RETRACTIVE TRANSFERS AND p-LOCAL FINITE GROUPS

KÁRI RAGNARSSON

Abstract. In this paper we explore the possibility of defining p-local finite groups ([6]) in terms of transfer properties of their classifying spaces. More precisely, we consider the question posed by Haynes Miller, whether an equivalent theory can be obtained by studying triples (f, t, X), where X is a p-complete, nilpotent space with finite fundamental group, f: BS → X is a map from the classifying space of a finite p-group, and t is a stable retraction of f satisfying Frobenius reciprocity at the level of stable homotopy. We refer to t as a retractive transfer of f and to (f, t, X) as a retractive transfer triple over S. In the case where S is elementary abelian, we answer this question in the affirmative by showing that a retractive transfer triple (f, t, X) over S does indeed induce a p-local finite group over S with X as its classifying space.

Using results from [20] we show that the converse is true for general finite p-groups. That is, for a p-local finite group (S, F, L), the natural inclusion θ: BS → X has a retractive transfer t, making (θ, t, |L|_{p}^{∧}) a retractive transfer triple over S. This also requires a proof, obtained jointly with Ran Levi, that |L|_{p}^{∧} is a nilpotent space, which is of independent interest.

Introduction

Defined by Broto-Levi-Oliver in [6], p-local finite groups are the culmination of a program initiated by Puig [18, 19] to find a formal framework for the plocal structure of a finite group. To a finite group G, one associates a fusion system (at a prime p) consisting of all p-subgroups of G and the homomorphims between them induced by conjugation in G. Puig formalized fusion systems and identified an important subclass of fusion systems, which we now call saturated fusion systems. Fusion systems of finite groups are contained in this class, but saturated fusion systems also arise in other important contexts, most notably in modular representation theory through Brauer subpairs of blocks of group algebras, and more recently as Chevalley groups of p-compact groups [7].

The fusion system of a group G can be considered as an algebraic interpretation of the p-local structure of the group. One can also take a topological approach, and think of the p-local structure of G as being the p-completed classifying space BG_{p}^{∧}. By Bob Oliver’s solution [16, 17] of the Martino-Priddy conjecture [14], these approaches are the same. That is, two groups induce the same fusion system if and only if their p-completed classifying spaces are homotopy equivalent. In fact the fusion system can be recovered from the classifying space via a homotopy theoretic construction, which is presented in Section 1.3. Therefore, one can in some sense regard BG_{p}^{∧} as a classifying space of the fusion system. This suggests that, more generally, each saturated fusion system may have a unique classifying space.

A p-local finite group consists of a saturated fusion system and an associated centric linking system, a category which contains just enough information to construct a classifying space associated to the fusion system. Thus one can think of a p-local finite group as a saturated fusion system with a chosen classifying space.

Date: October 19th, 2005.

The definition of p-local finite groups is rather complicated and has the drawback that there is no straightforward concept of morphisms between p-local finite groups, so they have not yet been made to form a category in any sensible way. In this paper, we adopt the approach used by Dwyer and Wilkerson for p-compact groups [10], and try to develop the theory of p-local finite groups in terms of classifying spaces.

Specifically, we consider a homotopy monomorphism f: BS → X from the classifying space of a finite p-group S to a p-complete, nilpotent space X with finite fundamental group, that is endowed with a stable retraction t satisfying Frobenius reciprocity at the level of stable homotopy. We will refer to such a triple (f, t, X) as a retractive transfer triple over S. For a retractive transfer triple (f, t, X), we ask whether X is the classifying space of a p-local finite group. This question will be addressed in Sections 3 to 5, where we answer the question in the affirmative in the case where S is elementary abelian (Theorem 2.4). Conversely we ask whether a p-local finite group (S, F, L) gives rise to a Frobenius transfer triple. This is indeed the case for any finite p-group S, as we show in Section 6. This involves joint work with Ran Levi where we show that the classifying space of a p-local finite group is a torsion space and (since its fundamental group is a finite p-group) consequently a nilpotent space.

Acknowledgements: The majority of the work in this paper was done as part of my Ph.D. thesis and I would like to thank my thesis advisor Haynes Miller for suggesting this problem and for his many helpful suggestions during the course of this project. I also thank Ran Levi for his help on the material in Section 6.1.

1. A quick review of p-local finite groups

In this section we give a brief overview of the theory of p-local finite groups. Most of this material is found in [6]. In this section and throughout the paper, p is a fixed prime.

1.1. Some definitions and terminology. We begin by recalling some terminology regarding p-local finite groups. Definition 1.1. 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.

Here Hom_{S}(P, Q) is the set of group homomorphims induced by conjugation by elements in S.

Before stating the next definition, we need to introduce some additional terminology and notation. 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 which is F-conjugate to P. Similarly P is fully normalized in F if |N_{S}(P)| ≥ |N_{S}(P′)| for every P′ ≤ S which is F-conjugate to P.

Definition 1.2. 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, and

Aut_{S}(P) is a Sylow subgroup of 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))}.

There is a class of subgroups of S of special interest to us, defined as follows.

Definition 1.3. Let F be a fusion system over a p-group S. A subgroup P ≤ S is F-centric if C_{S}(P′) ≤ P′ for every P′ that is F-conjugate to P. Let F^{c} denote the full subcategory of F whose objects are the F-centric subgroups of S.

Remark 1.4. The condition C_{S}(P′) ≤ P′ in the previous definition is equivalent to the condition C_{S}(P′) = Z(P′).

Definition 1.5. Let F be a fusion system over the p-group S. A centric linking system associated to F is a category L whose objects are the F-centric subgroups of S, together with a functor

π : L → F^{c},

and distinguished monomorphisms P ⟶^{δ_{P}} Aut_{L}(P), for each F-centric subgroup P ≤ S, which satisfy the following conditions.

(A) The functor π is the identity on objects and surjective on morphisms. More precisely, for each pair of objects P, Q ∈ L, the centre Z(P) acts freely on Mor_{L}(P, Q) by composition (upon identifying Z(P) with δ_{P} (Z(P)) ≤ Aut_{L}(P)), and π induces a bijection

Mor_{L}(P, Q)/Z(P) ⟶^{≅} Hom_{F}(P, Q).

