The integral cohomology of the group of loops

Abstract

Let PΣ 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 PΣ n is determined, establishing a conjecture of Brownstein and Lee. 20J06; 57M07

Topological complexity of basis-conjugating automorphism groups

“…Theorem 3.1 [Jensen et al 2006]. The rational cohomology algebra H * (PΣ n ) is isomorphic to E/I , where E is the exterior algebra over ‫ޑ‬ generated by degree one elements a i, j for 1 ≤ i = j ≤ n, and I is the homogeneous ideal generated by the degree two elements a i, j a j,i for i, j distinct, and…”

“…The basis-conjugating automorphism group of the free group F n is the subgroup of Aut(F n ) generated by the elements α i, j from (2-1) with 1 ≤ i, j ≤ n, and i = j. Following [Jensen et al 2006], we denote this group by PΣ n .…”

Resonance of basis-conjugating automorphism groups

“…The cohomology of PΣ n was computed by Jensen, McCammond, and Meier [8], resolving positively a conjecture of Brownstein and Lee [2].…”

Diagonal complexes and the integral homology of the automorphism group of a free product

“…The Euler characteristic of Wh(Z * n ) ∼ = FR(Z * n ), was computed in [17], and the homology was calculated in [14]. Recalling that the Hilbert-Poincaré series of Z is 1 + t we may reobtain these results.…”