A SEGAL CONJECTURE FOR p-COMPLETED CLASSIFYING SPACES

KÁRI RAGNARSSON

Abstract. We formulate and prove a new variant of the Segal Conjecture describing the group of homotopy classes of stable maps from the p-completed classifying space of a finite group G to the classifying space of a compact Lie group K as the p-adic completion of the Grothendieck group A_{p}(G, K) of principal (G, K)-bundles whose isotropy groups are p-groups. Collecting the result for different primes p, we get a new and simple description of the group of homotopy classes of stable maps between (uncompleted) classifying spaces of groups. This description allows us to determine the kernel of the map from the Grothendieck group A(G, K) of principal (G, K)-bundles to the group of homotopy classes of stable maps from BG to BK.

Introduction

Inspired by Atiyah’s description [3] of the representable complex periodic K-theory of a finite group G as the completion of the complex representation ring R[G] at its augmentation ideal,

R̂[G] ⟶^{≅} KU^{0}(BG),

Segal conjectured that the zeroth stable cohomotopy group of a finite group G could analogously be described as the completion of the Burnside ring A(G) of isomorphism classes of virtual finite G-sets ([14]), with respect to its augmentation ideal I(G),

A(G)_{I(G)}^{∧} ⟶^{≅} π_{S}^{0}(BG_{+}).

After many partial results by various authors, this conjecture was eventually settled in the affirmative by Carlsson in [7]. In this account we mention only the three major contributions which in the end combined to form a solution of the conjecture, but the reader is encouraged to look up [7] for an account of the history of the Segal conjecture. Adams-Gunawardena-Miller proved the conjecture for elementary abelian p-groups in [2]. May-McClure simplified the conjecture in [10], where they proved that if the conjecture holds for all finite p-groups, then it holds for all finite groups. The solution of the conjecture was completed by Carlsson in [7], where he supplied an inductive proof showing that if the Segal conjecture is true for elementary abelian p-groups then it is true for all finite p-groups.

Date: February 13, 2007.

The author was supported by EPSRC grant GR/S94667/01 during most of this work.

The Segal conjecture was extended by Lewis–May–McClure in [8] to describe the group of homotopy classes of stable maps between classifying spaces of finite groups, and later by May–Snaith–Zeliski in [11] to allow the classifying space of a compact Lie group in the target. For a finite group G and a compact Lie group K, they consider the Burnside module A(G, K). This is the Grothendieck group completion of isomorphism classes of principal (G, K)-bundles, which are (K × G)–bundles such that the K-action is free with finitely many orbits. The Burnside module A(G, K) is an A(G)-module which as a ℤ-module is free with one basis element [H, φ] for each conjugacy class of pairs (H, φ) consisting of a subgroup H ≤ G and a group homomorphism φ: H → K. Letting {BG_{+}, BK_{+}} denote the group of homotopy classes of stable maps Σ^{∞}BG_{+} → Σ^{∞}BK_{+}, there is a natural homomorphism

α: A(G, K) ⟶ {BG_{+}, BK_{+}},

which sends a basis element [H, φ] to the stable map Σ^{∞}Bφ_{+} ∘ tr_{H}, where tr_{H} is the transfer map discussed in Section 1.2. In [8] and [11] it is shown that the Segal conjecture implies that this map is a completion with respect to I(G). The original Segal conjecture is the special case when K is the trivial group.

The I(G)-adic completion of A(G, K) is in general difficult to describe, but in the special case where G is a finite p-group S, May-McClure [10] gave a simple description. Letting Ã(S, K) denote the quotient module obtained from A(S, K) by quotienting out basis elements of the form [P, ct] where ct is a constant homomorphism, they noticed that the I(S)-adic topology on Ã(S, K) coincides with the padic topology, and therefore the induced map

α̃: Ã(S, K) ⟶ {BS, BK}

is a p-adic completion.

The main result of this paper can be regarded as an extension of this result of May-McClure, where we drop the condition that G be a p-group but p-complete BG instead. In other words we describe {(BG)_{p}^{∧}, BK} in similarly simple terms as a p-completion of a certain submodule of Ã(G, K). Let A_{p}(G, K) be the submodule of A(G, K) generated by those (G, K)-bundles whose isotropy groups are p-groups, and let Ã_{p}(G, K) denote the corresponding submodule of A_{p}(G, K). We prove the following theorem, which appears in the text as Theorem 3.1.

Theorem A. For a finite group G and a compact Lie group K, the homomorphism

Ã_{p}(G, K) \underrightarrow{α̃_{p}} \{BG, BK\} ⟶^{ι_{p}^{∗}} \{(BG)_{p}^{∧}, BK\}

induced by α is a p-adic completion.

Here ι_{p}: 𝔹G_{p}^{∧} ↪ 𝔹G is the natural wedge summand inclusion obtained from the natural splitting Σ^{∞}BG ≃ ⋁_{q} Σ^{∞}BG_{q}^{∧}, as discussed in Section 1.1.

Since α is natural in G and finite K we can state the finite group version of this theorem in terms of isomorphisms of categories. This is done as Corollary 3.3 after we have developed the necessary framework.

As a consequence of the stable splitting of the classifying space of a finite group into its q-completed components, where q runs over all primes, one easily obtains a new description of {BG, BK}. The result is similar in spirit to Minami’s description of A(G)_{I}^{∧} for a compact Lie group G in [12]. Indeed, when K is the trivial group one recovers Minami’s result for finite G. This result is Theorem 3.4 in the text.

Theorem B. For a finite group G and a compact Lie group K, the homomorphism α induces an isomorphism of ℤ-modules

⨁_{p} Ã_{p}(G, K)_{p}^{∧} ⟶^{≅} {BG, BK}.

This description is much simpler than the I(G)-adic completion, but it has the drawback that it is not obvious how to decompose an element in {BG, BK} into p-completed parts in Ã_{p}(G, K)_{p}^{∧}. This matter is taken up in Section 5 where we describe the map

Ã(G, K) ⟶ {BG, BK} ⟶ {(BG)_{p}^{∧}, (BK)_{p}^{∧}} ⟶^{≅} Ã_{p}(G, K)_{p}^{∧}

for every prime p. Loosely speaking, this map sends an element X of A(G, K) to an element of Ã_{p}(G, K)_{p}^{∧} whose H-fixed point sets for subgroups H ≤ K × G agree with the H-fixed point sets of X if H is a finite p-group, and are empty otherwise. (This description should not be taken too literally as elements of A(G, K) and Ã_{p}(G, K)_{p}^{∧} are not actual (K × G)-bundles and do not have fixed point sets. A precise statement can be found in Section 5.) As a consequence we deduce the following description of the kernel of α.

Theorem C. Let G be a finite group and K a compact Lie group. (a) The kernel of the map

α: A(G, K) ⟶ {BG_{p}^{∧}, BK}

consists of those virtual (G, K)-bundles [X − Y] whose fixedpoint sets under the action of any finite p-subgroup P ≤ K × G satisfy

|W (P)\X^{P} | = |W (P)\Y^{P} |,

where W (P) = N_{K×G}(P)/P.

(b) The kernel of the map

α: A(G, K) ⟶ {BG, BK}

consists of those virtual (G, K)-bundles [X − Y] such that for every prime q and every q-subgroup Q ≤ K × G, the Q-fixed-point sets satisfy

|W (P)\X^{P} | = |W (P)\Y^{P} |.

A more informative version of this theorem appears in the text as Corollary 5.8 after some relevant concepts and notation have been introduced. In particular, not all p-subgroups P need be considered, and there is a justification why the quotients of the fixed-point sets appearing are finite sets. Note also that when K is finite one does not need to take the quotient by the W (P)-action (cf. Remark 5.9).

The Segal conjecture has been presented here in its weak form. There is a stronger form of the conjecture, also due to Segal, describing higher stable cohomotopy groups of classifying spaces of finite groups as a completion of certain equivariant stable cohomotopy groups, constructed by Segal in [13]. The proofs of the Segal conjecture mentioned above are in fact proofs of the stronger version. Indeed, Carlsson’s inductive argument would not have been possible without the presence of higher homotopy groups. The Lewis-May-McClure generalization of the conjecture also extends the stronger version of the Segal conjecture to describe spectra of stable maps between classifying spaces. We refer the reader to [8] for this description. This raises an obvious, interesting question. Namely, whether the results in this paper can be extended to also describe higher homotopy groups of spectra of stable maps between p-completed classifying spaces.

This paper is divided into four sections. In Section 1 we discuss some background material necessary for the main discussion. In Section 2 we introduce subconjugacy, which gives a convenient filtration of Burnside modules. In Section 3 we state and prove the new variants of the Segal conjecture for p-completed classifying spaces. Finally, in Section 5 we discuss how to decompose a stable map into its p-completed components and describe the kernel of the I(G)-adic completion map.

1. Preliminaries

In this section we discuss background material which will be needed in later sections. We begin by giving an overview of notation and conventions which will be used throughout this paper.

