An algebraic model for chains on $\Omega BG{}^{^\wedge }_p$
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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 $n\ge 2$: \begin{align*} H_n(\Omega BG {}^{^\wedge }_p;k) &\cong \mathrm {Tor}_{n-1}^{e.kG.e}(kG.e,e.kG), H^n(\Omega BG{}^{^\wedge }_p;k) &\cong \mathrm {Ext}^{n-1}_{e.kG.e}(e.kG,e.kG). \end{align*} Further algebraic structure is examined, such as products and coproducts, restriction and Steenrod operations.References
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Additional Information
- Dave Benson
- Affiliation: Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland
- MR Author ID: 34795
- Email: bensondj@maths.abdn.ac.uk
- Received by editor(s): July 25, 2007
- Published electronically: November 19, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2465835