Definition 31.2:

The augmentation map $\epsilon : S_0(X)\longrightarrow \mathbb{Z}$ is the group homomorphism given by

$$c_1p_1 + c_2p_2 + \dots + c_kp_k \mapsto c_1 + c_2 + \dots + c_k.$$

If $X$ is non-empty, the augmentation map is surjective. Since by definition, $\partial _0$ is the zero map and $Z_0(X) = S_0(X)$, we have to determine $B_0(X)$. The following theorem provides the answer.