {1,2} to the zero-dimensional simplex {4}.

If ${\displaystyle f}$ is bijective, and its inverse ${\displaystyle f^{-1}}$ is a simplicial map of L into K, then ${\displaystyle f}$ is called a simplicial isomorphism. Isomorphic simplicial complexes are essentially "the same", up ro a renaming of the vertices. The existence of an isomorphism between L and K is usually denoted by ${\displaystyle K\cong L}$.^[2]: 14 The function f defined above is not an isomorphism since it is not bijective. If we modify the definition to f(1)=4, f(2)=5, f(3)=6, then f is bijective but it is still not an isomorphism, since ${\displaystyle f^{-1}}$ is not simplicial: ${\displaystyle f^{-1}(\{4,5,6\})=\{1,2,3\}}$, which is not a simplex in K. If we modify L by removing {4,5,6}, that is, L is the ASC containing only the sets {4,5},{5,6},{6,4} and their subsets, then f is an isomorphism.

Let K and L be two geometric simplicial complexes (GSC). A simplicial map of K into L is a function ${\displaystyle f:K\to L}$ such that the images of the vertices of a simplex in K span a simplex in L. That is, for any simplex ${\displaystyle \sigma \in K}$, ${\displaystyle \operatorname {conv} (f(V(\sigma )))\in L}$. Note that this implies that vertices of K are mapped to vertices of L. ^[1]

Equivalently, one can define a simplicial map as a function from the underlying space of K (the union of simplices in K) to the underlying space of L, ${\displaystyle f:|K|\to |L|}$, that maps every simplex in K linearly to a simplex in L. That is, for any simplex ${\displaystyle \sigma \in K}$, ${\displaystyle f(\sigma )\in L}$, and in addition, ${\displaystyle f\vert _{\sigma }}$ (the restriction of ${\displaystyle f}$ to ${\displaystyle \sigma }$) is a linear function.^{[3]: 16 [4]: 3} Every simplicial map is continuous.

Simplicial maps are determined by their effects on vertices. In particular, there are a finite number of simplicial maps between two given finite simplicial complexes.

A simplicial map between two ASCs induces a simplicial map between their geometric realizations (their underlying polyhedra) using barycentric coordinates. This can be defined precisely.^{[2]: 15, Def.1.5.3} Let K, L be two ASCs, and let ${\displaystyle f:V(K)\to V(L)}$ be a simplicial map. The affine extension of ${\displaystyle f}$ is a mapping ${\displaystyle |f|:|K|\to |L|}$ defined as follows. For any point ${\displaystyle x\in |K|}$, let ${\displaystyle \sigma }$ be its support (the unique simplex containing x in its interior), and denote the vertices of ${\displaystyle \sigma }$ by ${\displaystyle v_{0},\ldots ,v_{k}}$. The point ${\displaystyle x}$ has a unique representation as a convex combination of the vertices, ${\displaystyle x=\sum _{i=0}^{k}a_{i}v_{i}}$ with ${\displaystyle a_{i}\geq 0}$ and ${\displaystyle \sum _{i=0}^{k}a_{i}=1}$ (the ${\displaystyle a_{i}}$ are the barycentric coordinates of ${\displaystyle x}$). We define ${\displaystyle |f|(x):=\sum _{i=0}^{k}a_{i}f(v_{i})}$. This |f| is a simplicial map of |K| into |L|; it is a continuous function. If f is injective, then |f| is injective; if f is an isomorphism between K and L, then |f| is a homeomorphism between |K| and |L|.^{[2]: 15, Prop.1.5.4}

Let ${\displaystyle f\colon |K|\to |L|}$ be a continuous map between the underlying polyhedra of simplicial complexes and let us write ${\displaystyle {\text{st}}(v)}$ for the star of a vertex. A simplicial map ${\displaystyle f_{\triangle }\colon K\to L}$ such that ${\displaystyle f({\text{st}}(v))\subseteq {\text{st}}(f_{\triangle }(v))}$, is called a simplicial approximation to ${\displaystyle f}$.

A simplicial approximation is homotopic to the map it approximates. See simplicial approximation theorem for more details.

Let K and L be two GSCs. A function ${\displaystyle f:|K|\to |L|}$ is called piecewise-linear (PL) if there exist a subdivision K' of K, and a subdivision L' of L, such that ${\displaystyle f:|K'|\to |L'|}$ is a simplicial map of K' into L'. Every simplicial map is PL, but the opposite is not true. For example, suppose |K| and |L| are two triangles, and let ${\displaystyle f:|K|\to |L|}$ be a non-linear function that maps the leftmost half of |K| linearly into the leftmost half of |L|, and maps the rightmost half of |K| linearly into the rightmostt half of |L|. Then f is PL, since it is a simplicial map between a subdivision of |K| into two triangles and a subdivision of |L| into two triangles. This notion is an adaptation of the general notion of a piecewise-linear function to simplicial complexes.

A PL homeomorphism between two polyhedra |K| and |L| is a PL mapping such that the simplicial mapping between the subdivisions, ${\displaystyle f:|K'|\to |L'|}$, is a homeomorphism.