Some non-Koszul algebras from rational homotopy theory

The McCool group, denoted $P{\mathrm{\Sigma }}_n$, is the group of pure symmetric automorphisms of a free group of rank $n$. The cohomology algebra $H^{\ast }(P{\mathrm{\Sigma }}_n,\mathbb{Q})$ was determined by Jensen, McCammond and Meier. We prove that $H^{\ast }(P{\mathrm{\Sigma }}_n,\mathbb{Q})$ is a non-Koszul algebra for $n\ge 4$, which answers a question of Cohen and Pruidze. We also study the enveloping algebra of the graded Lie algebra associated to the lower central series of $P{\mathrm{\Sigma }}_n$, and prove that it has two natural decompositions as a smash product of algebras.