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 ≥ 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 ΩS 2n+1 {p} implies the existence of a p-primary Kervaire invariant one element of order p in π S 2n(p−1)−2 . We prove the converse of this last implication and observe that the homotopy decomposition problem for ΩS 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 θ 3 to give a new decomposition of ΩS 55 {3} analogous to Selick's decomposition of ΩS 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 −→ Ω 2 S 2n+1 is homotopy equivalent to the double loop space of Anick's space.