P-local Finite Groups Research Articles

Subgroup collections controlling the homotopy type of a p -local compact group

Let (S,F,L) be a p-local compact group. We prove that the (uncompleted) homotopy type of the nerve of the linking system L is determined by the collection of subgroups of S that are F-centric and F-radical. This generalizes work of Broto, Grodal, Levi, Oliver, and the second author, who prove an analogous result for p-local finite groups.

Higher limits over the fusion orbit category

The fusion orbit category F‾C(G) of a discrete group G over a collection C is the category whose objects are the subgroups H in C, and whose morphisms H→K are given by the G-maps G/H→G/K modulo the action of the centralizer group CG(H). We show that the higher limits over F‾C(G) can be computed using the hypercohomology spectral sequences coming from the Dwyer G-spaces for centralizer and normalizer decompositions for G.If G is the discrete group realizing a saturated fusion system F, then these hypercohomology spectral sequences give two spectral sequences that converge to the cohomology of the centric orbit category Oc(F). This allows us to apply our results to the sharpness problem for the subgroup decomposition of a p-local finite group. We prove that the subgroup decomposition for every p-local finite group is sharp (over F-centric subgroups) if it is sharp for every p-local finite group with nontrivial center. We also show that for every p-local finite group (S,F,L), the subgroup decomposition is sharp if and only if the normalizer decomposition is sharp.

Local Gorenstein duality for cochains on spaces

We investigate when a commutative ring spectrum R satisfies a homotopical version of local Gorenstein duality, extending the notion previously studied by Greenlees. In order to do this, we prove an ascent theorem for local Gorenstein duality along morphisms of k-algebras. Our main examples are of the form R=C⁎(X;k), the ring spectrum of cochains on a space X for a field k. In particular, we establish local Gorenstein duality in characteristic p for p-compact groups and p-local finite groups as well as for k=Q and X a simply connected space which is Gorenstein in the sense of Dwyer, Greenlees, and Iyengar.

A completion theorem for fusion systems

We show that the twisted K-theory of the classifying space of a p-local finite group is isomorphic to the completion of the Grothendieck group of twisted representations of the fusion system with respect to the augmentation ideal of the representation ring of the fusion system. We use this result to compute the K-theory of the Ruiz-Viruel exotic 7-local finite groups.

Stratification and duality for homotopical groups

We generalize Quillen's F-isomorphism theorem, Quillen's stratification theorem, the stable transfer, and the finite generation of cohomology rings from finite groups to homotopical groups. As a consequence, we show that the category of module spectra over C⁎(BG,Fp) is stratified and costratified for a large class of p-local compact groups G including compact Lie groups, connected p-compact groups, and p-local finite groups, thereby giving a support-theoretic classification of all localizing and colocalizing subcategories of this category. Moreover, we prove that p-compact groups admit a homotopical form of Gorenstein duality.

Vector bundles over classifying spaces of p‐local finite groups and Benson–Carlson duality

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 in terms of representation rings of subgroups of its Sylow. 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 map for the cochains of the classifying space of a p-local finite group is Gorenstein in the sense of Dwyer-Greenlees-Iyengar and obtain some consequences about the cohomology ring of these classifying spaces.

Cohomology with twisted coefficients of the classifying space of a fusion system

We study the cohomology with twisted coefficients of the geometric realization of a linking system associated to a saturated fusion system F. More precisely, we extend a result due to Broto, Levi and Oliver to twisted coefficients. We generalize the notion of F-stable elements to Fc-stable elements in a setting of cohomology with twisted coefficients by an action of the fundamental group. We then study the problem of inducing an idempotent from an F-characteristic (S,S)-biset and we show that, if the coefficient module is nilpotent, then the cohomology of the geometric realization of a linking system can be computed by Fc-stable elements. As a corollary, we show that for any coefficient module, the cohomology of the classifying space of a p-local finite group can be computed by these Fc-stable elements.

