The higher structure of unstable homotopy groups

Abstract

We construct certain unstable higher-order homotopy operations indexed by the simplex categories of $\hat{I}{”}^n$ 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.

We would like to thank George Peschke for pointing out that the treatment of rational spaces was missing in Definition 0.2; we have accordingly added this case also to the proof of Theorem 3.1