Narrow Results

To help study the double suspension 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 W_n of E² has an integral classifying space BW_n and there is a homotopy fibration . In this paper we correspondingly filter BW_n in a manner compatible with Selick's filtration and the homotopy fibration , 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 BW_n.

The following sections are included:

• CONTENTS
• FOREWORD
• PREFACE