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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 ) → V ( L ) {\displaystyle f:V(K)\to V(L)} , that maps every simplex in K to a simplex in L. That is, for any σ ∈ K {\displaystyle \sigma \in K} , f ( σ ) ∈ L {\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 f {\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 f {\displaystyle f} is bijective, and its inverse f − 1 {\displaystyle f^{-1}} is a simplicial map of L into K, then f {\displaystyle 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 ≅ L {\displaystyle K\cong L} .³: 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 f − 1 {\displaystyle f^{-1}} is not simplicial: f − 1 ( { 4 , 5 , 6 } ) = { 1 , 2 , 3 } {\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.

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 → L {\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 σ ∈ K {\displaystyle \sigma \in K} , conv ⁡ ( f ( V ( σ ) ) ) ∈ L {\displaystyle \operatorname {conv} (f(V(\sigma )))\in L} . Note that this implies that vertices of K are mapped to vertices of L. ⁴

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 | → | L | {\displaystyle f:|K|\to |L|} , that maps every simplex in K linearly to a simplex in L. That is, for any simplex σ ∈ K {\displaystyle \sigma \in K} , f ( σ ) ∈ L {\displaystyle f(\sigma )\in L} , and in addition, f | σ {\displaystyle f\vert _{\sigma }} (the restriction of f {\displaystyle f} to σ {\displaystyle \sigma } ) is a linear function.⁵: 16 ⁶: 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.⁷: 15, Def.1.5.3  Let K, L be two ASCs, and let f : V ( K ) → V ( L ) {\displaystyle f:V(K)\to V(L)} be a simplicial map. The affine extension of f {\displaystyle f} is a mapping | f | : | K | → | L | {\displaystyle |f|:|K|\to |L|} defined as follows. For any point x ∈ | K | {\displaystyle x\in |K|} , let σ {\displaystyle \sigma } be its support (the unique simplex containing x in its interior), and denote the vertices of σ {\displaystyle \sigma } by v 0 , … , v k {\displaystyle v_{0},\ldots ,v_{k}} . The point x {\displaystyle x} has a unique representation as a convex combination of the vertices, x = ∑ i = 0 k a i v i {\displaystyle x=\sum _{i=0}^{k}a_{i}v_{i}} with a i ≥ 0 {\displaystyle a_{i}\geq 0} and ∑ i = 0 k a i = 1 {\displaystyle \sum _{i=0}^{k}a_{i}=1} (the a i {\displaystyle a_{i}} are the barycentric coordinates of x {\displaystyle x} ). We define | f | ( x ) := ∑ i = 0 k a i f ( v i ) {\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|.⁸: 15, Prop.1.5.4 

Simplicial approximation

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

References

1. Munkres, James R. (1995). Elements of Algebraic Topology. Westview Press. ISBN 978-0-201-62728-2. 978-0-201-62728-2 ↩

2. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3 978-3-540-00362-5 ↩

3. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3 978-3-540-00362-5 ↩

4. Munkres, James R. (1995). Elements of Algebraic Topology. Westview Press. ISBN 978-0-201-62728-2. 978-0-201-62728-2 ↩

5. Colin P. Rourke and Brian J. Sanderson (1982). Introduction to Piecewise-Linear Topology. New York: Springer-Verlag. doi:10.1007/978-3-642-81735-9. ISBN 978-3-540-11102-3. 978-3-540-11102-3 ↩

6. Bryant, John L. (2001-01-01), Daverman, R. J.; Sher, R. B. (eds.), "Chapter 5 - Piecewise Linear Topology", Handbook of Geometric Topology, Amsterdam: North-Holland, pp. 219–259, ISBN 978-0-444-82432-5, retrieved 2022-11-15 978-0-444-82432-5 ↩

7. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3 978-3-540-00362-5 ↩

8. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3 978-3-540-00362-5 ↩