Artin’s Braid Groups, Free Groups, and the Loop Space of the 2-sphere

The purpose of this article is to describe connections between the loop space of the 2-sphere, and Artin’s braid groups. 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. 1. A tale of two groups plus one more In 1924 E. Artin [1, 2] defined the n-th braid group Bn together with the n-th pure braid group Pn, the kernel of the natural map of Bn to Σn the symmetric group on n-letters. It is the purpose of this article to derive additional connections of these groups to homotopy theory, as well as some overlaps with algebraic, and topological properties of braid groups. This article gives certain relationships between free groups on n generators Fn, and braid groups which serve as a bridge between different structures. These connections, at the interface of homotopy groups of spheres, braids, and homotopy links, admit a common thread given by a simplicial group. Recall that a simplicial group Γ∗ is a collection of groups Γ0, Γ1, · · · ,Γn, · · · together with face operations di : Γn → Γn−1, and degeneracy operations si : Γn → Γn+1, for 0 ≤ i ≤ n. These homomorphisms are required to satisfy the standard simplicial identities. One example is Milnor’s free group construction F [K] for a pointed simplicial set K with base-point ∗ in degree zero. The simplicial group F [K] in degree n is the free group generated by the n simplices Kn modulo the single relation that s0 (∗) = 1. In case K is reduced, that is K consists of a single point in degree zero, the geometric realization of F [K] is homotopy equivalent to ΩΣ|K| [24]. Theorem 1.1 below addresses one property concerning the simplicial group given by F [∆[1]] where ∆[1] is the simplicial 1-simplex. This theorem gives one connection between F [∆[1]], and a simplicial group described next which arises naturally from Artin’s braid groups.