The assignment of classifying spectra to saturated fusion systems was suggested by Linckelmann and Webb and has been carried out by Broto, Levi and Oliver. A more rigid (but equivalent) construction of the classifying spectra is given in this paper. It is shown that the assignment is functorial for fusion-preserving homomorphisms in a way which extends the assignment of stable p-completed classifying spaces to finite groups, and admits a transfer theory analogous to that for finite groups. Furthermore the group of homotopy classes of maps between classifying spectra is described, and in particular it is shown that a fusion system can be reconstructed from its classifying spectrum regarded as an object under the stable classifying space of the underlying p-group.
We prove that the Díaz-Park's sharpness conjecture holds for saturated fusion systems defined on a Sylow $p$-subgroup of the group ${\rm G}_2(p)$, for $p\geq 5$.
In this short note we prove that a saturated fusion system admitting some special type of automorphism is nilpotent. This generalizes classical results by J.G. Thompson.
We say that a fusion system is the composition product of two subsystems if every morphism can be factored as a morphism in one fusion system followed by a morphism in the other. We establish a relationship between the characteristic idempotent of a saturated fusion system that is the composition product of saturated subsystems and the characteristic idempotents of the component systems. Consequently we obtain a compatibility result for transfer through the composition product and transfer through the component systems.
One way to understand the mod p homotopy theory of classifying spaces of finite groups is to compute their BZ/p-cellularization. In the easiest cases this is a classifying space of a finite group (always a finite p-group). If not, we show that it has infinitely many non-trivial homotopy groups. Moreover they are either p-torsion free or else infinitely many of them contain p-torsion. By means of techniques related to fusion systems we exhibit concrete examples where p-torsion appears.
Let $p$ be a prime number, $G$ a finite group, $P$ a $p$-subgroup of $G$ and $k$ an algebraically closed field of characteristic $p$. We study the relationship between the category $\Ff_P(G)$ and the behavior of $p$-permutation $kG$-modules with vertex $P$ under the Brauer construction. We give a sufficient condition for $\Ff_P(G)$ to be a saturated fusion system. We prove that for Scott modules with abelian vertex, our condition is also necessary. In order to obtain our results, we prove a criterion for the categories arising from the data of $(b, G)$-Brauer pairs in the sense of Alperin-Brou…
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 Hay…
We define here two new classes of saturated fusion systems, reduced fusion systems and tame fusion systems. These are motivated by our attempts to better understand and search for exotic fusion systems: fusion systems which are not the fusion systems of any finite group. Our main theorems say that every saturated fusion system reduces to a reduced fusion system which is tame only if the original one is realizable, and that every reduced fusion system which is not tame is the reduction of some exotic (nonrealizable) fusion system.
We construct a well-behaved transfer map from the p-local Burnside ring of the underlying p-group S to the p-local Burnside ring of a saturated fusion system F. Using this transfer map, we give new results on the characteristic idempotent of F -- the unique idempotent in the p-local double Burnside ring of S satisfying properties of Linckelmann and Webb. We describe this idempotent explicitly both in terms of fixed points and as a linear combination of transitive bisets. Additionally, using fixed points we determine the map for Burnside rings given by multiplication with the characteristic ide…
For a prime number $p$, a finite $p$-group of order $p^n$ has maximal class if it has nilpotency class $n-1$. Here we examine saturated fusion systems on maximal class $p$-groups and, in particular, we describe all the reduFor a prime number $p$, a finite $p$-group of order $p^n$ has maximal class if and only if it has nilpotency class $n-1$. Here we examine saturated fusion systems $\mathcal F$ on maximal class $p$-groups $S$ of order at least $p^4$. The Alperin-Goldschmidt Theorem for saturated fusion systems yields that $\mathcal F$ is entirely determined by the $\mathcal F$-automorphisms o…
We study the cohomology with twisted coefficients of the geometric realization of a linking system associated to a saturated fusion system $\mathcal{F}$. More precisely, we extend a result due to Broto, Levi and Oliver to twisted coefficients. We generalize the notion of $\mathcal{F}$-stable elements to $\mathcal{F}^c$-stable elements in a setting of cohomology with twisted coefficients by an action of the fundamental group.% or, in other word, with locally constant coefficients. We then study the problem of inducing an idempotent from an $\mathcal{F}$-characteristic $(S,S)$-biset and we show …
Complementing and extending the Inventiones work of Benson, Grodal, Henke [Group cohomology and control of p-fusion, Invent. Math. 197 (2014), 491--507] we give criteria for a space to have cohomology (strongly) F-isomorphic in the sense of Quillen to the stable elements. We extend results about groups models to fusion systems over discrete $p$-toral groups and profinite groups and provide various applications to the Kan-Thurston Theorem.
