Properties of Selick's filtration of the double suspension E2

To help study the double suspension [Formula: see text] when localised at a prime p, Selick filtered Ω2S2n+1 by H-spaces which geometrically realise a natural Hopf algebra filtration of H*(Ω2S2n+1;ℤ/p). Later, Gray showed that the fiber Wn of E2 has an integral classifying space BWn and there is a homotopy fibration [Formula: see text]. In this paper we correspondingly filter BWn in a manner compatible with Selick's filtration and the homotopy fibration [Formula: see text], study the multiplicative properties and homotopy exponents of the spaces in the filtrations, and use the filtrations to filter exponent information for the homotopy groups of S2n+1. Our results link three seemingly different in nature classical homotopy fibrations given by Toda, Selick and Gray and make them special cases of a systematic whole. In addition we construct a spectral sequence which converges to the homotopy groups of BWn.