(B) For each F-centric subgroup P ≤ S and each g ∈ P, π sends δ_{P} (g) ∈

Aut_{L}(P) to c_{g} ∈ Aut_{F}(P). (C) For each f ∈ Mor_{L}(P, Q) and each g ∈ P, the following square commutes

in L:

P ⟶^{f} Q
↓^{δ_{P} (g)}  ↓^{δ_{Q}(π(f)(g))}
P ⟶^{f} Q.

We can now finally define our objects of study.

Definition 1.6. A p-local finite group is a triple (S, F, L), where F is a saturated fusion system over a finite p-group S and L is a centric linking system associated to F. The classifying space of the p-local finite group is the p-completed geometric realisation |L|_{p}^{∧}.

We remark that a p-local finite group comes equipped with a natural inclusion

θ: BS ⟶ |L|_{p}^{∧}.

One of the main questions in the theory of p-local finite groups concerns the existence and uniqueness of a centric linking system associated to a given saturated fusion system. In Section 3 of [6], Broto-Levi-Oliver have developed an obstruction theory to address this question.

1.2. The fusion system of a group. In this section we will discuss the fusion system arising from a Sylow subgroup inclusion S ≤ G. This section serves as motivation for the discussion in the previous section as well as being of independent interest.

Definition 1.7. Let G be a finite group. The fusion system of G is the category F(G), whose objects are the p-subgroups of G and whose morphism sets are given by

Hom_{F(G)}(P, Q) = Hom_{G}(P, Q)

for all p-subgroups P, Q ≤ G.

For a p-subgroup S ≤ G, the fusion system of G over S is the full subcategory F_{S}(G) ⊆ F(G), whose objects are the subgroups of S.

If S is a Sylow subgroup of G, then the inclusion of F_{S}(G) in F(G) is an equivalence of categories, since every p-subgroup of G is conjugate to a subgroup of S.

Proposition 1.8. [6, Proposition 1.3] Let G be a finite group and let S be a p-subgroup. Then the fusion system F_{S}(G) of G over S is saturated if and only if S is a Sylow subgroup.

The centric linking system of a finite group was initially introduced in [5] as a powerful tool to study homotopy equivalences between p-completed classifying spaces of finite groups. The p-centric subgroups of a finite group G are the psubgroups P ≤ G whose centre Z(P) is a p-Sylow subgroup of the centraliser C_{G}(P). This notion of centricity is equivalent to the one introduced in 1.3 in the sense that if S ≤ G is a Sylow subgroup, then a subgroup P ≤ S is p-centric if and only if it is F_{S}(G)-centric.

For the following definition, we recall that if a group P ≤ G is p-centric, then one can write

C_{G}(P) = Z(P) × C′_{G}(P),

where C′_{G}(P) ≤ G has order prime to p. The notation C′_{G}(P) will be used in the definition. In addition, for subgroups P, Q ≤ G, we will let N_{G}(P, Q) denote the transporter

N_{G}(P, Q) = {g ∈ G|gP g^{−1} ≤ Q}.

Definition 1.9. Let G be a finite group. The centric linking system of G is the category L(G), whose objects are the p-centric subgroups of G and whose morphism sets are given by

Mor_{L(G)}(P, Q) = N_{G}(P, Q)/C′_{G}(P)

for all p-subgroups P, Q ≤ G.

For a p-subgroup S ≤ G, the centric linking system of G over S is the full subcategory L_{S}(G) ⊆ L(G) whose objects are the subgroups of S, that are p-centric in G.

In the case of a Sylow inclusion S ≤ G, the centric linking system L_{S}(G) is a centric linking system associated to the saturated fusion system F_{S}(G), and we have the following proposition, which serves as a motivating example for the definition of a p-local finite group.

Proposition 1.10. Let S be a Sylow subgroup of a finite group G. Then the triple (S, F_{S}(G), L_{S}(G)) is a p-local finite group over S. Furthermore, the natural map

θ: BS → |L_{S}(G)|_{p}^{∧}

is equivalent to the p-completed inclusion

BS → BG_{p}^{∧}

as a space under BS.

1.3. Homotopy theoretic constructions of fusion systems. In this section we recall how a map f: BS → X, from the classifying space of a finite p-group S to a space X, induces a fusion system F_{S,f}(X) over S. In general this fusion system is not saturated.

The following definition is motivated by the fact that two group homomorphisms φ, ψ: G → H between finite groups are H-conjugate if and only if the induced maps of classifying spaces are freely homotopic.

Definition 1.11. For any space X, any p-group S, and any map f: BS → X, define F_{S,f}(X) to be the category whose objects are the subgroups of S, and whose morphisms are given by

Hom_{F_{S,f}} (P, Q) = {φ ∈ Inj(P, Q) | f|_{BP} ≃ f|_{BQ} ∘ Bφ}

for each P, Q ≤ S.

It is easy to see that F_{S,f} is indeed a fusion system, although it need not be saturated. In the case where F_{S,f} is saturated however, one can get a candidate L_{S,θ}^{c}(|L|_{p}^{∧}) for an associated centric linking system by retaining information about the homotopies giving the equivalence f|_{BP} ≃ f|_{BQ} ∘ Bφ in the definition above. See [6] for details.

Theorem 1.12. [6, Theorems 7.4,7.5] For a p-local finite group (S, F, L), the fusion system F_{S,θ}(|L|_{p}^{∧}) is saturated and L_{S,θ}(|L|_{p}^{∧}) is a centric linking system associated to F_{S,θ}(|L|_{p}^{∧}). Furthermore, the p-local finite groups (S, F, L) and (S, F_{S,θ}(|L|_{p}^{∧}), L_{S,θ}^{c}(|L|_{p}^{∧})) are isomorphic.

1.4. p-local finite groups over abelian groups. We conclude this review by classifying the p-local finite groups over an abelian p-group S. The resulting classification shows that the strict equivalence classes of p-local finite groups over S are in a bijective correspondence with the subgroups W ≤ Aut(S) of order prime to p, under the assignment

W ↦ (S, F_{S}(W ⋉ S), L_{S}(W ⋉ S)),

where W ⋉ S is the semi-direct product. Here we say that two p-local finite groups are strictly equivalent if they have the same fusion system and their linking systems are isomorphic. In particular, there are no exotic p-local finite groups over abelian p-groups.

