Abstract
Gray showed that the homotopy fiber W n of the double suspension S 2n-1 → E2 Ω 2 S 2n+1 has an integral classifying space BW n , which fits in a homotopy fibration S 2n-1 → E2 Ω 2 S 2n+1 → υ BW n . In addition, after localizing at an odd prime p, BW n is an H-space and if p ≥ 5, then BW n is homotopy associative and homotopy commutative, and v is an H-map. We positively resolve a conjecture of Gray's that the same multiplicative properties hold for p = 3 as well. We go on to give some exponent consequences.
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