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| ''simplicial scheme, abstract simplicial complex'' | | ''simplicial scheme, abstract simplicial complex'' |
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− | A set, whose elements are called vertices, in which a family of finite non-empty subsets, called simplexes or simplices, is distinguished, such that every non-empty subset of a simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853601.png" /> is a simplex, called a face of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853602.png" />, and every one-element subset is a simplex. | + | A set, whose elements are called vertices, in which a family of finite non-empty subsets, called simplexes or simplices, is distinguished, such that every non-empty subset of a simplex $s$ is a simplex, called a face of $s$, and every one-element subset is a simplex. |
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− | A simplex is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853604.png" />-dimensional if it consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853605.png" /> vertices. The maximal dimension of its simplices (which may be infinite) is called the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853606.png" /> of a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853607.png" />. A simplicial complex is called locally finite if each of its vertices belongs to only finitely many simplices. A simplicial complex is called ordered if its vertices admit a partial ordering that is linear on every simplex. | + | A simplex is called $q$-dimensional if it consists of $q+1$ vertices. The maximal dimension of its simplices (which may be infinite) is called the dimension $\dim k$ of a simplicial complex $K$. A simplicial complex is called locally finite if each of its vertices belongs to only finitely many simplices. A simplicial complex is called ordered if its vertices admit a partial ordering that is linear on every simplex. |
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− | Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853608.png" /> be a set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s0853609.png" /> be a family of non-empty subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536010.png" />. A non-empty finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536011.png" /> is called a simplex if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536012.png" /> is non-empty. The resulting simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536013.png" /> is called the nerve of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536014.png" /> (cf. [[Nerve of a family of sets|Nerve of a family of sets]]). | + | Example. Let $X$ be a set and let $U = \{U_\alpha : \alpha \in A\}$ be a family of non-empty subsets of $X$. A non-empty finite subset $\alpha \in A$ is called a simplex if the set $\cap_{\alpha \in A} U_\alpha$ is non-empty. The resulting simplicial complex $A$ is called the nerve of the family $U$ (cf. [[Nerve of a family of sets|Nerve of a family of sets]]). |
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− | A simplicial mapping of a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536015.png" /> into a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536016.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536017.png" /> such that for every simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536019.png" />, its image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536020.png" /> is a simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536021.png" />. Simplicial complexes and their simplicial mappings form a category. | + | A simplicial mapping of a simplicial complex $K_1$ into a simplicial complex $K_2$ is a mapping $f: K_1\to K_2$ such that for every simplex $s$ in $K_1$, its image $f(x)$ is a simplex in $K_2$. Simplicial complexes and their simplicial mappings form a category. |
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− | If a simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536022.png" /> is an inclusion, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536023.png" /> is called a simplicial subcomplex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536024.png" />. All simplices of a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536025.png" /> of dimension at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536026.png" /> form a simplicial subcomplex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536027.png" />, which is written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536028.png" /> and is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536030.png" />-dimensional (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536031.png" />-) skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536033.png" />. A simplicial subcomplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536034.png" /> of a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536035.png" /> is called full if every simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536036.png" /> whose vertices all belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536037.png" /> is itself in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536038.png" />. | + | If a simplicial mapping $f : L \to K$ is an inclusion, then $L$ is called a simplicial subcomplex of $K$. All simplices of a simplicial complex $K$ of dimension at most $n$ form a simplicial subcomplex of $K$, which is written $K^n$ and is called the $n$-dimensional (or $n$-) skeleton of $K$. A simplicial subcomplex $L$ of a simplicial complex $K$ is called full if every simplex in $K$ whose vertices all belong to $L$ is itself in $L$. |
