Simplicial map explained

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.

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 .^[2] As an example, let K be 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 ^{-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, 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{conv}(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 _{\sigma} (the restriction of f to \sigma ) is a linear function.^[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. 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} _{i =0} a _{i} v _{i} with a _{i} \geq 0 and \sum ^{k} _{i =0} a _{i} = 1 (the a _{i} are the barycentric coordinates of x ). We define |f|(x) : = \sum ^{k} _{i =0} 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 st (v) for the star of a vertex. A simplicial map f _{\triangle\colon} K \to L such that f( st (v)) \subseteq st (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 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,

f: |K'|\to |L'| , is a homeomorphism.

Notes and References

1. Book: Munkres, James R. . James Munkres

   . Elements of Algebraic Topology . Westview Press . 1995 . 978-0-201-62728-2 . James Munkres.

2. , Section 4.3
3. Book: Colin P. Rourke and Brian J. Sanderson . Introduction to Piecewise-Linear Topology . Springer-Verlag . 1982 . New York . en . 10.1007/978-3-642-81735-9. 978-3-540-11102-3 .

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