Simplicial complex

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