Classifying spectra of saturated fusion systems

Kári Ragnarsson

Department of Mathematical Sciences University of Aberdeen Aberdeen AB24 3UE United Kingdom

                 Email: kari@maths.abdn.ac.uk

Abstract 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.

AMS Classification 55R35; 20D20, 55P42

Keywords fusion systems, p-local finite groups, stable homotopy, transfer

Introduction

Saturated fusion systems were introduced by Puig in [20, 21] as a formalization of fusion systems of groups. To a finite group G with Sylow p-subgroup S one associates a fusion system F_{S}(G) over S. This is the category whose objects are the subgroups of S, and whose morphisms are the conjugations induced by elements in G. Puig axiomatized this construction, thus allowing abstract fusion systems without requiring the presence, or indeed existence, of an ambient group G. He also identified important properties enjoyed by those fusion systems that are induced by groups. Puig called fusion systems with these properties full Frobenius systems. These definitions were later simplified by Broto-Levi-Oliver, who introduced the term saturated fusion systems in [7] (see Definition 1.3 below). A further simplification has been obtained by Kessar-Stancu in [11].

A useful tool for the study of saturated fusion systems would be a functor assigning a classifying space to each saturated fusion system. Exactly what a classifying space means in this context is made precise by the theory of p-local finite groups developed by Broto-Levi-Oliver in [7]. They define a p-local finite group as a triple (S, F, L), where S is a finite p-group, F is a saturated fusion system over S, and L is a centric linking system associated to F, a category which contains just enough information to construct a classifying space |L|_{p}^{∧} for F.

The motivating example for the definition of a p-local finite group comes from finite groups. In [6], Broto-Levi-Oliver give an algebraic construction for a centric linking system L_{S}^{c}(G) associated to the fusion system F_{S}(G) of a finite group G, and show that |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}.

Given the classifying space |L|_{p}^{∧}, one can by [7] reconstruct the fusion system via the following homotopy-theoretic construction:

Hom_{F} (P, Q) = {φ ∈ Hom (P, Q) | θ ∘ Bι_{Q} ∘ Bφ ≃ θ ∘ Bι_{P} },

where ι_{P} and ι_{Q} are the inclusions of the subgroups P and Q in S, and θ is the natural “inclusion” BS → |L|_{p}^{∧}. This construction was first applied by Martino-Priddy in [15] to show that if the p-completed classifying spaces of two finite groups have the same homotopy types, then their fusion systems are isomorphic.

The passage from saturated fusion systems to classifying spaces is more problematic. In general it is not known whether a saturated fusion system has an associated centric linking system, and if so, whether it is unique. Broto-Levi-Oliver have developed an obstruction theory to address these questions of existence and uniqueness. Oliver has shown in [18, 19] that these obstructions vanish for fusion systems of finite groups. Therefore L_{S}^{c}(G) is, up to equivalence, the unique centric linking system associated to the fusion system F_{S}(G) of a finite group G. Moreover, Oliver concludes that the p-completed classifying spaces of two finite groups are homotopy equivalent if their fusion systems over chosen Sylow subgroups are isomorphic via a fusion-preserving isomorphism of these Sylow subgroups, thus proving the Martino-Priddy conjecture [15].

By Oliver’s result the fusion system F_{S}(G) of a finite group G has a unique associated centric linking system. But even when we restrict our attention to fusion systems coming from groups, we do not have an expedient method to reconstruct the linking system L_{S}^{c}(G) (and consequently BG_{p}^{∧}) from the fusion data F_{S}(G). Nor do we know whether this assignment is functorial, that is whether a morphism between fusion systems of groups induces a map between their p-completed classifying spaces.

A classifying space functor is not yet within our reach, but the stable analogue presents a more tractable problem. When calculating the cohomology of a p-local finite group (S, F, L) in [7], Broto-Levi-Oliver construct a characteristic biset for F. This is an (S, S)-biset Ω with properties, suggested by Linckelmann-Webb, that guarantee that the induced stable selfmap of BS is an idempotent in cohomology with 𝔽_{p}-coefficients (see Proposition 4.2). Broto-Levi-Oliver noted that the stable summand of Σ^{∞}BS induced by a characteristic biset Ω is independent of the particular choice of Ω, and agrees with the suspension spectrum of the classifying space |L|_{p}^{∧}. Furthermore, they observed that the construction of Ω depends only on the saturated fusion system F and not on the centric linking system L, and that therefore the induced summand 𝔹F can be considered as a classifying spectrum for the saturated fusion system F.

In this paper we take their idea further. We give a different formulation of the construction of a classifying spectrum of a saturated fusion system F, which allows us to retain more information associated to F. More precisely, we refine the construction of the biset Ω in [7] to produce an idempotent ω̃ in {BS, BS} with the following stable idempotent analogues of the Linckelmann-Webb properties:

(a) ω̃ is a ℤ_{p}^{∧}-linear combination of homotopy classes of maps of the form Σ^{∞}Bφ ∘ tr_{P}, where P is a subgroup of S, φ ∈ Hom_{F} (P, S) and tr_{P} denotes the reduced transfer of the inclusion P ≤ S.

(b1) For each subgroup P ≤ S and each φ ∈ Hom_{F} (P, S), the restrictions ω̃ ∘ Σ^{∞}Bι_{P} and ω̃ ∘ Σ^{∞}Bφ are homotopic as maps Σ^{∞}BP → Σ^{∞}BS.

(b2) For each P ≤ S and each φ ∈ Hom_{F} (P, S), the compositions tr_{P} ∘ ω̃ and tr_{φ} ∘ ω̃, where tr_{φ} is the reduced transfer of the monomorphism φ: P → S, are homotopic as maps Σ^{∞}BS → Σ^{∞}BP.

(c) ϵ̃(ω) = 1, where ϵ̃: {BS, BS} → ℤ_{p}^{∧} is a morphism of modules derived from an augmentation of {BS_{+}, BS_{+}} (see Lemma 2.5 and Section 6).

We show that ω̃ is the unique idempotent in {BS, BS} with these properties and that Property (b1) characterizes morphisms in the fusion system F. Therefore we write ω̃_{F} and refer to ω̃_{F} as the stable characteristic idempotent of F.

The homotopy type of the stable summand of Σ^{∞}BS induced by ω̃_{F} agrees with the homotopy type of the classifying spectrum 𝔹F constructed by Broto-Levi-Oliver, so this construction offers nothing new in itself. It is the careful study of the characteristic idempotent which allows us to exercise control over its mapping telescope 𝔹F. We refer to the structure map σ_{F}: Σ^{∞}BS → 𝔹F of the mapping telescope as the structure map of F, and when regarded as an object under Σ^{∞}BS, we refer to the pair (σ_{F}, 𝔹F) as the structured classifying spectrum of F. The structure map σ admits a transfer map t_{F}, which is, up to homotopy, the unique map 𝔹F → BS such that t_{F} ∘ σ_{F} ≃ ω̃_{F} and σ_{F} ∘ t_{F} ≃ id_{𝔹F}.

The reward for taking this point of view is the following result, which further justifies the use of the term “classifying spectrum”. It appears in the text as Theorem 7.3.

Theorem A If F is a saturated fusion system over a finite p-group S, then F can be recovered from its structured classifying spectrum (σ_{F}, 𝔹F) by the following homotopy-theoretic construction:

Hom_{F} (P, Q) = {φ ∈ Hom (P, Q) | σ_{F} ∘ Σ^{∞}Bι_{Q} ∘ Σ^{∞}Bφ ≃ σ_{F} ∘ Σ^{∞}Bι_{P} },

where ι_{P} and ι_{Q} are the inclusions of the subgroups P and Q in S.

By [14, Example 5.2] the fusion system can not be recovered from the homotopy type of the classifying spectrum alone; it must be regarded as an object under Σ^{∞}BS. When applied to fusion systems of groups, this theorem gives an alternative stable classification of p-completed classifying spaces of finite groups, which is in some sense finer than the one in [14]. Combined with the Martino-Priddy conjecture, this shows that the unstable p-completed classifying space of a finite group is determined by the stable p-completed classifying space, regarded as an object under the stable classifying space of its p-Sylow subgroup. This matter is taken up in [24].

The central result in this paper, which allows us to conduct the necessary analysis of characteristic idempotents, is the calculation of an explicit ℤ_{p}^{∧}-basis for the submodule

ω̃_{F_{2}} ∘ {BS_{1}, BS_{2}} ∘ ω̃_{F_{1}} ⊂ {BS_{1}, BS_{2}},

for saturated fusion systems F_{1} and F_{2} over finite p-groups S_{1} and S_{2}, respectively. This module is naturally isomorphic to the group of stable maps between the classifying spectra of the fusion systems involved and so we get the following theorem, a more concise version of which appears later as Theorem 7.2, as an immediate consequence.

Theorem B Let F_{1} and F_{2} be saturated fusion systems over finite p-groups S_{1} and S_{2}, respectively. Then the group of homotopy classes of stable maps from 𝔹F_{1} to 𝔹F_{2} is a free ℤ_{p}^{∧}-module with one basis element σ_{F_{2}} ∘ (Σ^{∞}Bψ ∘ tr_{P}) ∘ t_{F_{1}} for every conjugacy class of pairs (P, ψ) consisting of a subgroup P ≤ S_{1} and a nontrivial homomorphism ψ: P → S_{2}. Conjugacy here means that F_{1}-conjugacy is taken in the source and F_{2}-conjugacy is taken in the target.

When F_{1} and F_{2} are fusion systems of groups, this theorem can be applied to give a new variant of the Segal conjecture describing the group of homotopy classes of stable maps between p-completed classifying spaces of finite groups. This discussion is taken up in [23].

If F_{1} and F_{2} are saturated fusion systems over finite p-groups S_{1} and S_{2}, respectively, the obstruction to restricting a homomorphism γ: S_{1} → S_{2} to a map between classifying spectra respecting their structure maps is the compatibility of γ with the stable characteristic idempotents. This compatibility is achieved when γ is a (F_{1}, F_{2})-fusion-preserving homomorphism, which means that γ induces a functor F_{γ}: F_{1} → F_{2} such that F_{γ}(P) = γ(P) for all P ≤ S_{1} and γ|_{Q} ∘ φ = F_{γ}(φ) ∘ γ|_{P} for all φ ∈ Hom_{F_{1}} (P, Q). Letting SFS denote the category whose objects are the saturated fusion systems and whose morphisms are fusion-preserving homomorphisms, we get the following result which follows from Theorem 7.9 in the text.

Theorem C There is a classifying spectrum functor

𝔹: SFS⟶ Spectra

acting on objects by sending a saturated fusion system to its classifying spectrum and on morphisms by sending a (F_{1}, F_{2})-fusion-preserving morphism γ to the map

𝔹γ_{F_{1}}^{F_{2}} := σ_{F_{2}} ∘ Σ^{∞}Bγ ∘ t_{F_{1}} : 𝔹F_{1} ⟶ 𝔹F_{2},

which satisfies

𝔹γ_{F_{1}}^{F_{2}} ∘ σ_{F_{1}} ≃ σ_{F_{2}} ∘ Σ^{∞}Bγ.

It is an important property of this functor that when γ: S_{1} → S_{2} is the restriction of a homomorphism \overline{γ}: G_{1} → G_{2} to Sylow subgroups, the map 𝔹F_{S_{1}}(G_{1}) → 𝔹F_{S_{2}}(G_{2}) induced by γ is equivalent to the map Σ^{∞}BG_{1}_{p}^{∧} → Σ^{∞}BG_{2}_{p}^{∧} induced by \overline{γ}, as maps of objects under the stable classifying spaces of their Sylow subgroups. This is proved in Section 10.

A monomorphism γ: S_{1} → S_{2} admits a transfer map tr_{γ}: Σ^{∞}BS_{2} → Σ^{∞}BS_{1}, which restricts to a map of classifying spectra that preserves transfer maps when γ is fusion-preserving. Collecting Theorem 8.6 and Propositions 9.5 and 9.6, we get the following result.

Theorem D There is an assignment of a transfer map

T r (γ_{F_{1}}^{F_{2}}) := σ_{F_{1}} ∘ tr_{γ} ∘ t_{F_{2}} : 𝔹F_{2} → 𝔹F_{1},

to every (F_{1}, F_{2})-fusion-preserving monomorphism γ: S_{1} → S_{2}. The assignment has the following properties:

(i) t_{F_{1}} ∘ T r (γ_{F_{1}}^{F_{2}}) ≃ tr_{γ} ∘ t_{F_{2}}.

(ii) T r (γ_{F_{1}}^{F_{2}}) ∘ T r (γ_{F_{2}}^{F_{3}}) ≃ T r (γ_{F_{1}}^{F_{3}}).

(iii) The composition 𝔹γ_{F_{1}}^{F_{2}} ∘ T r (γ_{F_{1}}^{F_{2}}) acts on H^{∗}(𝔹F_{2}; 𝔽_{p}) as multiplication by |S_{2}|/|S_{1}|.

(iv) The transfer T r (γ_{F_{1}}^{F_{2}}) satisfies the Frobenius reciprocity relation

T r (γ_{F_{1}}^{F_{2}})^{∗} (𝔹γ_{F_{1}}^{F_{2}}^{∗} (x) · y) = x · T r (γ_{F_{1}}^{F_{2}})^{∗} (y)

for x ∈ H^{∗}(𝔹F_{2}; 𝔽_{p}) and y ∈ H^{∗}(𝔹F_{1}; 𝔽_{p}).

The motivation for the work in this paper comes from a question of Miller regarding an alternative formulation of p-local finite groups in terms of homotopy subgroup inclusions satisfying a certain transfer property. The author has obtained a partial answer to this question, but on the way to doing so, has discovered results about classifying spectra of saturated fusion systems which are most likely of interest to a wider audience than the original question, and are therefore presented separately in this paper. These results do not depend on centric linking systems, and to emphasize this we mostly avoid mentioning centric linking systems in this paper. Implications for p-local finite groups will be discussed in a subsequent paper [22], where Miller’s question will be addressed.

Notational conventions: Throughout this paper, p is a fixed prime. Cohomology is always taken with 𝔽_{p}-coefficients. For a space X we let X_{+} be the pointed space obtained by adding a disjoint basepoint to X, and we let X_{p}^{∧} denote the Bousfield-Kan p-completion [5].

The category of finite groups and homomorphisms is denoted by Gr. For an element g of a group G, we let c_{g} denote the conjugation x ↦ gxg^{−1}. When H is a subgroup of G we write ^{g}H for the conjugate c_{g}(H) and H^{g} for the inverse conjugate c_{g}^{−1}(H) = g^{−1}Hg. For subgroups H and K of G we let N_{G}(H, K) denote the transporter

N_{G}(H, K) := {g ∈ G | ^{g}H ≤ K},

and write

Hom_{G} (H, K) := {c_{g} : H → K | g ∈ N_{G}(H, K)}
= N_{G}(H, K)/C_{G}(H)

for the set of homomorphisms from H to K induced by conjugation in G.

The inclusion of a subgroup H into a supergroup is denoted by ι_{H}, specifying the supergroup when there is danger of confusion. For the convenience of the reader we use the letters S, P and Q to refer to finite p-groups, while G and H refer to general finite groups. Moreover we use φ to denote homomorphisms belonging to fusion systems, while ψ and ρ denote general homomorphisms.

All stable homotopy takes place in the homotopy category of spectra which we denote by Spectra . A discussion of the stable homotopy category can be found for example in [1]. We will use the shorthand notation

Σ_{+}^{∞}X := Σ^{∞}(X_{+})

for the suspension spectrum of X_{+}. Since we often have cause to work with stable p-completed classifying spaces, we adopt the shorthand notation

𝔹(−) := Σ^{∞}B(−)_{p}^{∧},

regarded as functors

Gr ⟶ Spectra .

As is usual, for spaces X and Y we let {X, Y } denote the group of homotopy classes of stable maps Σ^{∞}X → Σ^{∞}Y , and for spectra E and F we let [E, F] denote the group of homotopy classes of (degree 0) maps E → F. All homotopies are unpointed.

Overview: In the first section we recall the definition of saturated fusion systems. The second section treats Burnside modules and the Segal conjecture relating them to stable maps between classifying spaces of groups. In addition we develop some tools and notation we will use throughout the paper. In Section 3 we introduce the notion of fusion subconjugacy. For fusion systems F_{1} and F_{2} over finite p-groups S_{1} and S_{2}, this gives a useful fusion-invariant filtration of the Burnside module A(S_{1}, S_{2}). In Section 4 we assign a characteristic idempotent ω_{F} in the p-completed double Burnside ring A(S, S)_{p}^{∧} to a saturated fusion system F over S. In Section 5 we perform a careful analysis of the inherent properties of this idempotent, and in Section 6 we interpret these results for the stable idempotent ω̃_{F} of 𝔹S induced by ω_{F}. In Section 7 we define the classifying spectrum of F as the summand 𝔹F of Σ^{∞}BS given by ω̃_{F}, and prove that this assignment is functorial. In Section 8 we develop the theory of transfers for classifying spectra, and in Section 9 we look at the behaviour of classifying spectra and their transfers in cohomology. We conclude this paper in Section 10 by showing that the theory of classifying spectra of saturated fusion systems developed here agrees with existing theories of stable classifying spaces of saturated fusion systems.

