The fibre of the degree $3$ map, Anick spaces and the double suspension

Abstract

Let $S^{2n+1}\{p\}$ denote the homotopy fibre of the degree $p$ self map of $S^{2n+1}$. For primes $p \ge 5$, work of Selick shows that $S^{2n+1}\{p\}$ admits a nontrivial loop space decomposition if and only if $n=1$ or $p$. Indecomposability in all but these dimensions was obtained by showing that a nontrivial decomposition of $\Omega S^{2n+1}\{p\}$ implies the existence of a $p$-primary Kervaire invariant one element of order $p$ in $\pi_{2n(p-1)-2}^S$. We prove the converse of this last implication and observe that the homotopy decomposition problem for $\Omega S^{2n+1}\{p\}$ is equivalent to the strong $p$-primary Kervaire invariant problem for all odd primes. For $p=3$, we use the $3$-primary Kervaire invariant element $\theta_3$ to give a new decomposition of $\Omega S^{55}\{3\}$ analogous to Selick's decomposition of $\Omega S^{2p+1}\{p\}$ and as an application prove two new cases of a long-standing conjecture stating that the fibre of the double suspension $S^{2n-1} \longrightarrow \Omega^2S^{2n+1}$ is homotopy equivalent to the double loop space of Anick's space.