The integral cohomology of the group of loops

TL;DR

This paper determines the integral cohomology ring of the group PSigma_n, which models motions of trivial links and automorphisms of free groups, confirming a prior conjecture.

Contribution

It explicitly computes the integral cohomology ring of PSigma_n, resolving a conjecture by Brownstein and Lee.

Findings

Integral cohomology ring of PSigma_n is fully determined.

Confirms the conjecture of Brownstein and Lee.

Provides algebraic structure insights into the group of motions of trivial links.

Abstract

Let PSigma_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 PSigma_n is determined, establishing a conjecture of Brownstein and Lee.