Namespaces
Variants
Actions

Simplicial complex

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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

Comments

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)

Comments

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
How to Cite This Entry:
Simplicial complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simplicial_complex&oldid=55462
This article was adapted from an original article by S.N. MalyginM.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article