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

Abstract

We use Richter's $2$-primary proof of Gray's conjecture to give a homotopy decomposition of the fibre ${\Omega }^3{S}^{17}\{2\}$ 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 $E^2:S^{2n-1}\to {\Omega }^2{S}^{2n+1}$ 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 $16$.

arXiv comment13 pages. Added references. To appear in Homology, Homotopy and Applications