# An algebraic model for chains on ΩBG^p

Dave Benson

Research output: Contribution to journalArticle

5 Citations (Scopus)

### 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 language English 2225-2242 18 Transactions of the American Mathematical Society 361 4 19 Nov 2008 https://doi.org/10.1090/S0002-9947-08-04728-4 Published - Apr 2009

### Fingerprint

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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