We begin with the following lemma which describes precisely how the conditions in definition 1.2 are simplified under the assumption that S is abelian. As this result is obvious to the experienced reader and proving it straight from the definition is an excellent exercise for those new to p-local finite groups, the proof is left to the reader.

Lemma 1.13. Let F be a fusion system over a finite abelian p-group S. Then F is saturated if and only if the following two conditions are satisfied:

(I_{ab}) Aut_{F}(S) has order prime to p. (II_{ab}) Every φ ∈ Hom_{F}(P, Q) is the restriction of some φ̃ ∈ Aut_{F}(S).

The following proposition follows easily.

Proposition 1.14. If S is an abelian finite p-group, then the assignment W ↦ F_{S}(W ⋉ S) gives a bijective correspondence between subgroups W ≤ Aut(S) of order prime to p and saturated fusion systems over S.

Being the fusion system of a group, the fusion system F_{S}(W ⋉S) has an obvious associated centric linking system L_{S}^{c}(W ⋉ S). This is in fact the only associated centric linking system and we have the following classification result.

Proposition 1.15. If S is an abelian finite p-group, then the assignment

W ↦ (S, F(W ⋉ S), L_{S}^{c}(W ⋉ S))

gives a bijective correspondence between subgroups W ≤ Aut(S) of order prime to p and strict equivalence classes of p-local finite groups over S. In particular, there are no exotic p-local finite groups over S.

Proof. When S is abelian, there are no proper centric subgroups. Therefore the obstruction to uniqueness of centric linking systems ([6, Section 3]) simplifies to the cohomology group H^{2}(W; S). Now use a transfer argument to show that H^{∗}(W; S) vanishes for ∗ > 0. □

2. Retractive transfer triples

First we make precise the setting we are working in. Cohomology will always be taken to be with 𝔽_{p}-coefficients unless otherwise specified. The following definition is a homotopy generalization of group monomorphisms.

Definition 2.1. A map f: Y → X between two topological spaces is a homotopy monomorphism at p it its induced map in cohomology makes H^{∗}(Y) a finitely generated H^{∗}(X)-module. In the special case where Y = BP is the classifying space of a finite p-group, we say that f is a p-subgroup inclusion.

The analogy with group monomorphisms is that a group homomorphism φ: P → G from a finite p-group P to a finite group G is a monomorphism if and only Bφ is a homotopy monomorphism at p. There are other definitions of homotopy monomorphisms in the literature, but these are equivalent in the setting in which we are working. Since the finite generation hypothesis is the only one we need, we avoid complication by considering only this definition. Since the prime p is fixed throughout we will refer to these concepts simply as “homotopy monomorphism” and “subgroup inclusion.”

We will demand some additional structure on our subgroup inclusions, namely that they allow a transfer with properties similar to that of the transfer of a Sylow subgroup inclusion.

Definition 2.2. Let f: Y → X be a map of spaces. A retractive transfer of f is a stable map t : Σ_{+}^{∞}X → Σ_{+}^{∞}Y such that Σ_{+}^{∞}f ∘ t ≃ id_{Σ_{+}^{∞}X}, and the following diagram commutes up to homotopy

Σ_{+}^{∞}X ⟶^{Δ} Σ_{+}^{∞}X ∧ Σ_{+}^{∞}X
↓^{t}  ↓^{1∧t}
Σ_{+}^{∞}Y ⟶^{(f∧1)∘Δ} Σ_{+}^{∞}X ∧ Σ_{+}^{∞}Y.

(1)

The objects that will be the focus of our attention are defined as follows.

Definition 2.3. A retractive transfer triple over a finite p-group S is a triple (f, t, X), where X is a connected, p-complete, nilpotent space with finite fundamental group, f is a subgroup inclusion BS → X and t is a retractive transfer of f.

Since the space X in the above definition is p-complete with finite fundamental group, it follows that π_{1}(X) is a finite p-group [10].

For a retractive transfer triple (f, t, X) over a finite p-group S, we ask the following questions:

• Is the fusion system F_{S,f}(X) saturated?

• If so, does there exist an associated centric linking system L? Is it unique?

• If an associated centric linking system exists, then what is the relation between the classifying space |L|_{p}^{∧} and X? Are they equivalent as objects under BS?

In the course of the following sections we will answer these questions affirmatively in the case when S is an elementary abelian p-group, proving the following theorem.

Theorem 2.4. Let S be a finite elementary abelian p-group, let (f, t, X) be a retractive transfer triple over S, and let W be the group of automorphisms of S that act trivially on H^{∗}(X) when regarded as a subring of H^{∗}(BS) via f^{∗}. Then the following hold

(i) W has order prime to p.

(ii) F_{S,f}(X) is equal to the saturated fusion system F_{S}(W ⋉ S)

(iii) F_{S,f}(X) has an associated centric linking system, which is unique up to

isomorphism, with classifying space B(W ⋉ S)_{p}^{∧}. (iv) There is a natural equivalence B(W ⋉ S)_{p}^{∧} ⟶^{≃} X of objects under BS.

Thus the triple (f, t, X) induces a p-local finite group (S, F_{S,f}(X), L_{S,f}^{c}(X_{p}^{∧})) over S with classifying space X.

Proof. The proof is by forward referencing. Part (i) is shown in Proposition 3.12. Part (ii) is Proposition 4.2. Part (iii) then follows from 1.15. Finally, part (iv) is Proposition 5.2. □

3. Cohomological structure of retractive transfer triples

In this section we discuss the cohomological structure of a retractive transfer triple (f, t, X) over a finite p-group S. We first discuss general properties in 3.1. We then specialize to the case where S is elementary abelian in 3.2 and show that in this case H^{∗}(X) is a ring of invariants of H^{∗}(BS) under the action of a group of order prime to p.

3.1. The general case. Applying the cohomology functor H^{∗}(−; 𝔽_{p}) to (1) we get maps

H^{∗}(X) ⟶^{f^{∗}} H^{∗}(BS) ⟶^{t^{∗}} H^{∗}(X)

with the following properties:

CohI t^{∗} ∘ f^{∗} = id. CohII t^{∗} is H^{∗}(X)-linear (Frobenius reciprocity). CohIII t^{∗} is a morphism of unstable modules over the Steenrod algebra. CohIV f^{∗} is a morphism of unstable algebras over the Steenrod algebra.

