3.2 Simplexes and simplicial complexes

Let us recall how the Euler characteristic of a surface is calculated. We first construct a polyhedron homeomorphic to the given surface, then count the numbers of vertices, edges and faces. The Euler characteristic of the polyhedron, and hence of the surface, is then given by equation (2.31). We abstract this procedure so that we may represent each part of a figure by some standard object. We take triangles and their analogues in other dimensions, called simplexes, as the standard objects. By this standardization, it becomes possible to assign to each figure Abelian group structures.

3.2.1 Simplexes

Simplexes are building blocks of a polyhedron. A 0-simplex $⟨p_0⟩$ is a point, or a vertex, and a 1-simplex $⟨p_0{p}_1⟩$ is a line, or an edge. A 2-simplex $⟨p_0{p}_1{p}_2⟩$ is defined to be a triangle with its interior included and a 3-simplex $⟨p_0{p}_1{p}_2{p}_3⟩$ is a solid tetrahedron (figure 3.2). It is common to denote a 0-simplex without the bracket; $⟨p_0⟩$ may be also written as $p_0$ . It is easy to continue this construction to any $r$ -simplex $⟨p_0{p}_1\dots p_r⟩$ . Note that for an $r$ -simplex to represent an $r$ -dimensional object, the vertices $p_i$ must be geometrically independent, that is, no $(r-1)$ -dimensional hyperplane contains all the $r+1$ points. Let $p_0,\dots ,p_r$ be points geometrically independent in ${\mathbb{R}}^m$ where $m\ge r$ . The $r$ -simplex ${\sigma }_r=⟨p_0,\dots ,p_r⟩$ is expressed as

Figure 3.2. 0-, 1-, 2- and 3-simplexes.

$${\sigma }^r=\left\{x\in {\mathbb{R}}^m\mid x=\sum _{i=0}^r{c}_i{p}_i,c_i\ge 0,\sum _{i=0}^r{c}_i=1\right\}.\text{\hspace{1em}(3.11)}$$

$(c_0,\dots ,c_r)$ is called the barycentric coordinate of $x$ . Since ${\sigma }_r$ is a bounded and closed subset of ${\mathbb{R}}^m$ , it is compact.

Let $q$ be an integer such that $0\le q\le r$ . If we choose $q+1$ points $p_{i_0},\dots ,p_{i_q}$ out of $p_0,\dots ,p_r$ , these $q+1$ points define a $q$ -simplex ${\sigma }_q=⟨p_{i_0},\dots ,p_{i_q}⟩$ , which is called a $q$ -face of ${\sigma }_r$ . We write ${\sigma }_q\le {\sigma }_r$ if ${\sigma }_q$ is a face of

${\sigma }_r$ . If ${\sigma }_q\ne {\sigma }_r$ , we say ${\sigma }_q$ is a proper face of ${\sigma }_r$ , denoted as ${\sigma }_q<{\sigma }_r$ . Figure 3.3 shows a 0-face $p_0$ and a 2-face $⟨p_1{p}_2{p}_3⟩$ of a 3-simplex $⟨p_0{p}_1{p}_2{p}_3⟩$ . There are one 3-face, four 2-faces, six 1-faces and four 0-faces. The reader should verify that the number of $q$ -faces in an $r$ -simplex is $\left(\genfrac{}{}{0ex}{}{r+1}{q+1}\right)$ . A 0-simplex is defined to have no proper faces.

Figure 3.3. A 0-face $p_0$ and a 2-face $⟨p_1{p}_2{p}_3⟩$ of a 3-simplex $⟨p_0{p}_1{p}_2{p}_3⟩$ .

3.2.2 Simplicial complexes and polyhedra

Let $K$ be a set of finite number of simplexes in ${\mathbb{R}}^m$ . If these simplexes are nicely fitted together, $K$ is called a simplicial complex. By ‘nicely’ we mean:

(i) an arbitrary face of a simplex of $K$ belongs to $K$ , that is, if $\sigma \in K$ and ${\sigma }^{\prime }\le \sigma$ then ${\sigma }^{\prime }\in K$ ; and
(ii) if $\sigma$ and ${\sigma }^{\prime }$ are two simplexes of $K$ , the intersection $\sigma \cap {\sigma }^{\prime }$ is either empty or a common face of $\sigma$ and ${\sigma }^{\prime }$ , that is, if $\sigma ,{\sigma }^{\prime }\in K$ then either $\sigma \cap {\sigma }^{\prime }=\varnothing$ or $\sigma \cap {\sigma }^{\prime }\le \sigma$ and $\sigma \cap {\sigma }^{\prime }\le {\sigma }^{\prime }$ .

For example, figure 3.4(a) is a simplicial complex but figure 3.4(b) is not. The dimension of a simplicial complex $K$ is defined to be the largest dimension of simplexes in $K$ .

Figure 3.4. (a) is a simplicial complex but (b) is not.

Example 3.4

Let ${\sigma }_r$ be an $r$ -simplex and $K=\{{\sigma }^{\prime }|{\sigma }^{\prime }\le {\sigma }_r\}$ be the set of faces of ${\sigma }_r$ . $K$ is an $r$ -dimensional simplicial complex. For example, take

${\sigma }_3=⟨p_0{p}_1{p}_2{p}_3⟩$ (figure 3.3). Then

$$\begin{array}{rl}& K=\{p_0,p_1,p_2,p_3,⟨p_0{p}_1⟩,⟨p_0{p}_2⟩,⟨p_0{p}_3⟩,\\ & ⟨p_1{p}_2⟩,⟨p_1{p}_3⟩,⟨p_2{p}_3⟩,⟨p_0{p}_1{p}_2⟩,⟨p_0{p}_1{p}_3⟩,\\ & ⟨p_0{p}_2{p}_3⟩,⟨p_1{p}_2{p}_3⟩,⟨p_0{p}_1{p}_2{p}_3⟩\}.\end{array}\text{\hspace{1em}(3.12)}$$

A simplicial complex $K$ is a set whose elements are simplexes. If each simplex is regarded as a subset of ${\mathbb{R}}^m$ ( $m\ge \dim K$ ), the union of all the simplexes becomes a subset of ${\mathbb{R}}^m$ . This subset is called the polyhedron $|K|$ of a simplicial complex $K$ . The dimension of $|K|$ as a subset of ${\mathbb{R}}^m$ is the same as that of $K$ ; $\dim |K|=\dim K$ .

Let $X$ be a topological space. If there exists a simplicial complex $K$ and a homeomorphism $f:|K|\to X$ , $X$ is said to be triangulable and the pair $(K,f)$ is called a triangulation of $X$ . Given a topological space $X$ , its triangulation is far from unique. We will be concerned with triangulable spaces only.

Example 3.5

Figure $3.5(a)$ is a triangulation of a cylinder $S^1\times [0,1]$ . The reader might think that somewhat simpler choices exist, figure $3.5(b)$ , for example. This is, however, not a triangulation since, for ${\sigma }_2=⟨p_0{p}_1{p}_2⟩$ and ${\sigma }_2^{\prime }=⟨p_2{p}_3{p}_0⟩$ , we find ${\sigma }_2\cap {\sigma }_2^{\prime }=⟨p_0⟩\cup ⟨p_2⟩$ , which is neither empty nor a simplex.

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

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