arXiv: Algebraic Topology | 2019
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.