Let \(p\) be an odd prime and assume all spaces and maps have been localised at \(p\). For \(m \geq 2\), let \(P^{m}(p^{r})\) denote the fibre of the map of degree \(p^{r}\) on \(S^{m-1}\). Let \(H: \Omega S^{2n+1} \rightarrow \Omega S^{2n+1}\) denote the James-Hopf map and let \(C(n)\) denote the fibre of the double suspension map \(E^{2} : S^{2n-1} \rightarrow \Omega^{2}S^{2n+1}\). The author describes a series of four long-standing conjectures posed in [F. R. Cohen, J. C. Moore and J. A. Neisendorfer, Ann. Math. (2) 110, 549-565 (1979; Zbl 0443.55009)] of which the central one is that \(C(n)\) is a double loop space of the form \(C(n) = \Omega^{2} T_{\infty}^{2np - 1}(p)\), where \(\{ T_{k}^{2n-1}(p^{r})\); \(0 \leq k \leq \infty \}\) is a sequence of H-spaces constructed in [S. D. Theriault, Properties of Anick’s spaces, Trans. Am. Math. Soc. 353, No. 3, 1009-1037 (2000)]. For \(p \geq 5\) these spaces originated in [D. Anick, Differential algebras in topology, Res. Notes Math. 3, A. K. Peters, Ltd., Wellesley, MA (1993)]. Writing \(T_{k}\) for \(T_{k}^{2n-1}(p^{r})\), the author shows that \(\Omega T_{\infty}\) is a retract of \(\Omega T_{k}\) if \(k \geq 1\). The proof uses the spaces \(\{ G_{k} \}\) introduced by Anick and the fact that \(\Omega T_{k}\) is a retract of \(\Omega G_{k}\). As a corollary the author reproves some of the results of [J. A. Neisendorfer, Topology 38, No. 6, 1293-1311 (1999; Zbl 0935.55008)].