NoteNextra-origin/content/Math4201/Math4201_L21.md

Math4201 Topology I (Lecture 21)

Simplicial complexes

Recall from last lecture

Let \sigma=\{a_0,a_1,\dots,a_n\} be a finite set. The $n$-dimensional simplex determined by \tau is given as:

\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\}

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.

\Delta^n is the topologized by the subspace topology inherited by the standard topology on Euclidean space \mathbb{R}^n.

Note that there are different ways to of embedding and all give the same topological space.

Abstract simplicial complexes

Definition for abstract simplicial complex

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.

1. \sigma\in K and \tau\subseteq \sigma, then \tau\in K.
2. For any v\in V, \{v\}\in K.

Then \tilde{X_k}=\bigsqcup_{\sigma\in K}\Delta_\sigma.

\Delta_\sigma is a simplex of dimension |\sigma|-1.

X_K is the topological realization of K.

Define an equivalence relation on \tilde{X_k} as follows:

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'}.

This just means that the two points have the same barycentric coordinates in the simplex.

Definition of barycentric coordinates

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:

x=t_0a_0+t_1a_1+\cdots+t_na_n

and t_i\geq 0 and \sum_{i=0}^n t_i=1.

The point x is in the simplex \Delta_\sigma if and only if t_i\geq 0 for all i.

Example of abstract simplicial complex

Let V=\{v_1,v_2,v_3,v_4,v_5\}.

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\}.

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\}.

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.

Defining maps on abstract simplicial complexes

Let K be an abstract simplicial complex. V=\{v_1,v_2,\dots,v_m\}

A map \pi:\tilde{X_k}\to X_K is a quotient map

X_K is equipped with the quotient topology.

Let f:V\to \mathbb{R}^m, then u_i=f(v_i).

\tilde{X_k}=\bigsqcup_{\sigma\in K}\Delta_\sigma is the disjoint union of all simplices in K.

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].

This is well-defined because the coefficients t_j are uniquely determined by the vertices v_{i_0},\dots,v_{i_k}.

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.

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.

In fact, this condition is satisfied and there is such a map G.

Example of map on abstract simplicial complexes

Consider the previous example of abstract simplicial complex.

Let f:V\to \mathbb{R} by f(v_i)=i.

Then f(\Delta_{\{v_1,v_2,v_3,v_4\}})=[1,4]

Then f(\Delta_{\{v_1,v_3\}})=[1,3]