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− | Every simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536039.png" /> canonically determines a [[Simplicial set|simplicial set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536040.png" />, whose simplices of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536041.png" /> are all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536042.png" />-tuples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536043.png" /> of vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536044.png" /> with the property that there is a simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536045.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536046.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536047.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536048.png" />. The boundary operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536049.png" /> and the degeneracy operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536050.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536051.png" /> are given by the formulas | + | Every simplicial complex $K$ canonically determines a [[Simplicial set|simplicial set]] $O(K)$, whose simplices of dimension $n$ are all $(n+1)$-tuples $(x_0, \ldots, x_n)$ of vertices of $K$ with the property that there is a simplex $s$ in $K$ such that $x_i \in s$ for each $i=0,\ldots,n$. The boundary operators $d_i$ and the degeneracy operators $s_i$ of $O(K)$ are given by the formulas |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536052.png" /></td> </tr></table>
| + | $$ |
| + | \begin{gathered} |
| + | d_i(x_0, \ldots, x_n) = (x_0, \ldots, \widehat{x_i}, \ldots, x_n),\\ |
| + | s_i(x_0, \ldots, x_n) = (x_0, \ldots, x_i, x_i, x_{i+1}, \ldots, x_n), |
| + | \end{gathered} |
| + | $$ |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536053.png" /></td> </tr></table>
| + | where $\widehat{(-)}$ denotes the omission of the symbol beneath it. When $K$ is ordered one can define a simplicial subset $O^+(K) \subset O(K)$, consisting of those simplices $(x_0, \ldots, x_n)$ for which $x_0\le \cdots \le x_n$. The (co)homology groups of $O(K)$ are isomorphic to the (co)homology groups of $O^+(K)$ and called the (co)homology groups of $K$. |
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− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536054.png" /> denotes the omission of the symbol beneath it. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536055.png" /> is ordered one can define a simplicial subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536056.png" />, consisting of those simplices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536057.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536058.png" />. The (co)homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536059.png" /> are isomorphic to the (co)homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536060.png" /> and called the (co)homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536061.png" />.
| + | To every triangulation ([[Simplicial space|simplicial space]]) $X$ corresponds a simplicial complex, whose vertices are the vertices of $X$ and whose simplices are those non-empty finite sets of vertices which span a simplex in $X$. For every simplicial complex $K$ there is a triangulation, unique up to an isomorphism, whose simplicial complex is $K$. It is called the geometric realization (or body, or geometric simplicial complex) of $K$, and is denoted by $|K|$. This yields the geometric model in the sense of Giever–Hu (see [[Simplicial set|Simplicial set]]) $\|O(K)\|$ of the simplicial set $O(K)$, and when $K$ is ordered, the geometric model in the sense of Milnor $|O^+(K)|$ of the simplicial set $O^+(K)$. The correspondence $K\mapsto\|O(K)\|$ is a covariant functor from the category of simplicial complexes to the category of cellular spaces. A topological space $X$ homeomorphic to the body $|K|$ of some simplicial complex $K$ is called a polyhedron (or a triangulated space, cf. [[Polyhedron, abstract|Polyhedron, abstract]]) and the pair $(K,f)$, where $f:|K|\to X$ is the homeomorphism, is called a triangulation of $X$. |
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− | To every triangulation ([[Simplicial space|simplicial space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536062.png" /> corresponds a simplicial complex, whose vertices are the vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536063.png" /> and whose simplices are those non-empty finite sets of vertices which span a simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536064.png" />. For every simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536065.png" /> there is a triangulation, unique up to an isomorphism, whose simplicial complex is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536066.png" />. It is called the geometric realization (or body, or geometric simplicial complex) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536067.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536068.png" />. This yields the geometric model in the sense of Giever–Hu (see [[Simplicial set|Simplicial set]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536069.png" /> of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536070.png" />, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536071.png" /> is ordered, the geometric model in the sense of Milnor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536072.png" /> of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536073.png" />. The correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536074.png" /> is a covariant functor from the category of simplicial complexes to the category of cellular spaces. A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536075.png" /> homeomorphic to the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536076.png" /> of some simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536077.png" /> is called a polyhedron (or a triangulated space, cf. [[Polyhedron, abstract|Polyhedron, abstract]]) and the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536078.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536079.png" /> is the homeomorphism, is called a triangulation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536080.png" />.
