Let \(X\) be a topological space, \(x_0\in X\), and \(n\geq 1\), and let \(I^n = [0,1]^n\) with boundary \(\del I^n\). The \(n\)th homotopy group \(\pi_n(X, x_0)\) is the set of homotopy classes of maps \(f\colon(I^n, \del I^n)\to(X, x_0)\), where the homotopies \(f_t\) are required to satisfy \(f_t(\del I^n) = \ts{x_0}\) for all \(t\), with the operation \([f]+[g]\coloneqq[f+g]\) given by \[ (f+g)(s_1, s_2, \ldots, s_n)\coloneqq \begin{cases} f(2s_1, s_2, \ldots, s_n) & 0\leq s_1\leq 1/2, \\ g(2s_1-1, s_2, \ldots, s_n) & 1/2\leq s_1\leq 1. \end{cases} \]
For \(n\geq 1\) the operation on \(\pi_n(X, x_0)\) is well defined and makes \(\pi_n(X, x_0)\) a group, and \(\pi_1(X, x_0)\) is the fundamental group. For \(n\geq 2\) the group \(\pi_n(X, x_0)\) is abelian. Since \(I^n/\del I^n\cong S^n\), the set \(\pi_n(X, x_0)\) is in bijection with the set \([(S^n, s_0), (X, x_0)]\) of based homotopy classes of based maps \((S^n, s_0)\to(X, x_0)\).
See [@Hat02].