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

We use Richter’s $2$-primary proof of Gray’s conjecture to give a homotopy decomposition of the fibre $\Omega^3 S^{17} \lbrace 2 \rbrace$ 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 $\pi_{*} (S^{17}; \mathbb{Z} / 2 \mathbb{Z})$ 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 $2n = 2, 4, 8 \: \mathrm{or} \: 16$.

Keywords

loop space decomposition, Moore space, Whitehead product

2010 Mathematics Subject Classification

55P10, 55P35, 55Q15

Full Text (PDF format)

Received 20 July 2017

Received revised 31 August 2017

Published 24 January 2018