For a prime $p$, we describe a protocol for handling a specific type of fusion system on a $p$-group by computer. These fusion systems contain all saturated fusion systems. This framework allows us to computationally determine whether or not two subgroups are conjugate in the fusion system for example. We describe a generation procedure for automizers of every subgroup of the $p$-group. This allows a computational check of saturation. These procedures have been implemented using MAGMA. We describe a program to search for saturated fusion systems $\mathcal{F}$ on $p$-groups with $O_p(\mathcal{F…
In this paper we complete the description of the $B\mathbb{Z} /p$-cellularization of the classifying spaces of all finite groups, for all primes $p$. The techniques are based in a careful analysis of the $p$-fusion structure of the groups involved -with special attention to their strongly closed subgroups- and Chachólski's description of the $A$-cellular approximation.
This paper introduces the notion of fusion action system, an abstraction of the $p$-local data of a finite group acting on a finite set. Fusion action systems are closely connected with the theory of fusion systems; we detail the relationship of this new definition to the existing literature. We also develop a theory of classifying spaces of fusion action systems, analogous to the relationship between by $p$-local finite groups and fusion systems. This allows us to discuss the covering space theory arising from a group action in a fusion-theoretic context.
A p-local finite group is an algebraic structure with a classifying space which has many of the properties of p-completed classifying spaces of finite groups. In this paper, we construct a family of 2-local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of Spin_7(q) (q an odd prime power) shown by Solomon not to occur as the 2-fusion in any actual finite group. Thus, the resulting classifying spaces are not homotopy equivalent to the 2-completed classifying space of any finite group. As predicted by Benson, these class…
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.
We give an alternative to the stable classification of p-completed homotopy types of classifying spaces of finite groups offered by Martino-Priddy. For a finite group G with Sylow subgroup S, we regard the stable p-completed classifying space of G as an object under the stable p-completed classifying space of S via the canonical inclusion map. Thus we get a classification in terms of induced fusion systems. Applying Oliver's solution to the Martino-Priddy conjecture, we obtain the surprising result that the unstable p-completed homotopy type of BG is determined by the stable p-completed inclus…
We extend Dwyer's sharp subgroup homology decomposition of the classifying space of a finite group to arbitrary saturated fusion systems and arbitrary Mackey functors.
The Segal conjecture describes stable maps between classifying spaces in terms of (virtual) bisets for the finite groups in question. Along these lines, we give an algebraic formula for the p-completion functor applied to stable maps between classifying spaces purely in terms of fusion data and Burnside modules.
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 …
We show that the abelian monoid of isomorphism classes of G-stable finite S-sets is free for a finite group G with Sylow p-subgroup S; here a finite S-set is called G-stable if it has isomorphic restrictions to G-conjugate subgroups of S. These G-stable S-sets are of interest, e.g., in homotopy theory. We prove freeness by constructing an explicit (but somewhat non-obvious) basis, whose elements are in one-to-one correspondence with the G-conjugacy classes of subgroups in S. As a central tool of independent interest, we give a detailed description of the embedding of the Burnside ring for a sa…
We show that every saturated fusion system $\mathcal{F}$ has a unique minimal $\mathcal{F}$-characteristic biset $Λ_\mathcal{F}$. We examine the relationship of $Λ_\mathcal{F}$ 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 $\mathcal{F}$-characteristic biset, and more generally, any centric linking system can be identified with the $\mathcal{F}$-centric part of $Λ_\mathcal{F}$ as bisets. We explore the grouplike properties of $Λ_\mathcal{F}$, and conjecture an identification of normalizer subsyste…
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.
Self equivalences of classifying spaces of $p$-local compact groups are well understood by means of the algebraic structure that gives rise to them, but explicit descriptions are lacking. In this paper we use a construction of Robinson of an amalgam $G$, realizing a given fusion system, to produce a split epimorphism from the outer automorphism group of $G$ to the group of homotopy classes of self homotopy equivalences of the classifying space of the corresponding $p$-local compact group.
