Let \(X\) be a topological space and \(A\subseteq X\) a subspace, and regard the singular chain group \(C_n(A)\) as a subgroup of \(C_n(X)\) via the inclusion. The group of relative chains is \(C_n(X,A) \coloneqq C_n(X)/C_n(A)\). A relative cycle is a chain \(\alpha \in C_n(X)\) with \(\del_n \alpha \in C_{n-1}(A)\).
A chain \(\alpha\in C_n(X)\) is a relative cycle if and only if its image in \(C_n(X,A)\) is a cycle for the boundary map induced by \(\del_n\).
[@Hat02].