Acknowledgements: I would like to thank my thesis advisor Haynes Miller for suggesting the problem out of which this work grew and for his enthusiasm and helpful advice during its progress. I also thank Bob Oliver for his emailed suggestions for the proof of convergence in Section 4 and Ran Levi for many lively and encouraging discussions on this subject. Finally I thank the referee on behalf of both the reader and myself for a very accurate and helpful report which has improved the exposition and clarity of the paper. The method of proof of the central result in this paper is, at least subconsciously, inspired by Nishida’s work in [17] and a preliminary version thereof.

1 Saturated fusion systems

In this section we recall the definition of a saturated fusion system. We begin by presenting the motivating example.

Definition 1.1 Let G be a finite group with Sylow p-subgroup S. The fusion system of G over S is the category F_{S}(G) whose objects are the subgroups of S, and whose morphisms are the homomorphisms induced by conjugation in G:

Hom_{F_{S}(G)} (P, Q) = Hom_{G} (P, Q) .

Puig [20, 21] axiomatized this construction as follows.

Definition 1.2 A fusion system F over a finite p-group S is a category, whose objects are the subgroups of S, and whose morphism sets Hom_{F} (P, Q) satisfy the following conditions:

(a) Hom_{S} (P, Q) ⊆ Hom_{F} (P, Q) ⊆ Inj (P, Q) for all P, Q ≤ S.

(b) Every morphism in F factors as an isomorphism in F followed by an inclusion.

From the definition it is clear that every fusion system over S contains the fusion system F_{S}(S) of S. We denote this fusion system by F_{S} for short.

Fusion systems at this level of generality are not particularly useful or interesting, so we restrict to a certain subclass of fusion systems introduced by Puig in [21]. Puig identified important properties enjoyed by fusion systems of groups, and called fusion systems with these properties full Frobenius systems. His definitions were later simplified by Broto-Levi-Oliver in [7], where they suggested the name saturated fusion systems. A further simplification has been obtained by Kessar-Stancu in [11].

We present the Broto-Levi-Oliver version below, but before stating the definition, we need to introduce some additional terminology. We say that two subgroups P, P′ ≤ S are F -conjugate if they are isomorphic in F. A subgroup P ≤ S is fully centralized in F if |C_{S}(P)| ≥ |C_{S}(P′)| for every P′ ≤ S that is F -conjugate to P. Similarly P is fully normalized in F if |N_{S}(P)| ≥ |N_{S}(P′)| for every P′ ≤ S that is F -conjugate to P.

Definition 1.3 A fusion system F over a p-group S is saturated if the following two conditions hold:

(I) If P ≤ S is fully normalized in F, then P is also fully centralized in F, and p does not divide the index of Aut_{S} (P) in Aut_{F} (P).

(II) If P ≤ S and φ ∈ Hom_{F} (P, S) are such that φ(P) is fully centralized, then φ extends to φ̄ ∈ Hom_{F} (N_{φ}, S), where

N_{φ} = {g ∈ N_{S}(P) | φ ∘ c_{g} ∘ φ^{−1} ∈ Aut_{S}(φ(P))}.

This definition is rather technical, and as the conditions in the definition are not used explicitly in this paper, it suffices for the reader to keep in mind that Condition I is a “prime to p” or “Sylow” property, analogous to the fact that the index of a Sylow subgroup in a finite group is not divisible by p. Condition II is a “maximal extension property” which (in a non-precise sense and when combined with Condition I) can be thought of as an axiomatic replacement of Sylow’s Second and Third Theorems.

The role of saturated fusion systems in the theory of classifying spectra developed in this paper is as follows. In Section 4 we construct a characteristic idempotent ω for a fusion system F with a characteristic biset Ω. These objects are defined precisely in Section 4, and for now it suffices to say that characteristic bisets are finite (S, S)-bisets with properties stipulated by Linckelmann-Webb. The classifying spectrum of F is then constructed using ω in Section 7. A construction of characteristic bisets for saturated fusion systems is given by Broto-Levi-Oliver in [7]. This allows us to develop the theory of classifying spectra of saturated fusion system. But existence of a classifying spectrum for a fusion system F depends only on the existence of a characteristic biset for F, and the properties of classifying spectra follow from the Linckelmann-Webb properties without using the saturation axioms. The theory therefore extends automatically to all fusion systems that have characteristic bisets. It is an interesting question whether the existence of a characteristic biset for a fusion system F implies saturation of F. The author believes this is true, which is why the results in this paper are only presented for saturated fusion systems.

2 Burnside Modules and the Segal conjecture

In this section we give a brief discourse about how stable maps between classifying spaces of finite groups G_{1} and G_{2} are related to (G_{1}, G_{2})-bisets.

For finite groups G_{1} and G_{2}, let A^{+}(G_{1}, G_{2}) be the set of isomorphism classes of finite sets with a right G_{1}-action and a free left G_{2}-action. The disjoint union operation makes A^{+}(G_{1}, G_{2}) into a commutative monoid. We denote the Grothendieck group completion by A(G_{1}, G_{2}) and refer to it as the Burnside module of G_{1} and G_{2}. The reader should beware that this is not standard terminology. The group structure of A(G_{1}, G_{2}) is easy to describe. It is a free abelian group with one generator corresponding to each transitive (G_{1}, G_{2})- biset. We proceed to describe and parametrize these basis elements below.

Definition 2.1 Let G_{1} and G_{2} be finite groups. A (G_{1}, G_{2})-pair is a pair (H, ψ) consisting of a subgroup H ≤ G_{1} and a homomorphism

ψ: H → G_{2}.

We say that two (G_{1}, G_{2})-pairs (H_{1}, ψ_{1}) and (H_{2}, ψ_{2}) are (G_{1}, G_{2})-conjugate if there exist elements g ∈ G_{1} and h ∈ G_{2} such that c_{g}(H_{1}) = H_{2} and the following diagram commutes

H_{1} ⟶^{ψ_{1}} G_{2}
≅↓^{c_{g}}  ↓^{c_{h}}
H_{2} ⟶^{ψ_{2}} G_{2}.

Remark 2.2 Define the graph of a (G_{1}, G_{2})-pair (H, ψ) by

Δ_{H}^{ψ} := {(h, ψ(h) | h ∈ H} ≤ G_{1} × G_{2}.

It is easy to check that (G_{1}, G_{2})-pairs (H_{1}, ψ_{1}) and (H_{2}, ψ_{2}) are (G_{1}, G_{2})- conjugate if and only if their graphs are conjugate in G_{1} × G_{2}.

We denote the (G_{1}, G_{2})-conjugacy class of a (G_{1}, G_{2})-pair (H, ψ) by [H, ψ]_{G_{1}}^{G_{2}} or, when there is no danger of confusion, just [H, ψ]. With a slight abuse of notation we will also let [H, ψ]_{G_{1}}^{G_{2}} (or [H, ψ]) denote the basis element of A(G_{1}, G_{2}) corresponding to the conjugacy class of the (G_{1}, G_{2})-pair (H, ψ). Thus [H, ψ] represents the isomorphism class of the (G_{1}, G_{2})-biset

G_{2} ×_{(H,ψ)} G_{1} := (G_{2} × G_{1})/ ∼,

with the obvious right G_{1}-action and left G_{2}-action, where

(x, gy) ∼ (xψ(g), y)

for x ∈ G_{2} , y ∈ G_{1} and g ∈ H.

Given three finite groups G_{1}, G_{2}, and G_{3}, we get a morphism of monoids

− ∘ − : A^{+}(G_{2}, G_{3}) × A^{+}(G_{1}, G_{2}) → A^{+}(G_{1}, G_{3})

by

(Ω, Λ) ↦ Ω ∘ Λ := Ω ×_{G_{2}} Λ,

which extends to a bilinear map

A(G_{2}, G_{3}) × A(G_{1}, G_{2}) → A(G_{1}, G_{3}).  (1)

This pairing can be described in terms of the basis elements using the double coset formula.

[K, ρ]_{G_{2}}^{G_{3}} ∘ [H, ψ]_{G_{1}}^{G_{2}} = ∑_{x∈K\G_{2}/ψ(H)} [ψ^{−1} (ψ (H) ∩ K^{x}) , ρ ∘ c_{x} ∘ ψ]_{G_{1}}^{G_{3}} . (2)

We pay special attention to the simple case where K = G_{2}, so ρ and ψ are composable morphisms. In this case the double coset formula simplifies to

[G_{2}, ρ]_{G_{2}}^{G_{3}} ∘ [H, ψ]_{G_{1}}^{G_{2}} = [H, ρ ∘ ψ]_{G_{1}}^{G_{3}}.  (3)

For a finite group G the pairing of (1) makes A(G, G) into a ring which we call the double Burnside ring of G. This should not be confused with the Burnside ring A(G) [25]. The latter is the Grothendieck group completion of the monoid of isomorphism classes of finite left G-sets. As a ℤ-module, A(G) is free ℤ-module with one generator [G/H] for each conjugacy class of subgroups H ≤ G. As a ring, the multiplicative structure on A(G) is induced by Cartesian product and linear extension.

Bisets relate to stable maps via the Becker-Gottlieb transfer [3]. We recall some basic properties of transfers here, and refer the reader to [2] for a more thorough discussion. Given a finite covering f: X → Y , where Y is connected, Becker-Gottlieb constructed a stable map tr_{f}: Σ_{+}^{∞}Y → Σ_{+}^{∞}X, called the transfer of f. (Actually, a more general transfer for fibrations with compact fibres has been constructed by Dwyer in [10] but we need not consider that here.) We will use the following important properties of transfers:

Contravariant functoriality If f: X → Y and g: Y → Z are finite coverings of connected spaces, then

tr_{g∘f} ≃ tr_{f} ∘ tr_{g}.

Normalization If f: X → Y is an n-fold cover of a connected space, then the induced map in singular cohomology (with any coefficients)

tr_{f}^{∗} ∘ Σ_{+}^{∞}f^{∗} : H^{∗}(Y) → H^{∗}(X) → H^{∗}(Y)

is multiplication by n.

Frobenius reciprocity If f: X → Y is a finite cover of a connected space, then the following diagram, where Δ_{X} and Δ_{Y} denote the respective diagonals of X and Y , commutes:

Σ_{+}^{∞}Y  ⟶^{Σ_{+}^{∞}Δ_{Y}}  Σ_{+}^{∞}Y ∧ Σ_{+}^{∞}Y
↓^{tr_{f}}  ↓^{1∧tr_{f}}
Σ_{+}^{∞}X ⟶^{(Σ_{+}^{∞}f∧id)∘Σ_{+}^{∞}Δ_{X}} Σ_{+}^{∞}Y ∧ Σ_{+}^{∞}X.

In particular,

tr_{f}^{∗}(f^{∗}(y) · x) = y · tr_{f}^{∗}(x)

for x ∈ H^{∗}(X) and y ∈ H^{∗}(Y).

Since Σ_{+}^{∞}X ≃ Σ^{∞}X ∨ 𝕊^{0}, the transfer tr_{f}: Σ_{+}^{∞}Y → Σ_{+}^{∞}X of a finite cover f: X → Y restricts to a reduced transfer Σ^{∞}X → Σ^{∞}Y . As there is no danger of confusion we also denote the reduced transfer by tr_{f}, and sometimes refer to it as transfer.

A monomorphism of groups ψ: G → G′ induces a fibration G′/ψ(G) ↪ BG ⟶^{Bψ} BG′. If [G′ : ψ(G)] is finite, which is always the case if G and G′ are finite, Bψ therefore admits a transfer map, which we denote tr_{ψ} for short. In the special case of the inclusion H ≤ G of a subgroup of finite index we denote the transfer by tr_{H}.

Given a finite (G_{1}, G_{2})-biset Ω ∈ A^{+}(G_{1}, G_{2}), we now get a stable map α(Ω) ∈ {BG_{1}_{+}, BG_{2}_{+}} as follows. Let Λ := G_{2}\Ω. Since the left G_{2}-action on Ω is free, we get a principal fibre sequence

G_{2} → Ω ×_{G_{1}} EG_{1} → Λ ×_{G_{1}} EG_{1}.

Let

ξ: Λ ×_{G_{1}} EG_{1} → BG_{2}

be the classifying map of this fibration. The projection map

Λ ×_{G_{1}} EG_{1} → BG_{1}

is a finite covering. Let τ: Σ_{+}^{∞}BG_{1} → Σ_{+}^{∞}(Λ ×_{G_{1}} EG_{1}) be the associated transfer map. The map α(Ω) is now defined as

α(Ω) := Σ_{+}^{∞}ξ ∘ τ.

This assignment extends to a homomorphism

α: A(G_{1}, G_{2}) → {BG_{1}_{+}, BG_{2}_{+}}

of abelian groups. Although it may not be immediate from the definition, it is shown for example in [4] that the map α is natural in the sense that it sends the pairing of (1) to the composition pairing for stable maps:

α(Ω ∘ Λ) = α(Ω) ∘ α(Λ).

Thus α is a ring homomorphism when G_{1} = G_{2}. One can check that the value of α on a basis element [H, ψ] is

α([H, ψ]) = Σ_{+}^{∞}Bψ ∘ tr_{H}.

The homomorphism α gives a way to relate A(G_{1}, G_{2}) to the group of homotopy classes of stable maps {BG_{1}_{+}, BG_{2}_{+}}. Lewis-May-McClure have made this relationship precise in [13]. As a consequence of the Segal conjecture (proved by Carlsson in [8]), they show that α is completion with respect to the augmentation ideal I(G_{1}) of the Burnside ring A(G_{1}). In the case where G_{1} is a p-group, May-McClure [16] showed that, after getting rid of basepoints, this completion takes a simple form, which we will describe below.

Definition 2.3 For finite groups G_{1} and G_{2}, we say that a (G_{1}, G_{2})-pair (H, ψ) is trivial if ψ is the trivial homomorphism. In this case we also say that the conjugacy class [H, ψ] is trivial. When ψ is not the trivial homomorphism, we say that the pair (H, ψ) and the conjugacy class [H, ψ] are non-trivial.

Let Ã(G_{1}, G_{2}) be the quotient module obtained from A(G_{1}, G_{2}) by quotienting out all trivial basis elements [H, ψ]. Recalling that Σ_{+}^{∞}BG ≃ Σ^{∞}BG ∨ 𝕊^{0}, where 𝕊^{0} = Σ^{∞}S^{0} is the suspension sphere spectrum, one can check that there is an induced map

α: Ã(G_{1}, G_{2}) ⟶ {BG_{1}_{+}, BG_{2}_{+}}/{BG_{1}_{+}, S^{0}} ≅ {BG_{1}, BG_{2}}.

May-McClure proved that when G_{1} is a p-group, I(G_{1})-adic completion coincides with p-adic completion on Ã(G_{1}, G_{2}), and deduced the following version of the Segal conjecture.

Theorem 2.4 (Segal conjecture [8, 13, 16]) If S is a finite p-group and G any finite group, then α induces an isomorphism

α̃: Ã(S, G)_{p}^{∧} ⟶^{≅} {BS, BG},

where (−)_{p}^{∧} = (−) ⊗ ℤ_{p}^{∧} is p-adic completion.

For finite p-groups S_{1} and S_{2}, we will, in view of the Segal conjecture, have reason to p-complete the Burnside module A(S_{1}, S_{2}). The resulting ℤ_{p}^{∧}-module A(S_{1}, S_{2})_{p}^{∧} is a free ℤ_{p}^{∧}-module with one basis element for each conjugacy class of (S_{1}, S_{2})-pairs, and by a further, yet slight, abuse of notation, we will also let [P, ψ]_{S_{1}}^{S_{2}} (or [P, ψ]) denote the basis element of A(S_{1}, S_{2})_{p}^{∧} corresponding to the conjugacy class of the (S_{1}, S_{2})-pair (P, ψ).

We conclude this section by adapting some “bookkeeping” tools for (S_{1}, S_{2})- bisets to keep track of elements of A(S_{1}, S_{2})_{p}^{∧}. First we note that the structure of A(S_{1}, S_{2})_{p}^{∧} allows us to define a collection of homomorphisms

χ_{[P,ψ]}: A(S_{1}, S_{2})_{p}^{∧} → ℤ_{p}^{∧},

one for each conjugacy class of (S_{1}, S_{2})-pairs, by demanding that

Ω = ∑_{[P,ψ]} χ_{[P,ψ]}(Ω) · [P, ψ],

for all Ω ∈ A(S_{1}, S_{2})_{p}^{∧}.

