The higher structure of unstable homotopy groups

We construct certain unstable higher-order homotopy operations indexed by the simplex categories of ${\mathrm{\Delta }}^n$ for $n\ge 2$ and prove that all elements in the homotopy groups of a wedge of spheres are generated under such operations by Whitehead products and the group structure. This provides a stronger unstable analogue of Cohen’s theorem on the decomposition of stable homotopy.