Unless otherwise specified, p is a fixed prime and all cohomology is taken with 𝔽_{p}-coefficients. For a space or spectrum X, we let X_{p}^{∧} denote the Bousfield-Kan p-completion [6], and for a ℤ-module M, we let M_{p}^{∧} denote the p-adic completion of M. Since we only consider finitely generated modules in this paper are finitely generated, we have M ≅ M ⊗ ℤ_{p}^{∧}, where ℤ_{p}^{∧} are the p-adic integers.

For a space X, let X_{+} denote the pointed space obtained by adding a disjoint basepoint to X. We use the shorthand notation Σ_{+}^{∞}X := Σ^{∞}X_{+} and recall that there is a natural equivalence Σ_{+}^{∞}X ≃ Σ^{∞}X ∨ 𝕊, where 𝕊 denotes the sphere spectrum. All stable homotopy will take place in the homotopy category of spectra, which we denote by Spectra .

Let Gr denote the category of finite groups. We will use ι_{H} to denote a subgroup inclusion H ↪ G, specifying the supergroup G when there is danger of confusion. Conjugations will come up frequently. For a group element g ∈ G we let c_{g} denote the conjugation isomorphism x ↦ gxg^{−1}. For a subgroup H ≤ G, we let ^{g}H denote c_{g}(H), and H^{g} denote c_{g}^{−1}(H).

We use the shorthand notations 𝔹: = Σ^{∞}B(−) and 𝔹_{+}: = Σ_{+}^{∞}(B(−)), regarded as functors Gr → Spectra . Since Σ^{∞}(BG_{p}^{∧}) ≃ (Σ^{∞}BG)_{p}^{∧} for a finite group G, we will write 𝔹G_{p}^{∧} without danger of confusion. We denote by 𝔹Gr the category whose objects are the finite groups and whose morphisms are homotopy classes of stable maps between classifying spaces,

Mor_{𝔹Gr} (G_{1}, G_{2}) = {BG_{1}, BG_{2}}.

Similarly we let 𝔹Gr_{p}^{∧} be the category with the same objects, but whose morphisms are homotopy classes of stable maps between p-completed classifying spaces.

Mor_{𝔹Gr}_{p}^{∧} (G_{1}, G_{2}) = {(BG_{1})_{p}^{∧}, (BG_{2})_{p}^{∧}}.

1.1. Bousfield-Kan p-completion of 𝔹G. In this section we list the basic properties of Bousfield-Kan completion of spectra at a prime. We recall how the suspension spectrum of the classifying space of a finite group decomposes as a wedge sum of its p-completions as p runs over all primes and see that this splitting is natural.

Bousfield-Kan completion at a prime p is an endofunctor (−)_{p}^{∧} defined on either the category of spaces or spectra, depending on the context. In either case its defining property is that for a map f: X → Y, the p-completed map f_{p}^{∧}: X_{p}^{∧} → Y_{p}^{∧} is a weak homotopy equivalence if and only if f induces an isomorphism in homology with 𝔽_{p} coefficients.

The p-completion functor comes with a natural transformation η_{p}: Id ⇒ (−)_{p}^{∧}. We say that a space or spectrum X is p-complete if the map η_{p}: X → X_{p}^{∧} is a weak equivalence. We say that X is p-good if X_{p}^{∧} is p-complete. Classifying spaces of finite groups and their suspension spectra are p-good. Classifying spaces of finite p-groups and their suspension spectra are p-complete.

As noted earlier, we have (Σ^{∞}BG)_{p}^{∧} ≃ Σ^{∞}(BG_{p}^{∧}) for a finite group G, and so we can denote both of these by 𝔹G_{p}^{∧} without danger of confusion. However, p-completion does not commute with suspension in general, as we have

Σ^{∞}((BG_{+})_{p}^{∧}) ≃ Σ^{∞}(BG_{p}^{∧})_{+} ≃ 𝕊 ∨ Σ^{∞}BG_{p}^{∧}

while

(Σ^{∞}BG_{+})_{p}^{∧} ≃ 𝕊_{p}^{∧} ∨ Σ^{∞}BG_{p}^{∧}.

This difference is one of the underlying reasons for the basepoint issues one has to circumvent in the formulation of the p-completed Segal conjectures.

When G is a finite group, 𝔹G is a torsion spectrum, so by Sullivan’s arithmetic square the natural maps η_{q}: 𝔹G → 𝔹G_{q}^{∧} induce a natural homotopy equivalence

h := ⋁_{q} η_{q}: 𝔹G ⟶^{≃} ⋁_{q} 𝔹G_{q}^{∧},

where the wedge sum runs over all primes q. We obtain a natural inclusion of 𝔹G_{p}^{∧} as a wedge summand of 𝔹G, defined as the composite

ι_{p}: 𝔹G_{p}^{∧} ↪ ⋁_{q} 𝔹G_{q}^{∧} ⟶^{h^{−1}} 𝔹G.

with left homotopy inverse η_{p}.

Lemma 1.1. For a finite group G and a compact Lie group K, the pcompletion functor sends f ∈ {BG, BK} to η_{p} ∘ f ∘ ι_{p} ∈ {BG_{p}^{∧}, BK_{p}^{∧}}.

Proof. By naturality of η_{p} we have f_{p}^{∧}∘η_{p} = η_{p}∘f, from which it follows that f_{p}^{∧} ≃ f_{p}^{∧} ∘ η_{p} ∘ ι_{p} = η_{p} ∘ f ∘ ι_{p}. □

Bousfield–Kan p-completion can also be thought of as Bousfield localization with respect to the homology theory H_{∗}(−, 𝔽_{p}). Bousfield localization is discussed in [5]. Taking this point of view, one of Bousfield’s results in [5] is that when X is a connective spectrum, the p-completion of X is homotopy equivalent to the function spectrum F (S^{−1}ℤ/p^{∞}, X), where S^{−1}ℤ/p^{∞} is the desuspension of the Moore spectrum Sℤ/p^{∞}, with Z/p^{∞} = ℤ[1/p]/ℤ.

Lemma 1.2. Let X and Y be spectra such that Y and F (X, Y) are both connective. Then there is a natural homotopy equivalence

Ad: F (X, Y_{p}^{∧}) ⟶^{≃} F (X, Y)_{p}^{∧}

Proof. The map is just the adjunction

F (X, F (S^{−1}ℤ/p^{∞}, Y)) ≅ F (S^{−1}ℤ/p^{∞}, F (X, Y)).
□

In this paper we study stable maps from the classifying space of a finite group to the classifying space of a compact Lie group. These exhibit very unusual behaviour under p-completion as seen in the following lemma.

Lemma 1.3. For a finite group G and a compact Lie group K the maps in the sequence below are all homotopy equivalences.

F (𝔹G_{p}^{∧}, 𝔹K) ⟶^{η_{p}∘−} F (𝔹G_{p}^{∧}, 𝔹K_{p}^{∧}) ⟶^{−∘η_{p}} F (𝔹G, 𝔹K_{p}^{∧}) ⟶^{Ad} F (𝔹G, 𝔹K)_{p}^{∧}

In particular they induce isomorphisms of groups of homotopy classes of maps.

Proof. We obtain the homotopy equivalence Ad because 𝔹K is connective by construction as a suspension spectrum, and F (𝔹G, 𝔹K) is connective by [8]. The map − ∘ η_{p} is a homotopy equivalence since the cofibre of η_{p} is H_{∗}(−, 𝔽_{p})-acyclic ([5]).

The map η_{p} ∘ − factors as the composition

F (𝔹G_{p}^{∧}, 𝔹K) ⟶^{η_{p}} F (𝔹G_{p}^{∧}, 𝔹K)_{p}^{∧} ⟶^{Ad} F (𝔹G_{p}^{∧}, 𝔹K_{p}^{∧})

Let S be a Sylow subgroup of G. As explained in the next subsection, the spectrum F (𝔹G_{p}^{∧}, 𝔹K) is a retract of the spectrum F (𝔹S, 𝔹K), which is p-complete and connective by [8]. Hence F (𝔹G_{p}^{∧}, 𝔹K) is p-complete and connective, and so the two maps in the factorization are both homotopy equivalences. □

When working with “pointed” classifying spectra one similarly has a sequence of homotopy equivalences

F ((𝔹_{+}G)_{p}^{∧}, (𝔹_{+}K)_{p}^{∧}) ⟶^{−∘η_{p}} F (𝔹_{+}G, (𝔹_{+}K)_{p}^{∧}) ⟶^{Ad} F (𝔹_{+}G, 𝔹_{+}K)_{p}^{∧}.

The function spectrum F ((𝔹_{+}G)_{p}^{∧}, 𝔹_{+}K) is no longer homotopy equivalent because it contains an uncompleted wedge summand F (𝕊_{p}^{∧}, 𝕊) (but p-completing this summand makes it homotopy equivalent).

1.2. Transfers. Recall (see for example [1]) that a subgroup inclusion H ↪ G, has a transfer map

tr_{H}: 𝔹_{+}G ⟶ 𝔹_{+}H

such that the composition 𝔹_{+}ι_{H} ∘ tr_{H} acts as multiplication by [G : H] in singular cohomology (with any coefficients). We will for the most part prefer to work with the reduced transfer map 𝔹G → 𝔹H, which we also denote by tr_{H} to reduce notation. The composition 𝔹ι ∘ tr_{H} involving the reduced transfer also acts as multiplication by [G : H] in singular cohomology.