| + | The points of the topological space $|K|$ can be identified with the functions $\alpha : K \to [0,1]$ for which the set $\{x\in K: \alpha(x) \ne 0\}$ is a simplex in $K$ and |
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− | The points of the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536081.png" /> can be identified with the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536082.png" /> for which the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536083.png" /> is a simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536084.png" /> and
| + | $$ |
| + | \sum_{x\in K} \alpha(x) = 1. |
| + | $$ |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536085.png" /></td> </tr></table>
| + | The number $\alpha(x)$ is called the $x$-th barycentric coordinate of $\alpha$. The formula |
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− | The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536086.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536087.png" />-th barycentric coordinate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536088.png" />. The formula
| + | $$ |
| + | d(\alpha, \beta) = \sqrt{\sum_{x\in K} (\alpha(x) - \beta(x))^2} |
| + | $$ |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536089.png" /></td> </tr></table>
| + | defines a metric on $|K|$, but the corresponding metric topology is, in general, stronger than the original one. The set $|K|$ equipped with this metric topology is written as $|K|_d$. |
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− | defines a metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536090.png" />, but the corresponding metric topology is, in general, stronger than the original one. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536091.png" /> equipped with this metric topology is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536092.png" />.
| + | A simplicial complex $K$ is isomorphic to the nerve of the family of stars of vertices of the space $|K|$, that is, to the nerve of the family of open subsets $\operatorname{St} x = \{\alpha \in |K|: \alpha(x) \ne 0\}$, where $x \in K$. |
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− | A simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536093.png" /> is isomorphic to the nerve of the family of stars of vertices of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536094.png" />, that is, to the nerve of the family of open subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536095.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536096.png" />.
| + | The following statements are equivalent: 1) the simplicial complex $K$ is locally finite; 2) the space $|K|$ is locally compact; 3) $|K| = |K|_d$; 4) $|K|$ is metrizable; and 5) $|K|$ satisfies the [[First axiom of countability|first axiom of countability]]. Moreover, the space $|K|$ is separable (compact) if and only if $K$ is at most countable (finite). |
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− | The following statements are equivalent: 1) the simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536097.png" /> is locally finite; 2) the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536098.png" /> is locally compact; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s08536099.png" />; 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360100.png" /> is metrizable; and 5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360101.png" /> satisfies the [[First axiom of countability|first axiom of countability]]. Moreover, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360102.png" /> is separable (compact) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360103.png" /> is at most countable (finite). | + | The cells of the complex $|K|$ are in one-to-one correspondence with the simplices of $K$, and the closure $|s|$ of the cell corresponding to a simplex $s$ is given by |
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− | The cells of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360104.png" /> are in one-to-one correspondence with the simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360105.png" />, and the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360106.png" /> of the cell corresponding to a simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360107.png" /> is given by
| + | $$ |
− | | + | |s| = \{\alpha \in |K| : \alpha(x) \ne 0 \implies x \in s \}. |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360108.png" /></td> </tr></table>
| + | $$ |
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− | It is homeomorphic to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360109.png" />-dimensional closed ball, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360110.png" />, so that the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360111.png" /> is regular. In addition, each set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360112.png" /> has a canonical linear (affine) structure, with respect to which it is isomorphic to the [[Standard simplex|standard simplex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360113.png" />. Because of this, and the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360114.png" /> for all simplices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360115.png" />, it turns out that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360116.png" /> can be mapped homeomorphically (can be imbedded) into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360117.