Next, we extend the notion of counting the number of S_{2}-orbits of (S_{1}, S_{2})- bisets to obtain a form of augmentation for Burnside modules. The resulting assignment is natural in that it sends the pairing of (1) to multiplication in ℤ_{p}^{∧}.

Lemma 2.5 For every pair of finite p-groups S_{1} and S_{2}, the assignment

A^{+}(S_{1}, S_{2}) → ℤ, Ω ↦ |S_{2}\Ω|

extends to a homomorphism

ϵ: A(S_{1}, S_{2})_{p}^{∧} → ℤ_{p}^{∧},

sending composition to multiplication.

Proof Recalling that bisets Ω ∈ A^{+}(S_{1}, S_{2}) have a free S_{2}-action, we see that each assignment

A^{+}(S_{1}, S_{2}) → ℤ, Ω ↦ |S_{2}\Ω|

is a morphism of monoids, and so we get an induced homomorphism ϵ: A(S_{1}, S_{2}) → ℤ and, after p-completion, an induced homomorphism ϵ: A(S_{1}, S_{2})_{p}^{∧} → ℤ_{p}^{∧}.

Using the freeness of the left action for bisets Ω ∈ A^{+}(S_{2}, S_{3}) and Λ ∈ A^{+}(S_{1}, S_{2}) again, we get

|S_{3}\(Ω ∘ Λ)| = |Ω ×_{S_{2}} Λ|/|S_{3}| = (|Ω| · |Λ|/|S_{2}|)/|S_{3}| = |S_{3}\Ω| · |S_{2}\Λ|.

The collection of homomorphisms ϵ: A(S_{1}, S_{2})_{p}^{∧} → ℤ_{p}^{∧} therefore sends composition to multiplication.

A useful, well known result states that for a finite group G, two finite G-sets Ω and Λ are isomorphic if and only if they have the same number of fixed points for every subgroup of G. Since the number of fixed points depends only on the conjugacy class of the subgroup, an alternative formulation is that there is an injective ℤ-module homomorphism

A(G) → ∏_{[H]} ℤ, Ω ↦ ∏_{[H]} |Ω^{H}|,

where the product is taken over conjugacy classes of subgroups H ≤ G.

For finite groups S_{1} and S_{2} we regard a (S_{1}, S_{2})-biset Ω as a left (S_{1} × S_{2})-set by putting (g, h)x := hxg^{−1} for g ∈ S_{1}, h ∈ S_{2} and x ∈ Ω. This assignment preserves isomorphism classes and we obtain an injection

A(S_{1}, S_{2}) ⟶ A(S_{1} × S_{2})

sending a basis element [P, ψ] to [(S_{1} × S_{2})/Δ_{P}^{ψ}]. For a subgroup Q ≤ S_{1} × S_{2}, this allows us to define Ω^{Q} as the fixed-point set of Ω under the action of Q. By linear extension and p-completion we get a well defined ℤ_{p}^{∧}-module homomorphism

A(S_{1}, S_{2})_{p}^{∧} ⟶ ℤ_{p}^{∧}, Ω ↦ |Ω^{Q}|,

depending only on the conjugacy class of Q. On basis elements we have

|[P, ψ]^{Q}| = |((S_{1} × S_{2})/Δ_{P}^{ψ} )^{Q}| = |Δ_{P}^{ψ} \N_{S_{1}×S_{2}}(Q, Δ_{P}^{ψ})| = \frac{|N_{S_{1}×S_{2}}(Q, Δ_{P}^{ψ})|}{|P |}.

Lemma 2.6 Let S_{1} and S_{2} be finite p-groups. Then the ℤ_{p}^{∧}-module homomorphism

A(S_{1}, S_{2})_{p}^{∧} ⟶ ∏_{[P,ψ]} ℤ_{p}^{∧}, Ω ↦ ∏_{[P,ψ]} |Ω^{Δ_{P}^{ψ}} |,

where the product is taken over conjugacy classes of (S_{1}, S_{2})-pairs, is injective.

Proof Being a composition of two injective homomorphisms, the ℤ-module homomorphism

A(S_{1}, S_{2}) ⟶ A(S_{1} × S_{2}) ⟶ ∏_{[Q]} ℤ, Ω ↦ ∏_{[Q]} |Ω^{Q}| ,

where the product runs over conjugacy classes of subgroups Q ≤ S_{1} × S_{2}, is itself injective. Noting that the collection of graphs of (S_{1}, S_{2})-pairs is closed under conjugation in S_{1} × S_{2} and taking subgroups, we see that for a (S_{1}, S_{2})- pair (P, ψ), we have

N_{S_{1}×S_{2}}(Q, Δ_{P}^{ψ}) = ∅,

and consequently

|[P, ψ]^{Q}| = 0,

if Q is not the graph of an (S_{1}, S_{2})-pair. We conclude that the restriction to a ℤ-module homomorphism

A(S_{1}, S_{2}) ⟶ ∏_{[P,ψ]} ℤ, Ω ↦ ∏_{[P,ψ]} |Ω^{Δ_{P}^{ψ}} |,

is injective, and it remains so after p-completion.

3 Fusion subconjugacy

In this section we introduce the notion of fusion subconjugacy for subgroups of a finite p-group S and for (S_{1}, S_{2})-pairs. This induces a filtration on the p-completed Burnside module A(S_{1}, S_{2})_{p}^{∧} and consequently of the group {BS_{1}, BS_{2}} of homotopy classes of stable maps. By studying this filtration we will obtain useful information about how homotopy classes of stable maps between classifying spaces of finite p-groups behave under composition with stable maps arising from fusion systems over those p-groups. The material in this section is presented for p-completed Burnside modules because we are mostly interested in that setting. However the analogous results still hold in the uncompleted or p-localized case.

Definition 3.1 Let F be a fusion system over a finite p-group S, and let P and Q be subgroups of S.

• We say that Q is F -subconjugate to P, and write Q ≾_{F} P, if there exists

a morphism φ ∈ Hom_{F} (Q, P).

• We say that Q is F -conjugate to P, and write Q ∼_{F} P if Q is isomorphic to P in F.

• We say that Q is strictly F -subconjugate to P, and write Q ⋨_{F} P if Q is

F -subconjugate to P, but not F -conjugate to P.

When there is no danger of confusion, we will write ≾, ⋨, and ∼ instead of ≾_{F}, ⋨_{F} and ∼_{F}. For the fusion system F_{S} of S, F_{S} -subconjugacy coincides with

S -subconjugacy. We make a similar definition for pairs.

Definition 3.2 Let F_{1} and F_{2} be fusion systems over finite p-groups S_{1} and S_{2}, respectively. Let (P, ψ) and (Q, ρ) be two (S_{1}, S_{2})-pairs.

• We say that (Q, ρ) is (F_{1}, F_{2})-subconjugate to (P, ψ), and write

(Q, ρ) ≾_{(F_{1},F_{2})}(P, ψ),

if  there  exist  morphisms  φ_{1} ∈ Hom_{F_{1}} (Q, P)  and

φ_{2} ∈ Hom_{F_{2}} (ρ(Q), ψ(P)) such that the following diagram commutes

Q ⟶^{ρ} ρ(Q)
φ_{1}↓  ↓^{φ_{2}}
P ⟶^{ψ} ψ(P).

• We say that (Q, ρ) is (F_{1}, F_{2})-conjugate to (P, ψ), and write

(Q, ρ) ∼_{(F_{1},F_{2})}(P, ψ),

if

(Q, ρ) ≾_{(F_{1},F_{2})}(P, ψ)  and  (P, ψ) ≾_{(F_{1},F_{2})}(Q, ρ).

• We say that (Q, ρ) is strictly (F_{1}, F_{2})-subconjugate to (P, ψ), and write

(Q, ρ) ⋨_{(F_{1},F_{2})}(P, ψ),

if (Q, ρ) is (F_{1}, F_{2})-subconjugate to (P, ψ), but not (F_{1}, F_{2})-conjugate

to (P, ψ).

When there is no danger of confusion, we will write ≾, ⋨, and ∼ instead of ≾_{(F_{1},F_{2})}, ⋨_{(F_{1},F_{2})} and ∼_{(F_{1},F_{2})}. As before, (F_{S_{1}}, F_{S_{2}})-conjugacy agrees with the

notion of (S_{1}, S_{2})-conjugacy defined in Section 2.

Remark 3.3 It is easy to check that (F_{1}, F_{2})-subconjugacy is preserved by (S_{1}, S_{2})-conjugacy. Therefore we will often say that an isomorphism class of pairs [Q, ρ] is (F_{1}, F_{2})-subconjugate to an isomorphism class [P, ψ] and write

[Q, ρ] ≾_{(F_{1},F_{2})}[P, ψ] (or [Q, ρ] ≾ [P, ψ])

if the subconjugacy relation

(Q, ρ) ≾_{(F_{1},F_{2})}(P, ψ)

holds between any (and hence all) representatives of the classes. Furthermore, we will use the same terminology when we regard [P, ψ] and [Q, ρ] as basis elements of A(S_{1}, S_{2}) or A(S_{1}, S_{2})_{p}^{∧}. The analogous remark applies to (F_{1}, F_{2})- conjugacy and strict (F_{1}, F_{2})-subconjugacy.

Remark 3.4 Subconjugacy among (S_{1}, S_{2})-pairs can in fact be regarded as a special case of subconjugacy among subgroups of S_{1} × S_{2}. Recall from [7, Section 1], that in the setting of the definition above, the fusion system F_{1} × F_{2} over S_{1} × S_{2} is defined by setting Hom_{F_{1}×F_{2}} (P, Q) to be the morphism set

{(φ_{1}, φ_{2})|_{P} ∈ Hom (P, Q) | φ_{i} ∈ Hom_{F_{i}} (P_{i}, S_{i}) , P ≤ P_{1} × P_{2}}

for all P, Q ≤ S_{1} × S_{2}. By [7, Lemma 1.5], the fusion system F_{1} × F_{2} is saturated if the fusion systems F_{1} and F_{2} are both saturated. For (S_{1}, S_{2})- pairs (P, ψ) and (P′, ψ′), one can check that (P′, ψ′) is (F_{1}, F_{2})-subconjugate to (P, ψ) if and only Δ_{P′}^{ψ′} is (F_{1} × F_{2})-subconjugate to Δ_{P}^{ψ}.

It is easy to check that (F_{1}, F_{2})-subconjugacy is a transitive relation. Therefore the (F_{1}, F_{2})-conjugacy classes of (S_{1}, S_{2})-pairs form a poset under (F_{1}, F_{2})- subconjugacy. Since (S_{1}, S_{2})-conjugacy classes of (S_{1}, S_{2})-pairs form a ℤ_{p}^{∧}- basis for the ℤ_{p}^{∧}-module A(S_{1}, S_{2})_{p}^{∧}, this leads us to a poset-indexed filtration as defined below.

Definition 3.5 Let F_{1} and F_{2} be fusion systems over finite p-groups S_{1} and S_{2}, respectively. Let (P, ψ) be a (S_{1}, S_{2})-pair.

• Let M (≾ [P, ψ], F_{1}, F_{2}) denote the submodule of A(S_{1}, S_{2})_{p}^{∧} generated by the basis elements [Q, ρ] such that

[Q, ρ] ≾_{(F_{1},F_{2})}[P, ψ].

• Let M (∼ [P, ψ], F_{1}, F_{2}) denote the submodule of A(S_{1}, S_{2})_{p}^{∧} generated by the basis elements [Q, ρ] such that

[Q, ρ] ∼_{(F_{1},F_{2})}[P, ψ].

• Let M (⋨ [P, ψ], F_{1}, F_{2}) denote the submodule of A(S_{1}, S_{2})_{p}^{∧} generated by the basis elements [Q, ρ] such that

[Q, ρ] ⋨_{(F_{1},F_{2})}[P, ψ].

When the fusion systems F_{1} and F_{2} are clear from the context, and there is no danger of confusion, we will write M (≾ [P, ψ]) , M (∼ [P, ψ]) and M (⋨ [P, ψ]) instead of M (≾ [P, ψ], F_{1}, F_{2}) , M (∼ [P, ψ], F_{1}, F_{2}) and M (⋨ [P, ψ], F_{1}, F_{2}).

The stable selfmaps of a finite p-group arising from morphisms in a fusion system are of special importance in this paper. We therefore consider the corresponding subring of A(S, S)_{p}^{∧}.

Definition 3.6 Let F be a fusion system over a finite p-group S. We denote by A_{F}(S, S) the submodule of A(S, S) generated by the basis elements [P, φ] where φ ∈ Hom_{F} (P, S).

After p-completion we obtain a submodule A_{F} (S, S)_{p}^{∧} of A(S, S)_{p}^{∧}, again generated by the basis elements [P, φ] where φ ∈ Hom_{F} (P, S).

Remark 3.7 One can check that

A_{F} (S, S)_{p}^{∧} = M (≾ [S, id], F_{S}, F) = M (≾ [S, id], F, F) = M (≾ [S, id], F, F_{S}) .

Under composition, the ℤ_{p}^{∧}-module A(S_{1}, S_{2})_{p}^{∧} becomes a left A(S_{2}, S_{2})_{p}^{∧}- module and a right A(S_{1}, S_{1})_{p}^{∧}-module. The filtration in Definition 3.5 is useful to us mainly because of the following lemma.

Lemma 3.8 Let F_{1} and F_{2} be fusion systems over the finite p-groups S_{1} and S_{2}, respectively. The following hold for every (S_{1}, S_{2})-pair (P, ψ):

(a) A_{F_{2}} (S_{2}, S_{2})_{p}^{∧} ∘ M (≾ [P, ψ]) ⊆ M (≾ [P, ψ]) ,

(b) A_{F_{2}} (S_{2}, S_{2})_{p}^{∧} ∘ M (⋨ [P, ψ]) ⊆ M (⋨ [P, ψ]) ,

(c) M (≾ [P, ψ]) ∘ A_{F_{1}} (S_{1}, S_{1})_{p}^{∧} ⊆ M (≾ [P, ψ]) ,

(d) M (⋨ [P, ψ]) ∘ A_{F_{1}} (S_{1}, S_{1})_{p}^{∧} ⊆ M (⋨ [P, ψ]) .

Proof We prove parts (a) and (b), and leave the proofs of (c) and (d), which are similar, to the reader.

First we show that for any (S_{1}, S_{2})-pair (Q, ρ) and any basis element [T, φ] of A_{F_{2}} (S_{2}, S_{2})_{p}^{∧}, we have

[T, φ] ∘ [Q, ρ] ∈ M (≾ [Q, ρ]) .

Indeed, by the double coset formula,

[T, φ] ∘ [Q, ρ] = ∑_{x∈T \S_{2}/ρ(Q)} [ρ^{−1} (ρ (Q) ∩ T^{x}) , φ ∘ c_{x} ∘ ρ] ,

and it suffices to prove that

[ρ^{−1} (ρ (Q) ∩ T^{x}) , φ ∘ c_{x} ∘ ρ] ≾ [Q, ρ]

for each x ∈ S_{2}. But this is clear by the diagram

ρ^{−1} (ρ (Q) ∩ T^{x}) ⟶^{φ∘c_{x}∘ρ} φ (^{x}ρ (Q) ∩ T)
ι↓  ↓_{c_{x}^{−1}∘φ^{−1}}
Q  ⟶^{ρ}  ρ (Q) .

To prove part (a), let [Q, ρ] ≾ [P, ψ]. By the preceding observation we get

[T, φ] ∘ [Q, ρ] ∈ M (≾ [Q, ρ]) ⊆ M (≾ [P, ψ]) .

Letting [T, φ] and [Q, ρ] vary over all basis elements of A_{F_{2}} (S_{2}, S_{2})_{p}^{∧} and M (≾ [P, ψ]), we get the desired result.

Similarly, part (b) follows upon noting that for a basis element [Q, ρ] of M (⋨ [P, ψ]) and a basis element [T, φ] of A_{F_{2}} (S_{2}, S_{2})_{p}^{∧}, we have

[T, φ] ∘ [Q, ρ] ∈ M (≾ [Q, ρ]) ⊆ M (⋨ [P, ψ]) .

We have the following structural corollaries.

Corollary 3.9 Let F be a fusion system over a finite p-group S. Then A_{F} (S, S)_{p}^{∧} is a subring of A(S, S)_{p}^{∧}. Similarly A_{F}(S, S) is a subring of A(S, S).

Proof This follows from Remark 3.7 and Lemma 3.8. The same proof works for the last statement.

