The $3$-primary classifying space of the fiber of the double suspension

Abstract:

Gray showed that the homotopy fiber $W_{n}$ of the double suspension $S^{2n-1}\overset {E^{2}}{\longrightarrow } \Omega ^{2}S^{2n+1}$ has an integral classifying space $BW_{n}$, which fits in a homotopy fibration $S^{2n-1}\overset {E^{2}}{\longrightarrow } \Omega ^{2} S^{2n+1}\overset {\nu }{\longrightarrow }BW_n$. In addition, after localizing at an odd prime $p$, $BW_{n}$ is an $H$-space and if $p\geq 5$, then $BW_{n}$ is homotopy associative and homotopy commutative, and $\nu$ 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.