ARTIN'S BRAID GROUPS, FREE GROUPS, AND THE LOOP SPACE OF THE 2-SPHERE

Abstract

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 with Vassiliev invariants in the work of Kohno (Linear representations of braid groups and classical Yang-Baxter equations, Cont. Math. 78 (1988), 339–369 and Vassiliev invariants and de Rham complex on the space of knots, Symplectic Geometry and Quantization, Contemp. Math. 179 (1994), Am. Math. Soc. Providence, RI, 123–138), and provides connections between these various topics. Two consequences are as follows: the homotopy groups of spheres are identified as ‘natural’ sub-quotients of free products of pure braid groups, and an axiomatization of certain simplicial groups arising from braid groups is shown to characterize the homotopy types of connected CW-complexes.