Introduction
One of the most useful computational tools in homology is the long exact sequence of a pair \((X, A)\text{,}\) which relates \(H_*(A)\text{,}\) \(H_*(X)\text{,}\) and the relative groups \(H_*(X, A)\text{.}\) In this lecture we construct the analogue for homotopy groups. Given a pair \((X, A)\) and a basepoint \(x_0 \in A\text{,}\) we define relative homotopy groups \(\pi_n(X, A, x_0)\) and show that they fit into a long exact sequence
\begin{equation*}
\cdots \to \pi_n(A, x_0) \to \pi_n(X, x_0) \to \pi_n(X, A, x_0) \to \pi_{n-1}(A, x_0) \to \cdots.
\end{equation*}
Informally, \(\pi_n(X, A, x_0)\) measures the failure of \(\pi_n(A, x_0) \to \pi_n(X, x_0)\) to be an isomorphism.
