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

Author: Stephen D. Theriault
Journal: Proc. Amer. Math. Soc. 136 (2008), 1489-1499
MSC (2000): Primary 55P45; Secondary 55R35
Published electronically: December 21, 2007
MathSciNet review: 2367123
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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.

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Stephen D. Theriault
Affiliation: Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

Keywords: Double suspension, $H$-space, exponent
Received by editor(s): October 30, 2006
Published electronically: December 21, 2007
Communicated by: Paul Goerss
Article copyright: © Copyright 2007 American Mathematical Society

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