Abstract simplicial complex


In combinatorics, an abstract simplicial complex is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles, their edges, and their vertices.
In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems.
An abstract simplex can be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra.

Definitions

A collection of non-empty finite subsets of a set S is called a set-family.
A set-family is called an abstract simplicial complex if, for every set in, and every non-empty subset, also belongs to.
The finite sets that belong to are called faces of the complex, and a face is said to belong to another face if, so the definition of an abstract simplicial complex can be restated as saying that every face of a face of a complex is itself a face of. The vertex set of is defined as, the union of all faces of. The elements of the vertex set are called the vertices of the complex. For every vertex v of, the set is a face of the complex, and every face of the complex is a finite subset of the vertex set.
The maximal faces of are called facets of the complex. The dimension of a face in is defined as : faces consisting of a single element are zero-dimensional, faces consisting of two elements are one-dimensional, etc. The dimension of the complex is defined as the largest dimension of any of its faces, or infinity if there is no finite bound on the dimension of the faces.
The complex is said to be finite if it has finitely many faces, or equivalently if its vertex set is finite. Also, is said to be pure if it is finite-dimensional and every facet has the same dimension. In other words, is pure if is finite and every face is contained in a facet of dimension.
One-dimensional abstract simplicial complexes are mathematically equivalent to simple undirected graphs: the vertex set of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspond to undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that do not have any incident edges.
A subcomplex of is an abstract simplicial complex L such that every face of L belongs to ; that is, and L is an abstract simplicial complex. A subcomplex that consists of all of the subsets of a single face of is often called a simplex of..
The d-skeleton of is the subcomplex of consisting of all of the faces of that have dimension at most d. In particular, the 1-skeleton is called the underlying graph of. The 0-skeleton of can be identified with its vertex set, although formally it is not quite the same thing.
The link of a face in, often denoted or, is the subcomplex of defined by
Note that the link of the empty set is itself.
Given two abstract simplicial complexes, and, a simplicial map is a function that maps the vertices of to the vertices of and that has the property that for any face of, the image is a face of. There is a category SCpx with abstract simplicial complexes as objects and simplicial maps as morphisms. This is equivalent to a suitable category defined using non-abstract simplicial complexes.
Moreover, the categorical point of view allows us to tighten the relation between the underlying set S of an abstract simplicial complex and the vertex set of : for the purposes of defining a category of abstract simplicial complexes, the elements of S not lying in are irrelevant. More precisely, SCpx is equivalent to the category where:
We can associate to an abstract simplicial complex K a topological space, called its geometric realization, which is the carrier of a simplicial complex. The construction goes as follows.
First, define as a subset of consisting of functions satisfying the two conditions:
Now think of the set of elements of with finite support as the direct limit of where A ranges over finite subsets of S, and give that direct limit the induced topology. Now give the subspace topology.
Alternatively, let denote the category whose objects are the faces of and whose morphisms are inclusions. Next choose a total order on the vertex set of and define a functor F from to the category of topological spaces as follows. For any face X in K of dimension n, let be the standard n-simplex. The order on the vertex set then specifies a unique bijection between the elements of and vertices of, ordered in the usual way. If is a face of dimension, then this bijection specifies a unique m-dimensional face of. Define to be the unique affine linear embedding of as that distinguished face of, such that the map on vertices is order-preserving.
We can then define the geometric realization as the colimit of the functor F. More specifically is the quotient space of the disjoint union
by the equivalence relation which identifies a point with its image under the map, for every inclusion.
If K is finite, then we can describe more simply. Choose an embedding of the vertex set of K as an affinely independent subset of some Euclidean space of sufficiently high dimension N. Then any face X in K can be identified with the geometric simplex in spanned by the corresponding embedded vertices. Take to be the union of all such simplices.
If K is the standard combinatorial n-simplex, then can be naturally identified with.

Examples

1. Let V be a finite set of cardinality. The combinatorial n-simplex with vertex-set V is an ASC whose faces are all subsets of V. If then this ASC is called the standard combinatorial n-simplex.
2. Let G be an undirected graph. The clique complex of G is an ASC whose faces are all cliques of G. The independence complex of G is an ASC whose faces are all independent sets of G. Clique complexes are the prototypical example of flag complexes. A flag complex is a complex K with the property that every set of elements that pairwise belong to faces of K is itself a face of K.
3. Let H be a hypergraph. A matching in H is a set of edges of H, in every two edges have an empty intersection. The matching complex of H is an ASC whose faces are all matchings in H. It is the independence complex of the line graph of H.
4. Let P be a partially ordered set. The order complex of P is an ASC whose faces are all finite chains in P. Its homology groups and other topological invariants contain important information about the poset P.
5. Let M be a metric space and δ a real number. The Vietoris–Rips complex is an ASC whose faces are the finite subsets of M with diameter at most δ. It has applications in homology theory, hyperbolic groups, image processing, and mobile ad hoc networking. It is another example of a flag complex.
6. Let be a square-free monomial ideal in a polynomial ring . Then the exponent vectors of those square-free monomials of that are not in determine an abstract simplicial complex via the map. In fact, there is a bijection between abstract simplicial complexes on vertices and square-free monomial ideals in. If is the square-free ideal corresponding to the simplicial complex then the quotient is known as the Stanley–Reisner ring of.

Enumeration

The number of abstract simplicial complexes on up to n elements is one less than the nth Dedekind number. These numbers grow very rapidly, and are known only for ; they are :
The number of abstract simplicial complexes whose vertices are exactly n labeled elements is given by the sequence "1, 2, 9, 114, 6894, 7785062, 2414627396434, 56130437209370320359966" , starting at n = 1. This corresponds to the number of antichain covers of a labeled n-set; there is a clear bijection between antichain covers of an n-set and simplicial complexes on n elements described in terms of their maximal faces.
The number of abstract simplicial complexes on exactly n unlabeled elements is given by the sequence "1, 2, 5, 20, 180, 16143" , starting at n = 1.

Relation to other concepts

An abstract simplicial complex with an additional property called the augmentation property or the exchange property yields a matroid. The following expression shows the relations between the terms:
HYPERGRAPHS = SET-FAMILIES ⊃ INDEPENDENCE-SYSTEMS = ABSTRACT-SIMPLICIAL-COMPLEXES ⊃ MATROIDS.