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HOME > Table of Contents and Abstracts > Vol. 68, No. 1
Tohoku Mathematical Journal
2016
March
SECOND SERIES VOL. 68, NO. 1
Tohoku Math. J.
68 (2016), 139-159
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Title
A NOTE ON RHODES AND GOTTLIEB-RHODES GROUPS
Author
Kyoung Hwan Choi, Jang Hyun Jo and Jae Min Moon
(Received April 25, 2014, revised October 6, 2014) |
Abstract.
The purpose of this paper is to give positive answers to some questions which are related to Fox, Rhodes, Gottlieb-Fox, and Gottlieb-Rhodes groups. Firstly, we show that for a compactly generated Hausdorff based $G$-space $(X,x_0,G)$ with free and properly discontinuous $G$-action, if $(X,x_0,G)$ is homotopically $n$-equivariant, then the $n$-th Gottlieb-Rhodes group $G\sigma_n(X,x_0,G)$ is isomorphic to the $n$-th Gottlieb-Fox group $G\tau_n(X/G,p(x_0))$. Secondly, we prove that every short exact sequence of groups is $n$-Rhodes-Fox realizable for any positive integer $n$. Finally, we present some positive answers to restricted realization problems for Gottlieb-Fox groups and Gottlieb-Rhodes groups.
Mathematics Subject Classification.
Primary 55Q05; Secondary 55Q70.
Key words and phrases.
Fox homotopy group, Gottlieb group, Gottlieb-Fox group, Gottlieb-Rhodes group, Rhodes group.
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