A homotopy decomposition of the fibre of the squaring map on \(\Omega^3 S^{17}\) (Q1707283)

For a based space loop space \(\Omega X\), let \(\Omega X \{k\}\) denote the homotopy fibre of the \(k\)th power map \(k:\Omega X\to\Omega X\). Let \(W_n\) and \(BW_n\) denote the homotopy fibre of the double suspension \(E^2:S^{2n-1}\to \Omega^2S^{2n+1}\) and its classifying space constructed by \textit{B. Gray} [Topology 27, No. 3, 301--310 (1988; Zbl 0668.55005)]. Let \(P^{2n+1}(p)\) denote the mod \(p\) Moore space \(P^{2n+1}(p)=S^{2n}\cup_p e^{2n+1}\) and let \(\iota_{2n}:S^{2n}\to P^{2n+1}(p)\) be the inclusion of the bottom cell. Recall that there is a \(p\)-local homotopy equivalence \(\Omega^2S^{2p+1}\{p\}\simeq_p\Omega^2S^3\langle 3\rangle\times W_p\) for an odd prime \(p\geq 3\) proved by \textit{P. Selick} [ibid. 17, 407--412 (1978; Zbl 0403.55021) and ibid. 20, 175--177 (1981; Zbl 0453.55006)] and that there is a (\(2\)-primary analogue) \(2\)-local homotopy equivalence \(\Omega^2S^5\{2\}\simeq_2\Omega^2S^3\langle 3\rangle\times W_2\) obtained by \textit{F. R. Cohen} [ibid. 23, 401--421 (1984; Zbl 0567.55012)]. It is also known that there is a \(2\)-local homotopy equivalence \(\Omega^2S^9\{2\}\simeq_2BW_2\times W_4\) proved by \textit{F. R. Cohen} and \textit{P. S. Selick} [Q. J. Math., Oxf. II. Ser. 41, No. 162, 145--153 (1990; Zbl 0695.55006)]. On the other hand, \textit{H. E. A. Campbell} et al. [Ann. Math. Stud. 113, 72--100 (1987; Zbl 0699.55005)] proved that \(\Omega^2S^{2n+1}\{2\}\) is atomic (and so that it is indecomposable) if \(2n+1\not= 3,5,9\) or 17. By the above investigations, Cohen and Selick [loc. cit.] conjectured that there is a \(2\)-local homotopy equivalence \(\Omega^2S^{17}\{2\}\simeq_2 BW_4\times W_8\). In this paper the author considers this conjecture and he proves that this conjecture is true after looping once. More precisely, he proves that there is a \(2\)-local homotopy equivalence \(\Omega^3S^{17}\{2\}\simeq_2 W_4\times\Omega W_8\). As an application, he also proves that the Whitehead square \([\iota_{2n},\iota_{2n}]\in \pi_{4n-1}(P^{2n+1}(2))\) is divisible by \(2\) iff \(2n=2,4,8\) or \(16\).