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

Authors:	Stephen D Theriault

Institution:	Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

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 , $BW_{n}$ is an -space and if $p\geq 5$ , then $BW_{n}$ is homotopy associative and homotopy commutative, and $\nu$ is an -map. We positively resolve a conjecture of Gray's that the same multiplicative properties hold for as well. We go on to give some exponent consequences.

Keywords:	Double suspension $H$-space exponent

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