The set of basepoint-preserving homotopy classes of maps (S^n, *) → (X, x_0), with a group structure for n ≥ 1 (abelian for n ≥ 2). π_1 is the fundamental group; higher π_n encode higher-dimensional 'holes'.
The set of basepoint-preserving homotopy classes of maps (S^n, *) → (X, x_0), with a group structure for n ≥ 1 (abelian for n ≥ 2). π_1 is the fundamental group; higher π_n encode higher-dimensional 'holes'.