The loop space decomposition problem for Ω S^{2n+1}{p} is equivalent to the strong odd-primary Kervaire invariant problem, yielding new p=3 decompositions.
Amelotte, A homotopy decomposition of the fibre of the squaring map on Ω 3S17, Homology, Homotopy and Applications 20 (2018), 141–154
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The fibre of the degree $3$ map, Anick spaces and the double suspension
The loop space decomposition problem for Ω S^{2n+1}{p} is equivalent to the strong odd-primary Kervaire invariant problem, yielding new p=3 decompositions.