An algebraic model for finite loop spaces

Abstract

The theory of p-local compact groups, developed in an earlier paper by the same authors, is designed to give a unified framework in which to study the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups, as well as some other families of a similar nature. It also includes, and in many aspects generalizes, the earlier theory of p-local finite groups. In this paper we show that the theory extends to include classifying spaces of finite loop spaces. Our main theorem is in fact more general and states that in a fibration whose base spaces if the classifying space of a finite group, and whose fibre is the classifying space of a p-local compact group, the total space is, up to p-completion the classifying space of a p-local compact group

Unitary embeddings of finite loop spaces

“…A recent preprint [10] shows that p-completions of finite loop spaces are also modeled by p-local compact groups. More generally, if f : X → Y is a finite regular covering space, where X is the classifying space of a p-local compact group, then Y ∧ p is the classifying space of a p-local compact group.…”

“…Let X = | L| ∧ p for some p-local compact group ( S, F , L) and let G be the fundamental group of Y . By Proposition 7.1 of [10], there is a fibration | L| → E → BG such that its fiberwise p-completion is the fibration X → Y → BG associated to the covering f . If Γ = Aut L ( S), then the classifying map of the fibration | L| → E → BG defines an extension of groups…”

“…In this article we deal with the same question for the combinatorial structures called p-local compact groups, which encode the p-local information of some spaces at a prime p. They were introduced in [9] to model p-completed classifying spaces of compact Lie groups, p-compact groups, as well as linear torsion groups, and they have been shown recently to model p-completions of classifying spaces of finite loop spaces [10] and other exotic examples [20].…”

Fusion systems

“…In fact, the above statement is a shortened version of the main theorem in [O3], where conditions are also given to be able to extend L 0 to a linking system L associated to F and containing L 0 as a normal linking subsystem in the sense of [O3,Definition 8] or [AOV1, Definition 1.27]. More generally, the proof of the theorem as stated above involves constructing (in all cases) a transporter system T associated to F such that L 0 T ; this is stated and proven explicitly in [BLO5,Theorem 5.4]. …”

“…The main difference is that an extra "continuity" condition on the set of morphisms in the fusion system is needed. By analogy with the finite case, a classifying space of a saturated fusion system over a discrete p-toral group F is the p-completed realization |L| ∧ p of some associated linking system L. We refer to [BLO3, Definitions 2.2 and 4.1] for precise definitions, and to [BLO3] and [BLO5] for some of the properties of fusion and linking systems in this setting. More recently, Levi and Libman [LL] have proven the existence and uniqueness of linking systems (and hence classifying spaces) associated to saturated fusion systems over a discrete p-toral group.…”