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. 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.

Definitions

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

Abstract simplicial complexs

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, $f:V(K)\to V(L)$, that maps every simplex in K to a simplex in L. That is, for any $\sigma \in K$, $f(\sigma )\in L$. As an example, let K be 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 $f$ is bijective, and its inverse $f^{-1}$ is a simplicial map of L into K, then $f$ 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 $K\cong L$. 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 $f^{-1}$ is not simplicial: $f^{-1}(\{4,5,6\})=\{1,2,3\}$, 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.

Geometric simplicial complexes

Let K and L be two geometric simplicial complexes. A simplicial map of K into L is a function $f:K\to L$ such that the images of the vertices of a simplex in K span a simplex in L. That is, for any simplex $\sigma \in K$, $\operatorname {conv} (f(\sigma ))\in L$. Note that this implies that vertices have vertices for images. Simplicial maps are thus determined by their effects on vertices. In particular, there are a finite number of simplicial maps between two given finite simplicial complexes. Simplicial maps induce continuous maps between the underlying polyhedra of the simplicial complexes: one simply extends linearly using barycentric coordinates.

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, $f:|K|\to |L|$, that maps every simplex in K linearly to a simplex in L. That is, for any simplex $\sigma \in K$, $f(\sigma )\in L$, and in addition, $f\vert _{\sigma }$ (the restriction of $f$ to $\sigma$) is a linear function.

Simplicial approximation

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

A simplicial approximation is homotopic to the map it approximates.