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Cellular approximations of $$p$$-local compact groups. (English) Zbl 1445.55008
Given pointed spaces $$A$$ and $$X$$, $$X$$ is said to be $$A$$-cellular if it can be constructed out of $$A$$ by means of (iterated) pointed homotopy colimits. E. Dror Farjoun [Cellular spaces, null spaces and homotopy localization. Berlin: Springer-Verlag (1995; Zbl 0842.55001)] proved the existence of an augmented and idempotent functor $$CW_A : \mathbf{Spaces}_*\to\mathbf{Spaces}_*$$ such that $$CW_AX$$ is $$A$$-cellular, and the augmentation $$\eta : CW \to \mbox{Id}$$ induces a weak equivalence $$\mbox{map}_*(A,CW_AX)\to \mbox{map}_*(A,X)$$, for every pointed space $$X$$. The space $$CW_AX$$, which is unique up to homotopy, is called the $$A$$-cellular approximation or $$A$$-cellularization of $$X$$.
Let $$BG$$ be the classifying space of an abelian $$p$$-torsion group $$G$$. The present paper is a part of a long-term program designed to understand the $$BG$$-cellular approximations of classifying spaces of groups and their mod $$p$$ homotopical analogues. The first objects studied in this context were classifying spaces of finite groups. The culminating result of R. J. Flores and R. M. Foote [Isr. J. Math. 184, 129–156 (2011; Zbl 1271.55013)] uses a previous classification of strongly closed subgroups of finite groups [R. J. Flores and R. M. Foote, Adv. Math. 222, No. 2, 453–484 (2009; Zbl 1181.20014)], emphasizing the tight relationship between the $$B\mathbb{Z}/p$$-cellular approximation of $$BG$$ and the $$p$$-local information of $$G$$ (or fusion system at the prime $$p$$). This key observation opened the way to the description of the $$B\mathbb{Z}/p$$-cellular approximation of classifying spaces of $$p$$-local finite groups, the mod $$p$$ homotopical analogues of classifying spaces of finite groups defined by C. Broto et al. [J. Am. Math. Soc. 16, No. 4, 779–856 (2003; Zbl 1033.55010)].
The main goal of this paper is to generalize the mentioned results to the most general framework of $$p$$-local compact groups, defined by C. Broto et al. [Geom. Topol. 11, 315–427 (2007; Zbl 1135.55008)] with special emphasis in the cases which arise from compact Lie groups.
##### MSC:
 55P60 Localization and completion in homotopy theory 55P47 Infinite loop spaces 55P65 Homotopy functors in algebraic topology
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