Artin braid groups and homotopy groups

Abstract

We study the Brunnian subgroups and the boundary Brunnian subgroups of the Artin braid groups. The general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the Artin braid groups modulo the boundary Brunnian braids, as well as given as summands of the centres of the quotient groups of Artin pure braid groups modulo boundary Brunnian braids. The results give new connections between the braid groups and the general higher homotopy groups of spheres. © 2009 London Mathematical Society.