Corollary 3.10 Let F_{1} and F_{2} be fusion systems over the finite p-groups S_{1} and S_{2}, respectively. For every (S_{1}, S_{2})-pair (P, ψ), the ℤ_{p}^{∧}-modules M (≾ [P, ψ]) and M (⋨ [P, ψ]) are left modules over A_{F_{2}} (S_{2}, S_{2})_{p}^{∧} and right modules over A_{F_{1}} (S_{1}, S_{1})_{p}^{∧}.

Definition 3.11 Let F be a fusion system over a finite p-group S, and let Ω ∈ A(S, S)_{p}^{∧}. We say that Ω is right F -stable if for every P ≤ S and every φ ∈ Hom_{F} (P, S) we have

Ω ∘ [P, φ]_{P}^{S} = Ω ∘ [P, ι_{P}]_{P}^{S}

in A(P, S)_{p}^{∧}. Similarly we say that Ω is left F -stable if for every P ≤ S and φ ∈ Hom_{F} (P, S) we have

[φ(P), φ^{−1}]_{S}^{P} ∘ Ω = [P, id_{P}]_{S}^{P} ∘ Ω

in A(S, P)_{p}^{∧}.

When Ω is represented by a (S, S)-biset X, the right F -stability condition means that the restriction of X to a (P, S)-biset via φ is isomorphic to the restriction of X via the inclusion P ↪ S, while left stability means that the restriction of X to a (S, P)-biset via φ is isomorphic to the restriction of X via the inclusion.

We will later define a similar notion of fusion stability for maps between stable classifying spaces of p-groups.

Lemma 3.12 Let F_{1} and F_{2} be fusion systems over the finite p-groups S_{1} and S_{2}, respectively, let Ω_{1} ∈ A(S_{1}, S_{1})_{p}^{∧} be left F_{1}-stable, and let Ω_{2} ∈ A(S_{2}, S_{2})_{p}^{∧} be right F_{2}-stable. If the (S_{1}, S_{2})-pairs (P, ψ) and (Q, ρ) are (F_{1}, F_{2})- conjugate, then

Ω_{2} ∘ [Q, ρ] ∘ Ω_{1} = Ω_{2} ∘ [P, ψ] ∘ Ω_{1}.

Proof Let ψ̃: P → ψ(P) denote the restriction of ψ to its image. Since (P, ψ) and (Q, ρ) are conjugate, there exist isomorphisms φ_{1} ∈ Hom_{F_{1}} (P, Q) and φ_{2} ∈ Hom_{F_{2}} (ψ(P), ρ(Q)) such that

ρ ∘ φ_{1} = ι_{ρ(Q)} ∘ φ_{2} ∘ ψ̃.

Using stability, and recalling the simple description of the double coset formula for composable morphisms in (3), we now obtain

Ω_{2} ∘ [Q, ρ]_{S_{1}}^{S_{2}} ∘ Ω_{1}
= Ω_{2} ∘ [Q, ι_{ρ(Q)} ∘ φ_{2} ∘ ψ̃ ∘ φ_{1}^{−1}]_{S_{1}}^{S_{2}} ∘ Ω_{1}
= Ω_{2} ∘ ([ψ (P) , ι_{ρ(Q)} ∘ φ_{2}]_{ψ(P)}^{S_{2}} ∘ [P, ψ̃]_{P}^{ψ(P)} ∘ [Q, φ_{1}^{−1}]_{S_{1}}^{P}) ∘ Ω_{1}
= (Ω_{2} ∘ [ψ (P) , ι_{ρ(Q)} ∘ φ_{2}]_{ψ(P)}^{S_{2}}) ∘ [P, ψ̃]_{P}^{ψ(P)} ∘ ([φ(P), φ_{1}^{−1}]_{S_{1}}^{P} ∘ Ω_{1})
= (Ω_{2} ∘ [ψ (P) , ι_{ψ(P)}]_{ψ(P)}^{S_{2}}) ∘ [P, ψ̃]_{P}^{ψ(P)} ∘ ([P, id_{P}]_{S_{1}}^{P} ∘ Ω_{1})
= Ω_{2} ∘ ([ψ (P) , ι_{ψ(P)}]_{ψ(P)}^{S_{2}} ∘ [P, ψ̃]_{P}^{ψ(P)} ∘ [P, id_{P}]_{S_{1}}^{P})) ∘ Ω_{1}
= Ω_{2} ∘ [P, ψ]_{S_{1}}^{S_{2}} ∘ Ω_{1}.

4 Characteristic idempotents

In this section, and for the rest of the paper, we restrict our attention to saturated fusion systems. For a saturated fusion system F over a finite p-group S, we will prove the existence of an idempotent ω ∈ A(S, S)_{p}^{∧}, related to F through properties made precise in Definition 4.3 below. These properties, and their importance, were originally recognized by Linckelmann-Webb for bisets. It is the careful analysis of ω which will allow us to produce the main results of this paper. In later sections we will see that ω is uniquely determined by F and that it characterizes the fusion system F, thus justifying the term characteristic idempotent.

In [7, Section 5], Broto-Levi-Oliver determined the cohomological structure of a p-local finite group (S, F, L). In short, they proved that in cohomology, the natural inclusion θ: BS → |L|_{p}^{∧} induces an isomorphism

H^{∗}(|L|_{p}^{∧}) ⟶^{≅} H^{∗}(F) ⊆ H^{∗}(BS),

where

H^{∗}(F) := lim \overline{}_{F} H^{∗}(B(−))

is the “ring of stable elements for F ”, regarded as a subring of H^{∗}(BS), via the identification

H^{∗}(F) ≅ {x ∈ H^{∗}(BS) | Bφ^{∗}(x) = Bι_{P}^{∗} (x) for all P ≤ S, φ ∈ Hom_{F} (P, S)}.

One of the key ingredients in their proof is the construction of a characteristic biset Ω ∈ A^{+}(S, S), as defined below. We take advantage of their construction and produce our characteristic idempotent by showing the convergence of a judiciously chosen subsequence of the sequence

[Ω], [Ω]^{2}, [Ω]^{3}, …

Definition 4.1 Let F be a fusion system over a finite p-group S. We say that an element Ω ∈ A(S, S) is a virtual characteristic biset for F if it has the following properties:

(a’) Ω ∈ A_{F}(S, S). (b1’) Ω is right F -stable. (b2’) Ω is left F -stable. (c’) ϵ(Ω) ≡ 1 (mod p).

If in addition Ω ∈ A^{+}(S, S) then we say that Ω is a characteristic biset for F.

We refer to these properties as the Linckelmann-Webb properties as they were first suggested in unpublished work of Linckelmann-Webb [12], although Property (b2’) did not feature there. We refer to Properties (b1’) and (b2’) collectively as Property (b’). The Linckelmann-Webb properties mimic the properties of a finite group G with Sylow subgroup S regarded as an (S, S)-biset, although some scaling may be required to obtain Property (c’). The importance of the Linckelmann-Webb properties is apparent in the following result.

Proposition 4.2 [12, 7] Let F be a fusion system over a finite p-group S. If Ω is a virtual characteristic biset for F, then the induced map α(Ω)^{∗} in cohomology is an idempotent in End(H^{∗}(BS)), is H^{∗}(F)-linear and a homomorphism of modules over the Steenrod algebra; and

Im[H^{∗}(BS) ⟶^{α(Ω)^{∗}} H^{∗}(BS)] = H^{∗}(F).

Proof See the proof of [7, Proposition 5.5].

A characteristic idempotent for a fusion system F over S is an idempotent in A(S, S)_{p}^{∧} with p-completed, idempotent analogues of the Linckelmann-Webb properties. This is stated precisely below.

Definition 4.3 Let F be a fusion system over a finite p-group S. A characteristic idempotent for F is an idempotent ω ∈ A(S, S)_{p}^{∧} with the following properties:

(a) ω ∈ A_{F} (S, S)_{p}^{∧}. (b1) ω is right F -stable. (b2) ω is left F -stable.

(c) ϵ(ω) = 1.

We again refer to Properties (b1) and (b2) collectively as Property (b).

The existence of characteristic bisets for saturated fusion systems was established by Broto-Levi-Oliver in [7] through a constructive argument. Although they, like Linckelmann-Webb, did not include Property (b2’) in their statement of the result, it is implicit in their construction.

Proposition 4.4 [7, Proposition 5.5] Every saturated fusion system F over a p-group S has a characteristic (S, S)-biset.

The preceding proposition is the only point in this paper where we rely on the saturation of fusion systems. If we were instead to assume that every fusion system in sight has a characteristic biset, then the construction of characteristic idempotents and classifying spectra, as well as the proof of their properties still go through. It is an interesting question whether this really amounts to weakening our hypothesis. That is, whether the existence of a characteristic biset for a fusion system F implies that F is saturated.

We now proceed by a sequence of lemmas about (S, S)-bisets to produce the characteristic idempotent.

Lemma 4.5 Let Ω and Λ be two (virtual) characteristic bisets for a fusion system F over a finite p-group S. Then Ω ∘ Λ is also a (virtual) characteristic biset for F. In particular, any power of Ω is a (virtual) characteristic biset for F.

Proof That Ω ∘ Λ has Property (a’) follows from Corollary 3.9. To see that Ω ∘ Λ has Property (b’), we note that for P ≤ S and φ ∈ Hom_{F} (P, S) we have

(Ω ∘ Λ) ∘ [P, φ]_{P}^{S} = Ω ∘ (Λ ∘ [P, φ]_{P}^{S}) = Ω ∘ (Λ ∘ [P, ι_{P}]_{P}^{S}) = (Ω ∘ Λ) ∘ [P, ι_{P}]_{P}^{S} ,

and similarly

[φ(P), φ^{−1}]_{S}^{P} ∘ (Ω ∘ Λ) = [P, id_{P}]_{S}^{P} ∘ (Ω ∘ Λ).

Property (c’) is clearly preserved since ϵ is multiplicative. The final statement now follows by induction.

Lemma 4.6 Let Ω ∈ A(S, S). Then there exists an M > 0 such that Ω^{M} is idempotent mod p.

Proof Let Ω̄ denote the image of Ω under the projection

A(S, S) → A(S, S)/pA(S, S).

It is equivalent to show that Ω̄^{M} is idempotent for some M > 0. Now, A(S, S) is a finitely generated ℤ-module and hence A(S, S)/pA(S, S) is finite. Consider the sequence

Ω̄, Ω̄^{2}, Ω̄^{3}, …

in A(S, S)/pA(S, S). By the pigeonhole principle there must be numbers N, t > 0 such that Ω̄^{N} = Ω̄^{N+t}. It follows that

Ω̄^{n} = Ω̄^{n+t}

for all n ≥ N. Now take m ≥ 0 such that mt > N and put M := mt. Then

Ω̄^{2M} = Ω̄^{M+mt} = Ω̄^{M+(m−1)t} = ⋯ = Ω̄^{M+t} = Ω̄^{M}.

The following two lemmas were suggested to the author by Bob Oliver. Although they hold for any p-torsion-free ring, we will state them only for A(S, S).

Lemma 4.7 If Ω ∈ A(S, S) is idempotent mod p^{k}, where k > 0, then Ω^{p} is idempotent mod p^{k+1}.

Proof Put q := p^{k}. By assumption we can write

Ω^{2} = Ω + qΛ  (4)

for some Λ ∈ A(S, S). It follows that

Ω^{2} + qΩΛ = Ω(Ω + qΛ) = Ω^{3} = (Ω + qΛ)Ω = Ω^{2} + qΛΩ,

so

qΩΛ = qΛΩ.

Since A(S, S) is torsion-free as a ℤ-module, we deduce that Ω and Λ commute. This allows us to apply the binomial formula to (4) and get

Ω^{2p} = Ω^{p} + \binom{p}{1}Ω^{p−1}qΛ + \binom{p}{2}Ω^{p−2}q^{2}Λ^{2} + ⋯ + \binom{p}{p − 1}Ωq^{p−1}Λ^{p−1} + q^{p}Λ^{p}.

A brief inspection of the coefficients occurring on the right hand side, taking into account that p divides q since k > 0, shows that we can therefore write

Ω^{2p} = Ω^{p} + pqΛ′

for some Λ′ ∈ A(S, S). Since pq = p^{k+1} we deduce that Ω^{p} is idempotent mod p^{k+1}.

Lemma 4.8 If Ω ∈ A(S, S) is idempotent mod p, then the sequence

Ω, Ω^{p}, Ω^{p^{2}}, …

converges in A(S, S)_{p}^{∧}. Furthermore the limit is idempotent.

Proof By Lemma 4.7 and induction, Ω^{p^{k}} is idempotent mod p^{k+1} for each k ≥ 0. That is to say,

Ω^{2p^{k}} − Ω^{p^{k}} ∈ p^{k+1}A(S, S)  (5)

for k ≥ 0. By induction it follows that

Ω^{np^{k}} − Ω^{p^{k}} ∈ p^{k+1}A(S, S)

for k ≥ 0, n > 0. In particular

Ω^{p^{l}} − Ω^{p^{k}} ∈ p^{k+1}A(S, S)

when l ≥ k > 0, so

Ω, Ω^{p}, Ω^{p^{2}}, …

is a Cauchy sequence in the p-adic topology of A(S, S). Hence it converges to a unique element ω ∈ A(S, S)_{p}^{∧}. Since the multiplication in A(S, S) is continuous with respect to the p-adic topology, ω^{2} is the limit of the sequence

Ω^{2}, Ω^{2p}, Ω^{2p^{2}}, …

Idempotence of ω now follows by taking the limit of (5) over k.

We can now prove the main result of this section.

Proposition 4.9 Every saturated fusion system has a characteristic idempotent.

Proof Let F be a saturated fusion system over a finite p-group S. Take a characteristic (S, S)-biset Ω as given by Proposition 4.4. By Lemmas 4.6 and 4.5 we may assume that Ω is idempotent mod p. By Lemma 4.8 the sequence

Ω, Ω^{p}, Ω^{p^{2}}, …

converges to an idempotent ω ∈ A(S, S)_{p}^{∧}. We show that ω has the Linckelmann-Webb properties.

By an induction similar to that in Lemma 4.7 one can show that ϵ(Ω) ≡ 1 (mod p) implies that ϵ(Ω^{p^{k}}) ≡ 1 (mod p^{k+1}) for k ≥ 0. It follows that ϵ(ω) = 1, proving (c).

It is not hard to see that A_{F}(S, S) is a closed subspace of A(S, S) in the p-adic topology and hence that A_{F}(S, S)_{p}^{∧} is a closed subspace of A(S, S)_{p}^{∧}. Since each Ω^{n} is in A_{F}(S, S) by Corollary 3.9, it follows that the limit ω is in A_{F}(S, S)_{p}^{∧}, proving (a).

Let P ≤ S and φ ∈ Hom_{F} (P, S). By Property (b1’) we have

Ω ∘ [P, φ]_{P}^{S} = Ω ∘ [P, ι_{P}]_{P}^{S}

and consequently

Ω^{p^{k}} ∘ [P, φ]_{P}^{S} = Ω^{p^{k}} ∘ [P, ι_{P}]_{P}^{S} ,

for all k ≥ 0. Since the pairing

∘: A(S, S) × A(P, S) → A(P, S)

is continuous in the p-adic topology on the relevant ℤ-modules, we can take limits to get

ω ∘ [P, φ]_{P}^{S} = ω ∘ [P, ι_{P}]_{P}^{S} ,

proving (b1). Property (b2) follows similarly from Property (b2’).

5 Properties of characteristic idempotents

In this section we perform a further study of the characteristic idempotents introduced in the previous section. We discover that the effect of multiplicaton by a characteristic idempotent on A(S_{1}, S_{2})_{p}^{∧} essentially amounts to quotienting out fusion conjugacy in the source or target, as appropriate. This allows us to glean important information about the structure of a characteristic idempotent in the [P, φ]-basis, which allows us to prove its uniqueness, and will also prove surprisingly useful for proving later naturality results.

Proposition 5.1 Let F_{1} and F_{2} be saturated fusion systems over the finite p-groups S_{1} and S_{2}, respectively, and let ω_{1} and ω_{2} be characteristic idempotents of F_{1} and F_{2}, respectively. If the (S_{1}, S_{2})-pairs (P, ψ) and (Q, ρ) are (F_{1}, F_{2})-conjugate, then

ω_{2} ∘ [Q, ρ] ∘ ω_{1} = ω_{2} ∘ [P, ψ] ∘ ω_{1}.

Proof Since ω_{1} is left F_{1}-stable and ω_{2} is right F_{2}-stable, this is a special case of Lemma 3.12.

Although the following proposition may be of limited interest in its own right, it is the central result of this paper.

