98 lines
3.9 KiB
Markdown
98 lines
3.9 KiB
Markdown
# Math4201 Topology I (Lecture 21)
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## Simplicial complexes
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### Recall from last lecture
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Let $\sigma=\{a_0,a_1,\dots,a_n\}$ be a finite set. The $n$-dimensional simplex determined by $\tau$ is given as:
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$$
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\Delta^n(a_0,a_1,\dots,a_n)=\left\{t_0a_0+t_1a_1+\cdots+t_na_n\mid t_i\geq 0, \sum_{i=0}^n t_i=1\right\}
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$$
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If we have vertices $\tau=\{a_0,a_1,\dots,a_k\}$, $\tau\subseteq \sigma$, the face of $\Delta^n$ is determined by $\tau$ with dimension $|\tau|-1$.
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$\Delta^n$ is the topologized by the subspace topology inherited by the standard topology on Euclidean space $\mathbb{R}^n$.
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Note that there are different ways to of embedding and all give the same topological space.
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### Abstract simplicial complexes
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#### Definition for abstract simplicial complex
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Let $V=\{v_0,v_1,\dots,v_n\}$ be a finite set (set of vertices of a simplicial complex). $K$ be the collection of subspaces of $V$.
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1. $\sigma\in K$ and $\tau\subseteq \sigma$, then $\tau\in K$.
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2. For any $v\in V$, $\{v\}\in K$.
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Then $\tilde{X_k}=\bigsqcup_{\sigma\in K}\Delta_\sigma$.
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$\Delta_\sigma$ is a simplex of dimension $|\sigma|-1$.
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$X_K$ is the topological realization of $K$.
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Define an equivalence relation on $\tilde{X_k}$ as follows:
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$x\in \Delta_\sigma\sim x'\in \Delta_{\sigma'}$ if and only if $x\in \Delta_{\sigma'\cap \sigma}^{|\sigma'\cap \sigma|-1}\subseteq \Delta_\sigma$ and $x'\in \Delta_{\sigma'\cap \sigma}^{|\sigma'\cap \sigma|-1}\subseteq \Delta_{\sigma'}$.
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This just means that the two points have the same barycentric coordinates in the simplex.
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#### Definition of barycentric coordinates
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Let $\sigma=\{a_0,a_1,\dots,a_n\}$ be a simplex. The barycentric coordinates of a point $x\in \Delta_\sigma$ are the coefficients $t_0,t_1,\dots,t_n$ such that:
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$$
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x=t_0a_0+t_1a_1+\cdots+t_na_n
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$$
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and $t_i\geq 0$ and $\sum_{i=0}^n t_i=1$.
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The point $x$ is in the simplex $\Delta_\sigma$ if and only if $t_i\geq 0$ for all $i$.
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<details>
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<summary>Example of abstract simplicial complex</summary>
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Let $V=\{v_1,v_2,v_3,v_4,v_5\}$.
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If we want to enclose $K=\{\{v_1,v_2,v_3,v_4\},\{v_3,v_4,v_5\}\}$, we need to fill all the singletons $\{v_1\},\{v_2\},\{v_3\},\{v_4\},\{v_5\}$, all the pairs in $K$, $\{v_1,v_2\},\{v_1,v_3\},\{v_1,v_4\},\{v_2,v_3\},\{v_2,v_4\},\{v_3,v_4\},\{v_3,v_5\},\{v_4,v_5\}$, and the triangle $\{v_1,v_2,v_3\}, \{v_1,v_2,v_4\}, \{v_1,v_3,v_4\}, \{v_2,v_3,v_5\}$.
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The final simplicial complex is $\tilde{X_k}=\bigsqcup_{\sigma\in K}\Delta(v_1,v_2,v_3,v_4)\sqcup \Delta(v_3,v_4,v_5)\sqcup \{v_1,v_2,v_3,v_4,v_5\}$.
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We use $\Delta(v_1,v_2,v_3,v_4)$ to denote the simplex with vertices $v_1,v_2,v_3,v_4$.
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</details>
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#### Defining maps on abstract simplicial complexes
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Let $K$ be an abstract simplicial complex. $V=\{v_1,v_2,\dots,v_m\}$
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A map $\pi:\tilde{X_k}\to X_K$ is a quotient map
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$X_K$ is equipped with the quotient topology.
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Let $f:V\to \mathbb{R}^m$, then $u_i=f(v_i)$.
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$\tilde{X_k}=\bigsqcup_{\sigma\in K}\Delta_\sigma$ is the disjoint union of all simplices in $K$.
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For $\sigma=\{v_{i_0},\dots,v_{i_k}\}$, we have a map $\Delta_\sigma\to \mathbb{R}^\ell$ given by $[t_{i_0}u_0+t_{i_1}u_1+\cdots+t_{i_k}u_k\mid t_j\geq 0, \sum_{j=0}^k t_j=1]$.
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This is well-defined because the coefficients $t_j$ are uniquely determined by the vertices $v_{i_0},\dots,v_{i_k}$.
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This induces $F:\tilde{X_k}\to \mathbb{R}^\ell$. This map is continuous because $F\vert_{\Delta_\sigma}$ is continuous for all $\sigma\in K$.
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Recall that if for any $x\in X_K$, the map $F$ restricted to $\pi^{-1}(x)$ is constant, then there is a unique continuous map $g$ satisfying $F=g\circ \pi$.
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In fact, this condition is satisfied and there is such a map $G$.
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<details>
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<summary>Example of map on abstract simplicial complexes</summary>
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Consider the previous example of abstract simplicial complex.
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Let $f:V\to \mathbb{R}$ by $f(v_i)=i$.
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Then $f(\Delta_{\{v_1,v_2,v_3,v_4\}})=[1,4]$
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Then $f(\Delta_{\{v_1,v_3\}})=[1,3]$
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</details> |