Minimal characteristic bisets for fusion systems

We show that every saturated fusion system F has a unique minimal F-characteristic biset ΛF. We examine the relationship of ΛF with other concepts in p-local finite group theory: In the case of a constrained fusion system, the model for the fusion system is the minimal F-characteristic biset, and more generally, any centric linking system can be identified with the F-centric part of ΛF as bisets. We explore the grouplike properties of ΛF, and conjecture an identification of normalizer subsystems of F with subbisets of ΛF.

Centralizers of subsystems of fusion systems

Abstract When (S,ℱ,ℒ) is a p-local finite group and (T,ℰ,ℒ0) is weakly normal in (S,ℱ,ℒ), we show that a definition of CS (ℰ) given by Aschbacher has a simple interpretation from which one can deduce existence and strong closure very easily. We also appeal to a result of Gross to give a new proof that there is a unique fusion system C ℱ(ℰ) on CS (ℰ).

Existence and uniqueness of classifying spaces for fusion systems over discretep-toral groups

A major question in the theory of p-local finite groups was whether any saturated fusion system over a finite p-group admits an associated centric linking system, and when it does, whether it is unique. Both questions were answered in the affirmative by Chermak, using the theory of partial groups and localities he developed. Using Chermak's ideas combined with the techniques of obstruction theory, Bob Oliver gave a different proof of Chermak's theorem. In this paper, we generalize Oliver's proof to the context of fusion systems over discrete p-toral groups, thus positively resolving the analogous questions in p-local compact group theory.

An algebraic model for finite loop spaces

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.

Saturated fusion systems as idempotents in the double Burnside ring

We give a new, unexpected characterization of saturated fusion systems on a p-group S in terms of idempotents in the p-local double Burnside ring of S that satisfy a Frobenius reciprocity relation, and reformulate fusion-theoretic phenomena in the language of idempotents. Interpreting our results in stable homotopy, we answer a long-standing question on stable splittings of classifying spaces of finite groups, and generalize the Adams--Wilkerson criterion for recognizing rings of invariants in the cohomology of an elementary abelian p-group. This work is partly motivated by a conjecture of Haynes Miller which proposes retractive transfer triples as a purely homotopy-theoretic model for p-local finite groups. We take an important step toward proving this conjecture by showing that a retractive transfer triple gives rise to a p-local finite group when two technical assumptions are made, thus reducing the conjecture to proving those two assumptions.

Fusion systems and localities

We introduce objective partial groups, of which the linking systems and p-local finite groups of Broto, Levi, and Oliver, the transporter systems of Oliver and Ventura, and the \({\mathcal{F}}\)-localities of Puig are examples, as are groups in the ordinary sense. As an application we show that if \({\mathcal{F}}\) is a saturated fusion system over a finite p-group then there exists a centric linking system \({\mathcal{L}}\) having \({\mathcal{F}}\) as its fusion system, and that \({\mathcal{L}}\) is unique up to isomorphism. The proof relies on the classification of the finite simple groups in an indirect and—for that reason—perhaps ultimately removable way.

Homology decompositions and groups inducing fusion systems

We relate the construction of groups which realize saturated fusion systems and signaliser functors with homology decompositions of p-local finite groups. We prove that the cohomology ring of Robinson's construction is in some precise sense very close to the cohomology ring of the fusion system it realizes.

Extensions of linking systems and fusion systems

We correct two errors in the statement and proof of a theorem in an earlier paper (2007), and at the same time extend that result to a more general theorem about extensions of p-local finite groups. Other special cases of this theorem have already been shown in two later papers, so we feel it will be useful to have this more general result in the literature.