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360118.png" /> may be transfinite), so that all closed cells <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360119.png" /> are (rectilinear) simplices. This means that the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360120.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360121.png" /> is a [[Simplicial space|simplicial space]] (a polyhedron), i.e. a union of closed simplices intersecting only on entire faces. This simplicial space is called a realization of the simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360122.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360123.png" />. | + | It is homeomorphic to the $q$-dimensional closed ball, where $q = \dim s$, so that the complex $K$ is regular. In addition, each set $|s|$ has a canonical linear (affine) structure, with respect to which it is isomorphic to the [[Standard simplex|standard simplex]] $\Delta^q$. Because of this, and the fact that $|s \cap s'| = |s| \cap |s'|$ for all simplices $s,s' \subset K$, it turns out that the space $|K|$ can be mapped homeomorphically (can be imbedded) into $\R^n$ (where $n$ may be transfinite), so that all closed cells $|s|$ are (rectilinear) simplices. This means that the image of $|K|$ in $\R^n$ is a [[Simplicial space|simplicial space]] (a polyhedron), i.e. a union of closed simplices intersecting only on entire faces. This simplicial space is called a realization of the simplicial complex $K$ in $\R^n$. |
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− | A simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360124.png" /> can only be realized in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360125.png" /> for finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360126.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360127.png" /> is locally finite, at most countable and of finite dimension. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360128.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360129.png" /> can be realized in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360130.png" />. A simplicial complex consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360131.png" /> vertices every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360132.png" />-element subset of which is a simplex cannot be realized in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360133.png" />. | + | A simplicial complex $K$ can only be realized in $\R^n$ for finite $n$ when $K$ is locally finite, at most countable and of finite dimension. Moreover, if $\dim K \le n$, then $K$ can be realized in $\R^{2n+1}$. A simplicial complex consisting of $2n+3$ vertices every $(n+1)$-element subset of which is a simplex cannot be realized in $\R^{2n}$. |
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− | From any simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360134.png" /> one can construct a new simplicial complex, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360135.png" />, whose vertices are the simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360136.png" /> and whose simplices are families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360137.png" /> of simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360138.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360139.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360140.png" /> is called the barycentric refinement (or subdivision) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360141.png" />. The cellular spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360143.png" /> are naturally homeomorphic (but not isomorphic). Under this homeomorphism, every vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360144.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360145.png" /> (that is, the zero-dimensional cell corresponding to the vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360146.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360147.png" />) is mapped onto the centre of gravity (the barycentre) of the closed simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360148.png" />. | + | From any simplicial complex $K$ one can construct a new simplicial complex, $\operatorname{Bd} K$, whose vertices are the simplices of $K$ and whose simplices are families $(s_0, \ldots, s_q)$ of simplices of $K$ such that $s_0 \subset \dots \subset s_q$. $\operatorname{Bd} K$ is called the barycentric refinement (or subdivision) of $K$. The cellular spaces $|\operatorname{Bd} K|$ and $|K|$ are naturally homeomorphic (but not isomorphic). Under this homeomorphism, every vertex $|s|$ of $|\operatorname{Bd} K|$ (that is, the zero-dimensional cell corresponding to the vertex $s$ of $\operatorname{Bd} K$) is mapped onto the centre of gravity (the barycentre) of the closed simplex $|s| \subset |K|$. |
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− | The simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360149.png" /> is ordered in a natural way. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360150.png" /> is ordered, then the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360151.png" /> (first vertex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360152.png" />) defines a simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360153.png" /> that preserves the ordering. It is called the canonical translation. Its geometric realization (which is a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360154.png" />) is homotopic to the natural homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360155.png" />. | + | The simplicial complex $\operatorname{Bd} K$ is ordered in a natural way. If $K$ is ordered, then the correspondence $s \mapsto$ (first vertex of $s$) defines a simplicial mapping $\operatorname{Bd} K\to K$ that preserves the ordering. It is called the canonical translation. Its geometric realization (which is a continuous mapping $|\operatorname{Bd} K| \to |K|$) is homotopic to the natural homeomorphism $|\operatorname{Bd} K| \to |K|$. |
