Singular $n$-simplex, unknown notation, homology

Question

I would like to know what is denoted in the singular simplices paragraph by $e_0,...,e_n$ here:

$$[p_0,p_1,...,p_n]=[\sigma(e_0),...,\sigma(e_n)]$$ ?

How this matches with simplicial identities $d_i$ and $s_i$ ?

Answer 1

$\Delta^n$ is the standard $n$-simplex in $\mathbb{R}^{n+1}$, i.e. the convex hull of the $n+1$ elements $e_0,\dots,e_n$ forming the standard basis of $\mathbb{R}^{n+1}$ ($e_k = (\delta_{k0},\dots,\delta_{kn})$ with $\delta_{kk} = 1$ and $\delta_{kj} = 0$ for $k \ne j$).

If $\sigma : \Delta^n \to X$ is a singular $n$-simplex, then the $p_i = \sigma(e_i)$ are (not necessarily distint) points of $X$. Hence the collection $[p_0,\dots,p_n]$ is not a complete description of $\sigma$, but only a symbolic notation. Defining $$\partial \sigma = \sum_{k=0}^n (-1)^k [p_0,\dots,p_{k-1},p_{k+1},\dots,p_n]$$ is therefore also only symbolic. In fact, $[p_0,\dots,p_{k-1},p_{k+1},\dots,p_n]$ denotes the $(n-1)$-simplex $\partial_k \sigma : \Delta^{n-1} \to X$ (the $k$-face of $\sigma$) defined by $$\partial_k \sigma = \sigma \circ s_k ,$$ where $s_k : \Delta^{n-1} \to \Delta^n$ is the restriction of the linear map determined by $s_k(e_j) = e_j$ for $j < k$, $s_k(e_j) = e_{j+1}$ for $j \ge k$. This map embeds $\Delta^{n-1}$ as the $k$-th face of $\Delta^n$, i.e. the convex hull of $e_0,\dots,e_{k-1},e_{k+1},\dots,e_n$.

Answer 2

The point $e_i$ is just $$ e_i=\begin{pmatrix} 0\\ \vdots\\ 0\\ 1\\ 0\\ \vdots\\ 0 \end{pmatrix}\in\mathbb R^{n+1} $$ with the entry $1$ in the $i$-th row. These points are exactly the vertices of the $n$-simplex $\Delta_n$.

Given this, you can compute that the $d_i$ and $s_i$ satisfy indeed the simplicial identities.