A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver

We extend the notion of a p-local finite group (defined in [BLO03]) to the notion of a p-local group. We define morphisms of p-local groups, obtaining thereby a category, and we introduce the notion of a representation of a p-local group via signalizer functors associated with groups. We construct a chain G = (G_0 → G_1 → ...) of 2-local finite groups, via a representation of a chain G^* = (G_0 → G_1 → ...) of groups, such that G_0 is the 2-local finite group of the third Conway sporadic group Co_3, and for n > 0, G_n is one of the 2-local finite groups constructed by Levi and Oliver in [LO02]. We show that the direct limit G of G exists in the category of 2-local groups, and that it is the 2-local group of the union of the chain G^*. The 2-completed classifying space of G is shown to be the classifying space B DI(4) of the exotic 2-compact group of Dwyer and Wilkerson [DW93].

On Oliver’s p-group conjecture: II

Let p be an odd prime and S a finite p-group. B. Oliver's conjecture arises from an open problem in the theory of p-local finite groups. It is the claim that a certain characteristic subgroup X(S) of S always contains the Thompson subgroup. In previous work the first two authors and M. Lilienthal recast Oliver's conjecture as a statement about the representation theory of the factor group S/X(S). We now verify the conjecture for a wide variety of groups S/X(S).

Wreath products and representations of p-local finite groups

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 p-local permutation representations.

On the homotopy type of the non-completed classifying space of a p-local finite group

We establish sufficient conditions for the nerve of the centric linking system of a p-local finite group (S, ℱ, ℒ) to have the homotopy type of an Eilenberg-MacLane space K(Γ, 1) for a group Γ which contains S. We prove that in this situation the entire p-local finite group can be reconstructed from Γ. Our theorem applies to many of the known examples of exotic p-local finite groups.

On two theorems of Flavell

We extend two theorems due to P. Flavell [6] to arbitrary fusion systems. The fusion system of a finite group G at a prime p encodes the structure of a Sylow-p-subgroup S together with extra information on G-conjugacy within S in category theoretic terms. From work of Alperin and Broue [1] it emerged that fusion systems of finite groups are particular cases of fusion systems of p-blocks of finite groups. In the early 1990’s, Puig introduced the notion of an abstract fusion system on a finite p-group S as a category whose objects are the subgroups of S and whose morphism sets satisfy a list of properties modelled around what one observes in the case of fusion systems of finite groups and blocks. There are ‘exotic’ fusion systems which do not arise as fusion system of a block. Benson [2] suggested that nonetheless any fusion system should give rise to a p-complete topological space which should coincide with the p-completion of the classifying space BG in case the fusion system does arise as fusion system of a finite group G. Broto, Levi and Oliver laid in [4] the homotopy theoretic foundations of such spaces called p-local finite groups and gave in particular a cohomological criterion for the existence and uniqueness of a p-local finite group associated with a given fusion system. While the existence and uniqueness of p-local finite groups for arbitrary fusion systems is still an open problem, there has been in recent years a steadily growing body of work by many authors trying to add to the understanding of fusion systems by extending classical concepts and results on the p-local structure of finite groups (some of which were relevant for the classification of finite simple groups) to all fusion systems. This is also the underlying philosophy of the present note. The following two theorems are Flavell’s Theorem A and Theorem B in [6], extended to arbitrary fusion systems. Our general terminology on fusion systems follows [7]; in particular, by a fusion system we always mean a saturated fusion system. Theorem 1. Let p be an odd prime, S a finite p-group, F a fusion system on S and α an automorphism of S acting freely on S−{1}, which stabilises F and whose order is a prime number r which does not divide the orders of the automorphism groups AutF(R) for all F-centric radical subgroups R of S. Then F = NF(S). The hypothesis that α stablises F means that for any two subgroups Q, R of S and any morphism φ : Q → R in F the morphism α ◦ φ ◦ α|α(Q) : α(Q) → α(R) is again a morphism in F . Moreover, a subgroup Q of S is called F -centric if CS(Q ) = Z(Q) for all subgroups Q of S such that Q ∼= Q in F ; a subgroup Q of S is called F -radical if Op(AutF (Q)) = AutQ(Q), the group of inner automorphisms of Q. By Alperin’s fusion theorem, F is completely determined by the automorphism groups in F of F -centric radical subgroups of S.