Abstract
In a 2009 paper, Dave Benson gave a description in purely algebraic terms of the mod $p$ homology of $\Omega(BG^\wedge_p)$, when $G$ is a finite group, $BG^\wedge_p$ is the $p$completion of its classifying space, and $\Omega(BG^\wedge_p)$ is the loop space of $BG^\wedge_p$. 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, $\mathcal{C}$ is the geometric realization of its nerve, $R$ is a commutative ring, and $\mathcal{C}^+_R$ is a "plus construction" for $\mathcal{C}$ in the sense of Quillen (taken with respect to $R$homology), then $H_*(\Omega(\mathcal{C}^+_R);R)$ 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 $\mathcal{C}^+_R=BG^\wedge_p$.
Original language  English 

Publisher  ArXiv 
Publication status  Submitted  6 Jul 2018 
Keywords
 math.AT
 55R35
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Profiles

Ran Levi
 Mathematical Sciences (Research Theme)
 School of Natural & Computing Sciences, Mathematical Science  Chair in Mathematical Sciences
Person: Academic