The $3$-primary classifying space of the fiber of the double suspension
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- by Stephen D. Theriault
- Proc. Amer. Math. Soc. 136 (2008), 1489-1499
- DOI: https://doi.org/10.1090/S0002-9939-07-09249-0
- Published electronically: December 21, 2007
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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.References
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Bibliographic Information
- Stephen D. Theriault
- Affiliation: Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
- MR Author ID: 652604
- Email: s.theriault@maths.abdn.ac.uk
- Received by editor(s): October 30, 2006
- Published electronically: December 21, 2007
- Communicated by: Paul Goerss
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 1489-1499
- MSC (2000): Primary 55P45; Secondary 55R35
- DOI: https://doi.org/10.1090/S0002-9939-07-09249-0
- MathSciNet review: 2367123