Localization of Eilenberg-MacLane G-spaces with respect to homology theory
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.
scientific article; zbMATH DE number 3846467