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− | A simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360156.png" /> (or its geometric realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360157.png" />) is called a simplicial approximation of a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360158.png" /> if, for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360159.png" />, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360160.png" /> belongs to the minimal closed simplex containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360161.png" />, or, equivalently, if for every vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360162.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360163.png" />. The mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360164.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360165.png" /> are homotopic. | + | A simplicial mapping $\phi : K \to L$ (or its geometric realization $|\phi| : |K| \to |L|$) is called a simplicial approximation of a continuous mapping $f : |K| \to |L|$ if, for every point $\alpha \in |K|$, the point $|\phi|(\alpha)$ belongs to the minimal closed simplex containing the point $f(\alpha)$, or, equivalently, if for every vertex $x \in K$, $f( \operatorname{St} x) \subset \operatorname{St} \phi(x)$. The mappings $f$ and $|\phi|$ are homotopic. |
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− | The simplicial approximation theorem states that if a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360166.png" /> is finite, then for every continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360167.png" /> there is an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360168.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360169.png" /> there is a simplicial approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360170.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360171.png" /> (regarded as a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085360/s085360172.png" />). | + | The simplicial approximation theorem states that if a simplicial complex $K$ is finite, then for every continuous mapping $f : |K| \to |L|$ there is an integer $N$ such that for all $n \ge N$ there is a simplicial approximation $\operatorname{Bd}^n : K \to L$ of $f$ (regarded as a mapping $|\operatorname{Bd}^n K| \to |L|$). |
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| ====References==== | | ====References==== |
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simplicial scheme, abstract simplicial complex
A set, whose elements are called vertices, in which a family of finite non-empty subsets, called simplexes or simplices, is distinguished, such that every non-empty subset of a simplex $s$ is a simplex, called a face of $s$, and every one-element subset is a simplex.
A simplex is called $q$-dimensional if it consists of $q+1$ vertices. The maximal dimension of its simplices (which may be infinite) is called the dimension $\dim k$ of a simplicial complex $K$. A simplicial complex is called locally finite if each of its vertices belongs to only finitely many simplices. A simplicial complex is called ordered if its vertices admit a partial ordering that is linear on every simplex.
Example. Let $X$ be a set and let $U = \{U_\alpha : \alpha \in A\}$ be a family of non-empty subsets of $X$. A non-empty finite subset $\alpha \in A$ is called a simplex if the set $\cap_{\alpha \in A} U_\alpha$ is non-empty. The resulting simplicial complex $A$ is called the nerve of the family $U$ (cf. Nerve of a family of sets).
A simplicial mapping of a simplicial complex $K_1$ into a simplicial complex $K_2$ is a mapping $f: K_1\to K_2$ such that for every simplex $s$ in $K_1$, its image $f(x)$ is a simplex in $K_2$. Simplicial complexes and their simplicial mappings form a category.
If a simplicial mapping $f : L \to K$ is an inclusion, then $L$ is called a simplicial subcomplex of $K$. All simplices of a simplicial complex $K$ of dimension at most $n$ form a simplicial subcomplex of $K$, which is written $K^n$ and is called the $n$-dimensional (or $n$-) skeleton of $K$. A simplicial subcomplex $L$ of a simplicial complex $K$ is called full if every simplex in $K$ whose vertices all belong to $L$ is itself in $L$.
Every simplicial complex $K$ canonically determines a simplicial set $O(K)$, whose simplices of dimension $n$ are all $(n+1)$-tuples $(x_0, \ldots, x_n)$ of vertices of $K$ with the property that there is a simplex $s$ in $K$ such that $x_i \in s$ for each $i=0,\ldots,n$. The boundary operators $d_i$ and the degeneracy operators $s_i$ of $O(K)$ are given by the formulas
$$
\begin{gathered}
d_i(x_0, \ldots, x_n) = (x_0, \ldots, \widehat{x_i}, \ldots, x_n),\\
s_i(x_0, \ldots, x_n) = (x_0, \ldots, x_i, x_i, x_{i+1}, \ldots, x_n),
\end{gathered}
$$
where $\widehat{(-)}$ denotes the omission of the symbol beneath it. When $K$ is ordered one can define a simplicial subset $O^+(K) \subset O(K)$, consisting of those simplices $(x_0, \ldots, x_n)$ for which $x_0\le \cdots \le x_n$. The (co)homology groups of $O(K)$ are isomorphic to the (co)homology groups of $O^+(K)$ and called the (co)homology groups of $K$.
