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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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 $ 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.


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Additional Information

Dave Benson
Affiliation: Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland
Email: bensondj@maths.abdn.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04728-4
PII: S 0002-9947(08)04728-4
Received by editor(s): July 25, 2007
Published electronically: November 19, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.