Homotopy group

Summary

The $n$-th homotopy group $\pi_n(X, x_0)$ of a pointed space consists of the homotopy classes of based maps from the $n$-sphere to $(X, x_0)$. For $n = 1$ it is the fundamental group, and for $n \ge 2$ it is abelian. Homotopy groups are invariants of homotopy type, and on CW complexes they detect homotopy equivalences, by the Whitehead theorem. They are easy to define and very hard to compute, even for spheres.