To every triangulation (simplicial space) $X$ corresponds a simplicial complex, whose vertices are the vertices of $X$ and whose simplices are those non-empty finite sets of vertices which span a simplex in $X$. For every simplicial complex $K$ there is a triangulation, unique up to an isomorphism, whose simplicial complex is $K$. It is called the geometric realization (or body, or geometric simplicial complex) of $K$, and is denoted by $|K|$. This yields the geometric model in the sense of Giever–Hu (see Simplicial set) $\|O(K)\|$ of the simplicial set $O(K)$, and when $K$ is ordered, the geometric model in the sense of Milnor $|O^+(K)|$ of the simplicial set $O^+(K)$. The correspondence $K\mapsto\|O(K)\|$ is a covariant functor from the category of simplicial complexes to the category of cellular spaces. A topological space $X$ homeomorphic to the body $|K|$ of some simplicial complex $K$ is called a polyhedron (or a triangulated space, cf. Polyhedron, abstract) and the pair $(K,f)$, where $f:|K|\to X$ is the homeomorphism, is called a triangulation of $X$.
The points of the topological space $|K|$ can be identified with the functions $\alpha : K \to [0,1]$ for which the set $\{x\in K: \alpha(x) \ne 0\}$ is a simplex in $K$ and
$$
\sum_{x\in K} \alpha(x) = 1.
$$
The number $\alpha(x)$ is called the $x$-th barycentric coordinate of $\alpha$. The formula
$$
d(\alpha, \beta) = \sqrt{\sum_{x\in K} (\alpha(x) - \beta(x))^2}
$$
defines a metric on $|K|$, but the corresponding metric topology is, in general, stronger than the original one. The set $|K|$ equipped with this metric topology is written as $|K|_d$.
A simplicial complex $K$ is isomorphic to the nerve of the family of stars of vertices of the space $|K|$, that is, to the nerve of the family of open subsets $\operatorname{St} x = \{\alpha \in |K|: \alpha(x) \ne 0\}$, where $x \in K$.
The following statements are equivalent: 1) the simplicial complex $K$ is locally finite; 2) the space $|K|$ is locally compact; 3) $|K| = |K|_d$; 4) $|K|$ is metrizable; and 5) $|K|$ satisfies the first axiom of countability. Moreover, the space $|K|$ is separable (compact) if and only if $K$ is at most countable (finite).
The cells of the complex $|K|$ are in one-to-one correspondence with the simplices of $K$, and the closure $|s|$ of the cell corresponding to a simplex $s$ is given by
$$
|s| = \{\alpha \in |K| : \alpha(x) \ne 0 \implies x \in s \}.
$$
It is homeomorphic to the $q$-dimensional closed ball, where $q = \dim s$, so that the complex $K$ is regular. In addition, each set $|s|$ has a canonical linear (affine) structure, with respect to which it is isomorphic to the standard simplex $\Delta^q$. Because of this, and the fact that $|s \cap s'| = |s| \cap |s'|$ for all simplices $s,s' \subset K$, it turns out that the space $|K|$ can be mapped homeomorphically (can be imbedded) into $\R^n$ (where $n$ may be transfinite), so that all closed cells $|s|$ are (rectilinear) simplices. This means that the image of $|K|$ in $\R^n$ is a simplicial space (a polyhedron), i.e. a union of closed simplices intersecting only on entire faces. This simplicial space is called a realization of the simplicial complex $K$ in $\R^n$.
A simplicial complex $K$ can only be realized in $\R^n$ for finite $n$ when $K$ is locally finite, at most countable and of finite dimension. Moreover, if $\dim K \le n$, then $K$ can be realized in $\R^{2n+1}$. A simplicial complex consisting of $2n+3$ vertices every $(n+1)$-element subset of which is a simplex cannot be realized in $\R^{2n}$.