When S is a Sylow subgroup of G, the index [G : S] is a unit in 𝔽_{p} and therefore 𝔹ι_{S} ∘ tr_{S} induces an isomorphism on H^{∗}(𝔹G) = H^{∗}(BG). After p-completion, 𝔹ι_{S} ∘ tr_{S} consequently becomes a homotopy equivalence

𝔹G_{p}^{∧} ⟶^{tr_{p}^{∧}} 𝔹S ⟶^{𝔹ι_{p}^{∧}} 𝔹G_{p}^{∧}

and so the p-completions of 𝔹ι_{S} and tr_{S} make 𝔹G_{p}^{∧} a wedge summand of 𝔹S_{p}^{∧} ≃ 𝔹S.

When H is a subgroup of G, we will often denote the p-completed transfer (tr_{H})_{p}^{∧} by tr_{H}. Similarly, when φ: H → G is a group homomorphism, we will often denote the p-completed map (𝔹φ)_{p}^{∧} by 𝔹φ. This is done to reduce notation. As we always distinguish clearly between 𝔹G_{p}^{∧} and 𝔹G (except when S is a p-group and the spectra are homotopy equivalent anyway), this should not cause any confusion.

1.3. Linear Categories. For a ring R, an R-linear category is a category in which the morphism sets are R-modules and composition is bilinear. It is perhaps more common to refer to ℤ-linear categories as pre-additive categories, but we use different terminology to emphasize the role of the ground ring R. We say that a functor F: C → D is a functor of R-linear categories if the categories C and D are R-linear and for every pair of objects c, c′ ∈ C the map

F: Mor_{C} (c, c′) → Mor_{D} (F (c), F (c′))

is a morphism of R-modules. An isomorphism of R-linear categories is a functor of R-linear categories that is also an isomorphism of categories.

Example 1.4. If R is a ring, then we can regard R as an R-linear category with one object ∘_{R} such that

Mor (∘_{R}, ∘_{R}) = R.

We will sometimes take this point of view for the rings ℤ and ℤ_{p}^{∧}.

1.4. The Burnside category. For a finite group G and a compact Lie group K, let Mor (G, K) be the set of isomorphism classes of principal (G, K)-bundles over finite G-sets. A (G, K)-bundle here means a Kbundle in the category of G-spaces, and principality means that the K-action is free. When K is finite these are just K-free, finite (K ×G)- sets. The set Mor (G, K) is an abelian monoid under the disjoint union operation and so we can consider its Grothendieck group completion which we call the Burnside module of G and K, and denote by A(G, K). The Burnside modules are abelian groups, and hence ℤ-modules, by construction. Their module structure is well understood and will be described below after a preliminary definition.

Definition 1.5. Let G be a finite group and K a compact Lie group. A (G, K)-pair is a pair (H, φ) consisting of a subgroup H ∈ G and a homomorphism φ: H → K. We say two (G, K)-pairs (H, φ) and (H′, φ′) are (G, K)-conjugate if there exist elements g ∈ G and k ∈ K such that c_{g}(H) = H′ and the following diagram commutes:

H ⟶^{φ} K
≅↓^{c_{g}}  ↓^{c_{k}}
H′ ⟶^{φ′} K.

Let C(G, K) denote the set of conjugacy classes of (G, K)-pairs.

When there is no danger of confusion we say conjugacy instead of (G, K)-conjugacy. We denote the conjugacy class of a (G, K)-pair (H, φ) by [H, φ]_{G}^{K}, or just [H, φ] when there is no danger of confusion.

Lemma 1.6. For a finite group G and a compact Lie group K, the number of conjugacy classes of (G, K)-pairs is finite.

Proof. One proves this easily using the folklore result (see for example [9, Theorem 2.3 (1)] for a proof) that for any finite group F there are only finitely many conjugacy classes of subgroups of K isomorphic to F. □

A (G, K)-pair (H, φ) gives rise to a principal (G, K)-bundle

(K × G)/Δ_{H}^{φ} → G/H,

where

Δ_{H}^{φ} := {(φ(h), h) | h ∈ H} ≤ K × G  (1)

is the (transposed) graph of φ: H → K. Two (G, K)-pairs give rise to isomorphic (G, K)-bundles if and only if they are conjugate.

The ring U(K × G) of finite (K × G)-complexes is introduced in [14] where a basis over ℤ is also described. One can identify A(G, K) with the submodule of U(G × K) generated by those (K × G)-complexes whose isotropy groups are of the form Δ_{H}^{φ}. This leads to the following description of A(G, K).

Proposition 1.7. [14] For a finite group G and a compact Lie group K, the Burnside module A(G, K) is a finitely generated, free ℤ-module with one basis element for each conjugacy class of (G, K)-pairs.

Proof. The ring U(K × G) is free as a ℤ-module with one basis element for every conjugacy class of subgroups of K × G. The Burnside module A(G, K) can be identified with the submodule generated by the subgroups of the form Δ_{H}^{φ} where (H, φ) is a (G, K)-pair. □

By a slight abuse of notation we also denote the basis element corresponding to a (G, K)-pair (H, φ) by [H, φ]_{G}^{K} (or [H, φ]). For each basis element [H, φ] we define a morphism

c_{[H,φ]} : A(G, K) ⟶ ℤ

by letting c_{[H,φ]}(X) be the coefficient at [H, φ] in the basis decomposition of X. In other words we demand that

X = ∑_{[H,φ]} c_{[H,φ]}(X) · [H, φ]

for all X ∈ A(G, K). When appropriate, we will also let c_{[H,φ]} denote the analogous homomorphism A(G, K)_{p}^{∧} → ℤ_{p}^{∧}.

The Burnside modules form the morphism sets of a certain linear category as described below. For finite groups G_{1} and G_{2}, and a compact Lie group K, there is a pairing

Mor (G_{2}, K) × Mor (G_{2}, G_{1}) ⟶ Mor (G_{1}, K),

given by

(X, Y) ⟼ G_{2}\(X × Y),

where G_{2} acts by the diagonal. This extends to a bilinear pairing

A(G_{2}, K) × A(G_{1}, G_{2}) ⟶ A(G_{1}, K),

suggestively denoted by

(X, Y) ⟼ X ∘ Y.

This pairing can be described on basis elements by the double coset formula

[H_{2}, φ_{2}] ∘ [H_{1}, φ_{1}] = ∑_{x∈H_{2}\G_{2}/φ_{1}(H_{1})} [φ_{1}^{−1} (φ_{1} (H_{1}) ∩ H_{2}^{x}) , φ_{2} ∘ c_{x} ∘ φ_{1}] .

It is easy to see that this composition pairing satisfies the associativity law and therefore we can make the following definition.

Definition 1.8. The Burnside category is the ℤ-linear category AGr whose objects are the finite groups and whose morphism sets are the Burnside modules,

Mor_{AGr} (G_{1}, G_{2}) = A(G_{1}, G_{2}),

where composition is given by the pairing described above.

Similarly, the p-completed Burnside category is the ℤ_{p}^{∧}-linear category AGr_{p}^{∧} whose objects are the finite groups and whose morphism sets are the p-completed Burnside modules,

Mor_{AGr}_{p}^{∧} (G_{1}, G_{2}) = A(G_{1}, G_{2})_{p}^{∧}.

Since we are focusing on p-local properties it will suit us, for a finite group G and a compact Lie group K, to study the submodule A_{p}(G, K) of the Burnside module A(G, K) obtained by considering only (G, K)- bundles whose isotropy groups are all p-groups. Alternatively, this is the submodule generated by those basis elements [P, φ], where P is a p-group. By the double coset formula, we see that these modules are preserved by the composition pairing as described in the following lemma.

Lemma 1.9. For finite groups G_{1} and G_{2}, and a compact Lie group K, we have

A(G_{2}, K) ∘ A_{p}(G_{1}, G_{2}) ⊂ A_{p}(G_{1}, K).

In particular

A_{p}(G_{2}, K) ∘ A_{p}(G_{1}, G_{2}) ⊂ A_{p}(G_{1}, K).

Therefore we can make the following definition.

Definition 1.10. The p-isotropy Burnside category is the ℤ-linear category A_{p}Gr whose objects are the finite groups and whose morphism sets are the p-Burnside modules,

Mor_{A_{p}Gr} (G_{1}, G_{2}) = A_{p}(G_{1}, G_{2}).

Similarly, the p-completed p-Burnside category is the ℤ_{p}^{∧}-linear category A_{p}Gr_{p}^{∧} whose objects are the finite groups and whose morphism sets are the p-completed p-Burnside modules,

Mor_{A_{p}Gr}_{p}^{∧} (G_{1}, G_{2}) = A_{p}(G_{1}, G_{2})_{p}^{∧}.

We conclude this discussion by introducing an augmentation for the Burnside category. This will prove very useful for understanding the filtration of Burnside modules introduced in Section 2.

