An algebraic model for chains on Ω𝐵𝐺^{^{∧}}_{𝑝}

Benson, Dave

doi:10.1090/S0002-9947-08-04728-4

An algebraic model for chains on $\Omega BG{}^{^\wedge}_p$

Author: Dave Benson
Journal: Trans. Amer. Math. Soc. 361 (2009), 2225-2242
MSC (2000): Primary 55P35, 55R35, 20C20; Secondary 55P60, 20J06, 13C40, 14M10
DOI: https://doi.org/10.1090/S0002-9947-08-04728-4
Published electronically: November 19, 2008
MathSciNet review: 2465835
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Abstract: We provide an interpretation of the homology of the loop space on the -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 is an idempotent in such that is the projective cover of the trivial module , and , 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.

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