Homotopy theory and generalized dimension subgroups

Abstract

Let $G$ be a group and $R,S,T$ its normal subgroups. There is a natural extension of the concept of commutator subgroup for the case of three subgroups $\Vert R,S,T\Vert$ as well as the natural extension of the symmetric product $\Vert \mathbf{r},\mathbf{s},t\Vert$ for corresponding ideals $\mathbf{r},\mathbf{s},\mathbf{t}$ in the integral group ring $\mathbb{Z}[G]$. In this paper, it is shown that the generalized dimension subgroup $G\cap (1+\Vert \mathbf{r},\mathbf{s},t\Vert )$ has exponent 2 modulo $\Vert R,S,T\Vert .$ The proof essentially uses homotopy theory. The considered generalized dimension quotient of exponent 2 is identified with a subgroup of the kernel of the Hurewicz homomorphism for the loop space over a homotopy colimit of classifying spaces.

18 pages