The Higher Structure of Unstable Homotopy Groups

Abstract

Abstract We construct certain unstable higher order homotopy operations indexed by the simplex categories of $\boldsymbol{\Delta }^{n}$ for $n\geq 2$, 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.