Proposition 5.2 Let F_{1} and F_{2} be saturated fusion systems over the finite p-groups S_{1} and S_{2}, respectively and let ω_{1} and ω_{2} be characteristic idempotents of F_{1} and F_{2}, respectively. Let I be the set of (F_{1}, F_{2})-conjugacy classes of (S_{1}, S_{2})-pairs, and pick a representative (P_{i}, ψ_{i}) for each i ∈ I. Then the collection

{ω_{2} ∘ [P_{i}, ψ_{i}] ∘ ω_{1} | i ∈ I}

forms a ℤ_{p}^{∧}-basis for ω_{2} ∘ A(S_{1}, S_{2})_{p}^{∧} ∘ ω_{1}.

Proof It follows from Proposition 5.1 that ω_{2} ∘ A(S_{1}, S_{2})_{p}^{∧} ∘ ω_{1} is spanned by the collection, so it suffices to prove linear independence. By Property (b) of characteristic idempotents and Lemma 3.8 we have

ω_{2} ∘ [P_{i}, ψ_{i}] ∘ ω_{1} ∈ M (≾ [P_{i}, ψ_{i}])

for each i ∈ I. Note however, that

ϵ (ω_{2} ∘ [P_{i}, ψ_{i}] ∘ ω_{1}) = ϵ(ω_{2}) · ϵ([P_{i}, ψ_{i}]) · ϵ(ω_{1}) = 1 · |S_{1}/P_{i}| · 1 = |S_{1}/P_{i}|,

whereas

ϵ (M (⋨ [P_{i}, ψ_{i}])) ⊆ p |S_{1}/P_{i}| ℤ_{p}^{∧}.  (6)

Therefore,

ω_{2} ∘ [P_{i}, ψ_{i}] ∘ ω_{1} ∈ M (≾ [P_{i}, ψ_{i}]) \ M (⋨ [P_{i}, ψ_{i}])  (7)

for each i ∈ I.

Now, let c_{i} ∈ ℤ_{p}^{∧} for each i ∈ I and assume that

∑_{i∈I} c_{i} · (ω_{2} ∘ [P_{i}, ψ_{i}] ∘ ω_{1}) = 0.  (8)

Put

I′ = {i ∈ I | c_{i} ≠ 0}.

If I′ is nonempty, then let j be a maximal element of I′ regarded as a poset under (F_{1}, F_{2})-subconjugacy. By (7) there is a (S_{1}, S_{2})-pair (Q, ρ) ∼ (P_{j}, ψ_{j}) such that

χ_{[Q,ρ]} (ω_{2} ∘ [P_{j}, ψ_{j}] ∘ ω_{1}) ≠ 0.

On the other hand, for i ∈ I′ \ {j} the maximality of j implies that [Q, ρ] is not (F_{1}, F_{2})-subconjugate to (P_{i}, ψ_{i}). Hence

χ_{[Q,ρ]} (M (≾ [P_{i}, ψ_{i}])) = 0

and in particular

χ_{[Q,ρ]} (ω_{2} ∘ [P_{i}, ψ_{i}] ∘ ω_{1}) = 0.

Now we get

χ_{[Q,ρ]} (∑_{i∈I} c_{i} · (ω_{2} ∘ [P_{i}, ψ_{i}] ∘ ω_{1})) = ∑_{i∈I} c_{i} · χ_{[Q,ρ]} (ω_{2} ∘ [P_{i}, ψ_{i}] ∘ ω_{1})

= ∑_{i∈I\I′} \underbrace{c_{i}}_{=0} ·χ_{[Q,ρ]} (ω_{2} ∘ [P_{i}, ψ_{i}] ∘ ω_{1})

+ ∑_{i∈I′\\{j\}} c_{i} · \underbrace{χ_{[Q,ρ]} (ω_{2} ∘ [P_{i}, ψ_{i}] ∘ ω_{1})}_{=0}

+ \underbrace{c_{j}}_{≠0} · \underbrace{χ_{[Q,ρ]} (ω_{2} ∘ [P_{j}, ψ_{j}] ∘ ω_{1})}_{≠0}

≠ 0,

contradicting (8). Therefore I′ must be empty and we conclude that the collection is linearly independent.

Remark 5.3 Since the multiplicative identity [S_{1}, id] of A(S_{1}, S_{1})_{p}^{∧} is a characteristic idempotent for F_{S_{1}} and [S_{2}, id] is a characteristic idempotent for F_{S_{2}} , Propositions 5.1 and 5.2 can also be used to obtain a basis for ω_{2} ∘ A(S_{1}, S_{2})_{p}^{∧} and A(S_{1}, S_{2})_{p}^{∧} ∘ ω_{1}.

We determine the structure of a characteristic idempotent ω by carefully analyzing the idempotence relation ω ∘ ω = ω. In light of the previous proposition, a convenient tool for doing this is the projection

A_{F} (S, S)_{p}^{∧} ⟶ ωA_{F} (S, S)_{p}^{∧},

given by left multiplication by ω. This projection can be easily described by

Ω = ∑_{[P,φ]≾[S,id]} χ_{[P,φ]}(Ω) · [P, φ] ⟼ ω ∘ Ω = ∑_{[P]≾[S]} χ_{[P]}^{F}(Ω) · (ω ∘ [P, ι_{P}]),

where the homomorphisms χ_{[P]}^{F} are as in the following definition. Definition 5.4 Let F be a fusion system over a finite p-group S. For each S - conjugacy class [P] of subgroups of S, let χ_{[P]}^{F} and χ_{F}^{[P]} be the homomorphisms A(S, S)_{p}^{∧} → ℤ_{p}^{∧} given by

χ_{[P]}^{F} = ∑_{[P,φ] ∼_{(F_{S} ,F)}[P,ι_{P}]} χ_{[P,φ]}

and

χ_{F}^{[P]} = ∑_{[Q,φ] ∼_{(F,F_{S})}[P,ι_{P}]} χ_{[Q,φ]}.

Note that similarly we have

Ω ∘ ω = ∑_{[P]≾[S]} χ_{F}^{[P]}(Ω) · ([P, ι_{P}] ∘ ω).

The following lemma now effectively allows us to determine the structure of characteristic idempotents. Lemma 5.5 Let F be a saturated fusion system over a finite p-group S and let ω be a characteristic idempotent of F. Then

ω = ω^{∼} + ω^{⋨},

where

ω^{∼} = \frac{1}{| Out_{F} (S) |} ∑_{φ∈Out_{F} (S)}[S, φ] ∈ M (∼ [S, id], F, F)

and

ω^{⋨} ∈ M (⋨ [S, id], F, F) .

Furthermore,

χ_{[P]}^{F}(ω) = 0

and

χ_{F}^{[P]}(ω) = 0

for all proper subgroups P < S.

Proof There is a direct sum of modules

A_{F} (S, S)_{p}^{∧} = M (≾ [S, id]) = M (∼ [S, id]) ⊕ M (⋨ [S, id]) ,

where M (⋨ [S, id]) is a two-sided ideal of A_{F} (S, S)_{p}^{∧} (by Lemma 3.8), and M (∼ [S, id]) is a subring of A_{F} (S, S)_{p}^{∧} (as can be easily checked). We can therefore uniquely write

ω = ω^{∼} + ω^{⋨},

where

ω^{∼} ∈ M (∼ [S, id])

and

ω^{⋨} ∈ M (⋨ [S, id]) .

From Properties (a) and (b) of characteristic idempotents, one deduces by standard techniques that ω^{∼} must be of the form

ω^{∼} = α ∑_{φ∈Out_{F} (S)}[S, φ],

where α ∈ ℤ_{p}^{∧}. Now,

ω^{∼} + ω^{⋨} = ω = ω^{2} = (ω^{∼})^{2} + (ω^{∼} ∘ ω^{⋨} + ω^{⋨} ∘ ω^{∼} + (ω^{⋨})^{2}),

where (ω^{∼})^{2} ∈ M (∼ [S, id]) and the second term is in the ideal M (⋨ [S, id]). By uniqueness of such decompositions ω^{∼} is therefore an idempotent, and since

(ω^{∼})^{2} = α · | Out_{F} (S) | · ω^{∼},

this implies that

α ∈ {0, | Out_{F} (S) |^{−1}}.

(Note that | Out_{F} (S) | is prime to p since F is saturated.) By Property (c) of characteristic idempotents we have

1 = ϵ(ω) = ϵ(ω^{∼}) + ϵ(ω^{⋨}).

As a special case of (6) we have ϵ(M (⋨ [S, id])) ⊆ p ℤ_{p}^{∧}, and it follows that

α · | Out_{F} (S) | = ϵ(ω^{∼}) ≡ ϵ(ω) ≡ 1 (mod p).

Hence α ≠ 0, leaving only the possibility

ω^{∼} = \frac{1}{| Out_{F} (S) |} ∑_{φ∈Out_{F} (S)}[S, φ].

To prove the second claim we start by deducing from Property (b) and the description of ω^{∼} above, that

ω ∘ ω^{∼} = ω.

Hence the idempotence of ω yields

ω = ω ∘ ω = ω ∘ (ω^{∼} + ω^{⋨}) = ω + ω ∘ ω^{⋨},

and we get

ω ∘ ω^{⋨} = 0.

By Property (a) and Remark 3.7, we can write

ω^{⋨} = ∑_{[P,φ]⋨[S,id]} χ_{[P,φ]}(ω) · [P, φ],

so

0 = ω ∘ ω^{⋨} = ∑_{[P]⋨[S]} χ_{[P]}^{F}(ω) · (ω ∘ [P, ι_{P}]) ,

and the result for χ_{[P]}^{F}(ω) follows upon noting that the collection

{ω ∘ [P, ι_{P}] | [P] ⋨ [S]}

is linearly independent over ℤ_{p}^{∧} by Proposition 5.2 and Remark 5.3.

The result for χ_{F}^{[P]}(ω) is proved similarly by performing the analogous simplifications of ω^{⋨} ∘ ω = 0. We omit the details.

The lemma has an interesting consequence.

Proposition 5.6 Every saturated fusion system has a unique characteristic idempotent.

Proof Let F be a saturated fusion system over a finite p-group S. By Proposition 4.9, F has a characteristic idempotent ω. We proceed to prove uniqueness. Recalling that we can write

ω = ∑_{[P,φ]≾[S,id]} χ_{[P,φ]}(ω) · [P, φ],

the method of proof is to set up a system of linear equations for the coefficients χ_{[P,φ]}(ω), and show that this system is fully determined and thus uniquely determines the coefficients. Note that we need not show that the system is solvable since the existence of a characteristic idempotent has already been established.

To present the argument it is helpful to index the basis of A_{F} (S, S)_{p}^{∧}. Take representatives P_{0}, P_{1}, …, P_{n} for the S -conjugacy classes of subgroups of S, labelled in such a way that

|P_{0}| ≥ |P_{1}| ≥ … ≥ |P_{n}|.

For each i ∈ {0, 1, …, n}, pick representatives φ_{i0}, φ_{i1}, …, φ_{im_{i}} for the S - conjugacy classes of homomorphisms in Hom_{F} (P, S), now labelled so that φ_{i0} = ι_{P_{i}}. The collection {[P_{i}, φ_{ij}] | 0 ≤ i ≤ n, 0 ≤ j ≤ m_{i}} is then a basis for A_{F} (S, S)_{p}^{∧}. We order the basis through the lexicographic order of the indexing set:

(0, 0) ≤ (0, 1) ≤ ⋯ ≤ (0, m_{0}) ≤ (1, 0) ≤ (1, 1) ≤ ⋯ ≤ (n, m_{n}).

This ordering has the property that [P_{i}, φ_{ij}] ≾_{(S,S)}[P_{k}, φ_{kl}] implies (k, l) ≤ (i, j).

For the remainder of this proof, a pair (i, j) is understood to satisfy 0 ≤ i ≤ n and 1 ≤ j ≤ m_{i}. For such a pair (i, j), write

Δ_{i}^{j} := Δ_{P_{i}}^{φ_{ij}}

and

c_{ij} := χ_{[P_{i},φ_{i}^{j}]}(ω)

for short. We will also use the shorthand notation ∑_{i,j} for the double sum ∑_{i=0}^{n} ∑_{j=0}^{m_{i}} . Using this notation we can write

ω = ∑_{i,j} c_{ij}[P_{i}, φ_{ij}].

Note that by Lemma 2.6, Property (b1) implies that

|ω^{Δ_{i}^{j}}| = |ω^{Δ_{i}^{0}}|

for all pairs (i, j). This can be rewritten as the equation

∑_{k,l} c_{kl} · (|[P_{k}, φ_{kl}]^{Δ_{i}^{j}}| − |[P_{k}, φ_{kl}]^{Δ_{i}^{0}}|) = 0,

which we refer to as Equation (i, j) when j ≠ 0. When j = 0, this equation becomes trivial. Instead we consider the equations given by Lemma 5.5. That is, we let Equation (0, 0) be the equation

∑_{j=0}^{m_{0}} c_{0j} = 1,

and for i = 1, …, n, we let Equation (i, 0) be the equation

∑_{j=0}^{m_{i}} c_{ij} = 0.

If we now write Equations (0, 0) to (i, m_{i}) one below the other going by the lexicographic order, we obtain a system of equations, which can be represented on matrix form as

Ac = b,

where c is a vector with entries c_{ij}, b is a vector with 1 in its first entry and 0 everywhere else, while A is a square matrix whose rows and columns are both indexed by pairs (i, j). The proof is completed by showing that A has nonzero determinant.

There is an obvious way to regard A as a (n×n) block matrix where the blocks are indexed by i. We show that A is in fact a lower triangular block matrix. Since

|[Q, ψ]^{Δ_{R}^{ρ}}| = |Δ_{Q}^{ψ}\N_{S×S}(Δ_{R}^{ρ}, Δ_{Q}^{ψ})| = \frac{|N_{S×S}(Δ_{R}^{ρ}, Δ_{Q}^{ψ})|}{|Q|}

for (S, S)-pairs (Q, ψ) and (R, ρ), we see by Remark 3.4 that

|[Q, ψ]^{Δ_{R}^{ρ}}| ≠ 0

if and only if

[R, ρ] ≾_{(S,S)}[Q, ψ].

The chosen order of the basis of A_{F} (S, S)_{p}^{∧} therefore guarantees that A is a lower triangular block matrix.

It now suffices to show that the matrices occurring on the diagonal of A have nonzero determinant. The i-th matrix on the diagonal has the form

A_{i} = ( \begin{matrix}1 & 1 & 1 & & 1 & 1 & 1 \\ −a_{i0} & a_{i1} & 0 & … & 0 & 0 & 0 \\ −a_{i0} & 0 & a_{i2} & & 0 & 0 & 0 \\ & ⋮ & & ... & & ⋮ \\ −a_{i0} & 0 & 0 & & a_{im_{i}−2} & 0 & 0 \\ −a_{i0} & 0 & 0 & … & 0 & a_{im_{i}−1} & 0 \\ −a_{i0} & 0 & 0 & & 0 & 0 & a_{im_{i}} \end{matrix} ) ,

where

a_{ij} = \frac{|N_{S×S}(Δ_{i}^{j})|}{|P_{i}|}.

Direct calculation shows that the determinant of A_{i} is

det(A_{i}) = ∑_{j=0}^{m_{i}} \frac{∏_{k=0}^{m_{i}} a_{ik}}{a_{ij}} ,

which is nonzero since all the a_{ij} ’s are positive integers.

The previous proposition allows us to speak of the characteristic idempotent of a saturated fusion system.

Definition 5.7 For a saturated fusion system F, let ω_{F} denote the characteristic idempotent of F.

Remark 5.8 We make the following observations about the proof of Proposition 5.6.

(1) As a byproduct of the proof we have produced an explicit system of equations which we can solve to produce ω_{F}.

(2) The coefficients in these equations are all integers. Therefore ω_{F} can be regarded as an element in the p-localized double Burnside ring:

ω_{F} ∈ A(S, S)_{(p)} = A(S, S) ⊗ ℤ_{(p)}.

(3) The proof actually shows that ω_{F} is the unique idempotent in A(S, S)_{p}^{∧} (or A(S, S)_{(p)}) with the Linckelmann-Webb Properties (a), (b1) and (c). A symmetric argument shows that ω_{F} is the unique idempotent in A(S, S)_{p}^{∧} with Properties (a), (b2) and (c). We are therefore in the surprising situation that for an idempotent in A(S, S)_{p}^{∧} with Properties (a) and (c), the presence of either stability Property (b1) or (b2) implies the presence of the other.

6 Stable characteristic idempotents

In the preceding sections we have undertaken a thorough study of the structure and properties of characteristic idempotents of saturated fusion systems. Going by [7], we will later construct the classifying spectrum of a saturated fusion system F over a finite p-group S as the mapping telescope of the stable characteristic idempotent ω̃_{F} of 𝔹S induced by the characteristic idempotent ω_{F}. In this section we briefly study the stable characteristic idempotent and interpret the consequences of the results of the preceding two sections in terms of stable maps, with a particular view towards fusion stability properties as defined below.

