### Abstract

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

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

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

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

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 |

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### Cite this

*Transactions of the American Mathematical Society*,

*361*(4), 2225-2242. https://doi.org/10.1090/S0002-9947-08-04728-4

**An algebraic model for chains on ΩBG^p.** / Benson, Dave.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 361, no. 4, pp. 2225-2242. https://doi.org/10.1090/S0002-9947-08-04728-4

}

TY - JOUR

T1 - An algebraic model for chains on ΩBG^p

AU - Benson, Dave

PY - 2009/4

Y1 - 2009/4

N2 - 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.

AB - 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.

U2 - 10.1090/S0002-9947-08-04728-4

DO - 10.1090/S0002-9947-08-04728-4

M3 - Article

VL - 361

SP - 2225

EP - 2242

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 4

ER -