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A Cell Complexes A Cell Complexes A.1 Simplicial Complexes A.1.1 Definitions Recall that a simplicial complex with vertex set V is a nonempty collection ∆ of finite subsets of V (called simplices) such that every singleton {v} is a simplex and every subset of a simplex A is a simplex (called a face of A). The cardinality r of A is called the rank of A,andr−1 is called the dimension of A. There are different conventions in the literature as to whether the empty set is considered a simplex. Our convention, as forced by the definition above, is that the empty set is a simplex; it has rank 0 and dimension −1. A subcomplex of ∆ is a subset ∆ that contains, for each of its elements A, all the faces of A; thus ∆ is a simplicial complex in its own right, with vertex set equal to some subset of V. Note that ∆ is a poset, ordered by the face relation. As a poset, it has the following formal properties: (a) Any two elements A, B ∈ ∆ have a greatest lower bound A ∩ B. (b) For any A ∈ ∆, the poset ∆≤A of faces of A is isomorphic to the poset of subsets of {1,...,r} for some integer r ≥ 0. Note. We will often express (b) more concisely by saying that ∆≤A is a Boolean lattice of rank r. Conditions (a) and (b) actually characterize simplicial complexes, in the sense that any nonempty poset ∆ satisfying (a) and (b) can be identified with the poset of simplices of a simplicial complex. Namely, take the vertex set V to be the set of rank-1 elements of ∆ [where the rank of A is defined to be the rank of the Boolean lattice ∆≤A]; then we can associate to each A ∈ ∆ the set A := {v ∈V|v ≤ A}. It is easy to check that A → A defines a poset isomorphism of ∆ onto a simplicial complex with vertex set V. We will therefore extend the standard terminology as follows: 662 A Cell Complexes Definition A.1. Any nonempty poset ∆ satisfying (a) and (b) will be called a simplicial complex. The elements of ∆ will be called simplices, and those of rank-1 will be called vertices. Note that in contrast to (a), two simplices A, B ∈ ∆ do not necessarily have a least upper bound (or any upper bound). We say that A and B are joinable if they have an upper bound; in this case they have a least upper bound A ∪ B, which is simply the set-theoretic union when ∆ is viewed as a set of subsets of its vertex set. More generally, an arbitrary set of simplices in a simplicial complex is said to be joinable if it has an upper bound, in which case it has a least upper bound. We visualize a simplex A of rank r as a geometric (r − 1)-simplex, the convex hull of its r vertices. One makes this precise by forming the geomet- ric realization |∆| of ∆, which is a topological space partitioned into (open) simplices |A|, one for each nonempty A ∈ ∆. To construct this topological space, start with an abstract real vector space with V as a basis. Let |A| be the interior of the simplex in this vector space spanned by the vertices of A, | | i.e., A consists of the linear combinations v∈A λvv with λv > 0 for all v and v∈A λv = 1. We then set |∆| := |A| . A∈∆ If ∆ is finite, then all of this is going on in RN , where N is the number of vertices of ∆, and we simply topologize |∆| as a subspace of RN .The question of how to topologize |∆| in the general case is more subtle, but the most commonly used topology is the weak topology with respect to the closed simplices: One first topologizes each closed simplex as a subspace of Euclidean space, and one then declares a subset of |∆| to be closed if and only if its intersection with each closed simplex is closed. For more details see any standard topology text, such as [129, p. 103; 179, Sections 2 and 3; 224, Section 3.1]. Exercises A.2. (a) Why is ∆ required to be nonempty in Definition A.1? (b) What should we mean by the “empty simplicial complex”? A.3. Show that condition (a) in Definition A.1 can be replaced by the follow- ing two conditions: (a1) ∆ has a smallest element. (a2) If two elements A, B ∈ ∆ have an upper bound, then they have a least upper bound A ∪ B. A.1 Simplicial Complexes 663 A.1.2 Flag Complexes Let P be a set with a binary relation called “incidence,” which is reflexive and symmetric. For example, P might consist of the points and lines of a projective plane, with the usual notion of incidence; or P might consist of the points, lines, and planes of a 3-dimensional projective space; or P might be a poset, with x and y incident if they are comparable (i.e., if x ≤ y or y ≤ x). A flag in P is a set