On Braid Groups, Free Groups, and the Loop Space of the 2-Sphere

Abstract

Abstract. The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin's braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor [25], and Habegger-Lin [17,22] on "homotopy string links". The current article exploits Lie algebras associated to Vassiliev invariants in work of T. Kohno [19,20], and provides connections between these various topics.Two consequences are as follows: (1) the homotopy groups of spheres are identified as "natural" sub-quotients of free products of pure braid groups, and (2) an axiomatization of certain simplicial groups arising from braid groups is shown to characterize the homotopy types of connected CW -complexes.

Delta-structures on mapping class groups and braid groups

“…By Proposition 2.6, this coincides with Brun(P k+1 (M )) for all k ≥ 2. 4 The same comment applies in the mapping class group case where Proposition 2.12 tells us that the Brunnian mapping classes in the full mapping class group Γ (k+1) (M ) must lie in Γ k+1 (M ) when k ≥ 2.…”

“…In [12], this combinatorial model was related to the braid groups of the disc D 2 , resulting in a description of π n (S 2 ) as the fixed point set of a particular action of P n (D 2 ). In [8] and [4], simplicial structures on the sequences {P n+1 (D 2 )} n≥0 were studied and these were related to the homotopy groups of spheres, in particular giving descriptions of these homotopy groups as subquotients of the braid groups. Many of these ideas are discussed in the survey paper [5].…”

Configurations, braids, and homotopy groups

“…It was shown in [18] that Milnor's free group construction for the circle F [S 1 ] admits a faithful representation into a simplicial group arising from Artin's pure braid groups with a simplicial structure analogous to that above. That the representation is faithful arises from properties of Yang-Baxter Lie algebras.…”

“…That the representation is faithful arises from properties of Yang-Baxter Lie algebras. Since one of the definitions of braid groups is as the fundamental groups of unordered configuration spaces, the relations between Artin braids and homotopy groups given in [18,71] are extended in this article by studying connections between the topology of configuration spaces and variations of simplicial groups.…”

On maps from loop suspensions to loop spaces and the shuffle relations on the Cohen groups

“…This group is important for studying the classical exponent problem in homotopy theory, for instance the classical results in [2] can be obtained by some simple combinatorial computations in the group H. In [39,40,41,45,48], the group H has been successfully applied to solve some problems in the classical homotopy theory including the Cohen conjecture. As a combinatorial group, H has connections with homotopy string links studied by Milnor and Habegger-Lin in low dimensional topology [27,28,19,23], as well as braid groups and simplicial groups [3,13,46].…”