Let S2n+1{p} denote the homotopy fibre of the degree p self map of S2n+1. For primes p ≥ 5, work by Selick shows that S2n+1{p} admits a non-trivial loop space decomposition if and only if n = 1 or p. Indecomposability in all but these dimensions was obtained by showing that a non-trivial decomposition of ΩS2n+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 ΩS2n+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 θ3 to give a new decomposition of ΩS55{3} analogous to Selick's decomposition of ΩS2p+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.

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}

A simple polytope P is called B-rigid if its combinatorial type is determined by the cohomology ring of the moment-angle manifold $\mathcal {Z}_P$ over P. We show that any tensor product decomposition of this cohomology ring is geometrically realized by a product decomposition of the moment-angle manifold up to equivariant diffeomorphism. As an application, we find that B-rigid polytopes are closed under products, generalizing some recent results in the toric topology literature. Algebraically, our proof establishes that the Koszul homology of a Gorenstein Stanley–Reisner ring admits a nontrivial tensor product decomposition if and only if the underlying simplicial complex decomposes as a join of full subcomplexes.

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.