An Algebraic Model for Chains on Ωbg

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 ≥ 2: Hn(ΩBG ∧ p ; k) ∼= Tor n−1 (kG.e, e.kG), H(ΩBG ∧ p ; k) ∼= Extn−1 e.kG.e(e.kG, e.kG). Further algebraic structure is examined, such as products and coproducts, restriction and Steenrod operations.