The integral cohomology of the group of loops

Abstract

Let $\text{ P}{\Sigma }_n$ denote the group that can be thought of either as the group of motions of the trivial $n$–component link or the group of symmetric automorphisms of a free group of rank $n$. The integral cohomology ring of $\text{ P}{\Sigma }_n$ is determined, establishing a conjecture of Brownstein and Lee.