Definition 6.1 Let F be a fusion system over a finite group S and let X be a spectrum. We say a map f: 𝔹S → X is right F -stable if for every P ≤ S and φ ∈ Hom_{F} (P, S) we have

f ∘ 𝔹φ ≃ f ∘ 𝔹ι_{P} .

Similarly we say that a map f: X → 𝔹S is left F -stable if for every P ≤ S and φ ∈ Hom_{F} (P, S) we have

tr_{φ} ∘ f ≃ tr_{P} ∘ f.

We will often simply say that a map is F -stable. Unless X ≃ 𝔹S in the previous definition, it should be clear from the context whether this means left or right F -stability.

In Section 2 we defined Ã(S, S)_{p}^{∧} to be the quotient module of A(S, S)_{p}^{∧} obtained by quotienting out all trivial basis elements [P, ψ], and discussed the isomorphism

α̃ : Ã(S, S)_{p}^{∧} ⟶^{≅} {BS, BS}

of the Segal Conjecture. Since Ã(S, S)_{p}^{∧} is a free ℤ_{p}^{∧}-module with one basis element [P, ψ] for each conjugacy class of nontrivial (S, S)-pairs, we can also regard Ã(S, S)_{p}^{∧} as a submodule (but not subring) of A(S, S)_{p}^{∧} and we do so in this section. From this point of view we define a version of the augmentation ϵ by

ϵ̃:= ϵ ∘ α^{−1}: \{BS, BS\} \underrightarrow{α^{−1}} Ã(S, S)_{p}^{∧} ⊆ A(S, S)_{p}^{∧} ⟶^{ϵ} ℤ_{p}^{∧}.

This is not a ring homomorphism and therefore not an augmentation, although it plays a similar role.

Definition 6.2 Let F be a fusion system over a finite p-group S. A stable characteristic idempotent for F is an idempotent ω ∈ {BS, BS} with the following properties:

(a) ω̃ ∈ α̃(A_{F} (S, S)_{p}^{∧}),

(b1) ω̃ is right F -stable.

(b2) ω̃ is left F -stable.

(c) ϵ̃(ω̃) = 1.

Proposition 6.3 Every saturated fusion system F has a unique stable characteristic idempotent ω̃_{F}.

Proof If F is a saturated fusion system over a finite p-group S, then the image ω̃_{F} in {BS, BS} of the characteristic idempotent ω_{F} under the surjection

\xymatrix{A(S, S)_{p}^{∧} \ar@{->>}[r] & Ã(S, S)_{p}^{∧} \ar[r]^{α̃}_{≅} & \{BS, BS\}}

is a stable characteristic idempotent for F.

To prove uniqueness, suppose that ω̃ is a stable characteristic idempotent of F and consider ω := α^{−1}(ω̃), regarded as an element of A(S, S)_{p}^{∧}. It is enough to show that ω is a characteristic idempotent for F, because the result then follows from uniqueness of characteristic idempotents (Proposition 5.6).

It is immediate that ω ∈ A_{F} (S, S)_{p}^{∧} and ϵ(ω) = 1. We show that ω is idempotent. Indeed, idempotence of ω̃ implies that

ν := ω ∘ ω − ω

is a linear combination of trivial basis elements. But since ν is an element of A_{F} (S, S)_{p}^{∧}, and the only trivial basis element in A_{F} (S, S)_{p}^{∧} is [1, triv], where 1 denotes the trivial subgroup, ν must be of the form ν = a · [1, triv]. Applying ϵ we get

1 = ϵ(ω ∘ ω) = ϵ(ω + ν) = 1 + a · |S|.

We conclude that a = 0, whence ν = 0 and ω is idempotent.

It only remains to prove that ω has Property (b). This is deduced from Property (b) for ω̃ in a similar way.

As a consequence of Lemma 5.5 we now show that stable characteristic idempotents completely characterize fusion-stable maps.

Corollary 6.4 Let F be a saturated fusion system over a finite p-group S and let X be a spectrum. A map f : 𝔹S → X is F -stable if and only if

f ∘ ω̃_{F} ≃ f.

Similarly, a map f: X → 𝔹S is F -stable if and only if

ω̃_{F} ∘ f ≃ f.

Proof Suppose first that f : 𝔹S → X is F -stable. Note that we can write

ω_{F} ≃ ∑_{[P]≾[S]} ( ∑_{[P,φ] ∼_{(F_{S} ,F)}[P,ι_{P}]} χ_{[P,φ]}(ω_{F})[P, φ])

Now,

f ∘ ω̃_{F}≃f ∘ ∑_{[P]≾[S]} (  ∑_{[P,φ] ∼_{(F_{S} ,F)}[P,ι_{P}]} χ_{[P,φ]}(ω_{F}) · α̃([P, φ]))

≃f ∘ ∑_{[P]≾[S]} ( ∑_{[P,φ] ∼_{(F_{S} ,F)}[P,ι_{P}]} χ_{[P,φ]}(ω_{F}) · (𝔹φ ∘ tr_{P}))

≃ ∑_{[P]≾[S]} (  ∑_{[P,φ] ∼_{(F_{S} ,F)}[P,ι_{P}]} χ_{[P,φ]}(ω_{F}) · (f ∘ 𝔹φ ∘ tr_{P}))

≃ ∑_{[P]≾[S]} ((  ∑_{[P,φ] ∼_{(F_{S} ,F)}[P,ι_{P}]} χ_{[P,φ]}(ω_{F})) · (f ∘ 𝔹ι_{P} ∘ tr_{P}))

≃ ∑_{[P]≾[S]} (χ_{[P]}^{F}(ω_{F}) · (f ∘ 𝔹ι_{P} ∘ tr_{P})) .

By Lemma 5.5, we have

χ_{[P]}^{F}(ω_{F}) = \begin{cases}1 & if P = S, \\ 0 & if P < S. \end{cases}

We conclude that

f ∘ ω̃_{F} ≃ f ∘ 𝔹ι_{S} ∘ tr_{S} ≃ f.

The converse implication follows from F -stability of ω̃_{F}. The result for left F -stable maps is deduced similarly by writing

ω_{F} = ∑_{[P]≾[S]} ( ∑_{[Q,φ] ∼_{(F,F_{S})}[P,ι_{P}]} χ_{[Q,φ]}(ω_{F})[Q, φ]) .

If f is a left F -stable map f, we get the following after some manipulations

ω̃_{F} ∘ f ≃ ∑_{[P]≾[S]} (χ_{F}^{[P]}(ω_{F}) · (𝔹ι_{P} ∘ tr_{P} ∘ f))

and ω̃_{F} ∘ f ≃ f follows after appealing to Lemma 5.5.

7 Classifying spectra

In this section we develop the theory of classifying spectra for saturated fusion systems. The classifying spectra constructed here agree with those suggested in [7], but the added rigidity of the new construction allows us to prove that the assignment is functorial for fusion-preserving homomorphisms between fusion systems. We will also endow the classifying spectrum of a saturated fusion system F over a finite p-group S with additional structure by regarding it as an object under 𝔹S. We support this idea by proving that we can reconstruct the fusion system F from its classifying spectrum, when regarded as an object under 𝔹S.

Let F be a saturated fusion system over a finite p-group S and denote the infinite mapping telescope of the stable characteristic idempotent ω̃_{F} by 𝔹F. In other words,

𝔹F := HoColim \overline{}(𝔹S \underrightarrow{ω̃_{F}} 𝔹S \underrightarrow{ω̃_{F}} 𝔹S \underrightarrow{ω̃_{F}} …) .

We denote the structure map of the homotopy colimit by σ_{F}: 𝔹S → 𝔹F. Since ω̃_{F} is idempotent up to homotopy, we get a homotopy factorization of ω̃_{F} through the homotopy colimit

ω̃_{F}: 𝔹S ⟶^{σ_{F}} 𝔹F ⟶^{t_{F}} 𝔹S,

such that σ_{F} ∘ t_{F} ≃ id_{𝔹F}. Note that t_{F} is, up to homotopy, the unique map with these properties. Since 𝔹F is a retract of the p-complete spectrum 𝔹S, it is p-complete.

Definition 7.1 Let F be a saturated fusion system over a finite p-group S. The classifying spectrum of F is the spectrum 𝔹F, the structure map of F is the map σ_{F}, and the transfer of F is the map

t_{F}: 𝔹F → 𝔹S,

such that σ_{F} ∘ t_{F} ≃ id_{𝔹F} and t_{F} ∘ σ_{F} ≃ ω̃_{F}.

The structured classifying spectrum of F is the mapping telescope

σ_{F}: 𝔹S → 𝔹F,

regarded as an object under 𝔹S.

In the special case when F = F_{S} is the fusion system of S, the stable characteristic idempotent ω̃_{F_{S}} is just the identity of 𝔹S, so the structured classifying spectrum of F_{S} becomes a natural homotopy equivalence

σ_{F_{S}} : 𝔹S ⟶^{≃} 𝔹F_{S}.

We will therefore often replace 𝔹F_{S} by 𝔹S.

It has been shown in [7] that for a p-local finite group (S, F, L), the classifying spectrum 𝔹F is homotopy equivalent to the infinite suspension spectrum of the classifying space |L|_{p}^{∧}, thus partly justifying the use of the term “classifying spectrum”. In Section 10 we extend this observation to structured classifying spectra.

As an obvious consequence of Proposition 5.2, the group of homotopy classes of maps between classifying spectra of fusion systems has an appealingly simple description, analogous to Theorem 2.4. This is Theorem B of the introduction.

Theorem 7.2 Let F_{1} and F_{2} be saturated fusion systems over the finite p-groups S_{1} and S_{2}, respectively, and let Ĩ be the set of (F_{1}, F_{2})-conjugacy classes of nontrivial (S_{1}, S_{2})-pairs. Pick a representative (P_{i}, ψ_{i}) for each i ∈ Ĩ. Then the collection

{σ_{F_{2}} ∘ α̃([P_{i}, ψ_{i}]) ∘ t_{F_{1}} | i ∈ Ĩ}

forms a ℤ_{p}^{∧}-basis for [𝔹F_{1}, 𝔹F_{2}].

Proof The map

[𝔹F_{1}, 𝔹F_{2}] ⟶ {BS_{1}, BS_{2}}, f ↦ t_{F_{2}} ∘ f ∘ σ_{F_{1}}

and its left inverse

{BS_{1}, BS_{2}} ⟶ [𝔹F_{1}, 𝔹F_{2}], g ↦ σ_{F_{2}} ∘ g ∘ t_{F_{1}}

make [𝔹F_{1}, 𝔹F_{2}] isomorphic to the submodule

ω̃_{F_{2}} ∘ {BS_{1}, BS_{2}} ∘ ω̃_{F_{1}} ⊂ {BS_{1}, BS_{2}}.

Since

ω̃_{F_{2}} ∘ {BS_{1}, BS_{2}} ∘ ω̃_{F_{1}} = α̃ (ω_{F_{2}} ∘ Ã(S_{1}, S_{2})_{p}^{∧} ∘ ω_{F_{1}}) ,

the result now follows from the explicit description of the ℤ_{p}^{∧}-basis of ω_{F_{2}} ∘ A(S_{1}, S_{2})_{p}^{∧} ∘ ω_{F_{1}} given in Proposition 5.2.

This theorem can be applied to fusion systems arising from finite groups to give a new and simple description of the group of homotopy classes of stable maps between p-completed classifying spaces of finite groups. Since this result is of independent interest, it will be presented separately in [23]. The proof given there is similar but more direct, with the added advantage that the double coset formula is preserved, and that the target can be the classifying space of a compact Lie group.

The following theorem further justifies the use of the term classifying spectrum. This appears in the introduction as Theorem A.

Theorem 7.3 If F is a saturated fusion system over a finite p-group S, then

Hom_{F} (P, Q) = {φ ∈ Hom (P, Q) | σ_{F} ∘ 𝔹ι_{Q} ∘ 𝔹φ ≃ σ_{F} ∘ 𝔹ι_{P} }

for all subgroups P, Q ≤ S.

Proof When P is the trivial subgroup there is nothing to show, so we assume otherwise. Now, let φ ∈ Hom (P, Q). Since t_{F} ∘ σ_{F} ≃ ω̃_{F} and σ_{F} ∘ ω̃_{F} ≃ σ_{F}, we have

σ_{F} ∘ 𝔹ι_{Q} ∘ 𝔹φ ≃ σ_{F} ∘ 𝔹ι_{P}

if and only if

ω̃_{F} ∘ 𝔹ι_{Q} ∘ 𝔹φ ≃ ω̃_{F} ∘ 𝔹ι_{P} ,

which we rewrite as

ω̃_{F} ∘ α̃([P, ι_{Q} ∘ φ]_{P}^{S}) ≃ ω̃_{F} ∘ α̃([P, ι_{P}]_{P}^{S}).

Since 𝔹P is naturally equivalent to the classifying spectrum of F_{P}, the fusion system of P, we can apply Theorem 7.2 to see that the last equivalence holds if and only if the (P, S)-pairs (P, ι_{Q} ∘ φ) and (P, ι_{P}) are (F_{P} , F)-conjugate. By definition this means that there exist g ∈ P and φ′ ∈ Hom_{F} (P, φ(P)) such that the following diagram commutes

P ⟶^{ι_{P}} P
≅↓^{c_{g}}  ↓^{φ′}
P ⟶^{ι_{Q}∘φ} φ(P),

or in other words such that

φ(x) = φ′ ∘ c_{g}^{−1}(x)

for all x ∈ P. This is in turn true if and only φ ∈ Hom_{F} (P, Q).

This theorem shows in particular that the fusion system of a finite group G with Sylow subgroup S is determined by 𝔹G regarded as an object under 𝔹S. Example 5.2 of [14] shows that the homotopy type of 𝔹G alone does not determine the fusion system; the classifying spectrum must be regarded as an object under 𝔹S. As this result is also of independent interest, and can be proved directly using the Segal Conjecture, it is treated separately in [24].

We will show that the assignment of a classifying spectrum to a saturated fusion systems is functorial, but first we need to specify which notion of morphisms between fusion systems we are working with. The following definition appeared in [20] but using different terminology.

Definition 7.4 Let F_{1} and F_{2} be fusion systems over finite p-groups S_{1} and S_{2}, respectively. A (F_{1}, F_{2})-fusion-preserving homomorphism is a group homomorphism γ: S_{1} ⟶ S_{2} for which there exists a functor F_{γ}: F_{1} ⟶ F_{2} such that

F_{γ}(P) = γ(P)

for all subgroups P ≤ S_{1}, and

γ|_{Q} ∘ φ = F_{γ}(φ) ∘ γ|_{P}

for all φ ∈ Hom_{F_{1}} (P, Q).

Remark 7.5 When there is no danger of confusion, we will often say simply that γ is fusion-preserving. The functor F_{γ} is uniquely determined by the fusion-preserving morphism γ in the above definition and we will from now on let F_{γ} denote the functor defined by a fusion-preserving homomorphism γ.

As a motivation for this definition one may may consider the equation γ ∘ c_{g}(x) = c_{γ(g)} ∘ γ(x) for a group homomorphism γ: G → H and elements g, x ∈ G. Certainly it follows easily from this equation that a homomorphism between finite groups induces a fusion-preserving homomorphism between their fusion systems. Similarly it is not too difficult to prove that a map between classifying spaces of p-local finite groups induces a fusion-preserving homomorphism between their fusion systems (see Remark 7.12).

Definition 7.6 Let FSbe the category whose objects are pairs (S, F) consisting of a finite p-group S and a fusion system F over S, and whose morphisms are fusion-preserving homomorphisms. Let SFS be the full subcategory of FS whose objects are pairs (S, F) where F is saturated.

Since a given homomorphism γ: S_{1} → S_{2} can be fusion-preserving for many different fusion systems, we will write γ_{F_{1}}^{F_{2}} to specify that it is regarded as an element in the morphism set Mor_{FS}((S_{1}, F_{1}), (S_{2}, F_{2})).

Lemma 7.7 Let F_{1} and F_{2} be saturated fusion systems over finite p-groups S_{1} and S_{2}, respectively, and let γ: S_{1} → S_{2} be a fusion-preserving homomorphism. If f: 𝔹S_{2} → X is a F_{2}-stable map then

f ∘ 𝔹γ: 𝔹S_{1} → X

is F_{1}-stable, and

f ∘ 𝔹γ ∘ ω̃_{F_{1}} ≃ f ∘ 𝔹γ.

Proof Indeed, for P ≤ S_{1} and φ ∈ Hom_{F_{1}} (P, S_{1}) we have

f ∘ 𝔹γ ∘ 𝔹φ ≃ f ∘ 𝔹(γ ∘ φ)
≃ f ∘ 𝔹F_{γ} (φ) ∘ 𝔹γ|_{P}
≃ f ∘ 𝔹ι_{γ(P)} ∘ 𝔹γ|_{P}
≃ f ∘ 𝔹(γ ∘ ι_{P})
≃ f ∘ 𝔹γ ∘ 𝔹ι_{P} ,

proving F_{1}-stability. The second claim follows from Corollary 6.4.

