simplex

Definition

The $n$-simplex ${\Delta }^n$ is the topological space

${\Delta }^n=\left\{(x_0,\dots ,x_n)\in {\mathbb{R}}^{n+1}|x_i\le 0,{\sum }^n{x}_i=1\right\}.$

Terminology

The $i$th vertex of ${\Delta }^n$ is $e_i=(0,\dots ,1,\dots ,0)$ in the ith position.

The $i$th face of ${\Delta }^n$ is

${\partial }_i{\Delta }^n=\{(x_0,\dots ,x_n)|x_i=0\}\subset {\Delta }^n$

Important maps

Boundary:

$$\begin{array}{rl}{\partial }_i{\Delta }^n& \stackrel{\sim }{\to }{\Delta }^{n-1}\\ (x_0,\dots ,x_n)& ⟼(x_0,\dots ,\widehat{x_i},\dots ,x_n)\end{array}$$

Where $\widehat{x_i}$ denotes that entry being excluded.

Face map (inverse of boundary map):

$$\begin{array}{rl}{d}^i:{\Delta }^{n-1}& ⟶{\Delta }^n\\ (x_0,\dots ,x_{n-1})& ⟼(x_0,\dots ,x_{i-1},0,x_i,\dots ,x_n)\end{array}$$

There is an equality of maps

$d^i\circ d^j=d^j\circ d^{i-1}:{\Delta }^{n-1}⟶{\Delta }^{n+1}$