Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(n\geq 0\), and regard the singular chain group \(C_n(A)\) as a subgroup of \(C_n(X)\) via the inclusion. A relative boundary is a chain \(\alpha\in C_n(X)\) of the form \(\alpha = \del\beta + \gamma\) with \(\beta \in C_{n+1}(X)\) and \(\gamma \in C_n(A)\).
A chain \(\alpha\in C_n(X)\) is a relative boundary if and only if its image in \(C_n(X,A) = C_n(X)/C_n(A)\) is a boundary for the induced boundary map. The relative homology group \(H_n(X,A)\) is isomorphic to the group of relative cycles in \(C_n(X)\) modulo the subgroup of relative boundaries.
[@Hat02].