### Abstract

We provide an interpretation of the homology of the loop space on the $ p$-completion of the classifying space of a finite group in terms of representation theory, and demonstrate how to compute it. We then give the following reformulation. If $ f$ is an idempotent in $ kG$ such that $ f.kG$ is the projective cover of the trivial module $ k$, and $ e=1-f$, then we exhibit isomorphisms for $ n\ge 2$:

$\displaystyle H_n(\Omega BG {}^{^\wedge}_p;k)$ $\displaystyle \cong \mathrm{Tor}_{n-1}^{e.kG.e}(kG.e,e.kG),$

$\displaystyle H^n(\Omega BG{}^{^\wedge}_p;k)$ $\displaystyle \cong \mathrm{Ext}^{n-1}_{e.kG.e}(e.kG,e.kG).$

Further algebraic structure is examined, such as products and coproducts, restriction and Steenrod operations.

$\displaystyle H_n(\Omega BG {}^{^\wedge}_p;k)$ $\displaystyle \cong \mathrm{Tor}_{n-1}^{e.kG.e}(kG.e,e.kG),$

$\displaystyle H^n(\Omega BG{}^{^\wedge}_p;k)$ $\displaystyle \cong \mathrm{Ext}^{n-1}_{e.kG.e}(e.kG,e.kG).$

Further algebraic structure is examined, such as products and coproducts, restriction and Steenrod operations.

Original language | English |
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Pages (from-to) | 2225-2242 |

Number of pages | 18 |

Journal | Transactions of the American Mathematical Society |

Volume | 361 |

Issue number | 4 |

Early online date | 19 Nov 2008 |

DOIs | |

Publication status | Published - Apr 2009 |