An algebraic model for chains on Ω𝐵𝐺^{^{∧}}_{𝑝}

Benson, Dave

doi:10.1090/S0002-9947-08-04728-4

An algebraic model for chains on $\Omega BG{}^{^\wedge }_p$
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Trans. Amer. Math. Soc. 361 (2009), 2225-2242 Request permission

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.

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