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.^[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]^{: 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 $f$ 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 $f$ 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 $K\cong L$ .^[2]^: 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 $f^{-1}$ 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 (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. ^[1]

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]^: 16^[4]^: 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.^[2]^{: 15, Def.1.5.3} 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, $x=\sum _{i=0}^{k}a_{i}v_{i}$ with $a_{i}\geq 0$ and $\sum _{i=0}^{k}a_{i}=1$ (the $a_{i}$ are the barycentric coordinates of $x$ ). We define $|f|(x):=\sum _{i=0}^{k}a_{i}f(v_{i})$ . 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|.^[2]^{: 15, Prop.1.5.4}

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. 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.

References

^ ^a ^b Munkres, James R. (1995). Elements of Algebraic Topology. Westview Press. ISBN 978-0-201-62728-2.
^ ^a ^b ^c ^d 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
^ 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.
^ 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

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

[:0-2] 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

[3] 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.

[4] 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

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