Whitehead product

Summary

The Whitehead product is an operation $[\alpha, \beta] \in \pi_{p+q-1}(X)$ on homotopy groups, defined for $\alpha \in \pi_p(X)$ and $\beta \in \pi_q(X)$. It comes from the attaching map of the top cell of the product $S^p \times S^q$. For $p = q = 1$ it is the commutator in the fundamental group. With it, the homotopy groups of a space form a graded Lie algebra, up to a shift of degree and signs.