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 complexes

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 $\sigma\in K$, $f(\sigma)\in L$.'', Section 4.3'' As an example, let K be the ASC containing the sets ,, and their subsets, and let L be the ASC containing the set and its subsets. Define a mapping ''f'' by: ''f''(1)=''f''(2)=4, ''f''(3)=5. Then ''f'' is a simplicial mapping, since ''f''()= which is a simplex in L, ''f''()=f()= which is also a simplex in L, etc.

If $f$ 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 to the zero-dimensional simplex .

If $f$ is bijective, and its inverse $f^$ is a simplicial map of L into K, then $f$ is called a simplicial isomorphism. Isomorphic simplicial complexes are essentially "the same", up to 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 $f^$ is not simplicial: $f^(\)= \$, which is not a simplex in K. If we modify L by removing , that is, L is the ASC containing only the sets ,, and their subsets, then ''f'' is an isomorphism.

Geometric simplicial complexes

Let K and L be two geometric simplicial complexes (GSC). 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 $\sigma\in K$, $\operatorname(f(V(\sigma)))\in L$. Note that this implies that vertices of K are mapped to vertices of L.

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, $f\vert_$ (the restriction of $f$ to $\sigma$) is a linear function. 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.'''' Let K, L be two ASCs, and let $f: V(K)\to V(L)$ be a simplicial map. The affine extension of $f$ is a mapping $, f, : , K, \to , L,$ defined as follows. For any point $x\in , K,$, let $\sigma$ be its support (the unique simplex containing ''x'' in its interior), and denote the vertices of $\sigma$ by $v_0,\ldots,v_k$. The point $x$ has a unique representation as a convex combination of the vertices, $x = \sum_^k a_i v_i$ with $a_i \geq 0$ and $\sum_^k a_i = 1$ (the $a_i$ are the barycentric coordinates of $x$). We define $, f, (x) := \sum_^k a_i f(v_i)$. 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'', .''''

Simplicial approximation

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

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

Piecewise-linear maps

Let K and L be two GSCs. A function $f: , K, \to , L,$ is called piecewise-linear (PL) if there exist a subdivision ''K''' of ''K'', and a subdivision ''L''' of ''L'', such that $f: , K', \to , L',$ 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 $f: , K, \to , L,$ 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 rightmost 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, $f: , K', \to , L',$, is a homeomorphism.

References

{{Reflist
Algebraic topology
Simplicial homology
Simplicial sets