No abstract

Let F n = x 1 , . . . , x n denote the free group with generators {x 1 , . . . , x n }. Nielsen and Magnus described generators for the kernel of the canonical epimorphism from the automorphism group of F n to the general linear group over the integers. In particular among them are the automorphisms χ k,i which conjugate the generator x k by the generator x i leaving the x j fixed for j = k. A computation of the cohomology ring as well as the Lie algebra obtained from the descending central series of the group generated by χ k,i for i < k is given here. Partial results are obtained for the group generated by all χ k,i . 20F28; 20F40, 20J06

Abstract. We give a combinatorial description of homotopy groups of ΣK(π, 1). In particular, all of the homotopy groups of the 3-sphere are combinatorially given.

We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes K for which the corresponding moment-angle complex Z K has the homotopy type of a wedge of spheres or a connected sum of sphere products. When K is flag, we identify in algebraic and combinatorial terms those K for which Z K is homotopy equivalent to a wedge of spheres, and give a combinatorial formula for the number of spheres in the wedge. This extends results of Berglund and Jöllenbeck on Golod rings and homotopy theoretical results of the first and third authors. We also establish a connection between minimally non-Golod rings and moment-angle complexes Z K which are homotopy equivalent to a connected sum of sphere products. We go on to show that for any flag complex K the loop spaces ΩZ K and ΩDJ(K) are homotopy equivalent to a product of spheres and loops on spheres when localised rationally or at any prime p = 2.

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 a summand of the center of the quotient groups of Artin pure braid groups modulo boundary Brunnian braids. These results give some deep and fundamental connections between the braid groups and the general higher homotopy groups of spheres.