Hence H^{∗}(X) is a direct summand of H^{∗}(BS) as a H^{∗}(X)-module and as a module over the Steenrod algebra. CohI allows us to regard H^{∗}(X) as a subring of H^{∗}(BS) and we will often do so without further comment.

These properties are quite restrictive and the question of which unstable subalgebras R^{∗} ⊂ H^{∗}(BS) over the Steenrod algebra admit a splitting H^{∗}(BS) → R^{∗} as R^{∗}-modules and unstable modules over the Steenrod algebra is interesting in itself. However, we focus our attention on p-local finite groups.

The following finiteness properties of retractive transfer triples will be needed later. The first fact is a standard result, a proof of which can be found in [10, Sec. 12].

Fact 3.1. If S is a finite p-group, then H^{∗}(BS) is Noetherian.

Lemma 3.2. Let S be a finite p-group and (f, t, X) be a Frobenius transfer triple over S. Then H^{∗}(X) is Noetherian and in particular X is of finite 𝔽_{p}-type.

Proof. By [10, Lemma 2.6], this follows from CohI, CohII and the fact that H^{∗}(BS) is Noetherian. □

Lemma 3.3. Let S be a finite p-group and (f, t, X) be a Frobenius transfer triple over S. Then X is of Z_{(p)}-finite type.

Proof. By the universal coefficient theorem, it suffices to show that X is of finite 𝔽_{p}-type and of finite ℚ-type. The former is Lemma 3.2 above. The latter is deduced in a similar way: By a transfer argument, BS has trivial ℚ-cohomology. As in the 𝔽_{p}-coefficient case, H^{∗}(X; ℚ) is a direct summand of H^{∗}(BS; ℚ). Hence X also has trivial ℚ-cohomology and we are done. □

3.2. The elementary abelian case. In this subsection, we restrict ourselves to the case where S is an elementary abelian finite p-group. In this case, we use a theorem of Goerss-Smith-Zarati [11], based on the celebrated work of Adams-Wilkerson [2], to prove that if (f, t, X) is a retractive transfer triple over S, then H^{∗}(X) is a ring of invariants H^{∗}(BS)^{W} for a subgroup W ≤ Aut(S) of order prime to p. Furthermore we show that the group W may be taken to be the group of automorphisms of S that act trivially on H^{∗}(X). We consider only the case of an odd prime. The results still hold true at the prime 2 and the proofs proceed in more or less the same way, but are simpler at times.

In [2], Adams and Wilkerson study the following category.

Definition 3.4. Let AW be the category of evenly graded unstable algebras R over the Steenrod algebra, that are integral domains.

They also make precise the notions of “algebraic extension” and “algebraic closure” in this setting and prove the following.

Proposition 3.5. [2, Proposition 1.5] Every object R in AW has an algebraic closure H in AW. If R has finite transcendence degree, then so does H.

Theorem 3.6. [2, Theorem 1.6] The objects H in AW, that are algebraically closed and of finite transcendence degree are precisely the polynomial algebras 𝔽_{p}[x_{1}, …, x_{n}] on generators x_{i} of degree 2.

Furthermore, in [2, Theorem 1.2] they show that an algebra in R of finite transcendence degree is a ring of invariants in its algebraic closure if and only if it satisfies two conditions, which can be interpreted as an integral closure condition and an inseparable closure condition. Based on that work, Goerss-Smith-Zarati identified sufficient conditions for an unstable algebra over the Steenrod algebra to be a ring of invariants. Before stating their result we need some preparation.

Let A denote the mod p Steenrod algebra and let A′ be the subalgebra generated by the power operations P^{i}, i ≥ 0. Then we have a splitting of A′-modules

A = A′ ⊕ A′′,

where A′′ is the 𝔽_{p}-vector subspace of A generated by admissible sequences involving the Bockstein operation. Let U denote the category of unstable A-modules and K denote the category of unstable A-algebras. In both cases morphisms are of degree zero. Let U′ and K′ denote the corresponding full subcategories whose objects are evenly graded.

By [13], the forgetful functor θ: U′ → U has a right adjoint θ̃: U → U′, which sends an unstable A-module M to the submodule of elements x of even degree satisfying α(x) = 0 for all α ∈ A′′, and morphisms to restrictions to these submodules. For unstable A-algebras the same construction gives a right adjoint θ̃: K → K′ to the forgetful functor θ: K′ → K, and we have a commutative diagram of functors

K′ ⟶^{θ} K ⟶^{θ̃} K′
↓  ↓  ↓
U′ ⟶^{θ} U ⟶^{θ̃} U′,

where the vertical functors are forgetful functors. As a consequence we obtain the following lemma.

Lemma 3.7. Properties CohI to CohIV are preserved by θ̃.

Proof. This is mostly self-evident. The functor diagram is only needed to make sense of CohII. □

We need to recall some things about reduced U-injectives.

Definition 3.8. An A-module M is a reduced U-injective if it is an injective object in the category U, and

Hom_{U}(ΣN, M) = 0,

for every A-module N, where Σ denotes the suspension functor. An unstable Aalgebra R is a reduced U-injective if it is a reduced U-injective when regarded as an A-module.

When S is elementary abelian, the cohomology ring H^{∗}(BS) is a reduced U-injective by [13]. If (f, t, X) is a retractive transfer triple over S, then H^{∗}(X) is a direct summand of H^{∗}(BS) as A-modules, and hence H^{∗}(X) is also a reduced U-injective. This will allow us to apply the following theorem to show that H^{∗}(X) is a ring of invariants in H^{∗}(BS).

Theorem 3.9. [11, Theorem 1.3] Let R be an unstable A-algebra that is a reduced U-injective satisfying

(i) θ̃R is a Noetherian integral domain.

(ii) θ̃R is integrally closed in its field of fractions.

Then there exists an integer n and a subgroup W ≤ GL(n, ℤ/p) such that R is isomorphic to the ring of invariants H^{∗}(B(ℤ/p)^{n}; 𝔽_{p})^{W}. Furthermore, W has order prime to p.