Definition 1.11. For a finite group G and a compact Lie group K, the orbit augmentation of A(G, K) is the homomorphism

ϵ: A(G, K) ⟶ ℤ

obtained as the group completion of the monoid morphism

Mor (G, K) ⟶ ℤ, X ⟼ |K\X|.

The orbit augmentation of A(G, K)_{p}^{∧} is the homomorphism

ϵ: A(G, K)_{p}^{∧} → ℤ_{p}^{∧}

obtained upon p-completion.

When K is finite we have ϵ(X) = |X|/|K|, since X is K-free. Note also that

ϵ([H, φ]) = |G|/|H|

for a (G, K)-pair (H, φ). One easily checks that the orbit aumentations send the composition pairing to multiplication, and consequently we obtain a ℤ-linear functor

ϵ: AGr ⟶ ℤ

and a ℤ_{p}^{∧}-linear functor

ϵ: AGr_{p}^{∧} ⟶ ℤ_{p}^{∧}.

We refer to these as the orbit augmentation functors of the respective Burnside categories.

1.5. The Segal conjecture. For a finite group G and a compact Lie group K, there is a homomorphism of ℤ-modules

α: A(G, K) ⟶ {BG_{+}, BK_{+}}

sending a basis element [H, φ] to the stable map 𝔹_{+}φ ∘ tr_{H}. This assignment is functorial for G and finite K, and so we get a functor of ℤ-linear categories

α: AGr ⟶ 𝔹Gr,

which is the identity on objects.

For a finite group G and a compact Lie group K, we have an obvious bilinear map

A(G, 1) × A(G, K) ⟶ A(G, K),

where 1 denotes the trivial group, extending the map sending a G-set X and a (G, K)-bundle Y to the (G, K)-bundle X × Y, where G acts via the diagonal and K acts only on the second coordinate.

When K = 1, this gives a ring structure on A(G) = A(G, 1), and the resulting ring is called the Burnside ring of G. This can also be described as the Grothendieck group of isomorphism classes of left Gsets. For general compact Lie groups K, the Burnside module A(G, K) becomes a module over A(G) under the action described above.

The orbit augmentation functor ϵ induces a ℤ-algebra augmentation

ϵ: A(G) ⟶ ℤ,

which extends the “counting map” sending a G-set X to its cardinality |X|. We denote the augmentation ideal by I(G).

The Segal conjecture states that for finite group G, the homomorphism A(G) → π_{S}^{0}(BG_{+}), which we describe as the composite

A(G) = A(G, 1) ⟶^{α} {BG_{+}, B1_{+}} ≅ π_{S}^{0}(BG_{+}),

is a completion with respect to the ideal I(G). Lewis-May-McClure showed in [8] that an extended version of the conjecture, describing the homotopy classes of stable maps between classifying spaces of finite groups, follows from the original version. In a further generalization, May-Snaith-Zelewski showed in [11] that one can allow the target group to be a compact Lie group. Since the Segal conjecture was proved by Carlsson in [7], we can state these extensions as a theorem.

Theorem 1.12 (Segal Conjecture [7, 8, 11]). For a finite groups G and a compact Lie group K, the map

α: A(G, K) ⟶ {BG_{+}, BK_{+}}

induces an isomorphism

α_{I(G)}^{∧}: A(G, K)_{I(G)}^{∧} ⟶^{≅} {BG_{+}, BK_{+}},

where

A(G, K)_{I(G)}^{∧} = lim_{\overline{}}_{k} (A(G, K)/I(G)^{k}A(G, K))

denotes the I(G)-completion of A(G, K).

This is a magnificent result, but in general the I(G)-adic completions are difficult to calculate. However, when the groups involved are pgroups, the situation is simplified. For a finite group G and a compact Lie group K, let Ã(G, K) be the module obtained from A(G, K) by quotienting out all basis elements of the form [H, ct], where ct is the constant homomorphism. May-McClure noticed in [10] that when G is a p-group, the I(G)-adic topology on Ã(G, K) is equivalent to the p-adic topology on Ã(G, K). (This is not true on A(G, K).) The result is the following version of the Segal conjecture.

Theorem 1.13 (Segal Conjecture [7, 8, 10, 11]). Let S be a finite p-group and K be a compact Lie group. Then the map

α: A(S, K) ⟶ {BS_{+}, BK_{+}}

induces an isomorphism

α̃_{p}^{∧}: Ã(S, K)_{p}^{∧} ⟶ {BS, BK}.

An explanation of the target of this isomorphism is in order. Recall that Σ_{+}^{∞}BK ≃ Σ^{∞}BK ∨ 𝕊. Now, the submodule of {BS_{+}, BK_{+}} generated by the maps α([P, ct]), where ct is the constant homomorphism P → K, consists of the maps that factor through the 𝕊-term of Σ_{+}^{∞}BK. Therefore it is appropriate to replace {BS_{+}, BK_{+}} with

{BS, BK} ≅ {BS_{+}, BK_{+}}/{BS_{+}, S^{0}}

when passing from A(S, K) to Ã(S, K) to obtain the homomorphism α̃: Ã(S, K) → {BS, BK}. The homomorphism α̃_{p}^{∧} is the p-completion of α̃.

Since α and α̃ preserve composition when restricted to finite groups, both versions of the Segal conjecture described in this section can be formulated in terms of isomorphisms of linear categories. However, we refrain from doing so.

1.6. Restrictions and transfers. Let G be a finite group and K be a compact Lie group. If S is a subgroup of G, we can regard a (S, K)- pair as a (G, K)-pair, or restrict a (K ×G)-bundle to a (K ×S)-bundle. This gives us a way to traverse between A(S, K) and A(G, K), which will be useful later when we take S to be a Sylow subgroup. In this section we briefly describe these operations and interpret them in terms of stable maps.

Given a (S, K)-pair (H, φ), we can regard H as a subgroup of G to get a (G, K)-pair (H, φ)_{G}^{K}. This assignment extends to a homomorphism

Φ: A(S, K) ⟶ A(G, K).

It is easy to check that

[H, φ]_{G}^{K} = [H, φ]_{S}^{K} ∘ [S, id]_{G}^{S}

using the double coset formula. Furthermore, [S, id]_{G}^{S} corresponds to [G]_{G}^{S}, the isomorphism class of G, regarded as a (S × G)-set under the action (s, g).x = gxs^{−1}.

The homomorphism α sends [S, id]_{G}^{S} to the transfer map tr_{S}: 𝔹_{+}G → 𝔹_{+}S. Thus we get an induced homomorphism

Φ: {BS_{+}, BK_{+}} → {BG_{+}, BK_{+}}, f ↦ f ∘ tr_{S}.

Note that α commutes with Φ by construction.

A (G, K)-bundle X, can be regarded as a (S, K)-bundle X|_{(S,K)} via restriction. This induces a homomorphism

Γ: A(G, K) ⟶ A(S, K).

It is easy to see that

[X|_{(S,K)}] ≅ [X] ∘ [G]_{S}^{G},

where [X] denotes the isomorphism class of X, and [G]_{S}^{G} is the isomorphism class of G regarded as a (G × S)-set via the action (g, s).x = gxs^{−1}. Since [G]_{S}^{G} corresponds to [S, ι]_{S}^{G} in A(S, G) we see that Γ is given by

[X] ↦ [X] ∘ [S, ι]_{S}^{G}.

Applying α we get an induced homomorphism of stable maps

Γ: {BG_{+}, BK_{+}} → {BS_{+}, BK_{+}}, f ↦ f ∘ 𝔹_{+}ι_{S}.

Note that α also commutes with Γ by construction.

To accommodate the basepoint issue we need to consider the reduced operations Φ̃ and Γ̃ that fit into the diagrams

\xymatrix{A(S, K) \ar[r]^{Φ} \ar@{->>}[d]^{π} & A(G, K) \ar[r]^{Γ} \ar@{->>}[d]^{π} & A(S, K) \ar@{->>}[d]^{π} \\ Ã(S, K) \ar[r]^{Φ̃} & Ã(G, K) \ar[r]^{Γ̃} & Ã(S, K)}

and

\xymatrix{\{BS_{+}, BK_{+}\} \ar[r]^{Φ} \ar@{->>}[d]^{π} & \{BG_{+}, BK_{+}\} \ar[r]^{Γ} \ar@{->>}[d]^{π} & \{BS_{+}, BK_{+}\} \ar@{->>}[d]^{π} \\ \{BS, BK\} \ar[r]^{Φ̃} & \{BG, BK\} \ar[r]^{Γ̃} & \{BS, BK\},}

where the vertical arrows are the obvious projection maps. The reduced operations Φ̃ and Γ̃ commute with α̃.

Since we are working p-locally we need to consider the p-completed operations

Φ̃_{p}^{∧}: {BS_{p}^{∧}, BK} ⟶ {BG_{p}^{∧}, BK}, f ↦ f ∘ (tr_{S})_{p}^{∧}

and

Γ̃_{p}^{∧}: {BG_{p}^{∧}, BK} ⟶ {BS_{p}^{∧}, BK}, f ↦ f ∘ (𝔹ι_{S})_{p}^{∧}.

