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.
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.
We present a constructive method to compute the cellularization with respect to K(Z/p, m) for any integer m > 0 of a large class of H-spaces, namely all those which have a finite number of non-trivial K(Z/p, m)-homotopy groups (the pointed mapping space map(K(Z/p, m), X) is a Postnikov piece). We prove in particular that the K(Z/p, m)-cellularization of an H-space having a finite number of K(Z/p, m)-homotopy groups is a p-torsion Postnikov piece. Along the way we characterize the BZ/p^r-cellular classifying spaces of nilpotent groups.
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 partic…
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.
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 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…
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 …
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 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 extend Dwyer's sharp subgroup homology decomposition of the classifying space of a finite group to arbitrary saturated fusion systems and arbitrary Mackey functors.
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 …
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.
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…
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).
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.
In this paper, important concepts from finite group theory are translated to localities, in particular to linking localities. Here localities are group-like structures associated to fusion systems which were introduced by Chermak. Linking localities (by Chermak also called proper localities) are special kinds of localities which correspond to linking systems. Thus they contain the algebraic information that is needed to study $p$-completed classifying spaces of fusion systems as generalizations of $p$-completed classifying spaces of finite groups. Because of the group-like nature of localities…
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.
A. Chermak has recently proved that to each saturated fusion system over a finite $p$-group, there is a unique associated centric linking system. B. Oliver extended Chermak's proof by showing that all the higher cohomological obstruction groups relevant to unique existence of centric linking systems vanish. Both proofs indirectly assume the classification of finite simple groups. We show how to remove this assumption, thereby giving a classification-free proof of the Martino-Priddy conjecture concerning the $p$-completed classifying spaces of finite groups. Our main tool is a 1971 result of th…
We apply the yoga of classical homotopy theory to classification problems of G-extensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG to classifiying spaces of certain higher groupoids. In particular, to every fusion category C we attach the 3-groupoid BrPic(C) of invertible C-bimodule categories, called the Brauer-Picard groupoid of C, such that equivalence classes of G-extensions of C are in bijection with homotopy classes of maps from BG to the classifying space of BrPic(C). This gives…
Classifying endotrivial kG-modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite group G, has been a long-running quest. By deep work of Dade, Alperin, Carlson, Thevenaz, and others, it has been reduced to understanding the subgroup consisting of modular representations that split as the trivial module k direct sum a projective module when restricted to a Sylow p-subgroup. In this paper we identify this subgroup as the first cohomology group of the orbit category on non-trivial p-subgroups with values in the units k^x, viewed as a constant coefficien…
In 1993, Broué, Malle and Michel initiated the study of spetses on the Greek island bearing the same name. These are mysterious objects attached to non-real Weyl groups. In algebraic topology, a $p$-compact group $\mathbf{X}$ is a space which is a homotopy-theoretic $p$-local analogue of a compact Lie group. A connected $p$-compact group $\mathbf{X}$ is determined by its root datum which in turn determines its Weyl group $W_\mathbf{X}$. In this article we give strong numerical evidence for a connection between these two objects by considering the case when $\mathbf{X}$ is the exotic $2$-compac…
