Affine simplicies and barycentric subdivision:

Given points ${\bf v}_1, {\bf v}_2,\dots, {\bf v}_{p+1}$ in the standard $n-$simplex $\Delta_n$, the continuous map $\sigma:\Delta_p\longrightarrow \Delta_n$ given in terms of the barycentric coordinates

$$\sum_{i = 1}^{p+1}\lambda_i{\bf e}_{i}\mapsto \sum_{i = 1}^{p+1}\lambda_i{\bf v}_{i} \eqno(34.1)$$

is called an affine $p-$simplex and is denoted by $[{\bf v}_1, {\bf v}_2,\dots, {\bf v}_{p+1}]$. Note that the given need not be affinely independent. Each such $\sigma$ is an element of $S_p(\Delta_n)$ and the subgroup generated by them is called the group of affine $p-$simplicies in $\Delta_n$ denoted by $A_p(\Delta_n)$. Thus $A_p(\Delta_n)$ is the set of all formal linear combinations with integer coefficients of affine simplicies. Since the face maps (29.1) are affine maps we conclude from exercise 2 that the boundary homomorphism $\partial _p:S_p(\Delta_n)\longrightarrow S_{p-1}(\Delta_n)$ maps $A_p(\Delta_n)$ into $A_{p-1}(\Delta_n)$ and so we get a subcomplex $\{A_p(\Delta_n)/ p = 0, 1, 2, \dots\}$ with boundary maps as the restrictions of $\partial _p$ to $A_p(\Delta_n)$.

If ${\bf b} \in \Delta_n$ is a given point the cone over the affine simplex $\sigma = [{\bf v}_1, {\bf v}_2,\dots, {\bf v}_{p+1}]$ with vertex apex ${\bf b}$ is denoted by $K_{\bf b}\sigma$ and is defined as

$$K_{\bf b}\sigma = [{\bf b}, {\bf v}_1, {\bf v}_2,\dots, {\bf v}_{p+1}] \eqno(34.2)$$

The cone $K_{\bf b}\sigma$ is thus an affine $p+1$ simplex. If we start with a zero simplex namely, a point ${\bf v} \in S_n(\Delta_n)$, the cone over it is the line segment $[{\bf b}, {\bf v}]$. Since $A_p(\Delta_n)$ is a free abelian group generated by the affine $p$ simplicies, we obtain by extension a group homomorphism $K_{\bf b}:A_p(\Delta_n)\longrightarrow A_{p+1}(\Delta_n)$. As in the proof of theorem (29.7) it is easy to compute the boundary of the cone $K_{\bf b}\sigma$ for any affine $p$ simplex.

For a zero simplex $\sigma = [{\bf v}]$ we evidently have $\partial _1K_{\bf b}(\sigma) = \sigma - [{\bf b}]$. We now calculate the faces of the affine $p+1$ simplex $K_{\bf b}(\sigma)$. If $j \geq 1$,

$$(K_{\bf b}\sigma \circ \Phi_j^p)(\lambda_1,\lambda_2,\dots, \lambda_{p+1}) = K_{\bf b}\sigma(\lambda_1, \dots, \lambda_j, 0, \lambda_{j+1},\dots, \lambda_{p+1})$$

$$= [{\bf b}, {\bf v}_1,\dots, {\bf v}_{j-1}, {\bf v}_{j+1},\dots, {\bf v}_{p+1}].$$

This is the cone over the $j-$th face of $[{\bf v}_1, {\bf v}_2,\dots, {\bf v}_{p+1}]$. Turning to the case $j = 0$,

$$(K_{\bf b}\sigma \circ \Phi_0^p)(\lambda_1,\lambda_2,\dots, \lamb... ...}\sigma(0, \lambda_1,\dots, \lambda_{p+1}) = [{\bf v}_1,\dots, {\bf v}_{p+1}].$$

Using equation (29.4) we immediately get the following result.