Abstract

In a 2009 paper, Dave Benson gave a description in purely algebraic terms of the mod $p$ homology of $\Omega \left({\mathit{BG}}_p^{\wedge }\right)$, when $G$ is a finite group, ${\mathit{BG}}_p^{\wedge }$ is the $p$-completion of its classifying space, and $\Omega \left({\mathit{BG}}_p^{\wedge }\right)$ is the loop space of ${\mathit{BG}}_p^{\wedge }$. The main purpose of this work is to shed new light on Benson’s result by extending it to a more general setting. As a special case, we show that if $\mathcal{𝒞}$ is a small category, $\left|\mathcal{𝒞}\right|$ is the geometric realization of its nerve, $R$ is a commutative ring, and $|\mathcal{𝒞}{|}_R^{+}$ is a “plus construction” for $\left|\mathcal{𝒞}\right|$ in the sense of Quillen (taken with respect to $R$-homology), then $H_{\ast }\left(\Omega \left(|\mathcal{𝒞}{|}_R^{+}\right);R\right)$ can be described as the homology of a chain complex of projective $R\mathcal{𝒞}$-modules satisfying a certain list of algebraic conditions that determine it uniquely up to chain homotopy. Benson’s theorem is now the case where $\mathcal{𝒞}$ is the category of a finite group $G$, $R={\mathbb{𝔽}}_p$ for some prime $p$, and $|\mathcal{𝒞}{|}_R^{+}={\mathit{BG}}_p^{\wedge }$.

Keywords

Mathematical Subject Classification 2010

Milestones

Authors