The higher structure of unstable homotopy groups
Samik Basu, David Blanc, Debasis Sen
TL;DR
This work develops unstable higher order homotopy operations indexed by finite subcategories of $\Delta$ and shows that the unstable homotopy groups of wedges of spheres are generated, under these operations, by Whitehead products and the group structure, providing an unstable analogue of Cohen's stable decomposition. A key device is a framework of sequential simplicial-space approximations and the notion of atomic maps, yielding that for the $k$-connected cover of $S^k$ the atomic maps are lifts of $\eta_k$, $\nu_k$, $\sigma_k$ and $\alpha_1(p)$ and that $\pi_*(\mathbf X)$ is generated by these atoms under higher order operations. The paper then analyzes spheres and complex projective spaces, showing transgression properties of indecomposables in Serre spectral sequences and giving a rational description of Hopf fibrations for $\mathbb{C}\mathbf P^n$, thereby illustrating how unstable higher operations organize the entire unstable homotopy algebra in these settings. Overall, the results illuminate the structure of unstable homotopy groups through a combinatorial and spectral-sequence–driven operadic lens, with potential implications for explicit decompositions and computations.
Abstract
We construct certain unstable higher-order homotopy operations indexed by the simplex categories of $Δ^{n}$ 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.