An algebraic model for chains on $\Omega BG\phat$
Dave Benson
Department of Mathematics, University of Aberdeen,
Aberdeen AB24 3UE

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 $n\ge 2$:

H_n(\Omega BG\phat;k) \cong \Tor_{n-1}^{e.kG.e}(kG.e,e.kG)

H^n(\Omega BG\phat;k) \cong \Ext^{n-1}_{e.kG.e}(e.kG,e.kG).  

Further algebraic structure is examined, such as products and
coproducts, restriction and Steenrod operations.