### Abstract

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

Pages (from-to) | 1035-1043 |

Number of pages | 9 |

Journal | Proceedings of the American Mathematical Society |

Volume | 142 |

Issue number | 3 |

Early online date | 24 Sep 2013 |

DOIs | |

Publication status | Published - 2014 |

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### Cite this

*Proceedings of the American Mathematical Society*,

*142*(3), 1035-1043. https://doi.org/10.1090/S0002-9939-2013-11795-8

**The rational cohomology of a p-local compact group.** / Broto, C; Levi, R; Oliver, B.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 142, no. 3, pp. 1035-1043. https://doi.org/10.1090/S0002-9939-2013-11795-8

}

TY - JOUR

T1 - The rational cohomology of a p-local compact group

AU - Broto, C

AU - Levi, R

AU - Oliver, B

PY - 2014

Y1 - 2014

N2 - For any prime $ p$, the theory of $ p$-local compact groups is modelled on the $ p$-local homotopy theory of classifying spaces of compact Lie groups and $ p$-compact groups and generalises the earlier concept of $ p$-local finite groups. These objects have maximal tori and Weyl groups, although the Weyl groups need not be generated by pseudoreflections. In this paper, we study the rational $ p$-adic cohomology of the classifying space of a $ p$-local compact group and prove that just as for compact Lie groups, it is isomorphic to the ring of invariants of the Weyl group action on the cohomology of the classifying space of the maximal torus. This is applied to show that unstable Adams operations on $ p$-local compact groups are determined in the appropriate sense by the map they induce on rational cohomology.

AB - For any prime $ p$, the theory of $ p$-local compact groups is modelled on the $ p$-local homotopy theory of classifying spaces of compact Lie groups and $ p$-compact groups and generalises the earlier concept of $ p$-local finite groups. These objects have maximal tori and Weyl groups, although the Weyl groups need not be generated by pseudoreflections. In this paper, we study the rational $ p$-adic cohomology of the classifying space of a $ p$-local compact group and prove that just as for compact Lie groups, it is isomorphic to the ring of invariants of the Weyl group action on the cohomology of the classifying space of the maximal torus. This is applied to show that unstable Adams operations on $ p$-local compact groups are determined in the appropriate sense by the map they induce on rational cohomology.

U2 - 10.1090/S0002-9939-2013-11795-8

DO - 10.1090/S0002-9939-2013-11795-8

M3 - Article

VL - 142

SP - 1035

EP - 1043

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 3

ER -