46 III ComplexesIII.1 Simplicial ComplexesThere are many ways to represent a topological space, one being a collection ofsimplices that are glued to each other in a structured manner. Such a collectioncan easily grow large but all its elements are simple. This is not so convenientfor hand-calculations but close to ideal for computer implementations. In thisbook, we use simplicial complexes as the primary representation of topology.Simplices. Let u0, u1, . . . , ukbe points in Rd. A point x =Pki=0λiuiis anaffine combination of the uiif the λisum to 1. The affine hull is the set of affinecombinations. It is a k-plane if the k + 1 points are affinely independent bywhich we mean that any two affine combinations, x =Pλiuiand y =Pµiui,are the same iff λi= µifor all i. The k + 1 points are affinely independent iffthe k vectors ui− u0, for 1 ≤ i ≤ k, are linearly independent. In Rdwe canhave at most d linearly independent vectors and therefore at most d+1 affinelyindependent points.An affine combination x =Pλiuiis a convex combination if all λiare non-negative. The convex hull is the set of convex combinations. A k-simplex is theconvex hull of k + 1 affinely independent points, σ = conv {u0, u1, . . . , uk}. Wesometimes say the uispan σ. Its dimension is dim σ = k. We use special namesof the first few dimensions, vertex for 0-simplex, edge for 1-simplex, trianglefor 2-simplex, and tetrahedron for 3-simplex; see Figure III.1. Any subset ofFigure III.1: From left to right: a vertex, an edge, a triangle, and a tetrahedron.affinely independent points is again affinely independent and therefore alsodefines a simplex. A face of σ is the convex hull of a non-empty subset of theuiand it is proper if the subset is not the entire set. We sometimes write τ ≤ σif τ is a face and τ < σ if it is a proper face of σ. Since a set of size k + 1 has2k+1subsets, including the empty set, σ has 2k+1− 1 faces, all of which areproper except for σ itself. The boundary of σ, denoted as bd σ, is the union ofall proper faces, and the interior is everything else, int σ = σ − bd σ. A pointx ∈ σ belongs to int σ iff all its coefficients λiare positive. It follows that everyIII.1 Simplicial Complexes 47point x ∈ σ belongs to the interior of exactly one face, namely the one spannedby the points uithat correspond to positive coefficients λi.Simplicial complexes. We are interested in sets of simplices that are closedunder taking faces and that have no improper intersections.Definition. A simplicial complex is a finite collection of simplices K suchthat σ ∈ K and τ ≤ σ implies τ ∈ K, and σ, σ0∈ K implies σ ∩ σ0is eitherempty or a face of both.The dimension of K is the maximum dimension of any of its simplices. Theunderlying space, denotes as || K ||, is the union of its simplices together with thetopology inherited from Rd. A polyhedron is the underlying space of a simplicialcomplex. A triangulation of a topological space X is a simplicial complex Ktogether with a homeomorphism between X and || K ||. The topological spaceis triangulable if it has a triangulation. A subcomplex of K is a simplicialcomplex L ⊆ K. It is full if it contains all simplices in K spanned by verticesin L. A particular subcomplex is the j-skeleton consisting of all simplices ofdimension j or less, K(j)= {σ ∈ K | dim σ ≤ j}. The 0-skeleton is alsoreferred to as the vertex set, Vert K = K(0). Skeleta are generally not full.A subset of a simplicial complex useful in talking about local neighborhoodsis the star of a simplex τ consisting of all simplices that have τ as a face,St τ = {σ ∈ K | τ ≤ σ}. Generally, the star is not closed under taking faces.We can make it into a complex by adding all missing faces. The result isthe closed star,St τ , which is the smallest subcomplex that contains the star.The link consists of all simplices in the closed star that are disjoint from τ ,Lk τ = {υ ∈ St τ | υ ∩ τ = ∅}. It τ is a vertex then the link is just thedifference between the closed star and the star. More generally, it is the closedstar minus the stars of all faces of τ. For example if K triangulates a 2-manifoldwithout boundary then the link of an edge is a pair of points, a 0-sphere, andthe link of a vertex is a cycle of edges and vertices, a 1-sphere.Abstract simplicial complex. It is often easier to construct a complexabstractly and to worry abut how to put it into Euclidean space later.Definition. An abstract simplicial complex is a finite collection of sets Asuch that α ∈ A and β ⊆ α implies β ∈ A.The sets in A are its simplices. The dimension of a simplex is dim α = card α−1and the dimension of the complex is the maximum dimension of any of its48 III Complexessimplices. A face of α is a non-empty subset β ⊆ α, which is proper if β 6= α.The vertex set is the union of all simplices, Vert A =SA =Sα∈Aα. Asubcomplex is an abstract simplicial complex B ⊆ A. Two abstract simplicialcomplexes are isomorphic if there is a bijection b : Vert A → Vert B such thatα ∈ A iff b(α) ∈ B. The largest abstract simplicial complex with a vertex setof size n has cardinality 2n− 1. Given a (geometric) simplicial complex K, wecan construct an abstract simplicial complex A by throwing away all simplicesand retaining only their sets of vertices. We call A a vertex scheme of K.Symmetrically, we call K a geometric realization of A but also of every abstractsimplicial complex isomorphic to A. Constructing geometric realizations issurprisingly easy if the dimension of the ambient space is sufficiently high.Geometric Realization Theorem. An abstract simplicial complex of di-mension d has a geometric realization in R2d+1.Proof. Let f : Vert A → R2d+1be an injection whose image is a set of pointsin general position. Specifically, any 2d + 2 or fewer of the points are affinelyindependent. Let α and α0be simplices in A with k = dim α and k0= dim α0.The union of the two has size card (α ∪ α0) = card α+card α0−card (α ∩ α0) ≤k + k0+ 2 ≤ 2d + 2. The points in α ∪ α0are therefore affinely independent,which implies that every convex combination x of points in α ∪ α0is unique.Hence x belongs to σ = conv f(α) as well as to σ0= conv f (α0) iff x is a convexcombination of α ∩ α0. This implies that the intersection of σ and σ0is eitherempty of the simplex conv f(α ∩ α0), as required.Simplicial maps. Let K be a
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