Now, given saturated fusion systems F_{1} and F_{2} over finite p-groups S_{1} and S_{2}, respectively, and a fusion-preserving homomorphism γ: S_{1} → S_{2}, we define a map of spectra

𝔹γ_{F_{1}}^{F_{2}} : 𝔹F_{1} → 𝔹F_{2}

by

𝔹γ_{F_{1}}^{F_{2}} := σ_{F_{2}} ∘ 𝔹γ ∘ t_{F_{1}}.

We will show that this assignment is functorial below. The proof will first be presented in a setting where the target category keeps track of the structure maps of the classifying spectra. To this end we make the following definition.

Definition 7.8 Let C be the category whose objects are maps of spectra

σ: 𝔹S → X,

where S is a finite p-group and X is a p-complete spectrum, and where the set of morphisms from σ_{1}: 𝔹S_{1} → X_{1} to σ_{2}: 𝔹S_{2} → X_{2} consists of pairs (γ, g), where γ: S_{1} → S_{2} is a group homomorphism and g: X_{1} → X_{2} is a map of spectra, such that the following diagram commutes up to homotopy:

𝔹S_{1} ⟶^{σ_{1}} X_{1}
𝔹γ↓  ↓^{g}
𝔹S_{2} ⟶^{σ_{2}} X_{2}.

Theorem 7.9 There is a structured classifying spectrum functor

Υ : SFS⟶ C

defined on objects by

(S, F) ↦ (σ_{F}: 𝔹S → 𝔹F)

and on morphisms by

γ_{F_{1}}^{F_{2}} ↦ (γ, 𝔹γ_{F_{1}}^{F_{2}}) .

Proof There are three things to show. First, it is clear by construction that σ_{F}: 𝔹S → 𝔹F is an object of C.

Second, if γ_{F_{1}}^{F_{2}} ∈ Mor_{SFS}((S_{1}, F_{1}), (S_{2}, F_{2})), then we need to prove commutativity of the diagram

𝔹S_{1} ⟶^{σ_{F_{1}}} 𝔹F_{1}
𝔹γ↓  ↓^{𝔹γ_{F_{1}}^{F_{2}}}
𝔹S_{2} ⟶^{σ_{F_{2}}} 𝔹F_{2}.

Going around the top half of the square, we get

𝔹γ_{F_{1}}^{F_{2}} ∘ σ_{F_{1}} = σ_{F_{2}} ∘ 𝔹γ ∘ t_{F_{1}} ∘ σ_{F_{1}}
≃ σ_{F_{2}} ∘ 𝔹γ ∘ ω̃_{F_{1}}.

Going around the bottom half of the square, we get

σ_{F_{2}} ∘ 𝔹γ.

Since σ_{F_{2}} is F_{2}-stable, these are the same by Lemma 7.7.

Third, we need to show that Υ preserves compositions. For this, we let (S_{1}, F_{1}), (S_{2}, F_{2}) and (S_{3}, F_{3}) be saturated fusion systems, and

γ_{1}: S_{1} → S_{2}  and  γ_{2}: S_{2} → S_{3}

be fusion-preserving morphisms between them. Now,

𝔹γ_{2}_{F_{2}}^{F_{3}} ∘ 𝔹γ_{1}_{F_{1}}^{F_{2}} ≃ σ_{F_{3}} ∘ 𝔹γ_{2} ∘ \underbrace{t_{F_{2}} ∘ σ_{F_{2}}}_{=ω̃_{F_{2}}} ∘𝔹γ_{1} ∘ t_{F_{1}}

≃^{(7.7)} σ_{F_{3}} ∘ 𝔹γ_{2} ∘ 𝔹γ_{1} ∘ t_{F_{1}}

≃ σ_{F_{3}} ∘ 𝔹(γ_{2} ∘ γ_{1}) ∘ t_{F_{1}}

≃ 𝔹(γ_{2} ∘ γ_{1})_{F_{1}}^{F_{3}}.

Composing with the forgetful functor from C to Spectra , we obtain the following corollary.

Corollary 7.10 There is a classifying spectrum functor

𝔹:SFS⟶ Spectra

defined on objects by

(S, F) ↦ 𝔹F

and on morphisms by

γ_{F_{1}}^{F_{2}} ↦ 𝔹γ_{F_{1}}^{F_{2}}.

We conclude this section by illustrating that the fusion-preserving homomorphisms are the only homomorphisms inducing maps between classifying spectra that preserve structure maps.

Proposition 7.11 Let F_{1} and F_{2} be saturated fusion systems over finite p-groups S_{1} and S_{2}, respectively. If γ: S_{1} → S_{2} is a group homomorphism such that 𝔹γ: 𝔹S_{1} → 𝔹S_{2} restricts to a map g: 𝔹F_{1} → 𝔹F_{2} making the following diagram commute up to homotopy

𝔹S_{1} ⟶^{σ_{F_{1}}} 𝔹F_{1}
𝔹γ↓  ↓^{g}
𝔹S_{2} ⟶^{σ_{F_{2}}} 𝔹F_{2},

then γ is fusion-preserving.

Proof We will produce a functor F_{γ}: F_{1} → F_{2} that makes γ fusionpreserving. There are two things to check. First, that given a homomorphism φ ∈ Hom_{F_{1}} (P, Q) there is a unique induced homomorphism F_{γ}(φ): γ(P) → γ(Q) such that γ|_{Q} ∘ φ = F_{γ}(φ) ∘ γ|_{P}, and second, that F_{γ}(φ) is in F_{2}. Functoriality of F_{γ} follows from the uniqueness.

To prove the first claim, let K be the kernel of γ. Then K ∩ P is the kernel of γ|_{P}, and by standard group theory there exists a homomorphism F_{γ}(φ) fitting into the following commutative diagram

\xymatrix{K ∩ P \ar[d] \\ P \ar[r]^{φ} \ar@{->>}[d]^{γ|_{P}} & Q \ar@{->>}[d]^{γ|_{Q}} \\ γ(P) \ar@{-->}[r]^{F_{γ}(φ)} & γ(Q)}

if and only if the restriction of γ|_{Q} ∘ φ to K ∩ P is trivial. Furthermore, since γ|_{P} is surjective onto γ(P), this condition uniquely determines F_{γ}(φ) if it exists. Now,

σ_{F_{2}} ∘ 𝔹(γ|_{Q} ∘ φ|_{K∩P}) ≃ σ_{F_{2}} ∘ 𝔹γ ∘ 𝔹ι_{Q} ∘ 𝔹φ ∘ 𝔹ι_{K∩P}
≃ g ∘ σ_{F_{1}} ∘ 𝔹ι_{Q} ∘ 𝔹φ ∘ 𝔹ι_{K∩P}
≃ g ∘ σ_{F_{1}} ∘ 𝔹ι_{P} ∘ 𝔹ι_{K∩P}
≃ σ_{F_{2}} ∘ 𝔹γ ∘ 𝔹ι_{K∩P}
≃ ∗.

By applying Theorem 7.2 we conclude that (γ|_{Q} ∘ φ|_{K∩P}) is trivial.

The second claim is proved similarly by first performing the following manipulation

σ_{F_{2}} ∘ 𝔹ι_{γ(Q)} ∘ 𝔹F_{γ}(φ) ∘ 𝔹γ|_{P} ≃ σ_{F_{2}} ∘ 𝔹ι_{γ(Q)} ∘ 𝔹γ|_{Q} ∘ 𝔹φ
≃ g ∘ σ_{F_{1}} ∘ 𝔹ι_{Q} ∘ 𝔹φ
≃ g ∘ σ_{F_{1}} ∘ 𝔹ι_{P}
≃ σ_{F_{2}} ∘ 𝔹γ ∘ 𝔹ι_{P} ,

from which we conclude by Theorem 7.2, that the (P, S_{2})-pair (P, ι_{γ(Q)} ∘ F_{γ}(φ) ∘ γ|_{P}) is (F_{P} , F_{2})-conjugate to (P, γ ∘ ι_{P}). By definition this means that there exist g ∈ P and φ′ ∈ Hom_{F_{2}} (γ(P), γ(φ(P))) making the following diagram commute

P ⟶^{γ|_{P}}  γ(P)
≅↓^{c_{g}}  ↓^{φ′}
P ⟶^{F_{γ}(φ)∘γ|_{P}} γ(φ(P)).

This implies that

F_{γ}(φ) ∘ γ(x) = φ′ ∘ γ ∘ c_{g^{−1}}(x)
= φ′ ∘ c_{γ(g^{−1})} ∘ γ(x),

for all x ∈ P. Since γ is surjective onto γ(P) this implies that

F_{γ}(φ) = φ′ ∘ c_{γ(g^{−1})} ∈ Hom_{F_{2}} (γ(P), γ(Q)) .

Remark 7.12

(1) An unstable version of the preceding proof, using [7, Proposition 4.4] instead of Theorem 7.2, shows that a map between classifying spaces of p-local finite groups restricts to a fusion-preserving homomorphism of underlying p-groups.

(2) In the second paragraph of the proof we showed that

φ(Ker γ ∩ P) ≤ Ker γ

for all P, Q ≤ S and φ ∈ Hom_{F_{1}} (P, Q). In other words, Ker γ is strongly closed in F_{1}.

8 Transfer theory for classifying spectra

In the classical setting, the map of classifying spaces induced by an injective homomorphism ψ: G → H of finite groups admits a transfer tr_{ψ}: Σ_{+}^{∞}BH → Σ_{+}^{∞}BG (see for example [2]), which restricts to a reduced transfer Σ^{∞}BH → Σ^{∞}BG, also denoted by tr_{ψ}. In this section we adapt this theory to injective fusion-preserving homomorphisms and obtain a transfer theory between classifying spectra. The transfers we construct will not respect structure maps of saturated fusion systems and so do not fit into the framework of structured classifying spectra. They do however respect transfer maps of saturated fusion systems and we will formulate our results in terms of a functor analogous to the structured classifying space functor. One motivation for doing so is the following, rather surprising result.

Proposition 8.1 If F is a saturated fusion system over a finite p-group S, then

Hom_{F} (P, Q) = {φ ∈ Inj (P, Q) | tr_{φ} ∘ tr_{Q} ∘ t_{F} ≃ tr_{P} ∘ t_{F}}.

Proof The proof is similar to the proof of Theorem 7.3. The extra condition that φ be injective is required to ensure the existence of the transfer tr_{φ}.

Definition 8.2 Let F_{1} and F_{2} be saturated fusion systems over finite p-groups S_{1} and S_{2}, respectively. If γ is an injective, fusion-preserving homomorphism S_{1} → S_{2}, then the (F_{1}, F_{2})-transfer of γ is

T r (γ_{F_{1}}^{F_{2}}) := σ_{F_{1}} ∘ tr_{γ} ∘ t_{F_{2}} : 𝔹F_{2} → 𝔹F_{1}.

As in the previous section we show that this assignment is functorial, working first with a target category that keeps track of all the structure involved.

Definition 8.3 Let SFS_{0} be the subcategory of SFS, whose objects are saturated fusion systems, but whose morphisms are the injective fusionpreserving homomorphisms.

Definition 8.4 Let D be the category whose objects are maps of spectra

t: X → 𝔹S,

where S is a finite p-group and X is a p-complete spectrum, and where the set of morphisms from t_{2}: X_{2} → 𝔹S_{2} to t_{1}: X_{1} → 𝔹S_{1} consists of pairs (γ, g) where γ: S_{1} → S_{2} is a group monomorphism and g: X_{2} → X_{1} is a map of spectra such that the following diagram commutes up to homotopy:

X_{2} ⟶^{t_{2}} 𝔹S_{2}
g↓  ↓^{tr_{γ}}
X_{1} ⟶^{t_{1}} 𝔹S_{1}.

The proof of functoriality for transfers is very similar to the proof of Theorem 7.9 and we will need the following analogue of Lemma 7.7.

Lemma 8.5 Let F_{1} and F_{2} be saturated fusion systems over finite p-groups S_{1} and S_{2}, respectively, and let γ: S_{1} → S_{2} be a fusion-preserving homomorphism. If f: X → 𝔹S_{2} is a F_{2}-stable map, then

tr_{γ} ∘ f: X → 𝔹S_{1}

is F_{1}-stable and

ω̃_{F_{1}} ∘ tr_{γ} ∘ f = tr_{γ} ∘ f.

Proof Let P ≤ S_{1} and φ ∈ Hom_{F_{1}} (P, S). Recalling that the assignment of transfers to group monomorphisms contravariantly preserves compositions up to homotopy, we get

tr_{φ} ∘ tr_{γ} ∘ f ≃ tr_{(γ∘φ)} ∘ f
≃ tr_{(F_{γ}(φ)∘γ|_{P})} ∘ f
≃ tr_{γ|_{P}} ∘ tr_{F_{γ}(φ)} ∘ f
≃ tr_{γ|_{P}} ∘ tr_{γ(P)} ∘ f
≃ tr_{(ι_{γ(P)}∘γ|_{P})} ∘ f
≃ tr_{(γ∘ι_{P})} ∘ f
≃ tr_{P} ∘ tr_{γ} ∘ f,

proving F_{1}-stability. The second claim follows from Corollary 6.4.

Theorem 8.6 There is a structured fusion transfer functor

Ξ : SFS_{0} → D^{op}

defined on objects by

(S, F) ↦ (t_{F}: 𝔹F → 𝔹S)

and on morphisms by

γ_{F_{1}}^{F_{2}} ↦ (γ, T r (γ_{F_{1}}^{F_{2}})) .

Proof The proof is similar to the proof of Theorem 7.9. There are again three things to show. First, it is clear by construction that t_{F}: 𝔹F → 𝔹S is an object of D.

Second, if γ_{F_{1}}^{F_{2}} ∈ Mor_{SFS_{0}} ((S_{1}, F_{1}), (S_{2}, F_{2})), then we prove commutativity of the diagram

𝔹F_{2} ⟶^{t_{F_{2}}} 𝔹S_{2}
T r(γ_{F_{1}}^{F_{2}})↓  ↓^{tr_{γ}}
𝔹F_{1} ⟶^{t_{F_{1}}} 𝔹S_{1}.

by applying Lemma 8.5 and recalling that t_{F_{2}} is F_{2}-stable (since ω̃_{F_{2}} is F_{2}- stable).

The third part, that Ξ preserves compositions, also follows from Lemma 8.5 and functoriality of the classical transfer, just as in the proof of Theorem 7.9.

Composing with the forgetful functor from D^{op} to Spectra ^{op} we obtain the following corollary.

Corollary 8.7 There is a fusion transfer functor

SFS_{0} → Spectra ^{op}

defined on objects by

(S, F) ↦ 𝔹F

and on morphisms by

γ_{F_{1}}^{F_{2}} ↦ T r (γ_{F_{1}}^{F_{2}}) .

We conclude this section with an application of the transfer theory. If F_{1} and F_{2} are saturated fusion systems over a finite p-group S, then we say that a map

f: 𝔹F_{2} → 𝔹F_{1}

is structure-preserving if f ∘ σ_{F_{2}} ≃ σ_{F_{1}} . We say that 𝔹F_{2} is a summand of 𝔹F_{1} as objects under 𝔹S if there is a structure-preserving map f: 𝔹F_{1} → 𝔹F_{2} and a map r: 𝔹F_{2} → 𝔹F_{1} such that

f ∘ r ≃ id_{𝔹F_{2}}.

Proposition 8.8 Let F_{1} and F_{2} be saturated fusion systems over a finite p-group S. The following four conditions are equivalent:

(i) F_{1} is a subcategory of F_{2}; (ii) id_{S} is (F_{1}, F_{2})-fusion-preserving; (iii) 𝔹F_{2} is a summand of 𝔹F_{1} as objects under 𝔹S; (iv) There is a structure-preserving map f: 𝔹F_{1} → 𝔹F_{2}.

Proof (i) ⇒ (ii): Obvious.

(ii) ⇒ (iii): The map 𝔹id_{F_{1}}^{F_{2}} is structure-preserving and

𝔹id_{F_{1}}^{F_{2}} ∘ T r (id_{F_{1}}^{F_{2}}) = σ_{F_{2}} ∘ 𝔹id ∘ t_{F_{1}} ∘ σ_{F_{1}} ∘ tr_{id} ∘ t_{F_{2}}

= σ_{F_{2}} ∘ id ∘ ω̃_{F_{1}} ∘ id ∘ t_{F_{2}}

=^{(7.7)} σ_{F_{2}} ∘ t_{F_{2}}

= id_{𝔹F_{2}}.

