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 is a simplex, called a face of , and every one-element subset is a simplex.

A simplex is called -dimensional if it consists of vertices. The maximal dimension of its simplices (which may be infinite) is called the dimension of a simplicial complex . 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 be a set and let be a family of non-empty subsets of . A non-empty finite subset is called a simplex if the set is non-empty. The resulting simplicial complex is called the nerve of the family (cf. Nerve of a family of sets).

A simplicial mapping of a simplicial complex into a simplicial complex is a mapping such that for every simplex in , its image is a simplex in . Simplicial complexes and their simplicial mappings form a category.

If a simplicial mapping is an inclusion, then is called a simplicial subcomplex of . All simplices of a simplicial complex of dimension at most form a simplicial subcomplex of , which is written and is called the -dimensional (or -) skeleton of . A simplicial subcomplex of a simplicial complex is called full if every simplex in whose vertices all belong to is itself in .

Every simplicial complex canonically determines a simplicial set , whose simplices of dimension are all -tuples of vertices of with the property that there is a simplex in such that for each . The boundary operators and the degeneracy operators of are given by the formulas

where denotes the omission of the symbol beneath it. When is ordered one can define a simplicial subset , consisting of those simplices for which . The (co)homology groups of are isomorphic to the (co)homology groups of and called the (co)homology groups of .

To every triangulation (simplicial space) corresponds a simplicial complex, whose vertices are the vertices of and whose simplices are those non-empty finite sets of vertices which span a simplex in . For every simplicial complex there is a triangulation, unique up to an isomorphism, whose simplicial complex is . It is called the geometric realization (or body, or geometric simplicial complex) of , and is denoted by . This yields the geometric model in the sense of Giever–Hu (see Simplicial set) of the simplicial set , and when is ordered, the geometric model in the sense of Milnor of the simplicial set . The correspondence is a covariant functor from the category of simplicial complexes to the category of cellular spaces. A topological space homeomorphic to the body of some simplicial complex is called a polyhedron (or a triangulated space, cf. Polyhedron, abstract) and the pair , where is the homeomorphism, is called a triangulation of .

The points of the topological space can be identified with the functions for which the set is a simplex in and

The number is called the -th barycentric coordinate of . The formula

defines a metric on , but the corresponding metric topology is, in general, stronger than the original one. The set equipped with this metric topology is written as .

A simplicial complex is isomorphic to the nerve of the family of stars of vertices of the space , that is, to the nerve of the family of open subsets , where .

The following statements are equivalent: 1) the simplicial complex is locally finite; 2) the space is locally compact; 3) ; 4) is metrizable; and 5) satisfies the first axiom of countability. Moreover, the space is separable (compact) if and only if is at most countable (finite).

The cells of the complex are in one-to-one correspondence with the simplices of , and the closure of the cell corresponding to a simplex is given by

It is homeomorphic to the -dimensional closed ball, where , so that the complex is regular. In addition, each set has a canonical linear (affine) structure, with respect to which it is isomorphic to the standard simplex . Because of this, and the fact that for all simplices , it turns out that the space can be mapped homeomorphically (can be imbedded) into (where may be transfinite), so that all closed cells are (rectilinear) simplices. This means that the image of in 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 in .

A simplicial complex can only be realized in for finite when is locally finite, at most countable and of finite dimension. Moreover, if , then can be realized in . A simplicial complex consisting of vertices every -element subset of which is a simplex cannot be realized in .

From any simplicial complex one can construct a new simplicial complex, , whose vertices are the simplices of and whose simplices are families of simplices of such that . is called the barycentric refinement (or subdivision) of . The cellular spaces and are naturally homeomorphic (but not isomorphic). Under this homeomorphism, every vertex of (that is, the zero-dimensional cell corresponding to the vertex of ) is mapped onto the centre of gravity (the barycentre) of the closed simplex .

The simplicial complex is ordered in a natural way. If is ordered, then the correspondence (first vertex of ) defines a simplicial mapping that preserves the ordering. It is called the canonical translation. Its geometric realization (which is a continuous mapping ) is homotopic to the natural homeomorphism .

A simplicial mapping (or its geometric realization ) is called a simplicial approximation of a continuous mapping if, for every point , the point belongs to the minimal closed simplex containing the point , or, equivalently, if for every vertex , . The mappings and are homotopic.

The simplicial approximation theorem states that if a simplicial complex is finite, then for every continuous mapping there is an integer such that for all there is a simplicial approximation of (regarded as a mapping ).

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 -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