A homotopy decomposition of the fibre of the squaring map on $\Omega^3S^{17}$

TL;DR

This paper provides a homotopy decomposition of the fibre of the squaring map on the triple loop space of the 17-sphere, refining previous results and relating decompositions to Whitehead products and divisibility properties.

Contribution

It introduces a new homotopy decomposition of the fibre of the squaring map on a^3S^{17} using Richters proof, refining earlier decompositions for spheres and connecting to Whitehead products.

Findings

Decomposition of a^3S^{17}{2} using Richters proof.

Splitting of mod-2 homotopy groups in terms of double suspension fibre.

Whitehead square divisibility characterized for specific dimensions.

Abstract

We use Richter's $2$-primary proof of Gray's conjecture to give a homotopy decomposition of the fibre $\Omega^3S^{17}\{2\}$ of the $H$-space squaring map on the triple loop space of the $17$-sphere. This induces a splitting of the mod-$2$ homotopy groups $\pi_\ast(S^{17}; \mathbb{Z}/2\mathbb{Z})$ in terms of the integral homotopy groups of the fibre of the double suspension $E^2:S^{2n-1} \to \Omega^2S^{2n+1}$ and refines a result of Cohen and Selick, who gave similar decompositions for $S^5$ and $S^9$. We relate these decompositions to various Whitehead products in the homotopy groups of mod-$2$ Moore spaces and Stiefel manifolds to show that the Whitehead square $[i_{2n}, i_{2n}]$ of the inclusion of the bottom cell of the Moore space $P^{2n+1}(2)$ is divisible by $2$ if and only if $2n=2, 4, 8$ or $16$.