A p-local finite group consists of a finite p-group S, together with a pair of categories which encode "conjugacy" relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system F over a finite p-group S is saturated can be determined by just looking at smaller classes of subgroups of S. We also prove that the homotopy type of the classifying space of a given p-local finite group is independent of the family of subgroups used to define it, in the sense that it remains unchanged when that family ranges from the set of F -centric F -radical subgroups (at a minimum) to the set of F -quasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to p-constrained finite groups, and prove that they in fact all arise from groups.

Abstract. A p-local finite group consists of a finite p-group S, together with a pair of categories which encode "conjugacy" relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we study and classify extensions of p-local finite groups, and also compute the fundamental group of the classifying space of a p-local finite group.A p-local finite group consists of a finite p-group S, together with a pair of categories (F, L), of which F is modeled on the conjugacy (or fusion) in a Sylow subgroup of a finite group. The category L is essentially an extension of F and contains just enough extra information so that its p-completed nerve has many of the same properties as p-completed classifying spaces of finite groups. We recall the precise definitions of these objects in Section 1, and refer to [BLO2] and [5A1] for motivation for their study.In this paper, we study extensions of saturated fusion systems and of p-local finite groups. This is in continuation of our more general program of trying to understand to what extent properties of finite groups can be extended to properties of p-local finite groups, and to shed light on the question of how many (exotic) p-local finite groups there are. While we do not get a completely general theory of extensions of one p-local finite group by another, we do show how certain types of extensions can be described in a manner very similar to the situation for finite groups.From the point of view of group theory, developing an extension theory for plocal finite groups is related to the question of to what extent the extension problem for groups is a local problem, i.e., a problem purely described in terms of a Sylow p-subgroup and conjugacy relations inside it. In complete generality this is not the case. For example, strongly closed subgroups of a Sylow p-subgroup S of G need not correspond to normal subgroups of G. However, special cases where this does happen include the case of existence of p-group quotients (the focal subgroup theorems; see [Go,) and central subgroups (described via the Z * -theorem of Glauberman [Gl]).

We characterize Hopf spaces with finitely generated cohomology as an algebra over the Steenrod algebra. We "deconstruct" the original space into an H-space Y with finite mod p cohomology and a finite number of p-torsion Eilenberg-Mac Lane spaces. We give a precise description of homotopy commutative H-spaces in this setting.

Fix a prime p. Since their definition in the context of localization theory, the homotopy functors P BZ/p and CW BZ/p have shown to be powerful tools used to understand and describe the mod p structure of a space. In this paper, we study the effect of these functors on a wide class of spaces which includes classifying spaces of compact Lie groups and their homotopical analogues. Moreover, we investigate their relationship in this context with other relevant functors in the analysis of the mod p homotopy, such as Bousfield-Kan completion and Bousfield homological localization.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1218 NATÀLIA CASTELLANA AND RAMÓN FLORES book ([Far96, page 175]), where he asked about the cellularity of the p-completion of BG. This seemed a natural extension of the problems considered by the second author concerning BZ/p-homotopy of finite groups (see [Flo07], [FS07], and [FF]), so it was natural to investigate this structure with similar methods.We show that under certain hypotheses, we are able to characterize the effect of the nullification functor P BZ/p by means of a fibration.Theorem 4.1. Let X be a connected space with finite fundamental group and such that (P BZ/p (X 1 )) ∧ p * . Then there is a fibration, where X p is the covering of X whose fundamental group is T p (π 1 (X)), and L Z[ 1 p ] (X p ) denotes Bousfield homological localization of X p with respect to H * (−; Z[ 1 p ]). In particular, one can compute the homotopy groups of P BZ/p (X) in terms of those of X if X is good enough. This result is quite general and in fact describes in a single statement a phenomenon which was previously known for finite groups, p-compact groups and some compact Lie groups, but not for p-local compact or Kač-Moody groups (Corollary 4.13); so, it can then be read as a common property of a big family of homotopy meaningful spaces. Moreover, finite loop spaces also satisfy this property (Corollary 4.12).Another source of examples is the theory of infinite loop spaces. McGibbon [McG] shows that infinite loop spaces satisfy the hypothesis of Theorem 4.1 (see Corollary 4.14).We also obtain some interesting consequences of these results, including a detailed analysis of the relationship between the BZ/p-nullification and Z[1/p]-localization of these spaces -which is very much in the spirit of [Dwy96, Section 6]-and the commutation of nullity functors on them, a situation that was discussed in [RS00] in a general framework.The second part of the paper deals with the effect of the cellularization functor CW BZ/p on classifying spaces of compact Lie groups. We show a Serre-type dichotomy theorem. Theorem 6.9. Let G be a compact connected Lie group. If there exists a non-pcohomologically central element of order p, then the BZ/p-cellullarization of BG has infinitely many non-zero homotopy groups. Otherwise, it has the homotopy typeThis statement is in fact a consequence of a more general statement which extends Proposition 2.3 in [FS07].Theo...

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In this paper we obtain a description of the Grothendieck group of complex vector bundles over the classifying space of a p‐local finite group (S,F,L) in terms of representation rings of subgroups of S. We also prove a stable elements formula for generalized cohomological invariants of p‐local finite groups, which is used to show the existence of unitary embeddings of p‐local finite groups. Finally, we show that the augmentation C∗(|L|p∧;double-struckFpfalse)→double-struckFp is Gorenstein in the sense of Dwyer–Greenlees–Iyengar and obtain some consequences about the cohomology ring of false|scriptLfalse|p∧.

Abstract. In this paper we construct faithful representations of saturated fusion systems over discrete p-toral groups and use them to find conditions that guarantee the existence of unitary embeddings of p-local compact groups. These conditions hold for the Clark-Ewing and Aguadé-Zabrodsky p-compact groups as well as some exotic 3-local compact groups. We also show the existence of unitary embeddings of finite loop spaces. IntroductionIn the theory of compact Lie groups, the existence of a faithful unitary representation for every compact Lie group is a consequence of the Peter-Weyl theorem. This paper is concerned with the existence of analogous representations for several objects in the literature which are considered to be homotopical counterparts of compact Lie groups.In 1994, W.G. Dwyer and C.W. Wilkerson [17] introduced p-compact groups. They are loop spaces which satisfy some finiteness properties at a particular prime p. For example, if G is a compact Lie group such that its group of connected components is a finite p-group, then its p-completion G , where a bijective correspondence between connected p-compact groups and reflection data over the p-adic integers was established.Many ideas from the theory of compact Lie groups have a homotopical analogue for p-compact groups. Faithful unitary representations correspond to homotopy monomorphisms at the prime p from the classifying space BX of a pcompact group into BU (n) ∧ p for some n. A homotopy monomorphism at p is map g such that the homotopy fiber F of g ∧ p is BZ/p-null, that is, the evaluation map Map(BZ/p, F ) → F is a homotopy equivalence. For simplicity, we will call such maps BX → BU (n) 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 2010 Mathematics Subject Classification. 55R35, (primary), 20D20, 20C20 (secondary).

Given two finite p-local finite groups and a fusion preserving morphism between their Sylow subgroups, we study the question of extending it to a continuous map between their classifying spaces. The results depend on the construction of the wreath product of p-local finite groups which is also used to study plocal permutation representations.