Let \(X\) be a topological space, let \(C_n(X)\) be the group of singular $n$-chains of \(X\), and let \(\del_n\colon C_n(X)\to C_{n-1}(X)\) be the boundary map. An \(n\)-chain \(\alpha\in C_n(X)\) is an \(n\)-cycle if \(\del_n\alpha=0\) [@Hat02]. The \(n\)-cycles form the subgroup \(Z_n(X)\coloneqq\ker\del_n\).
With \(B_n(X)\coloneqq\im\del_{n+1}\) the group of \(n\)-boundaries, the relation \(\del_n\circ\del_{n+1}=0\) gives \(B_n(X)\subseteq Z_n(X)\), and the singular homology group is \(H_n(X)=Z_n(X)/B_n(X)\).