These fit into the commutative diagram

\xymatrix{Ã(S, K) \ar[r]^{Φ̃} \ar[d]^{ι_{p}^{∗}∘α̃} & Ã(G, K) \ar[r]^{Γ̃} \ar[d]^{ι_{p}^{∗}∘α̃} & Ã(S, K) \ar[d]^{ι_{p}^{∗}∘α̃} \\ \{BS_{p}^{∧}, BK\} \ar[r]^{Φ̃_{p}^{∧}} & \{BG_{p}^{∧}, BK\} \ar[r]^{Γ̃_{p}^{∧}} & \{BS_{p}^{∧}, BK\},}

where the vertical maps are the functor α̃ followed by precomposition with ι_{p}, the natural inclusion of wedge summands described in Section 1.1. The naturality of ι_{p} ensures the commutativity of the diagram and in fact this is the reason why ι_{p} was introduced so carefully.

Notice that

[G_{G}^{S}] ∘ [G_{S}^{G}] ≅ [G_{S}^{S}],

where [(G)_{S}^{S}] denotes the isomorphism class of G regarded as a (S ×S)- set via the action (s_{1}, s_{2}).x = s_{1}xs_{2}^{−1}. Therefore the composite

Γ ∘ Φ: A(S, K) ⟶ A(S, K).

is given by

Γ ∘ Φ(X) = X ∘ [G]_{S}^{S}.

This operation will be studied in Section 2.

We also consider the opposite composite

Φ ∘ Γ: A(G, K) ⟶ A(G, K).

Its corresponding operation on stable maps is given by

Φ ∘ Γ(f) = f ∘ 𝔹_{+}ι_{S} ∘ tr_{S}.

The interesting case is when S is a Sylow p-subgroup of G. Then 𝔹S ≃ 𝔹S_{p}^{∧} and the composite 𝔹ι_{S} ∘ tr_{S} is a homotopy equivalence after p-completion. Therefore Φ̃_{p}^{∧} ∘ Γ̃_{p}^{∧} is an automorphism of {BG_{p}^{∧}, (BK)} which factors through {BS, BK}. We conclude that {BG_{p}^{∧}, BK} is a direct summand of {BS, BK} isomorphic to

Γ̃_{p}^{∧} ∘ Φ̃_{p}^{∧} ({BS, BK}) = α̃_{p}^{∧} (π (A(S, K)_{p}^{∧} ∘ [G]_{S}^{S})) .

This observation gives us a strategy to prove Theorem A. Over the next two sections we show that A(S, K)_{p}^{∧} ∘ [G]_{S}^{S} is isomorphic to A_{p}(G, K)_{p}^{∧} in a way that is compatible with α̃_{p}.

2. Subconjugacy

From now on, fix a finite group G with Sylow p-subgroup S and a compact Lie group K. In this section we find an explicit basis for the submodule

A(S, K)_{p}^{∧} ∘ [G] ⊆ A(S, K)_{p}^{∧},

where [G] ∈ A(S, S)_{p}^{∧} is the isomorphism class of G regarded as a (S × S)-set under the action (s_{1}, s_{2}).x = s_{1}xs_{2}^{−1}. From this basis it is easy to obtain a basis for {BG_{p}^{∧}, BK} and prove Theorem A.

The main tool we will use is a filtration induced by the following preorder on conjugacy classes of (S, K)-pairs.

Definition 2.1. Let (P, φ) and (Q, ψ) be two (S, K)-pairs. We say that (Q, ψ) is subconjugate to (P, φ), and write

(Q, ψ) ≾ (P, φ),

if there exist elements g ∈ G and k ∈ K such that the following diagram commutes

Q ⟶^{ψ} ψ(Q)
c_{g}↓  ↓^{c_{k}}
P ⟶^{φ} φ(P).

It is clear that subconjugacy is preserved by (S, K)-conjugacy and therefore we can pass to conjugacy classes.

Definition 2.2. We say that a (S, K)-conjugacy class of pairs [Q, ψ] is subconjugate to an (S, K)-conjugacy class [P, φ] and write

[Q, ψ] ≾ [P, φ]

if the subconjugacy relation

(Q, ψ) ≾ (P, φ)

holds between any (and hence all) representatives of the classes.

It is also clear that subconjugacy is a transitive relation and so induces an equivalence relation as described in the following definition.

Definition 2.3. We say that [Q, ψ] is (G, K)-conjugate to [P, φ], and write

[Q, ψ] ∼ [P, φ],

if [Q, ψ] ≾ [P, φ] and [P, φ] ≾ [Q, ψ]. We say that [Q, ψ] is strictly subconjugate to [P, φ], and write

[Q, ψ] ⋨ [P, φ],

if [Q, ψ] is subconjugate to [P, φ], but not (G, K)-conjugate to [P, φ].

Let I denote the set of (G, K)-conjugacy classes of (S, K)-pairs. This is a poset under subconjugacy and we proceed to construct an I-indexed filtration of A(S, K)_{p}^{∧}.

Definition 2.4. For an (S, K)-pair (P, φ), let M (≾ [P, φ]) denote the submodule of A(S, K)_{p}^{∧} generated by the basis elements [Q, ψ] such that

[Q, ψ] ≾ (P, φ),

and let M (⋨ [P, φ]) denote the submodule generated by the basis elements [Q, ψ] such that

[Q, ψ] ⋨ (P, φ).

We will show that this filtration is preserved by composition with [G]. First note that by the double coset formula one has

[G] = ∑_{x∈S\G/S}[S ∩ S^{x}, c_{x}].

This prompts us to make a formalization.

Definition 2.5. Let R be the submodule of A(S, S)_{p}^{∧} generated by the basis elements [P, φ], where the homomorphism φ: P → S is induced by conjugation by an element in G.

Clearly [G] ∈ R. Using the double coset formula, one easily sees that R is in fact a subring of A(S, S)_{p}^{∧}. The next lemma shows that the modules M (≾ [P, φ]) and M (⋨ [P, φ]) defined above are right R-modules. They are in fact also left R-modules but this is not needed.

Lemma 2.6. The following hold for every (S, K)-pair (P, φ): (a) M (≾ [P, φ]) ∘ R ⊆ M (≾ [P, φ]) ,

(b) M (⋨ [P, φ]) ∘ R ⊆ M (⋨ [P, φ]) .

Proof. We begin by proving part (a). It suffices to show that for every basis element [Q, ψ] ≾ [P, φ] of M (≾ [P, φ]) and every basis element [T, c_{g}] of R, one has

[Q, ψ] ∘ [T, c_{g}] ∈ M (≾ [P, φ]) .

By the double coset formula we can write

[Q, ψ] ∘ [T, c_{g}] = ∑_{x∈Q\S/^{g}T}[T ∩ Q^{xg}, ψ ∘ c_{x} ∘ c_{g}]

so it suffices to note that

[T ∩ Q^{xg}, ψ ∘ c_{x} ∘ c_{g}] ≾ [Q, ψ] ≾ [P, φ]

for every x ∈ S.

The same argument proves part (b), for if we assume that [Q, ψ] is strictly subconjugate to [P, φ] at the beginning, we get

[T ∩ Q^{xg}, ψ ∘ c_{x} ∘ c_{g}] ≾ [Q, ψ] ⋨ [P, φ]

at the end. □

In particular this lemma shows that the operation

Γ ∘ Φ = (−) ∘ [G] : A(S, K)_{p}^{∧} ⟶ A(S, K)_{p}^{∧}

introduced in Section 1.6 preserves the subconjugacy filtration.

Lemma 2.7. If the basis elements [P, φ] and [Q, ψ] of A(S, K) are (G, K)-conjugate, then

Γ ∘ Φ([P, φ]) = Γ ∘ Φ([Q, ψ]).

Proof. Clearly (P, φ) and (Q, ψ) are also (G, K)-conjugate when regarded as (G, K)-pairs, so, using the notation of Section 1.6,

Φ ([P, φ]) = Φ ([Q, ψ]) .

Consequently

Γ (Φ ([P, φ])) = Γ (Φ ([Q, ψ])) .
□

Proposition 2.8. Pick a representative [P_{i}, φ_{i}] for each i ∈ I. The collection

C = {Γ ∘ Φ([P_{i}, φ_{i}]) | i ∈ I}

forms a ℤ_{p}^{∧}-basis for Γ ∘ Φ(A(S, K)_{p}^{∧}).

Proof. By Lemma 2.7 it is clear that C spans Γ ∘ Φ(A(S, K)_{p}^{∧}), so it suffices to prove linear independence.

For a (S, K)-pair (P, φ) we have

Γ ∘ Φ([P, φ]) = [P, φ] ∘ [G] ∈ M (≾ [P, φ])

by Lemma 2.6. But note that

ϵ(M (≾ [P, φ])) = ϵ([P, φ]) · ℤ_{p}^{∧}

and

ϵ(M (⋨ [P, φ])) = p · ϵ([P, φ]) · ℤ_{p}^{∧}.

