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, andan axiomatization of certain simplicial groups arising from braid groups is shown to characterize the homotopy types of connected CW-complexes. © 2010 Published by Oxford University Press. All rights reserved.