### Abstract

Gray showed that the homotopy fiber W-n of the double suspension S2n-1 <(E-2)under right arrow> Omega S-2(2n+1) has an integral classifying space BWn, which fits in a homotopy fibration S2n-1 <(E-2)under right arrow> Omega S-2(2n+1) <(nu)under right arrow>BWn. In addition, after localizing at an odd prime p, BWn is an H- space and if p >= 5, then BWn 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.

Original language | English |
---|---|

Pages (from-to) | 1489-1499 |

Number of pages | 11 |

Journal | Proceedings of the American Mathematical Society |

Volume | 136 |

Issue number | 4 |

Early online date | 21 Dec 2007 |

Publication status | Published - Apr 2008 |

### Keywords

- double suspension
- H-space
- exponent
- homotopy-groups
- exponents

## Cite this

Theriault, S. D. (2008). The 3-primary classifying space of the fiber of the double suspension.

*Proceedings of the American Mathematical Society*,*136*(4), 1489-1499.