Since

ϵ(Γ ∘ Φ([P, φ])) = ϵ([P, φ] ∘ [G]) = ϵ([P, φ]) · ϵ([G])

and p does not divide ϵ([G]) = |G|/|S|, we deduce that

Γ ∘ Φ([P, φ]) ∈ M (≾ [P, φ]) \ M (⋨ [P, φ]) .  (2)

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

∑_{i∈I} c_{i} · (Γ ∘ Φ([P_{i}, φ_{i}])) = 0.  (3)

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 subconjugacy. By (2) there is a (S, K)-pair

(Q, ψ) ∼_{(G,K)}(P_{j}, φ_{j}) such that

c_{[Q,ψ]} (Γ ∘ Φ([P_{j}, φ_{j}])) ≠ 0.

On the other hand, for i ∈ I′ \ {j}, the maximality of j implies that [Q, ψ] is not subconjugate to (P_{i}, φ_{i}). Hence

c_{[Q,ψ]} (M (≾ [P_{i}, φ_{i}])) = 0

and in particular

c_{[Q,ψ]} (Γ ∘ Φ([P_{i}, φ_{i}])) = 0.

Now we get

c_{[Q,ψ]} (∑_{i∈I} c_{i} · (Γ ∘ Φ([P_{i}, φ_{i}]))) = ∑_{i∈I} c_{i} · c_{[Q,ψ]} (Γ ∘ Φ([P_{i}, φ_{i}]))

= ∑_{i∈I\I′} \underbrace{c_{i}}_{=0} ·c_{[Q,ψ]} (Γ ∘ Φ([P_{i}, φ_{i}]))

+ ∑_{i∈I′\\{j\}} c_{i} · \underbrace{c_{[Q,ψ]} (Γ ∘ Φ([P_{i}, φ_{i}]))}_{=0}

+ \underbrace{c_{j}}_{≠0} · \underbrace{c_{[Q,ψ]} (Γ ∘ Φ([P_{i}, φ_{i}]))}_{≠0}

≠ 0,

contradicting (3). Therefore I′ must be empty and we conclude that the collection is linearly independent.

□

3. p-completed Segal conjectures

In this section we state and prove the new variants of the Segal conjecture promised in the introduction. We begin by describing the group of homotopy classes of stable maps from the p-completed classifying space of a finite group to the classifying space of a compact Lie group. The map (ι_{p}^{∗} ∘ α̃_{p})_{p}^{∧} in the statement of the theorem is the algebraic p-completion of the composite

Ã(G, K) \underrightarrow{α̃_{p}} \{BG, BK\} ⟶^{−∘ι_{p}} \{BG_{p}^{∧}, BK\},

where ι_{p}: 𝔹G_{p}^{∧} ↪ 𝔹G is the natural wedge-summand inclusion discussed in Section 1.1. Note that the target {BG_{p}^{∧}, BK} is p-complete because it is a direct summand of {BS, BK} which is p-complete by the Segal Conjecture.

Theorem 3.1. For a finite group G and a compact Lie group K, the map

(ι_{p}^{∗} ∘ α̃_{p})_{p}^{∧}: Ã_{p}(G, K)_{p}^{∧} ⟶ {BG_{p}^{∧}, BK}

is an isomorphism of ℤ_{p}^{∧}-modules, natural in G and finite K.

Proof. Let S be a Sylow subgroup of G, and let I be the set of (G, K)- conjugacy classes of (S, K)-pairs. For each i ∈ I, pick a representative (P_{i}, φ_{i}) and consider the submodule M_{I} ⊆ A(S, K)_{p}^{∧} generated by the collection {[P_{i}, φ_{i}] | i ∈ I}. Since every p-subgroup of G is conjugate to a subgroup of S, the homomorphism Φ: A(S, K) → A(G, K) described in Section 1.6 induces an isomorphism

Φ_{p}^{∧} : M_{I} ⟶^{≅} A_{p}(G, K)_{p}^{∧}.

Letting M̃_{I} denote the image of M_{I} in Ã(S, K)_{p}^{∧}, this descends to an isomorphism

Φ̃_{p}^{∧} : M̃_{I} ⟶^{≅} Ã_{p}(G, K)_{p}^{∧}.

Proposition 2.8 implies that there is an isomorphism

Γ̃_{p}^{∧} ∘ Φ̃_{p}^{∧}: M̃_{I} ⟶^{≅} Γ̃_{p}^{∧} ∘ Φ̃_{p}^{∧}(M̃_{I})

and that

Γ̃_{p}^{∧} ∘ Φ̃_{p}^{∧}(M̃_{I}) = Γ̃_{p}^{∧} ∘ Φ̃_{p}^{∧}(Ã(S, K)_{p}^{∧}).

By the Segal conjecture, and the commutativity of α with Γ̃_{p}^{∧} and Φ̃_{p}^{∧}, there is an isomorphism

α̃_{p}^{∧}: Γ̃_{p}^{∧} ∘ Φ̃_{p}^{∧}(Ã(S, K)_{p}^{∧}) ⟶^{≅} Γ̃_{p}^{∧} ∘ Φ̃ ({BS, BK})

Since the composition 𝔹ι_{S} ∘ tr_{S} is a homotopy self-equivalence of 𝔹G_{p}^{∧}, the homomorphisms Φ̃_{p}^{∧} and Γ̃_{p}^{∧} induce an automorphism

Φ̃_{p}^{∧} ∘ Γ̃_{p}^{∧}: {BG_{p}^{∧}, BK} ⟶^{≅} {BG_{p}^{∧}, BK}.

We conclude that Φ̃_{p}^{∧} is a surjection, so

{BG_{p}^{∧}, BK} = Φ̃_{p}^{∧} ({BS, BK}) ,

and that Γ̃_{p}^{∧} is an injection. Hence there is an isomorphism

Γ̃_{p}^{∧}: {BG_{p}^{∧}, BK} ⟶^{≅} Γ̃_{p}^{∧} ∘ Φ̃_{p}^{∧} ({BS, BK}) .

Now we have a commutative diagram

\xymatrix{M̃_{I} \ar[r]^{Γ∘Φ}_{≅} \ar[dr]^{Φ}_{≅} & Γ ∘ Φ(M̃_{I}) \ar[r]^{=} & Γ ∘ Φ(Ã(S, K)_{p}^{∧}) \ar[r]^{α_{p}^{∧}}_{≅} & Γ ∘ Φ(\{BS, BK\}) \\  & _{Ã_{p}(G, K){∧ \atop p}} \ar[rr]^{_{(ι_{p}^{∗}∘α̃_{p}){∧ \atop p}}} \ar[u]^{Γ} &  & \{BG_{p}^{∧}, BK\}, \ar[u]^{Γ}_{≅}}

where we have written Φ and Γ instead of Φ̃_{p}^{∧} and Γ̃_{p}^{∧}. Concentrating first on the left side of this diagram, we see that since Φ and Γ ∘ Φ are both isomorphisms, Γ must be an isomorphism. Turning our attention to the right side of the diagram, we see that the bottom arrow forms the bottom part of a rectangular commutative diagram where all the other arrows are isomorphism. Hence the bottom arrow is itself an isomorphism. □

Using Lemma 1.6 we obtain an immediate reformulation.

Corollary 3.2. For a finite group G and a compact Lie group K, the map

(α̃_{p})_{p}^{∧}: Ã_{p}(G, K)_{p}^{∧} ⟶ {BG, BK}_{p}^{∧} ≃ {(BG)_{p}^{∧}, (BK)_{p}^{∧}},

is an isomorphism of ℤ_{p}^{∧}-modules, natural in G and finite K.

The naturality property in Corollary 3.2 can also be stated as follows.

Corollary 3.3. The functor

(α̃_{p})_{p}^{∧}: A_{p}Gr_{p}^{∧} → 𝔹Gr_{p}^{∧}

is an isomorphism of ℤ_{p}^{∧}-linear categories.

Finally we can collect our results for different primes p and prove Theorem B of the introduction.

Theorem 3.4. For a finite group G and a compact Lie group K, the map

⨁_{q} (ι_{q}^{∗} ∘ α̃_{q})_{q}^{∧}: ⨁_{q} Ã_{q}(G, K)_{q}^{∧} → ⨁_{q}{BG_{q}^{∧}, BK} ≅ {BG, BK},

where the sums run over all primes q, is an isomorphism of ℤ-modules, natural in G and finite K.

Proof. The stable splitting 𝔹G ≃ ⋁_{q} 𝔹G_{q}^{∧} induces a splitting

{BG, BK} ≅ ⨁_{q}{BG_{q}^{∧}, BK}.

The result now follows from Theorem 3.1. □

One can also restrict this theorem to finite groups and formulate a corollary about an isomorphism of ℤ-linear categories, but we refrain from doing so.

4. Alternative formulations

The basepoint issue mentioned before in this paper is more a technical nuisance for formulating statements than an actual problem in proving them. The main issue with the added basepoint is that for a finite group G, the suspension spectrum of the p-completion of BG_{+} is Σ^{∞}((BG_{+})_{p}^{∧}) ≃ 𝔹G_{p}^{∧}∨𝕊, which is not p-complete as 𝕊 is not p-complete. Consequently, for a compact Lie group K, the module {(BG_{+})_{p}^{∧}, BK_{+}} need not be p-complete as maps factoring through the sphere spectrum can contribute a non-completed part.

