Anick's spaces and the double loops of odd primary Moore spaces (Q2701670)

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 [\textit{F. R. Cohen, J. C. Moore} and \textit{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 [\textit{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 [\textit{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 [\textit{J. A. Neisendorfer}, Topology 38, No. 6, 1293-1311 (1999; Zbl 0935.55008)].