Homology 2: Simplexes and Simplicial Complexes

Definition. A 0-simplex $\langle p_0\rangle$ is a point or a vertex. A 1-simplex $\langle p_0p_1\rangle$ is a line or an edge. A 2-simplex $\langle p_0p_1p_2\rangle$ is a triangle with its interior included. A 3-simplex $\langle p_0p_1p_2p_3\rangle$ is a solid tetrahedron.

A 0-simplex $\langle p_0\rangle$ may be simply written as $p_0$.

Note that in order for an $r$-simplex to represent an $r$-dimensional object, the vertices $p_i$ must be geometrically independent, i.e. no $(r-1)$-dimensional hyperplane contains all the $r+1$ points. Let $p_0,\cdots,p_r$ be points geometrically independent in $\mathbb R^m$ ($m\geq r$). The $r$-simplex $$\sigma_r=\{x\in\mathbb R^m: x=\sum_{i=0}^r c_ip_i,\ c_i\geq 0,\ \sum_{i=0}^r c_i=1\}$$ has the points $p_0,\cdots,p_r$ as its vertices. The ordered $r+1$-tuple $(c_0,c_1,\cdots,c_r)$ is called the barycentric coordinate of $x$. The 3-simplex $\langle p_0p_1p_2p_3\rangle$ four 0-faces (vertices) $p_0,p_1,p_2,p_3$; six 1-faces (edges) $\langle p_0p_1\rangle$, $\langle p_0p_2\rangle$, $\langle p_0p_3\rangle$, $\langle p_1p_2\rangle$, $\langle p_1p_3\rangle$, $\langle p_2p_3\rangle$; four 2-faces (faces) $\langle p_0p_1p_2\rangle$, $\langle p_0p_2p_3\rangle$, $\langle p_0p_1p_3\rangle$, $\langle p_1p_2p_3\rangle$.

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 fitted together we mean that:

  1. An arbitrary face of a simplex of $K$ belongs to $K$.
  2. If $\sigma$ and $\sigma’$ are two simplexes of $K$, $\sigma\cap\sigma’$ is either empty or a face of $\sigma$ and $\sigma’$.

The dimension of a simplicial complex is defined to be the maximum dimension of simplexes in $K$.

Let $\sigma_r$ be an $r$-simplex and $K$ be the set of faces of $\sigma_r$. Then $K$ is an $r$-dimensional simplicial complex.For example, take $\sigma_3=\langle p_0p_1p_2,p_3\rangle$. Then $$\begin{array}{c}K=\{p_0,p_1,p_2,p_3,\langle p_0p_1\rangle,\langle p_0p_2\rangle,\langle p_0p_3\rangle,\langle p_1p_2\rangle,\langle p_1p_3\rangle,\langle p_2p_3\rangle,\\\langle p_0p_1p_2\rangle,\langle p_0p_1p_3\rangle,\langle p_0p_2p_3\rangle,\langle p_1p_2p_3\rangle,\langle p_0p_1p_2p_3\rangle\}.\end{array}$$

Definition. Let $K$ be a simplicial complex of simplexes in $\mathbb R^m$. The union of all the simplexes of $K$ is a subset of $\mathbb R^m$ called the polyhedron $|K|$ of a simplicial complex $K$. Note that $\dim |K|=\dim K$.

Let $X$ be a topological space. If there is a simplicial complex $K$ and a homeomorphism $f:|K|\longrightarrow X$, $X$ is said to be triangulable and the pair $(K,f)$ is called a triangulation of $X$.

Example. The following picture shows a triangulation of $S^1\times [0,1]$.

Example. The following example is not a triangulation of $S^1\times [0,1]$.

Let $\sigma_2=\langle p_0p_1p_2\rangle$ and $\sigma_2’=\langle p_2p_3p_0\rangle$. Then $\sigma_2\cap\sigma_2’=\langle p_0\rangle\cup\langle p_2\rangle$. This is neither $\emptyset$ nor a simplex.

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