Transfer and characteristic idempotents for saturated fusion systems

Algorithms for fusion systems with applications to $p$-groups of small order

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…

Classifying spectra of saturated fusion systems

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…

Reduced, tame and exotic fusion systems

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.

离散群对融合系统的可实现性：II

Realizability of fusion systems by discrete groups: II

该研究对比离散p环面群上饱和融合系统的四种可实现性类型，针对特定局部有限群给出相关融合系统与链接系统的显式构造，证明了饱和性判定条件并推广了Cartan-Eilenberg稳定元素定理。

Minimal characteristic bisets for fusion systems

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…

On the Basis of the Burnside Ring of a Fusion System

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.

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.

Representation rings for fusion systems and dimension functions

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.

融合稳定的tom Dieck同态

The fusion-stable tom Dieck homomorphism

本文证明了tom Dieck同态及其因子符号同态在限制到p-群的融合稳定子群时仍满射，进而解决了Mazza-Miller的问题，确定了Lefschetz同态满射当且仅当域为F₂。

融合系统的细胞近似

Cellular approximations of fusion systems

本文研究了融合系统分类空间的细胞化问题，提出了判断分类空间是否为细胞的代数准则，并对特定异常示例进行了计算。

The abelian monoid of fusion-stable finite sets is free

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…

All p-local finite groups of rank two for odd prime p

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.

Higher limits over the fusion orbit category

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 …

Control of fixed points over discrete $p$-toral groups, and existence and uniqueness of linking systems

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.

Subgroup families controlling p-local finite groups

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 …