of pairwise incident elements of P . For example, if P is a poset with incidence defined as above, a flag is simply a chain, i.e., a linearly ordered subset. Definition A.4. The flag complex associated to P is the simplicial com- plex ∆(P ) with P as vertex set and the finite flags as simplices. A flag complex of rank 2 (dimension 1) is also called an incidence graph. In the case of a poset, the flag complex is also called the order complex of the poset. One example of this construction appears naturally in the founda- tions of the theory of simplicial complexes: Example A.5. If P is the poset of nonempty simplices of a simplicial com- plex Σ, then ∆(P )isthebarycentric subdivision of Σ [179, Lemma 15.3; 224, 3.3]. See Figure A.1 for an example, where Σ consists of a 2-simplex and its faces, and the vertices of the barycentric subdivision are labeled with the corresponding element of P . Note how the six 2-simplices in the barycentric w w e g e g σ σ u v uv f f Fig. A.1. A 2-simplex and its barycentric subdivision. subdivision correspond to the maximal chains of (nonempty) simplices of Σ. From the point of view of geometric realizations, an element of the poset P is a simplex, and the corresponding vertex of ∆(P ) is its barycenter. Thus there are three types of barycenters, corresponding to the three types of simplices of Σ (vertex, edge, 2-simplex). These types are indicated by the three colors (black, white, gray) in the picture of ∆(P ). Remark A.6. The requirement that P be simplicial in this example is not necessary; P could just as easily be the poset of cells of a more general kind of complex, called a regular cell complex (see Section A.2), where the cells are not 664 A Cell Complexes necessarily simplices. The interested reader can draw some low-dimensional examples to illustrate this. Not all simplicial complexes are flag complexes. For example, the boundary of a triangle is not a flag complex. [If it were, then the three vertices would be pairwise incident and hence would be the vertices of a simplex of the complex.] Here is a useful characterization of flag complexes: Proposition A.7. The following conditions on a simplicial complex ∆ are equivalent: (i) ∆ is a flag complex. (ii) Every finite set of pairwise joinable simplices is joinable. (iii) Every set of three pairwise joinable simplices is joinable. (iv) Every finite set of pairwise joinable vertices is joinable. Proof. It is immediate that (i) =⇒ (ii) =⇒ (iv) =⇒ (i) and that (ii) =⇒ (iii). The proof that (iii) =⇒ (ii) is a straightforward induction andisleftasanexercise. A.1.3 Chamber Complexes and Type Functions Definition A.8. Let ∆ be a finite-dimensional simplicial complex. We say that ∆ is a chamber complex if all maximal simplices have the same dimension and any two can be connected by a gallery. Here a gallery, as usual, is a sequence of maximal simplices in which any two consecutive ones are adjacent, i.e., are distinct and have a common codimension-1 face. There is also a more general notion of pregallery,inwhich consecutive chambers are allowed to be equal, as in Chapter 1. The maximal simplices of a chamber complex are called chambers, and their codimension-1 faces will be called panels. A chamber complex is said to be thin if each panel is a face of exactly two chambers. Example A.9. Although many examples of thin chamber complexes occur throughout this book, we mention here the classical examples: If |∆| is a connected manifold without boundary, then ∆ is a thin chamber complex. Proofs can be found in many topology textbooks. For the convenience of the reader, we outline a proof in Exercise A.17 below, based on local homology calculations. [Note: In the topological literature, thin chamber complexes are often called pseudomanifolds.] Note that the set C = C(∆) of chambers has a well-defined distance func- tion d(−, −), defined to be the minimal length of a gallery joining two cham- bers. In other words, d(−, −) is the standard metric on the vertices of the chamber graph, the latter being defined in the obvious way. If there is any danger of confusion, we will also refer to the distance just defined as the gallery A.1 Simplicial Complexes 665 distance. The diameter of ∆, denoted by diam ∆, is the diameter of the metric space of chambers. We can also consider galleries between arbitrary simplices A and B, i.e., galleries C0,...,Cl with A ≤ C0 and B ≤ Cl, and we define d(A, B)tobethe minimal length of such a gallery.
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