(iii) ⇒ (iv): Obvious.

(iv) ⇒ (i): Let P ≤ S and φ ∈ Hom_{F_{1}} (P, S). Then

σ_{F_{1}} ∘ 𝔹φ ≃ σ_{F_{1}} ∘ 𝔹ι_{P} .

Consequently

σ_{F_{2}} ∘ 𝔹φ ≃ f ∘ σ_{F_{1}} ∘ 𝔹φ ≃ f ∘ σ_{F_{1}} ∘ 𝔹ι_{P} ≃ σ_{F_{2}} ∘ 𝔹ι_{P} .

By Theorem 7.3 it follows that φ ∈ Hom_{F_{2}} (P, S).

9 Cohomology of classifying spectra

In this section we apply the mod p cohomology functor to the theory of the previous sections and observe that the cohomology of fusion systems and their classifying spectra behaves like group cohomology in many important ways. Cohomology will always be taken with 𝔽_{p}-coefficients and therefore we will denote the functor H^{∗}(−; 𝔽_{p}) by H^{∗}(−).

We begin by recording the analogue of the last part of Proposition 4.2

Proposition 9.1 Let F be a saturated fusion system over a finite p-group S. Then the map ω_{F}^{∗} induced by its stable characteristic idempotent in cohomology is an idempotent in End(H^{∗}(BS)), is H^{∗}(F)-linear and a homomorphism of modules over the Steenrod algebra; and

Im[H^{∗}(BS) \underrightarrow{ω_{F}^{∗}} H^{∗}(BS)] = H^{∗}(F).

Proof The proof is essentially the same as in [7]. The reader is referred to [7, Proposition 5.5] for details.

The next proposition, which is arguably the most important result in this section, has already been observed by Broto-Levi-Oliver in [7].

Proposition 9.2 [7, Section 5] Let F be a saturated fusion system over a finite p-group S. Then the structure map σ_{F} induces a split monomorphism

H^{∗}(𝔹F) ↪ H^{∗}(BS)

in cohomology, with image H^{∗}(F).

Proof By construction it is clear that σ_{F} induces the inclusion

Im[H^{∗}(BS) \underrightarrow{ω_{F}^{∗}} H^{∗}(BS)] ↪ H^{∗}(BS)

in cohomology, and by Proposition 9.1 this has image H^{∗}(F).

Remark 9.3 Using Corollary 6.4, Castellana-Morales show in [9] that the analogous result holds for any generalized cohomology theory. That is, for a saturated fusion system F over a finite p-group S and a generalized cohomology theory E, the structure map σ_{F} induces a split monomorphism

E^{∗}(𝔹F) ↪ E^{∗}(BS)

with image

E^{∗}(F) := lim \overline{}_{F} E^{∗}(B(−)).

Since H^{∗}(F) is a subring of H^{∗}(BS), and is consequently an unstable algebra over the Steenrod algebra, we can now regard H^{∗}(𝔹F) as an unstable algebra over the Steenrod algebra via the isomorphism

σ_{F}^{∗} : H^{∗}(𝔹F) ⟶^{≅} H^{∗}(F).

With this understanding, the map

σ_{F}^{∗} : H^{∗}(𝔹F)⟶H^{∗}(𝔹S)

is obviously a map of unstable algebras over the Steenrod algebra.

The H^{∗}(F)-linearity of Proposition 9.1 is a version of what is commonly referred to as Frobenius reciprocity in the classical group cohomology setting. We record this property in the more familiar form.

Proposition 9.4 (Frobenius reciprocity) Let F be a saturated fusion system over a finite p-group S. If x ∈ H^{∗}(𝔹F) and y ∈ H^{∗}(BS), then

t_{F}^{∗} (σ_{F}^{∗} (x) · y) = x · t_{F}^{∗} (y) .

Proof Since there is only one fusion system in play, we will drop the subscript F and simply write σ, t and ω̃. Now, by using H^{∗}(F)-linearity of ω^{∗} and the fact that σ^{∗} preserves multiplication, we have

t^{∗} (σ^{∗} (x) · y) = t^{∗} ∘ ω^{∗}( \underbrace{σ^{∗} (x)}_{∈H^{∗}(F)} · y)

= t^{∗} (σ^{∗} (x) · ω^{∗} (y))

= t^{∗} (σ^{∗} (x) · (σ^{∗} ∘ t^{∗} (y)))

= \underbrace{t^{∗} ∘ σ^{∗}}_{id} (x · t^{∗} (y))

= x · t^{∗} (y) .

The usefulness of transfers in group cohomology lies to a large extent in the well known result, that for a group monomorphism γ: G → H, the effect of the composition tr_{γ}^{∗} ∘ 𝔹γ^{∗} in cohomology is simply multiplication by the index [H : γ(G)]. The situation for transfers between classifying spectra induced by fusion-preserving monomorphisms is similar. Although one might expect that the order of the outer automorphism groups of the fusion systems should come into play, it is only the order of the underlying p-groups that is important. The reason is that the characteristic idempotents, which are used to construct the transfer, have a “normalizing effect”. In some sense they divide out the order of the outer automorphism groups.

Proposition 9.5 Let F_{1} and F_{2} be saturated fusion systems over S_{1} and S_{2}, respectively, and let γ: S_{1} → S_{2} be a fusion-preserving monomorphism. Then the composition

T r (γ_{F_{1}}^{F_{2}})^{∗} ∘ 𝔹γ_{F_{1}}^{F_{2}}^{∗}

acts on H^{∗}(𝔹F_{1}) as multiplication by |S_{2}|/|S_{1}|.

Proof Indeed,

𝔹γ_{F_{1}}^{F_{2}} ∘ T r (γ_{F_{1}}^{F_{2}}) = σ_{F_{2}} ∘ Bγ ∘ t_{F_{1}} ∘ σ_{F_{1}} ∘ tr_{γ} ∘ t_{F_{2}}

= σ_{F_{2}} ∘ Bγ ∘ ω̃_{F_{1}} ∘ tr_{γ} ∘ t_{F_{2}}

=^{(7.7)} σ_{F_{2}} ∘ Bγ ∘ tr_{γ} ∘ t_{F_{2}}.

Now, if x ∈ H^{∗}(𝔹F_{1}), then

T r (γ_{F_{1}}^{F_{2}})^{∗} ∘ 𝔹γ_{F_{1}}^{F_{2}}^{∗} (x) = t_{F_{2}}^{∗} ∘ tr_{γ}^{∗} ∘ Bγ^{∗} ∘ σ_{F_{2}}^{∗} (x)

= t_{F_{2}}^{∗} (\frac{|S_{2}|}{|S_{1}|} · σ_{F_{2}}^{∗} (x))

= \frac{|S_{2}|}{|S_{1}|} · t_{F_{2}}^{∗} (σ_{F_{2}}^{∗} (x))

= \frac{|S_{2}|}{|S_{1}|} · x.

Transfers between classifying spectra also exhibit Frobenius reciprocity.

Proposition 9.6 (Frobenius reciprocity) Let F_{1} and F_{2} be saturated fusion systems over S_{1} and S_{2}, respectively, and let γ: S_{1} → S_{2} be a fusion-preserving monomorphism. If x ∈ H^{∗}(𝔹F_{2}) and y ∈ H^{∗}(𝔹F_{1}), then

T r (γ_{F_{1}}^{F_{2}})^{∗} (𝔹γ_{F_{1}}^{F_{2}}^{∗} (x) · y) = x · T r (γ_{F_{1}}^{F_{2}})^{∗} (y) .

Proof Using the facts that σ_{F_{1}}^{∗} is a ring homomorphism, and that t_{F_{2}}^{∗} and tr_{γ}^{∗} both exhibit Frobenius reciprocity, we get

T r (γ_{F_{1}}^{F_{2}})^{∗} (𝔹γ_{F_{1}}^{F_{2}}^{∗} (x) · y) = t_{F_{2}}^{∗} ∘ tr_{γ}^{∗} ∘ σ_{F_{1}}^{∗} (𝔹γ_{F_{1}}^{F_{2}}^{∗} (x) · y)

= t_{F_{2}}^{∗} ∘ tr_{γ}^{∗} ∘ ((σ_{F_{1}}^{∗} ∘ 𝔹γ_{F_{1}}^{F_{2}}^{∗} (x)) · σ_{F_{1}}^{∗} (y) )

= t_{F_{2}}^{∗} ∘ tr_{γ}^{∗} ((𝔹γ^{∗} ∘ σ_{F_{2}}^{∗} (x)) · σ_{F_{1}}^{∗} (y) )

= t_{F_{2}}^{∗} (σ_{F_{2}}^{∗} (x) · (tr_{γ}^{∗} ∘ σ_{F_{1}}^{∗} (y)))
= x · (t_{F_{2}}^{∗} ∘ tr_{γ}^{∗} ∘ σ_{F_{1}}^{∗} (y))
= x · T r (γ_{F_{1}}^{F_{2}})^{∗} (y) .

The reciprocity results in this section can be shown to hold at the level of stable homotopy, as is the case with transfers induced by finite covers, when the structure maps preserve diagonals (in particular the classifying spectra involved must have diagonal maps). Since the only known cases where this happens is for fusion systems belonging to p-local finite groups, in which case the classifying spectra are suspension spectra, this discussion is postponed for [22].

10 Comparison to stable classifying spaces

We conclude the paper by comparing the theory of classifying spectra of saturated fusion systems with the theory obtained by infinite suspension of classifying spaces of finite groups and p-local finite groups, and proving that the theory of classifying spectra extends both these theories.

It is shown in [7] that when F has an associated centric linking system L, the classifying spectrum 𝔹F is homotopy equivalent to the p-completed suspension spectrum Σ^{∞}|L|_{p}^{∧}. We extend this observation to structured classifying spectra. The reader is referred to [7] for the precise definition of centric linking systems and p-local finite groups.

Proposition 10.1 Let (S, F, L) be a p-local finite group. Then the infinite suspension

Σ^{∞}θ: Σ^{∞}BS ⟶ Σ^{∞}|L|_{p}^{∧}

of the natural inclusion θ: BS → |L|_{p}^{∧} is equivalent to the structure map

σ_{F}: 𝔹S ⟶ 𝔹F.

Proof First we recall that 𝔹S is homotopy equivalent to Σ^{∞}BS, and we may therefore identify the two via a chosen homotopy equivalence.

In [7] it is shown that θ is F -stable and it follows that Σ^{∞}θ is F -stable. By Corollary 6.4 it follows that

Σ^{∞}θ ∘ ω̃_{F} ≃ Σ^{∞}θ.

By construction of the structured classifying spectrum as a mapping telescope we get a map h: 𝔹F → Σ^{∞}|L|_{p}^{∧} such that

h ∘ σ_{F} ≃ Σ^{∞}θ.

In particular the corresponding equality holds for the induced maps in cohomology with 𝔽_{p}-coefficients. It is shown in [7] (see also Proposition 9.1) that in cohomology the maps σ_{F} and θ both induce injctions with image H^{∗}(F) in H^{∗}(BS), and therefore we conclude that h induces an isomorphism

h^{∗}: H^{∗}(Σ^{∞}|L|_{p}^{∧}) ⟶^{≅} H^{∗}(𝔹F).

Since the spectra involved are p-complete, we deduce that h is a homotopy equivalence.

We now turn our attention to p-completed classifying spaces of finite groups. This theory overlaps in parts with the theory of classifying spaces of p-local finite groups since the classifying space of the p-local finite group induced by a finite group G is homotopy equivalent to BG_{p}^{∧}. An additional aspect for the stable classifying spaces of groups is that the inclusion of a Sylow subgroup S into a finite group G has a stable transfer, which we compare with the transfer of a saturated fusion system.

Proposition 10.2 Let G be a finite group with Sylow subgroup S. Then the map

𝔹ι_{S}: 𝔹S ⟶ 𝔹G

induced by the inclusion S ≤ G is equivalent to the structure map

σ_{F_{S}(G)}: 𝔹S ⟶ 𝔹F_{S}(G).

Furthermore, if we let g be a homotopy inverse of the homotopy equivalence

𝔹ι_{S} ∘ tr_{S}: 𝔹G ⟶ 𝔹G,

then the map

t′ := tr_{S} ∘ g: 𝔹G ⟶ 𝔹S

is equivalent to the map

t_{F_{S}(G)}: 𝔹F_{S}(G) ⟶ 𝔹S.

Proof Write F := F_{S}(G). It was shown in [7] that for a finite p-group G with Sylow subgroup S, the map Bι_{S}: BS ⟶ BG is equivalent to the inclusion θ: BS ⟶ |L|_{p}^{∧} of BS into the classifying space of the corresponding p-local finite group. By Proposition 10.1 it follows that the map 𝔹ι_{S}: 𝔹S ⟶ 𝔹G is equivalent to the map σ_{F}: 𝔹S ⟶ 𝔹F.

When proving the second claim we can fix a homotopy equivalence 𝔹F → 𝔹G and regard t′ as a map 𝔹F → 𝔹S. Assume, for now, that ω̃_{F} ∘ tr_{S} ≃ tr_{S}. Then ω̃_{F} ∘ t′ ≃ t′ and we get

t_{F} ≃ t_{F} ∘ 𝔹ι_{S} ∘ t′ ≃ t_{F} ∘ σ_{F} ∘ t′ ≃ ω̃_{F} ∘ t′ ≃ t′,

which is what we want to show.

To prove that ω_{F} ∘ tr_{S} ≃ tr_{S} it suffices, by Corollary 6.4, to establish that tr_{S} is F -stable, which is actually quite well known. One way to convince oneself of this is to note that tr_{S} is the image in {BG, BS} of [S, id_{S}] ∈ A(G, S), and that for P ≤ S and φ ∈ Hom_{F} (P, S) one has

[φ(P), φ^{−1}]_{S}^{P} ∘ [S, id_{S}]_{G}^{S} = [φ(P), φ^{−1}]_{G}^{P} = [P, id_{P}]_{G}^{P} ∈ A(G, P),

since φ is a conjugation induced by an element of G.

With the notation of the preceding theorem, the stable characteristic idempotent of F_{S}(G) can be obtained as

ω̃_{F_{S}(G)} ≃ t′ ∘ 𝔹ι_{S},

and the characteristic idempotent is then

ω_{F_{S}(G)} = α^{−1}(ω̃_{F_{S}(G)}),

regarded as an element of A(S, S)_{p}^{∧}.

Another feature of the theory of p-completed classifying spaces of finite groups is that it is functorial. Namely, given a homomorphism γ̄: G_{1} → G_{2} between finite groups, one gets a map of p-completed classifying spaces Bγ̄_{p}^{∧}: BG_{1}_{p}^{∧} → BG_{2}_{p}^{∧}. Furthermore, the restriction γ: S_{1} → S_{2} to Sylow subgroups is fusion preserving for the fusion systems F_{S_{1}}(G_{1}) and F_{S_{2}}(G_{2}). Hence we get a map 𝔹γ_{F_{1}}^{F_{2}} of classifying spectra, which we can compare to 𝔹γ̄, the infinite suspension of Bγ̄_{p}^{∧}.

Proposition 10.3 Let γ̄: G_{1} → G_{2} be a homomorphism of finite groups with a restriction γ: S_{1} → S_{2} to Sylow subgroups. Then the diagram

𝔹S_{1} ⟶^{𝔹ι_{S_{1}}} 𝔹G_{1}
𝔹γ↓  ↓^{𝔹γ̄}
𝔹S_{2} ⟶^{𝔹ι_{S_{2}}} 𝔹G_{2}

is equivalent to the diagram

𝔹S_{1} ⟶^{σ_{F_{1}}} 𝔹F_{S_{1}}(G_{1})
𝔹γ↓  ↓^{𝔹γ_{F_{1}}^{F_{2}}}
𝔹S_{2} ⟶^{σ_{F_{2}}} 𝔹F_{S_{2}}(G_{2}).

Proof By Proposition 10.2 we can replace all but the map on the right side of the upper diagram with the corresponding maps and objects on the lower diagram. All that remains is to show is now that, when thus regarded as a map 𝔹F_{S_{1}}(G_{1}) → 𝔹F_{S_{2}}(G_{2}), the map 𝔹γ̄ is homotopic to the map 𝔹γ_{F_{1}}^{F_{2}} . Now, from the equivalence 𝔹γ̄ ∘ σ_{F_{1}} ≃ σ_{F_{2}} ∘ 𝔹γ it follows that

𝔹γ̄ ≃ 𝔹γ̄ ∘ σ_{F_{1}} ∘ t_{F_{1}} ≃ σ_{F_{2}} ∘ 𝔹γ ∘ t_{F_{1}} = 𝔹γ_{F_{1}}^{F_{2}}.

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Department of Mathematical Sciences University of Aberdeen Aberdeen AB24 3UE United Kingdom

Email: kari@maths.abdn.ac.uk