In this paper we have so far opted to remove the offending sphere spectrum, at the cost of disregarding maps factoring through the sphere spectrum, which results in introducing the aesthetically unpleasant quotients of double Burnside modules. However, there are other ways to smooth over the basepoint issue. In this section we consider two alternative approaches. The first is to p-complete the added sphere spectrum rather than to remove it, and thus force everything in sight to be p-complete. The second approach is to deal separately with the failure of {(BG_{+})_{p}^{∧}, BK_{+}} to be p-complete, which turns out to be precisely one copy of ℤ corresponding to maps from one sphere spectrum to the other.

For each approach we give the corresponding versions of of Theorem 1.12 and state the generalizations to p-completed classifying spaces of finite groups. These generalizations are obtained using the same filtration argument already presented and so the proofs are omitted.

4.1. p-completed suspension spectra. Instead of looking at the suspension spectrum of the p-completion of BG_{+}, for a finite group G, we can look at the p-completion of the suspension spectrum of BG_{+}. Thus we obtain the p-complete spectrum (𝔹_{+}G)_{p}^{∧} ≃ 𝔹G_{p}^{∧} ∨ 𝕊_{p}^{∧}. Taking this approach for a finite p-group S and a compact Lie group K, the appropriate version of Theorem 1.12 is that the map

α_{p}^{∧}: A(S, K)_{p}^{∧} ⟶^{≅} [(𝔹_{+}S)_{p}^{∧}, (𝔹_{+}K)_{p}^{∧}]

is an isomorphism of ℤ_{p}^{∧}-modules. Using the same transfer and filtration arguments as in the proof of Theorem 3.1 we obtain the following generalization.

Theorem 4.1. For a finite group G and a compact Lie group K, the map

(α_{p})_{p}^{∧}: A_{p}(G, K)_{p}^{∧} ⟶^{≅} [(𝔹_{+}G)_{p}^{∧}, (𝔹_{+}K)_{p}^{∧}]

is an isomorphism of ℤ_{p}^{∧}-modules

While this approach has the advantage of retaining more information from the double Burnside module A_{p}(G, K), it has the drawback that the p-completed suspension spectra involved are rather contrived. Furthermore, we obtain no analogue of Theorem 3.4 as 𝔹_{+}G is not a wedge sum of its completions (since the sphere spectrum is not a torsion spectrum).

4.2. Precise approach. The statement of the Segal Conjecture for stable maps from classifying spaces of p-groups presented in Theorem 1.12 is actually a simplified version of a more detailed result obtained by May-McClure in [10].

For a finite group G with Sylow subgroup S and a compact Lie group K, let I_{p}(G, K) be the kernel of the map ϵ: A_{p}(G, K) → ℤ. As a ℤ-module, I_{p}(G, K) has basis {[P, φ] − |S|/|P | · [S, ct]}. There is a splitting A_{p}(G, K) = ℤ ⊕ I_{p}(G, K), where the ℤ term is generated by [S, ct]. Since I(G) · A(G, K) = I(G, K) this induces a splitting A_{p}(G, K)_{I(G)}^{∧} = ℤ ⊕ I_{p}(G, K)_{I(G)}^{∧}.

May-McClure showed that for a finite p-group S, the I(S)-adic topology on I(S, K) coincides with the p-adic topology. Therefore the Segal Conjecture says that the map

(α_{p})_{I(S)}^{∧}: ℤ ⊕ I(S, K)_{p}^{∧} ⟶ {BS_{+}, BK_{+}}

is an isomorphism. Again, using the transfer and filtration arguments from Sections 2 and 3, one can prove the following generalization.

Theorem 4.2. For a finite group G and a compact Lie group K, the map

(α_{p})_{I(G)}^{∧}: ℤ ⊕ I_{p}(G, K)_{p}^{∧} ⟶ {(BG_{+})_{p}^{∧}, BK_{+}}

is an isomorphism of ℤ-modules.

Collecting this result for different primes one can obtain an isomorphism of ℤ-modules.

ℤ ⊕ ⨁_{p} I_{p}(G, K)_{p}^{∧} ⟶^{≅} {BG_{+}, BK_{+}}.

Unlike the isomorphism in 3.4 this isomorophism is not natural in G.

Taking K = 1 one recovers a special case of Minami’s description of the I(G)-adic completion of the Burnside ring in [12] (he allows G to be compact). It should be noted that even though one can regard A(G, K) as a submodule of A(K × G) when K is finite, the results in this paper do not follow from Minami’s results, as his result involves the I(K × G)-adic completion whereas we are interested in the I(G)-adic completion, and the two actions are hard to reconcile.

5. Decompositions

In this section we address the following question. Given a (G, K)- pair (H, φ), what is the element in Ã_{p}(G, K)_{p}^{∧} corresponding to the p-completed stable map α̃([H, φ])? In other words, we describe the homomorphism

π̃_{p}: Ã(G, K) ⟶^{(−)_{p}^{∧}∘α̃} {(BG)_{p}^{∧}, (BK)_{p}^{∧}} ⟶^{((α̃_{p})_{p}^{∧})^{−1}} Ã_{p}(G, K)_{p}^{∧}.

For technical reasons it is more convenient to first describe the homomorphism

π_{p}: A(G, K) ⟶^{(−)_{p}^{∧}∘α} [(𝔹_{+}G)_{p}^{∧}, (𝔹_{+}K)_{p}^{∧}] ⟶^{((α_{p})_{p}^{∧})^{−1}} A_{p}(G, K)_{p}^{∧}.

and then interpret the results for homomorphism π̃ obtained by removing the additional sphere spectrum.

Let 1_{p} ∈ A_{p}(G, G)_{p}^{∧} be the pre-image of the identity of (𝔹_{+}G)_{p}^{∧} under the isomorphism

(α_{p})_{p}^{∧}: A_{p}(G, G)_{p}^{∧} ⟶^{≅} [(𝔹_{+}G)_{p}^{∧}, (𝔹_{+}G)_{p}^{∧}].

This immediately gives us a way to describe π_{p}, for if X ∈ A(G, K) then we can regard X ∘ 1_{p} as an element of A_{p}(G, K)_{p}^{∧}, and we have

(α_{p})_{p}^{∧}(X ∘ 1_{p}) = α(X)_{p}^{∧} ∘ (α_{p})_{p}^{∧}(1_{p}) = α(X)_{p}^{∧} ∘ id = α(X)_{p}^{∧}.

Therefore

π_{p}(X) = X ∘ 1_{p}.

We proceed to determine 1_{p} using filtration methods similar to those in Section 2. Since many details are similar, they are left to the reader.

Fix a Sylow subgroup S of G. Like in Section 2, we say that a (G, G)-pair (F, ψ) is subconjugate to a (G, G)-pair (H, φ), and write (F, ψ) ≾ (H, φ) if there exist elements g_{1}, g_{2} ∈ G making the following diagram commute

F ⟶^{ψ} ψ(F)
c_{g_{1}} ↓  ↓^{c_{g_{2}}}
H ⟶^{φ} φ(H).

This time the induced equivalence relation is just (G, G)-conjugacy, and so subconjugacy induces an order relation on (G, G)-conjugacy classes of (G, G)-pairs.

For a basis element [P, φ] of A_{p}(G, G), let M (≾ [P, φ]) denote the submodule of A_{p}(G, G) generated by basis elements that are subconjugate to [P, φ], and similarly let M (⋨ [P, φ]) denote the submodule generated by strictly subconjugate basis elements. We make the following key observation, which is proved just like Proposition 2.8.

Lemma 5.1. For every (G, G)-pair (P, φ), where P is a p-group,

[P, φ] ∘ [S, ι_{S}] ∈ M (≾ [P, φ]) \ M (⋨ [P, φ]) .

Let R denote the subring of A_{p}(G, G) generated by basis elements of the form [P, ι_{P}], where ι_{P} denotes the inclusion P ≤ G. Note that R = M (≾ [S, ι]). The above lemma allows us to prove the following.

Proposition 5.2. 1_{p} ∈ R.

Proof. Take a maximal [Q, ψ] such that c_{[Q,ψ]}(1_{p}) ≠ 0. By Lemma 5.1 we have

c_{[Q,ψ]}([Q, ψ] ∘ [S, ι_{S}]) ≠ 0

(otherwise [Q, ψ] ∘ [S, ι_{S}] ∈ M (⋨ [Q, ψ])).

If [P, φ] is a basis element different from [Q, ψ] such that c_{[P,φ]}(1_{p}) ≠ 0, then, by maximality, [Q, ψ] is not subconjugate to [P, φ] and hence c_{[Q,ψ]}(X) = 0 for all X ∈ M (≾ [P, φ]). In particular,

c_{[Q,ψ]}([P, φ] ∘ [S, ι_{S}]) = 0.

