Homotopy Lie algebras, lower central series and the Koszul property

Abstract

Let $X$ and $Y$ be finite-type CW–complexes ($X$ connected, $Y$ simply connected), such that the rational cohomology ring of $Y$ is a $k$–rescaling of the rational cohomology ring of $X$. Assume $H^{\ast }\left(X,\mathbb{Q}\right)$ is a Koszul algebra. Then, the homotopy Lie algebra ${\pi }_{\ast }\left(\Omega Y\right)\otimes \mathbb{Q}$ equals, up to $k$–rescaling, the graded rational Lie algebra associated to the lower central series of ${\pi }_1\left(X\right)$. If $Y$ is a formal space, this equality is actually equivalent to the Koszulness of $H^{\ast }\left(X,\mathbb{Q}\right)$. If $X$ is formal (and only then), the equality lifts to a filtered isomorphism between the Malcev completion of ${\pi }_1\left(X\right)$ and the completion of $\left[\Omega S^{2k+1},\Omega Y\right]$. Among spaces that admit naturally defined homological rescalings are complements of complex hyperplane arrangements, and complements of classical links. The Rescaling Formula holds for supersolvable arrangements, as well as for links with connected linking graph.