Remark 3.10. Looking closely at the proof of the theorem in [11] and the tools from [2] used therein, one sees that θ̃H^{∗}(B(ℤ/p)^{n}; 𝔽_{p}) is in fact the algebraic closure of θ̃R^{∗} in AW and W is the group of automorphisms of (ℤ/p)^{n} acting trivially on R. The point is that, in their proof, Goerss-Smith-Zarati apply Theorem 1.2 of [2], which, as mentioned above, really gives necessary and sufficient conditions for when the embedding of an algebra of finite transcendence degree into its algebraic closure in AW is a Galois extension. This is made clear in the introduction of [2] although the authors chose to make the statement of the theorem less technical.

The following technical result will be needed.

Lemma 3.11. Let R and H be unstable A-algebras that are U-injectives and suppose f: R → H is a morphism of A-algebras making H finitely generated over R. Then θ̃f makes θ̃H finitely generated over θ̃R.

Proof. Recall from [11] that there are unique A′-algebra homomorphisms

π_{R}: R → θ̃R and π_{H}: H → θ̃H

such that

π_{R} ∘ ι_{R} = id_{θ̃R} and π_{H} ∘ ι_{H} = id_{θ̃H},

where

ι_{R}: θ̃R → R and ι_{H}: θ̃H → H

denote the natural inclusions. (Strictly speaking this is an abuse of notation and we should replace θ̃H and θ̃R by θθ̃H and θθ̃R, respectively.)

Now, suppose {h_{1}, …, h_{n}} is a set of generators for H over R. Let h ∈ θ̃H. Then we can write

ι_{H}(h) = ∑_{i=1}^{n} f(r_{i})h_{i},

for some r_{i} ∈ R Consequently,

h = π_{H} ∘ ι_{H}(h) = π_{H} (∑_{i=1}^{n} f(r_{i})h_{i})

= ∑_{i=1}^{n} π_{H}(f(r_{i}))π_{H}(h_{i}).

Therefore, if we can show that π_{H} ∘ f = θ̃f ∘ π_{R}, then we can deduce that {π_{H}(h_{1}), …, π_{H}(h_{n})} is a set of generators for θ̃H over θ̃R and we are done.

To prove this we first observe that by [11, Corollary 3.3(i)] there is a unique morphism of unstable A-algebras g: R → H such that

π_{H} ∘ g = θ̃f ∘ π_{R}.

Next we note that by construction of θ̃ we have

ι_{H} ∘ θ̃g = g ∘ ι_{R},

from which it follows that

θ̃g = π_{H} ∘ ι_{H} ∘ θ̃g = π_{H} ∘ g ∘ ι_{R} = θ̃f ∘ π_{R} ∘ ι_{R} = θ̃f.

But by [11, Corollary 3.3(ii)] the map

θ̃: Hom_{K}(R, H) ⟶ Hom_{K′}(θ̃R, θ̃H)

is a bijection, so g = f, and hence

π_{H} ∘ f = π_{H} ∘ g = θ̃f ∘ π_{R}.

□

Proposition 3.12. Let (f, t, X) be a retractive transfer triple over a finite elementary abelian p-group S and let W ≤ Aut(S) be the subgroup of automorphisms of S acting trivially on H^{∗}(X). Then

H^{∗}(X) = H^{∗}(BS)^{W}

and W has order prime to p.

Proof. Let us show that H^{∗}(X) satisfies the conditions of Theorem 3.9 above. By the remark after Definition 3.8, H^{∗}(X) is a reduced U-injective. By [13, 24] we have

θ̃H^{∗}(BS) ≅ 𝔽_{p}[x_{1}, …, x_{n}],  (2)

where n is the rank of S. In particular, θ̃H^{∗}(BS) is a Noetherian integral domain. A similar argument to the proof of Lemma 3.3 shows that θ̃H^{∗}(X) is also a Noetherian integral domain. It remains only to show that θ̃H^{∗}(X) is integrally closed in its field of fractions.

For this we first recall from [2], that θ̃H^{∗}(BS) is integrally closed in its field of fraction. Now, let x be in the field of fractions of θ̃H^{∗}(X) and suppose that x is integral over θ̃H^{∗}(X). Write x = a/b, with a, b ∈ θ̃H^{∗}(X). Then x is also integral over θ̃H^{∗}(BS), and since θ̃H^{∗}(BS) is integrally closed, this implies that x ∈ θ̃H^{∗}(BS). We now have the equation a = bx in θ̃H^{∗}(BS). Applying θ̃t^{∗} and using θ̃H^{∗}(X)-linearity (CohII), we get:

a = θ̃t^{∗}(a) = θ̃t^{∗}(bx) = b · θ̃t^{∗}(x).

Since θ̃H^{∗}(BS) is an integral domain, this implies that

x = a/b = θ̃t^{∗}(x) ∈ θ̃H^{∗}(X).

Before applying Theorem 3.9, we note that since H^{∗}(BS) is finitely generated over H^{∗}(X), Lemma 3.11 applies to show that θ̃H^{∗}(BS) is finitely generated over θ̃H^{∗}(X), and hence θ̃H^{∗}(BS) is an algebraic extension of θ̃H^{∗}(X). Since θ̃H^{∗}(BS) is algebraically closed in AW, we conclude that θ̃H^{∗}(BS) is the algebraic closure of θ̃H^{∗}(X) in AW.

Applying Theorem 3.9 along with Remark 3.10 and the observation in the preceding paragraph now completes the proof. □

4. Proving saturation of F_{f,S}(X)

We have now determined the cohomological structure of a retractive transfer triple (f, t, X) over an elementary abelian p-group S. In this section we extend the analysis to homotopy theory and prove that the fusion system F_{f,S} := F_{f,S}(X) is saturated. The key tool which allows us to draw conclusions in homotopy from the cohomological results in 3.2 is the following theorem of Lannes, which was originally proved by Miller in [15] in the case when Y is a space with cohomology of the type H^{∗}(Y) = U(M).

Theorem 4.1. [12] Let X be a connected space and V an elementary abelian p-group. Suppose that X is nilpotent, that π_{1}X is finite and that H^{∗}(X) is of finite type. Then the cohomology functor induces a natural bijection

[BV, X] ⟶^{≅} Hom_{K}(H^{∗}(X), H^{∗}(BV)).