Now we deduce that

c_{[Q,ψ]} (1_{p} ∘ [S, ι_{S}]) = c_{[Q,ψ]} (∑_{[P,φ]} c_{[P,φ]}(1_{p}) · [P, φ] ∘ [S, ι_{S}])

= ∑_{[P,φ]} c_{[P,φ]}(1_{p}) · c_{[Q,ψ]} ([P, φ] ∘ [S, ι_{S}])

= c_{[Q,ψ]}(1_{p}) · c_{[Q,ψ]} ([Q, ψ] ∘ [S, ι_{S}]) ≠ 0.

On the other hand, we have 1_{p} ∘ [S, ι_{S}] = [S, ι_{S}]. Therefore we must have [Q, ψ] = [S, ι_{S}]. We deduce that [S, ι_{S}] is the unique maximal element such that c_{[S,ι_{S}]}(1_{p}) ≠ 0, and hence 1_{p} ∈ M (≾ [S, ι_{S}]) = R. □

Let n be the number of conjugacy classes of p-subgroups of G and pick one representative P_{i} for each conjugacy class, labelled from 1 to n so that P_{1} = S, and i ≥ j if P_{j} is conjugate to a subgroup of P_{i}. Since 1_{p} ∈ R, we can write

1_{p} = ∑_{j=0}^{n} a_{j} · [P_{j}, ι_{P_{j}}].

For subgroups H and F of G, let N_{G}(F, H) denote the transporter

N_{G}(K, H) = {g ∈ G | K^{g} ≤ H}.

Proposition 5.3. The multiplicative identity of A_{p}(G, G)_{p}^{∧} is given by

1_{p} = ∑_{j=1}^{n} a_{j} · [P_{j}, ι_{P_{j}}],

where the coefficients a_{j} satisfy the equations

∑_{j=1}^{n} a_{j} · \frac{|N_{G}(P_{i}, P_{j})|}{|P_{j}|} = 1  (4)

for i = 1, …, n.

Proof. In this proof we will denote all inclusions of groups by ι. For every i we have 1_{p} ∘ [P_{i}, ι] = [P_{i}, ι]. In particular, c_{[P_{i},ι]}(1_{p} ∘ [P_{i}, ι]) = 1. By the double coset formula, we have

[P_{j}, ι] ∘ [P_{i}, ι] = ∑_{x∈P_{j}\G/P_{i}}[P_{i} ∩ P_{j}^{x}, c_{x}].

It is easy to check that [P_{i} ∩ P_{j}^{x}, c_{x}] = [P_{i} ∩ P_{j}^{x}, ι], and

c_{[P_{i},ι]}([P_{i} ∩ P_{j}^{x}, ι]) = \begin{cases}1 & if ^{x}P_{i} ≤ P_{j}, \\ 0 & otherwise. \end{cases}

Noting that ^{x}P_{i} ≤ P_{j} if and only if x ∈ N_{G}(P_{i}, P_{j}), we see that

c_{[P_{i},ι]}([P_{j}, ι] ∘ [P_{i}, ι]) = |P_{j}\N_{G}(P_{i}, P_{j})/P_{i}|

If x ∈ N_{G}(P_{i}, P_{j}) and g ∈ P_{i}, then there is a h ∈ P_{j} such that xg = hx. Therefore

P_{j}\N_{G}(P_{i}, P_{j})/P_{i} ≅ P_{j}\N_{G}(P_{i}, P_{j}),

and

c_{[P_{i},ι]}([P_{j}, ι] ∘ [P_{i}, ι]) = \frac{|N_{G}(P_{i}, P_{j})|}{|P_{j}|}.

Now we get

1 = c_{[P_{i},ι]}(1_{p} ∘ [P_{i}, ι])

= c_{[P_{i},ι]}(∑_{j=0}^{n} a_{j} · [P_{j}, ι] ∘ [P_{i}, ι])

= ∑_{j=1}^{n} a_{j} · c_{[P_{i},ι]}([P_{j}, ι] ∘ [P_{i}, ι])

= ∑_{j=1}^{n} a_{j} · \frac{|N_{G}(P_{i}, P_{j})|}{|P_{j}|}

□

Remark 5.4. Since |N_{G}(P_{i}, P_{j})| = 0 when i < j, the equations in (4) constitute a lower triangular matrix equation. The elements on the diagonal are |N_{G}(P_{i})/P_{i}| ≠ 0, so the matrix is of maximal rank and the equations suffice to determine 1_{p} uniquely. We do not need to prove that the equation is solvable (this could be an issue because the elements on the diagonal need not be units in ℤ_{p}^{∧}) since we already know that A_{p}(G, G)_{p}^{∧} has a multiplicative identity.

We now proceed to describe π_{p} in terms of fixed-point sets. For every (G, K)-pair (H, ψ), there is a homomorphism

χ_{[H,ψ]} : A(G, K) ⟶ ℤ,

depending only on the conjugacy class of (H, ψ), which sends a (G, K)- bundle X to

χ_{[H,ψ]}(X) = χ (W (Δ_{H}^{ψ})\X^{Δ_{H}^{ψ}}) ,

where χ is the Euler characteristic, X^{Δ_{H}^{ψ}} is the fixed-point space, and

W (Δ_{H}^{ψ}) = N_{K×G}(Δ_{H}^{ψ})/Δ_{H}^{ψ}.

Note that for a (G, K)-pair (H′, ψ′) we have

W (Δ_{H}^{ψ})\((K × G)/Δ_{H′}^{ψ′})^{Δ_{H}^{ψ}} ≅ N_{K×G}(Δ_{H}^{ψ})\N_{K×G}(Δ_{H}^{ψ}, Δ_{H′}^{ψ′})/Δ_{H′}^{ψ′}

and that N_{K×G}(Δ_{H}^{ψ})\N_{K×G}(Δ_{H}^{ψ}, Δ_{H′}^{ψ′}) is finite, so we in fact have

χ_{[H,ψ]}(X) = |W (Δ_{H}^{ψ})\X^{Δ_{H}^{ψ}} |

for a principal (G, K)-bundle X over a finite G-set.

Let C_{p}(G, K) be the set of conjugacy classes of (G, K)-pairs (P, ψ) where P is a p-group.

Proposition 5.5. [14] The homomorphism

χ_{p}: A_{p}(G, K) ⟶ ∏_{C_{p}(G,K)} ℤ, X ⟼ ∏_{C_{p}(G,K)} χ_{[P,ψ]}(X)

is an injection.

We will describe π_{p} in terms of this embedding. To this end we need the following reformulation of a result of Benson–Feschbach.

Lemma 5.6. [4] For X ∈ A(G, K), a (G, K)-pair (H, ψ) and a subgroup F ≤ G, we have

χ_{[H,ψ]}(X ∘ [F, ι_{F}]_{G}^{G}) = \frac{|N_{G}(H, F)|}{|F |}χ_{[H,ψ]}(X).

In particular χ_{[H,ψ]}(X ∘ [F, ι_{F}]_{G}^{G}) = 0 if H is not subconjugate to F.

Proof. This follows from Proposition 3.1 in [4] (which is actually proved in the case where K is finite and without taking the quotient by W (Δ_{H}^{ψ}) but the same argument works in this setting). □

Theorem 5.7. The homomorphism π_{p} sends X ∈ A(G, K) to the unique element X_{p} ∈ A_{p}(G, K)_{p}^{∧} such that

χ_{p}(X_{p}) = χ_{p}(X).

In other words, π_{p} is the unique homomorphism fitting into the commutative diagram.

[figure: A(G, K); π_{p}; χ_{p}; A_{p}(G, K)_{p}^{∧}; χ_{p}; ∏_{C_{p}(G,K)} ℤ_{p}^{∧}.]

The homomorphism π̃_{p} is the homomorphism obtained from π_{p} by quotienting out all trivial basis elements.

Proof. For X ∈ A(G, K) and a (G, K)-pair (Q, ψ) with Q a p-group we have

χ_{[Q,ψ]}(π_{p}(X)) = χ_{[Q,ψ]}(X ∘ 1_{p})

= ∑_{j=0}^{n} a_{j}χ_{[Q,ψ]}(X ∘ [P_{j}, ι_{P_{j}}])

= ∑_{j=0}^{n} a_{j} \frac{|N_{G}(Q, P_{j})|}{|P_{j}|}χ_{[Q,ψ]}(X)

= χ_{[Q,ψ]}(X),

where the last step follows from Proposition 5.3 since Q is conjugate to one of the groups P_{i}. □

Letting p run over all primes we obtain the following corollary, from which Theorem C in the introduction follows.

Corollary 5.8. The kernel of the map

α: A(G, K) ⟶ {BG_{+}, BK_{+}} ≅ A(G, K)_{I(G)}^{∧}

is the submodule consisting of elements X such that χ_{[H,ψ]}(X) = 0 for all (G, K)-pairs [H, ψ] such that the order of H is a prime power.

Remark 5.9. If K is finite one may replace the morphism χ_{[H,ψ]} with the morphism sending X ↦ |X^{Δ_{H}^{ψ}}| throughout this section and the results still hold true.

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

  E-mail address: kari@maths.abdn.ac.uk