Let \(X\) be a topological space, \(A\subseteq X\) a subspace, \(x_0 \in A\), and \(n\geq 1\). Let \(D^n\) be the closed unit ball, \(S^{n-1} = \del D^n\), and \(s_0\in S^{n-1}\) a basepoint. A map of triples \(f\colon (D^n, S^{n-1}, s_0) \to (X, A, x_0)\) is a continuous map \(f\colon D^n\to X\) with \(f(S^{n-1})\subseteq A\) and \(f(s_0) = x_0\). The relative homotopy group is \[ \pi_n(X, A, x_0) \coloneqq \ts{ f\colon (D^n, S^{n-1}, s_0) \to (X, A, x_0) } / \homotopic, \] the set of homotopy classes of maps of triples, where the homotopies are through maps of triples.
Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(x_0\in A\).
\(\pi_n(X, A, x_0)\) is a group for \(n\geq 2\) and an abelian group for \(n\geq 3\); \(\pi_1(X, A, x_0)\) is a pointed set.
The sequence \[ \cdots \to \pi_n(A, x_0) \to \pi_n(X, x_0) \to \pi_n(X, A, x_0) \xrightarrow{\del} \pi_{n-1}(A, x_0)\to \cdots \to \pi_0(X, x_0), \] whose unlabelled maps are induced by the inclusions \((A, x_0)\injects(X, x_0)\) and \((X, x_0, x_0)\injects (X, A, x_0)\) and where \(\del\) restricts a map of triples to \(S^{n-1}\), is exact.
[@Hat02].