Simplicial map
A simplicial map 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. A simplicial map 'of K into L is a function from the vertices of K'' to the vertices of L,, that maps every simplex in K to a simplex in L. That is, for any,.' 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=''f=4, f''=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 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 is bijective, and its inverse is a simplicial map of L into K, then 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. The function f defined above is not an isomorphism since it is not bijective. If we modify the definition to f=4, f=5, f=6, then f is bijective but it is still not an isomorphism, since is not simplicial:, 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. A simplicial map 'of K into L is a function such that the images of the vertices of a simplex in K span a simplex in L. That is, for any simplex, . 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 to the underlying space of L, , that maps every simplex in K linearly to a simplex in L. That is, for any simplex,, and in addition, 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 using barycentric coordinates. This can be defined precisely.' Let K, L be two ASCs, and let be a simplicial map. The affine extension of is a mapping defined as follows. For any point, let be its support, and denote the vertices of by. The point has a unique representation as a convex combination of the vertices, with and . We define. 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 be a continuous map between the underlying polyhedra of simplicial complexes and let us write for the star of a vertex. A simplicial map such that, is called a simplicial approximation to.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 is called piecewise-linear '' if there exist a subdivision KA PL homeomorphism between two polyhedra |K| and |L| is a PL mapping such that the simplicial mapping between the subdivisions,, is a homeomorphism.