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

The purpose of this article is to describe a connection between the single 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 [17, 18], and Habegger-Lin [11, 15] on “homotopy string links”. The novelty of the current article is a description of connections between these topics. 1. A tale of two groups 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 the n-th symmetric group. It is the purpose of this article to derive some additional connections of these groups to homotopy theory, as well as some overlaps with other subjects. This article gives certain new 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, knots, and links, 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. In case K is reduced (K is reduced provided K consists of a single point in degree 0), Milnor proved that the geometric realization of F [K], denoted |F [K]|,