Homology with coefficients

Definition

Let $X$ be a topological space and let $G$ be an abelian group. The group $C_n(X; G)$ of singular $n$-chains with coefficients in $G$ is the set of finite formal sums $\sum_i g_i \sigma_i$. Here $g_i \in G$ and $\sigma_i \colon \Delta^n \to X$ are singular $n$-simplices; equivalently, $C_n(X; G) = C_n(X) \otimes_{\Z} G$. The boundary map is $\partial\left(\sum_i g_i \sigma_i\right) = \sum_i \sum_{k=0}^{n} (-1)^k g_i \, \sigma_i|_{[v_0, \ldots, \hat v_k, \ldots, v_n]}$, where the hat means that the vertex is omitted. This gives a chain complex, and the homology of $X$ with coefficients in $G$, written $H_n(X; G)$, is its homology in degree $n$. For a subspace $A \subseteq X$, the relative groups $H_n(X, A; G)$ come from $C_n(X; G) / C_n(A; G)$ in the same way.