A homotopy decomposition of the fibre of the squaring map on $\Omega^3 S^{17}$

Abstract

We use Richter's 2-primary proof of Gray's conjecture to give a homotopy decomposition of the fibre Ω 3 S 17 {2} of the H-space squaring map on the triple loop space of the 17-sphere. This induces a splitting of the mod-2 homotopy groups π * (S 17 ; Z/2Z) in terms of the integral homotopy groups of the fibre of the double suspension E 2 : S 2n−1 → Ω 2 S 2n+1 and refines a result of Cohen and Selick, who gave similar decompositions for S 5 and S 9 . We relate these decompositions to various Whitehead products in the homotopy groups of mod-2 Moore spaces and Stiefel manifolds to show that the Whitehead square [i 2n , i 2n ] of the inclusion of the bottom cell of the Moore space P 2n+1 (2) is divisible by 2 if and only if 2n = 2, 4, 8 or 16. This will follow from a preliminary result (Proposition 3.1) equating the divisibility of [i 2n , i 2n ] with the vanishing of a Whitehead product in the mod-2 homotopy of the Stiefel manifold V 2n+1,2 , i.e., the unit tangent bundle over S 2n . It is shown in [17] that there do not exist maps S 2n−1 × P 2n (2) → V 2n+1,2 extending the inclusions of the bottom cell S 2n−1 and bottom Moore space P 2n (2) if 2n = 2, 4, 8 or 16.When 2n = 2, 4 or 8, the Whitehead product obstructing an extension is known to vanish for reasons related to Hopf invariant one, leaving only the boundary case 2n = 16 unresolved. We find that the Whitehead product is also trivial in this case.2. The decomposition of Ω 3 S 17 {2}

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

“…Explicit decompositions of Ω 2 S 5 {2}, Ω 2 S 9 {2} and Ω 3 S 17 {2} corresponding to the first three 2primary Kervaire invariant classes θ 1 = η 2 , θ 2 = ν 2 and θ 3 = σ 2 are given in [5], [9] and [1].…”

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

“…Since such elements are well known to exist only for n = 2, 4 or 8, these are the only dimensions for which Ω 2 S 2n+1 {2} can decompose non-trivially. Explicit decompositions of Ω 2 S 5 {2}, Ω 2 S 9 {2} and Ω 3 S 17 {2} corresponding to the first three 2-primary Kervaire invariant classes θ 1 = η 2 , θ 2 = ν 2 and θ 3 = σ 2 are given in [5], [9] and [1].…”