We consider the Burnside ring $A(\mathcal{F})$ of $\mathcal{F}$-stable $S$-sets for a saturated fusion system $\mathcal{F}$ defined on a $p$-group $S$. It is shown by S. P. Reeh that the monoid of $\mathcal{F}$-stable sets is a free commutative monoid with canonical basis $\{α_P\}$. We give an explicit formula that describes $α_P$ as an $S$-set. In the formula we use a combinatorial concept called broken chains which we introduce to understand inverses of modified Möbius functions.
We prove, for certain pairs G,G of finite groups of Lie type, that the p-fusion systems for G and G' are equivalent. In other words, there is an isomorphism between a Sylow p-subgroup of G and one of G' which preserves p-fusion. This occurs, for example, when G=H(q) and G'=H(q') for a simple Lie type H, and q and q' are prime powers, both prime to p, which generate the same closed subgroup of the p-adic units. Our proof uses homotopy theoretic properties of the p-completed classifying spaces of G and G', and we know of no purely algebraic proof of this result.
We define the representation ring of a saturated fusion system $\mathcal F$ as the Grothendieck ring of the semiring of $\mathcal F$-stable representations, and study the dimension functions of $\mathcal F$-stable representations using the transfer map induced by the characteristic idempotent of $\mathcal F$. We find a list of conditions for an $\mathcal F$-stable super class function to be realized as the dimension function of an $\mathcal F$-stable virtual representation. We also give an application of our results to constructions of finite group actions on homotopy spheres.
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.
We use the Steenrod algebra to study $CH^*BG$, the mod $p$ Chow ring of the classifying space of $G$. We describe a localization property which relates a given $G$ to its elementary abelian subgroups, and we study a number of particular cases, namely symmetric groups and Chevalley groups. It turns out that the Chow rings of these groups are completely determined by the abelian subgroups and their fusion.
In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analisis of the possible saturated fusion systems in terms of the outer automorphism group ant the proper F-radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p-local finite group, getting some new examples of these for p = 3.
For a finite group G of Lie type and a prime p, we compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic, with a very short list of exceptions. When p is different from the defining characteristic, the situation is much more complex, but can always be reduced to a case where the natural map from Out(G) to outer automorphisms of the fusion or linking system is split surjective. This work is motivated in part by questions involving extending the local structure of a …
The transporter systems of Oliver and Ventura and the localities of Chermak are classes of algebraic structures that model the $p$-local structures of finite groups. Other than the transporter categories and localities of finite groups, important examples include centric, quasicentric, and subcentric linking systems for saturated fusion systems. These examples are however not defined in general on the full collection of subgroups of the Sylow group. We study here punctured groups, a short name for transporter systems or localities on the collection of nonidentity subgroups of a finite $p$-grou…
Dwyer, Miller and Wilkerson proved that at the prime 2, the classifying spaces of SU(2) and SO(3) can be obtained as a homotopy pushout of the classifying spaces of certain subgroups. In this paper we show explicitly how these decompositions arise from the fusion systems of SU(2) and SO(3) over maximal discrete 2-toral subgroups.
The existence and uniqueness of linking systems associated to saturated fusion systems over discrete $p$-toral groups were proved by Levi and Libman. Their proof make indirectly use of the classification of the finite simple groups. Here we extend some results and arguments of Glauberman and Lynd to show how to remove this assumption.
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.
There exist spaces BSol(q) which are the classifying spaces of a family of 2-local finite groups based on certain fusion system over the Sylow 2-subgroups of Spin_7(q). In this paper we calculate the cohomology of BSol(q) as an algebra over the Steenrod algebra A_2. We also provide the calculation of the cohomology algebra over A_2 of the finite group of Lie type G_2(q).
The fusion orbit category $\overline{\mathcal F _{\mathcal C}} (G)$ of a discrete group $G$ over a collection $\mathcal C$ is the category whose objects are the subgroups $H$ in $\mathcal C$, and whose morphisms $H \to K$ are given by the $G$-maps $G/H \to G/K$ modulo the action of the centralizer group $C_G(H)$. We show that the higher limits over $\overline{\mathcal F _{\mathcal C}} (G)$ 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 …
In our previous paper, we constructed and studied a functorial extension of the evaluation map $S^1 \times \mathcal{L}X \to X$ to transfers along finite covers. In this paper, we show that this induces a natural evaluation map on the full subcategory of the homotopy category of spectra consisting of $p$-completed classifying spectra of finite groups. To do this, we leverage the close relationship between this full subcategory and the Burnside category of fusion systems.
