Abstract Simplicial Complexes

Definition of Abstract Simplicial Complexes and Maps Between Them... 

Definition 2.1. A finite abstract simplicial complex is a finite set A together with a collection A of subsets of A such that if X A and Y C then Y A. 

Given two finite abstract simplicial complexes Ai and A2 such that a Ai implies a A2, we say that Ai is an abstract simplicial subcomplex of A2, and write Zii C 2i2- If in addition there exists a A2 such that <7 Zli, we say that Zii is a proper subcomplex of A2. 

One also talks about the dimension of each simplex, which is 1 less than its cardinality as a set. When a simplex has dimension d, one says d-simplex. Vertices have dimension 0. The dimension is defined for finite abstract simplicial complexes as well it is equal to the maximum of the dimensions of its simplices. The dimension is denoted by dim. If zli is an abstract simplicial subcomplex of A2, then dim/ii < dim/l2-... 

Some words about degeneracies occurring in this context are in order. In Definition 2.1 we have allowed the empty collection of sets. The corresponding abstract simplicial complex is called the void abstract simplicial complex, and is denoted by 0, or by. The void abstract simplicial complex is the only one that does not contain the empty set as one of its simplices. 

The meticulous reader may have noticed that the term simplex was used by us in two ways, first, to denote any set that is in the collection of sets defining an abstract simplicial complex, and second to denote the abstract simplicial complex whose collection consists of all sets. The distinction is of course purely formal, and the reason one usually uses the same term for these two notions is that one can associate to each 6 A the simplex A, which in turn will be a subcomplex of A. 

The identity map on the set of vertices is always a simplicial map from the abstract simplicial complex onto itself. 

Whenever Z i and Z 2 are abstract simplicial complexes such that Ai is a subcomplex of Z 2, we have a natural simplicial inclusion map Ai A2-... 

For A — and any abstract simplicial complex A, there exists a unique simplicial map from A to A] this map takes all vertices of A to the vertex of A. 

Definition 2.8. Let A and A2 be two abstract simplicial complexes, and let f A —> A2 be a simplicial map between them. Then f is called an isomorphism of abstract simplicial complexes if the induced map f V A ) —> V(A2) is a bijection and its inverse induces a simplicial map as well. If such an isomorphism exists, then Ai and A2 are said to be isomorphic as abstract simplicial complexes. 

Clearly, the isomorphism is an equivalence relation. It is the equality relation for the abstract simplicial complexes. 

An important special class of simplicial maps are isomorphisms f A A, where A is an abstract simplicial complex. These maps are called automorphisms of A. The composition of two automorphisms is again an automorphism, and an inverse of each automorphism is also an automorphism hence the set of automorphisms forms a group by composition. This is the group of symmetries of the abstract simplicial complex A, and we denote it by... 

Aut (Z ). For example, the group of automorphisms of the abstract simplicial complex in Example 2.2 is the full symmetric group Ss. 

Until now, we have restricted ourselves to considering the finite abstract simplicial complexes. The natural question is, what happens if we drop the condition that the ground set should be finite. We invite the reader to check that all the definitions make sense, and that all the statements hold just as they do for the finite ones. For reasons that will become clear once we look at the notion of geometric realization, we keep the condition that the simplices have finite cardinality. We restate Definition 2.1 for future reference. 

Example 2.10. Let A be the set of natural numbers. Then we obtain an abstract simplicial complex by taking all finite subsets a A such that for any two elements from a, one of them must divide the other one. 

Remark 2.11. In Chapter 4 we shall see that the finite abstract simplicial complexes together with simplicial maps actually form a category, and that the same is true if one takes all abstract simplicial complexes. 

Since the abstract simplicial complexes are some of the main characters of Combinatorial Algebraic Topology, there is a large variety of concepts and constructions pertaining to them. We shall now describe some of these. 

In the degenerate cases, the deletion of the vertex v from the abstract simplicial complex 0, v will give the empty simplex, whereas the deletion of the empty set from any abstract simplicial complex will give the void simplex. Furthermore, if 5 is a set of simplices of A, then we define the deletion of S by setting... 

Another important concept in the context of abstract simplicial complexes is that of a Unk of a simplex. 

For example, in a hollow tetrahedron, the abstract simplicial complex A consisting of all subsets of 1,2,3,4 except for the set 1,2,3,4 itself, a link of an edge consists of the two vertices that do not belong to that edge. Note also that the void simplex can never be the link of anything, since any link contains an empty set. 

Even though the abstract simplicial complex Ax V A2 depends on the choice of vertices v and W2, in practice these are usually suppressed from notations. 

The following is one of the most fundamental constructions that allow one to produce new abstract simplicial complexes from old ones. 

Clearly, we have commutativity for arbitrary abstract simplicial complexes Ax and Z 2, the joins Ax A2 and Z 2 Ax are isomorphic. The join is also associative namely, for arbitrary abstract simplicial complexes Ax, A2, and As, the joins (Zi A2) As and Z i As As) are isomorphic. 

Another important property of the join is that for any abstract simplicial complex A and any simplex t A, the abstract simplicial complexes lk2i(r) r and star(r) are isomorphic. 

The join of an arbitrary abstract simplicial complex A with the empty simplex is equal to A. 

A standard combinatorial gadget that one associates to an abstract simplicial complex is that of a face poset. To start with, we have the following definition. 

Definition 2.19. Let A he an arbitrary abstract simplicial complex. A face poset of A is the poset J A) whose set of elements consists of all nonempty simplices of A and whose partial order relation is the inclusion relation on the set of simplices. 

For example, for an arbitrary abstract simplicial complex 2l, a standard linear extension of the face poset P(zi) is obtained by setting a >l t whenever dim <7 > dimr, and choosing an arbitrary order within each set of simplices of the same dimension. 

There are two standard ways to subdivide abstract simplicial complexes. 

Definition 2.21. Let A be an abstract simplicial complex. The barycentric subdivision of A is also an abstract simplicial complex, which is denoted by Bd i4 and defined by... 

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