On symmetric commutator subgroups, braids, links and homotopy groups

Abstract

Abstract. In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This gives a connection between links and homotopy groups. Similar results hold for braid and surface groups.

Brunnian braids and Lie algebras

“…, I n its ideals. We define the notion of the fat bracket sum and the symmetric bracket sum of ideals which is similar to the corresponding fat commutator product and symmetric commutator product in groups [1], [14]. Given a Lie algebra L, and a set of its ideals I 1 , .…”

Brunnian braids on surfaces

“…Roughly speaking, d i is obtained by sending x i to 1 and keeping other generators. The following lemma is a special case of [15,Theorem 4.3]. Let H be a normal subgroup of G. A set X of elements of H is called a set of normal generators for H in G if H is the normal closure of X in G. We say that H has finitely many normal generators in G if there is a finite set X such that H is the normal closure of X in G.…”

Brunnian subgroups of mapping class groups and braid groups

“…. , x t ) such that each x belongs to some R i and there is at least one x from each R i , see [13]. Fat commutator subgroups are important in homotopy theory, see for example [18] where they are used to describe the homotopy groups of spheres and of suspensions ΣK(π, 1).…”

“…We define the fat commutator subgroup [[R j : j ∈ J]] to be the subgroup of j∈J R j generated by all commutator representatives of {R j } j∈J [19]. Fat commutator subgroups are important in homotopy theory: see, for example, [28] where they are used to describe the homotopy groups of spheres and of suspensions ΣK(π, 1).…”