Let (f, t, X) be a retractive transfer triple over an elementary abelian finite p-group S. By definition, X is a nilpotent space and π_{1}(X) is finite. By Lemma 3.2, H^{∗}(X) is of finite type. Thus Theorem 4.1 applies to X.

Proposition 4.2. Let (f, t, X) be a retractive transfer triple over an elementary abelian group S, and let W ≤ Aut(S) be the subgroup of automorphisms of S acting trivially on H^{∗}(X). Then

F_{f,S} = F_{S}(W ⋉ S),

and in particular F_{f,S} is saturated.

Proof. Let V and V′ be two subgroups in S, and let ι_{V} and ι_{V′} be the inclusions V ↪ S and V′ ↪ S, respectively. By definition of F_{f,S}, a morphism φ: V → V′ is in Hom_{F_{f,S}}(V, V′) if and only if

f ∘ Bι_{V′} ∘ Bφ ≃ f ∘ Bι_{V} .

By Theorem 4.1, this is true if and only if the corresponding equality

Bφ^{∗} ∘ Bι_{V′}^{∗} ∘ f^{∗} ≃ Bι_{V}^{∗} ∘ f^{∗}  (3)

holds in cohomology.

Similarly, let θ be the natural map BS → B(W ⋉ S)_{p}^{∧}. Then a morphism φ: V → V′ is in Hom_{F_{S}(W ⋉S)}(V, V′) if and only if

θ ∘ Bι_{V′} ∘ Bφ ≃ θ ∘ Bι_{V} .

Again, by Theorem 4.1, this is true if and only if the corresponding equality

Bφ^{∗} ∘ Bι_{V′}^{∗} ∘ θ^{∗} ≃ Bι_{V}^{∗} ∘ θ^{∗}  (4)

holds in cohomology.

By the Stable Elements Theorem [9, Theorem XII.10.1], we have

H^{∗}(B(W ⋉ S)) ≅ H^{∗}(BS)^{W} ,

so the two maps f^{∗} and θ^{∗} are both the inclusion

H^{∗}(BS)^{W} ↪ H^{∗}(BS).

Therefore the two conditions (3) and (4) in cohomology agree and

φ ∈ Hom_{F_{f,S}}(V, V′)

if and only if

φ ∈ Hom_{F_{S}(W ⋉S)}(V, V′).

Saturation follows because W has order prime to p by Proposition 3.12. □

5. Comparing classifying spaces

So far we have proved that if (f, t, X) is a retractive transfer triple over an elementary abelian finite p-group S, then the induced fusion system F_{f,S} := F_{f,S}(X) agrees with the saturated fusion system F_{S}(W ⋉S), where W := Aut_{F}(S) has order prime to p. This fusion system has a unique associated centric linking system L_{S}(W ⋉ S) by the results in Section 1.4. Thus we have shown that to the triple (f, t, X) one can associate a unique p-local finite group (S, F_{f,S}, L_{S}(W ⋉ S)).

In this section we use Wojtkowiak’s obstruction theory to compare the classifying space |L_{S}(W ⋉ S)|_{p}^{∧} = B(W ⋉ S)_{p}^{∧} of the p-local finite group

(S, F_{f,S}, L_{S}(W ⋉ S)) to the space X. We will prove that BS ⟶^{f} X is homotopy equivalent to BS ⟶^{θ} B(W ⋉ S)_{p}^{∧} as spaces under BS, thus completing the proof of Theorem 2.4.

In [22, 23], Wojtkowiak develops an obstruction theory for the existence and uniqueness of maps from homotopy colimits to p-complete spaces. This obstruction theory is quite general, and it is interesting to note that, when applied to the classifying spaces of p-local finite groups, Wojtkowiak’s obstructions to the existence of maps are identical to the obstructions to uniqueness of centric linking systems in [6]. It is this phenomenon which we will use to our advantage in this section.

The particular result we wish to apply is reproduced below. For the sake of simplicity we state the result only in a special case, relevant to our needs.

Theorem 5.1. [22, 23] Let S be a finite abelian p group and W a group of order prime to p that acts on S. For any nilpotent p-local space X of Z_{(p)}-finite type with trivial W -action, the natural map

[B(W ⋉ S), X] ⟶^{−∘Bi_{S}} [BS, X]^{W}

is a bijection.

The application of the theorem allows us to prove the following.

Proposition 5.2. Let (f, t, X) be a retractive transfer triple over an elementary abelian group S and put W := Aut_{F_{f,S}}(S). Let θ be the p-completed inclusion BS ↪ B(W ⋉ S)_{p}^{∧}. Then there is a homotopy equivalence

h: (θ, B(W ⋉ S)_{p}^{∧}) ⟶^{≃} (f, X)

of spaces under BS.

Proof. Since X is p-complete, it is in particular p-local. By Lemma 3.3, X is of finite Z_{(p)}-type. Thus Theorem 5.1 applies to give a bijection

[B(W ⋉ S), X] ⟶^{−∘Bι} [BS, X]^{W} ,

where Bι is the inclusion BS ↪ B(W ⋉ S) (before completion).

Now, f ∈ [BS, X]^{W} by definition of W. Hence the bijection yields a map h: B(W ⋉ S) → X such that

h ∘ Bι ≃ f.

Upon applying the cohomology functor H^{∗}(−) we see, as in the proof of Proposition 4.2, that the induced maps f^{∗} and Bι^{∗} are both the inclusion

H^{∗}(BS)^{W} ↪ H^{∗}(BS).

From Bι^{∗} ∘ h^{∗} = f^{∗}, we can therefore deduce that h induces an isomorphism in cohomology. After p-completion we therefore get a homotopy equivalence

h_{p}^{∧}: B(W ⋉ S)_{p}^{∧} ⟶^{≃} X_{p}^{∧} ≃ X

such that

h_{p}^{∧} ∘ θ = h_{p}^{∧} ∘ ι_{p}^{∧} ≃ f.
□

6. p-local finite groups induce retractive transfer triples

In Section 2 the notion of a retractive transfer triple over a finite p-group S was introduced and in Sections 3 to 5 it was shown that, in the case where S is elementary abelian, such a triple induces a p-local finite group. In this section we consider the reverse implication and prove the following theorem.

