An algebraic model for chains on ΩBG^p

Dave Benson

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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.
Original languageEnglish
Pages (from-to)2225-2242
Number of pages18
JournalTransactions of the American Mathematical Society
Volume361
Issue number4
Early online date19 Nov 2008
DOIs
Publication statusPublished - Apr 2009

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Wedge
Coproducts
Loop Space
Classifying Space
Algebraic Structure
Representation Theory
Reformulation
Idempotent
Completion
Homology
Isomorphism
Trivial
Finite Group
Cover
Restriction
Module
Model
Demonstrate
Interpretation

Cite this

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

In: Transactions of the American Mathematical Society, Vol. 361, No. 4, 04.2009, p. 2225-2242.

Research output: Contribution to journalArticle

Benson, Dave. / An algebraic model for chains on ΩBG^p. In: Transactions of the American Mathematical Society. 2009 ; Vol. 361, No. 4. pp. 2225-2242.
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