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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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