Homotopy theory and generalized dimension subgroups

JOURNAL OF ALGEBRA

卷:484

Article

Ivanov, Sergei O.¹  Mikhailov, Roman^1,2  Wu, Jie³ 

[1] St Petersburg State Univ, Chebyshev Lab, 14th Line,29b, St Petersburg 199178, Russia

[2] Steklov Math Inst, St Petersburg Dept, St Petersburg, Russia

[3] Natl Univ Singapore, Dept Math, 10 Lower Kent Ridge Rd, Singapore 119076, Singapore

关键词: Homotopy groups;    Group ring;    Generalized dimension subgroups;    Simplicial groups;   

DOI  :  10.1016/j.jalgebra.2017.04.012

来源: Elsevier

【 摘 要 】

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 parallel to R, S, T parallel to as well as the natural extension of the symmetric product parallel to r, s, t parallel to for corresponding ideals r, s, t in the integral group ring Z[G]. In this paper, it is shown that the generalized dimension subgroup G boolean AND (1 + parallel to r, s, t parallel to) has exponent 2 modulo parallel to R, S, T parallel to. 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. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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