Localization of Eilenberg-MacLane G-spaces with respect to homology theory
type: Item
instance of: scholarly article
title: Localization of Eilenberg-MacLane G-spaces with respect to homology theory
zbMATH Open document ID: 0533.55008
author: Zen-ichi Yosimura
published in: Osaka Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: Let G be a compact Lie group and C(G) a collection of closed subgroups of G, one in each conjugacy class. Let \(h_*(\),A) be a homology theory with coefficients in the abelian group A defined on the category \({\mathcal C}\) of based CW complexes. Let \({\mathcal A}_{{\mathcal G}}=\{A_ H\}_{H\in C(G)}\) be a family of abelian groups. In a standard way (generalized equivariantly) one characterizes the notion of \((h_*,{\mathcal A}_{{\mathcal G}})\) localizations of G based CW complexes X, \(\eta_ X: X\to L_{(h_*,G)}X=LX.\) Based on the technique developed by Bousfield the author shows: Theorem: Let G be a compact Lie group, \(h_*\) be a homology theory defined on \({\mathcal C}\) and \({\mathcal A}_{{\mathcal G}}\) be a family of abelian groups. Given any based G CW complex X, there is a based G ma\(p\eta_ X: X\to LX\) of based G CW complexes such that \(\eta_ X\) is an \((h_*,{\mathcal A}_{{\mathcal G}})\) equivalence and LX is \((h_*,{\mathcal A}_{{\mathcal G}})\) local. In addition the author discusses G localization with respect to ordinary homology and complex homology \(K_*\). There is an overlap with results by May, McClure and Triantafillou who treat equivariant localization with respect to a set of primes.
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 57S15
zbMATH DE Number: 3846467
zbMATH Keywords: equivariant localization
zbMATH Keywords: generalized homology theories
reviewed by: Karl Heinz Dovermann
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3846467