Theorem 6.1. Let (S, F, L) be a p-local finite group. Then the natural inclusion θ: BS → |L|_{p}^{∧} has a retractive transfer t, and (θ, t, |L|_{p}^{∧}) is a retractive transfer triple.

There are two parts to the proof. In Subsection 6.1, which is joint work with Ran Levi, we show that |L|_{p}^{∧} and the inclusion θ: BS → |L|_{p}^{∧} of the Sylow subgroup satisfy the technical conditions of a retractive transfer triple. Most notably we show that the classifying space of a p-local finite group is both a torsion space and a nilpotent space, a result which is of independent interest.

In Subsection 6.2 we apply results from [20] to obtain a stable retraction t of the inclusion θ: BS → |L|_{p}^{∧}, and show that it satisfies Frobenius reciprocity. These results combine to complete the proof of Theorem 6.1

6.1. Technical conditions. Let (S, F, L) be a p-local finite group. In this subsection, which is joint work with Ran Levi, we verify that the space |L|_{p}^{∧} and the natural map

θ: BS → |L|_{p}^{∧}

satisfy the technical conditions of retractive transfer triples. It has already been shown in [6] that |L|_{p}^{∧} is p-complete, that the fundamental group of |L|_{p}^{∧} is finite and that θ is a homotopy monomorphism. We proceed to show that |L|_{p}^{∧} is nilpotent.

Lemma 6.2. For a p-local finite group (S, F, L), the groups H^{k}(|L|_{p}^{∧}; ℤ) and H_{k}(|L|_{p}^{∧}; ℤ) are finite p-groups for all k ≥ 1.

Proof. As is shown in [6], |L|_{p}^{∧} is a stable retract of BS. In particular H^{∗}(|L|_{p}^{∧}; ℤ) is a subring of H^{∗}(BS; ℤ). Since H^{k}(BS; ℤ) is a finite p-group for k ≥ 1 (a transfer argument shows that H^{k}(BS; ℤ) is p-torsion and finite generation is evident from the cell structure of BS,) it follows that H^{k}(|L|_{p}^{∧}; ℤ) is a finite p-group for k ≥ 1. Applying the universal coefficient theorem now allows us to deduce the result for homology groups. □

Proposition 6.3. For a p-local finite group (S, F, L), the homotopy groups π_{k}(|L|_{p}^{∧}) are finite p-groups for all k ≥ 1. In particular, |L|_{p}^{∧} is a torsion space.

Proof. We first reduce to the case where |L|_{p}^{∧} is simply connected. It is shown in [6] that π_{1}(X) is a finite p-group. Letting X̃ be a universal cover of |L|_{p}^{∧}, it therefore suffices to show that the homotopy groups of X̃ are all finite p-groups. But it is shown in [4] that X̃ is again the classifying space of a p-local finite group, so we can reduce to the simply connected case.

Assume therefore that |L|_{p}^{∧} is simply connected. By Lemma 6.2, H_{k}(|L|_{p}^{∧}; ℤ) is a finite p-group for all k ≥ 1. Since |L|_{p}^{∧} is simply connected, we can apply the Hurewicz theorem modulo the class of finite abelian p-groups, and deduce that the homotopy groups of |L|_{p}^{∧} are all finite p-groups. □

Corollary 6.4. The classifying space of a p-local finite group is nilpotent.

Proof. This follows from Proposition 6.3, the fact that any finite p-group is nilpotent and the fact that the action of any finite p-group on a finite abelian p-group is nilpotent. □

6.2. The retractive transfer of a p-local finite group. In this subsection we show that the natural inclusion of a Sylow subgroup into the classifying space of a p-local finite group has a retractive transfer. We will use results from [20], which were actually originally developed for this purpose but have turned out to be perhaps more interesting than their intended goal, and are therefore published separately.

Remark 6.5. There is a slight difference between the spectra appearing in this paper and the ones appearing in [20]. The difference arises because here we add a basepoint to our spaces before forming suspension spectra. The effect at the level of stable homotopy is to add a sphere wedge summand to all spectra in sight. It is easy to check that all the results quoted from [20] carry over in the form stated in this section.

Let (S, F, L) be a p-local finite group, and let ω̃ be the idempotent of Σ_{+}^{∞}BS induced by the characteristic idempotent of F, as defined in [20]. We refer to ω̃ as the pointed stable idempotent of F. It has the following 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 nontrivial subgroup of S, φ ∈ Hom_{F}(P, S) and tr_{P} denotes the transfer of the inclusion P ≤ S.

(b) 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.

In addition there is a non-triviality condition, which, combined with Properties (a) and (b), suffices to determine ω̃ uniquely. But as this property is not needed here we omit the statement. We refer to Property (b) as F-invariance.

The pointed classifying spectrum of F is the stable summand 𝔹_{+}F of Σ_{+}^{∞}BS induced by ω̃. This is the infinite mapping telescope of ω̃:

𝔹_{+}F = HoColim (Σ_{+}^{∞}BS ⟶^{ω̃} Σ_{+}^{∞}BS ⟶^{ω̃} Σ_{+}^{∞}BS ⟶^{ω̃} ⋯) ,

and as such it comes equipped with a pointed structure map, Σ_{+}^{∞}BS ⟶^{σ} 𝔹_{+}F, which is the structure map of the homotopy colimit, and a unique (up to homo-

topy) map 𝔹_{+}F ⟶^{t} Σ_{+}^{∞}BS such that σ ∘ t ≃ id_{𝔹_{+}F} and t ∘ σ ≃ ω̃.

Just as in [20], one can show that the pointed structure map Σ_{+}^{∞}BS ⟶^{σ} 𝔹_{+}F

is equivalent to the infinite pointed suspension Σ_{+}^{∞}BS ⟶^{Σ_{+}^{∞}θ} Σ_{+}^{∞}|L|_{p}^{∧} of the nat-

ural inclusion BS ⟶^{θ} |L|_{p}^{∧}, as objects under Σ_{+}^{∞}BS. We may therefore replace

Σ_{+}^{∞}BS ⟶^{σ} 𝔹_{+}F by Σ_{+}^{∞}BS ⟶^{Σ_{+}^{∞}θ} Σ_{+}^{∞}|L|_{p}^{∧} in the discussion above, and obtain a unique (up to homotopy) map

t : Σ_{+}^{∞}|L|_{p}^{∧} ⟶ Σ_{+}^{∞}BS,

such that

Σ_{+}^{∞}θ ∘ t ≃ id_{Σ_{+}^{∞}|L|_{p}^{∧}}  and  t ∘ Σ_{+}^{∞}θ = ω̃.

