Homotopy groups of spheres

Summary

The homotopy groups of spheres are the groups $\pi_k(S^n)$ of homotopy classes of based maps from the $k$-sphere to the $n$-sphere. Some are simple: $\pi_k(S^n) = 0$ for $k < n$, and $\pi_n(S^n) \cong \Z$, detected by the degree. For $k > n$ they are mostly finite but highly irregular, and no complete formula is known. Their computation has driven much of algebraic topology, through fibrations, spectral sequences and stable homotopy theory.