Some non-Koszul algebras from rational homotopy theory

TL;DR

This paper proves that the rational cohomology algebra of the McCool group is non-Koszul for ranks four and higher, and explores its algebraic decompositions, answering a question in rational homotopy theory.

Contribution

It demonstrates the non-Koszul property of the cohomology algebra of the McCool group for n ≥ 4 and analyzes its algebraic decompositions, providing new insights into its structure.

Findings

Cohomology algebra is non-Koszul for n ≥ 4

The enveloping algebra admits two natural smash product decompositions

Answers a previously open question in the field

Abstract

The McCool group, denoted $P\Sigma_n$, is the group of pure symmetric automorphisms of a free group of rank $n$. The cohomology algebra $H^*(P\Sigma_n, \mathbb{Q})$ was determined by Jensen, McCammond and Meier. We prove that $H^*(P\Sigma_n, \mathbb{Q})$ is a non-Koszul algebra for $n \geq 4$, 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 $P\Sigma_n$, and prove that it has two natural decompositions as a smash product of algebras.