Topology Seminar: Bob Oliver

In an article published in 2009, David Benson described, for a finite group $G$, the mod $p$ homology of the space $\Omega(BG^{\wedge}_p)$ -- the loop space of the $p$-completion of $BG$ -- in purely algebraic terms. In joint work with Carlos Broto and Ran Levi, we have tried to better understand Benson's result by generalizing it. Among other things, we showed that when $\mathcal{C}$ is a small category, $|\mathcal{C}|$ its geometric realization, $R$ a commutative ring, and $|\mathcal{C}|_R^+$ a plus construction of $|\mathcal{C}|$ with respect to homology with coefficients in $R$, then $H_*(\Omega(|\mathcal{C}|_R^+);R)$ is the homology of any chain complex of projective $R\mathcal{C}$-modules that satisfy certain conditions. Benson's theorem is then the special case where $\mathcal{C}$ is the category associated to a finite group $G$ and $R=\mathbb{F}_p$, so that $p$-completion appears as a special case of the plus construction.