From any simplicial complex $K$ one can construct a new simplicial complex, $\operatorname{Bd} K$, whose vertices are the simplices of $K$ and whose simplices are families $(s_0, \ldots, s_q)$ of simplices of $K$ such that $s_0 \subset \dots \subset s_q$. $\operatorname{Bd} K$ is called the barycentric refinement (or subdivision) of $K$. The cellular spaces $|\operatorname{Bd} K|$ and $|K|$ are naturally homeomorphic (but not isomorphic). Under this homeomorphism, every vertex $|s|$ of $|\operatorname{Bd} K|$ (that is, the zero-dimensional cell corresponding to the vertex $s$ of $\operatorname{Bd} K$) is mapped onto the centre of gravity (the barycentre) of the closed simplex $|s| \subset |K|$.
The simplicial complex $\operatorname{Bd} K$ is ordered in a natural way. If $K$ is ordered, then the correspondence $s \mapsto$ (first vertex of $s$) defines a simplicial mapping $\operatorname{Bd} K\to K$ that preserves the ordering. It is called the canonical translation. Its geometric realization (which is a continuous mapping $|\operatorname{Bd} K| \to |K|$) is homotopic to the natural homeomorphism $|\operatorname{Bd} K| \to |K|$.
A simplicial mapping $\phi : K \to L$ (or its geometric realization $|\phi| : |K| \to |L|$) is called a simplicial approximation of a continuous mapping $f : |K| \to |L|$ if, for every point $\alpha \in |K|$, the point $|\phi|(\alpha)$ belongs to the minimal closed simplex containing the point $f(\alpha)$, or, equivalently, if for every vertex $x \in K$, $f( \operatorname{St} x) \subset \operatorname{St} \phi(x)$. The mappings $f$ and $|\phi|$ are homotopic.
The simplicial approximation theorem states that if a simplicial complex $K$ is finite, then for every continuous mapping $f : |K| \to |L|$ there is an integer $N$ such that for all $n \ge N$ there is a simplicial approximation $\operatorname{Bd}^n : K \to L$ of $f$ (regarded as a mapping $|\operatorname{Bd}^n K| \to |L|$).
References
[1] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) |
[2] | P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960) |
[3] | J.H.C. Whitehead, "Simplicial spaces, nuclei and $M$-groups" Proc. London Math. Soc. , 45 (1939) pp. 243–327 |
In the West, the concept described here is usually called an (abstract) simplicial complex; the term simplicial scheme would normally be understood to mean a simplicial object in the category of schemes (cf. Simplicial object in a category).
References
[a1] | C.R.F. Maunder, "Algebraic topology" , v. Nostrand (1972) |
[a2] | S. Lefshetz, "Topology" , Chelsea, reprint (1956) |
[a3] | K. Lamotke, "Semisimpliziale algebraische Topologie" , Springer (1968) |
A facet of an abstract simplicial complex is a maximal face. A complex is pure if all facets have the same dimension.
For a face $F$ of a simplicial complex $K$, we let $F^\Delta$ denote all faces contained in $F$. A shelling is a linear order $\sqsubseteq$ on the facets of $K$, such that for a facet $F$,
$$
\bigcup_{G \sqsubset F} G^\Delta \cap F^\Delta
$$
is a subcomplex generated by the codimension 1 faces of $F$. A complex is shellable if it is pure and possesses a shelling (some authors omit the requirement to be pure). If a complex is shellable then its face ring is Cohen–Macaulay.
References
[b1] | Ezra Miller, Bernd Sturmfels, "Combinatorial commutative algebra" Graduate Texts in Mathematics 227 Springer (2005) ISBN 0-387-23707-0 Zbl 1090.13001 |
[b2] | Richard P. Stanley, "Combinatorics and commutative algebra" , (2nd ed.)mBirkhäuser (1996) ISBN 0-81764-369-9 Zbl 1157.13302 Zbl 0838.13008 |