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# Homology, Homotopy and Applications

## Volume 20 (2018)

### Number 1

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

Pages: 141 – 154

DOI: https://dx.doi.org/10.4310/HHA.2018.v20.n1.a9

#### Author

#### Abstract

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

Received 20 July 2017

Received revised 31 August 2017

Published 24 January 2018