We proceed to show that t satisfies the Frobenius reciprocity relation illustrated in (1) and thus t is a retractive transfer for θ.

Proposition 6.6. The idempotent ω̃ satisfies the Frobenius reciprocity relation

(ω̃ ∧ ω̃) ∘ Δ ≃ (ω̃ ∧ 1) ∘ Δ ∘ ω̃,

where Δ: Σ_{+}^{∞}BS → Σ_{+}^{∞}BS ∧ Σ_{+}^{∞}BS is the image of the diagonal of BS under the infinite suspension functor Σ_{+}^{∞}.

Proof. Recall that we can write ω̃ as a linear combination with Z_{p}^{∧}-coefficients of maps Σ_{+}^{∞}Bφ ∘ tr_{P} ∈ {BS_{+}, BS_{+}}, where P ≤ S and φ ∈ Hom_{F}(P, S). For such a map Σ_{+}^{∞}Bφ ∘ tr_{P} we have, by F-invariance of ω̃,

ω̃ ∘ Σ_{+}^{∞}Bφ ≃ ω̃ ∘ Σ_{+}^{∞}Bι_{P} ,  (5)

where ι_{P} is the inclusion P ≤ S. We will take advantage of this and the fact [1] that the transfer tr_{P} of the inclusion ι_{P} satisfies the Frobenius relation

(1 ∧ tr_{P}) ∘ Δ_{S} ≃ (Σ_{+}^{∞}Bι_{P} ∧ 1) ∘ Δ_{P} ∘ tr_{P} ,  (6)

where Δ_{P} and Δ_{S} are the diagonals of Σ_{+}^{∞}BP and Σ_{+}^{∞}BS respectively. We will also use the fact that, since Σ_{+}^{∞}Bφ has a desuspension, it commutes with the diagonals as follows

Δ_{S} ∘ Σ_{+}^{∞}Bφ ≃ (Σ_{+}^{∞}Bφ ∧ Σ_{+}^{∞}Bφ) ∘ Δ_{P} .  (7)

Now,

(ω̃ ∧ 1) ∘ Δ_{S} ∘ Σ_{+}^{∞}Bφ ∘ tr_{P} ≃^{(7)} (ω̃ ∧ 1) ∘ (Σ_{+}^{∞}Bφ ∧ Σ_{+}^{∞}Bφ) ∘ Δ_{P} ∘ tr_{P}

≃ ((ω̃ ∘ Σ_{+}^{∞}Bφ) ∧ Σ_{+}^{∞}Bφ) ∘ Δ_{P} ∘ tr_{P}

≃^{(5)} ((ω̃ ∘ Σ_{+}^{∞}Bι_{P}) ∧ Σ_{+}^{∞}Bφ) ∘ Δ_{P} ∘ tr_{P}

≃ (ω̃ ∧ Σ_{+}^{∞}Bφ) ∘ (Σ_{+}^{∞}Bι_{P} ∧ 1) ∘ Δ_{P} ∘ tr_{P}

≃^{(6)} (ω̃ ∧ Σ_{+}^{∞}Bφ) ∘ (1 ∧ tr_{P}) ∘ Δ_{S}

≃ (ω̃ ∧ (Σ_{+}^{∞}Bφ ∘ tr_{P})) ∘ Δ_{S}.

By summing over the different Σ_{+}^{∞}Bφ ∘ tr_{P}, we get the desired result. □

Corollary 6.7. Let (S, F, L) be a p-local finite group and let t: Σ_{+}^{∞}|L|_{p}^{∧} → Σ_{+}^{∞}BS be as constructed above. Then t is a retractive transfer of the inclusion θ: BS ⟶ |L|_{p}^{∧}.

Proof. We deduce the Frobenius reciprocity relation

(1 ∧ t) ∘ Δ_{|L|_{p}^{∧}} ≃ (Σ_{+}^{∞}θ ∧ 1) ∘ Δ_{S} ∘ t

from

(ω̃ ∧ ω̃) ∘ Δ_{S} ≃ (ω̃ ∧ 1) ∘ Δ_{S} ∘ ω̃  (8)

as follows. Applying (Σ_{+}^{∞}θ ∧ 1) ∘ − ∘ t to the left hand side of (8) and rewriting, we get

(Σ_{+}^{∞}θ ∧ 1) ∘ (ω̃ ∧ ω̃) ∘ Δ_{S} ∘ t ≃ (Σ_{+}^{∞}θ ∧ 1) ∘ ((t ∘ Σ_{+}^{∞}θ) ∧ (t ∘ Σ_{+}^{∞}θ)) ∘ Δ_{S} ∘ t
≃ ((Σ_{+}^{∞}θ ∘ t) ∧ t) ∘ (Σ_{+}^{∞}θ ∧ Σ_{+}^{∞}θ) ∘ Δ_{S} ∘ t
≃ (1 ∧ t) ∘ Δ_{|L|_{p}^{∧}} ∘ Σ_{+}^{∞}θ ∘ t
≃ (1 ∧ t) ∘ Δ_{|L|_{p}^{∧}}

Doing the same with the right hand side yields

(Σ_{+}^{∞}θ ∧ 1) ∘ (ω̃ ∧ 1) ∘ Δ_{S} ∘ ω̃ ∘ t ≃ ((Σ_{+}^{∞}θ ∘ t ∘ Σ_{+}^{∞}θ) ∧ 1) ∘ Δ_{S} ∘ (t ∘ Σ_{+}^{∞}θ ∘ t)
≃ (Σ_{+}^{∞}θ ∧ 1) ∘ Δ_{S} ∘ t.

By (8) we can take the last two equations together to form a new equation

(1 ∧ t) ∘ Δ_{|L|_{p}^{∧}} ≃ (Σ_{+}^{∞}θ ∧ 1) ∘ Δ_{S} ∘ t.
□

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

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