THE RATIONAL COHOMOLOGY OF A p-LOCAL COMPACT GROUP

C. BROTO, R. LEVI, AND B. OLIVER

Let p be a prime number. In [BLO3], we developed the theory of p-local compact groups. The theory is modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups, and generalises the earlier concept of p-local finite groups [BLO2]. It provides a coherent context in which classifying spaces of compact Lie groups and p-compact groups [DW] can be studied, and also gives rise to many exotic examples. In this paper, we study the rational p-adic cohomology

H_{ℚ_{p}}^{∗}(−) =^{def} H^{∗}(−, ℤ_{p}) ⊗ ℚ_{p}

of a p-local compact group. Our main result here is that, as one would expect, the p-adic rational cohomology of p-local compact groups behaves the same way as that of a compact Lie group.

Theorem A. Let G = (S, F, L) be a p-local compact group. Let S_{0} ≤ S be its maximal torus, and let W (G) =^{def} Aut_{F}(S_{0}) be its Weyl group. Then

H_{ℚ_{p}}^{∗}(BG) ≅ H_{ℚ_{p}}^{∗}(BS_{0})^{W (G)}.

Of course, the Weyl group of a p-local compact group need not be a pseudo-reflection group, and hence the rational cohomology of the classifying space is not in general a polynomial algebra.

Like compact Lie groups and p-compact groups, p-local compact groups admit unstable Adams operations, which are defined in [JLL], using the internal structure of the p-local group in question, rather than its rational cohomology. One application of Theorem A is Proposition 3.2, which states that under a mild condition, the obvious cohomological definition of an unstable Adams operation characterises the same family of maps as the one referred to in [JLL] as ”geometric unstable Adams operations”.

Another easy application of Theorem A is the observation that if G is a p-local compact group with maximal torus S_{0}, then the inclusion in G of the p-local subgroup given by the normaliser N_{G}(S_{0}) induces a rational p-adic cohomology isomorphism.

In Section 1, we recall the basic concepts in the theory of p-local compact groups which will be needed to prove Theorem A. Section 2 is dedicated to the proof of the theorem. Finally in Section 3 we discuss the applications described above.

2010 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D20.

Key words and phrases. Classifying space, p-completion, p-local compact groups, fusion.

C. Broto is partially supported by FEDER-MICINN grant MTM 2010-20692.

R. Levi is partially supported by EPSRC grant EP/I019073/1.

B. Oliver is partially supported by UMR 7539 of the CNRS, and by project ANR BLAN08-2 338236, HGRT.

1. Some basic concepts

We recall the definition and some basic properties of p-local compact groups. The reader is referred to [BLO3] for a comprehensive account of these objects.

We begin by defining discrete p-toral groups. By ℤ/p^{∞} we mean the union of all ℤ/p^{r} with respect to the natural inclusions. Definition 1.1. A discrete p-torus is a group isomorphic to (ℤ/p^{∞})^{r} for some positive integer r. A discrete p-toral group is a group S which contains a normal discrete p-torus S_{0}, with p-power index. The normal subgroup S_{0} will be referred to as the maximal torus or the identity component of S, and the quotient group Γ ≅ S/S_{0} will be called the group of components of S.

The identity component S_{0} of a discrete p-toral group S can be characterised as the subset of all infinitely p-divisible elements in S, and also as the unique minimal subgroup of finite index in S. Thus, S_{0} is a characteristic subgroup. The rank of S is the number r = rk(S) such that S_{0} ≅ (ℤ/p^{∞})^{r}.

Recall that for P, Q ≤ S, the transporter set T_{S}(P, Q) is the set of all elements g ∈ S such that gP g^{−1} ≤ Q. We denote by Hom_{S}(P, Q) the set of all homomorphisms c_{g}: P → Q , which are restrictions of an inner automorphism of S, and by Inj(P, Q) denote the set of all the injective homomorphisms P → Q. We are now ready to recall the definition of fusion systems over discrete p-toral groups. Definition 1.2. A fusion system F over a discrete p-toral 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.

Two subgroups P, P′ ≤ S are called F-conjugate if P and P′ are isomorphic as objects in F. A subgroup P ≤ S is said to be F-centric if for every subgroup P′ ≤ S which is F-conjugate to P, C_{S}(P′) = Z(P′).

All fusion systems considered in this paper are required to be saturated [BLO3, Definition 2.2]. Although the results we present here are based on properties of saturated fusion systems proved in [BLO3], we do not explicitly use the saturation axioms, and thus we will not repeat them here.

Next, we briefly recall what are centric linking systems and p-local compact groups. The full definition can be found in [BLO3, Definitions 4.1, 4.2] Definition 1.3. Let F be a fusion system over a discrete p-toral 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) 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:

\xymatrix{P \ar[r]^{f} \ar[d]_{δ_{P} (g)} & Q \ar[d]^{δ_{Q}(π(f)(g))} \\ P \ar[r]^{f} & Q.}

A p-local compact group is a triple G = (S, F, L), where S is a discrete p-toral group, F is a saturated fusion system over S, and L is a centric linking system associated to F. The classifying space of G is the p-completed nerve |L|_{p}^{∧}, which we will generally denote by BG.

In [BLO3], the authors show that compact Lie groups and p-compact groups give rise to particular examples of p-local compact groups. Another large family of examples arises from linear torsion groups. In each case, the respective classifying space coincides up to homotopy (after p-completion in the case of genuine groups) with the classifying space of the p-local compact group it gives rise to.

Definition 1.4. Let G = (S, F, L) be a p-local compact group. Then the Weyl group W (G) of G is defined to be the automorphism group in F of the maximal torus S_{0} ≤ S.

Notice that H_{ℚ_{p}}^{∗}(X) =^{def} H^{∗}(X, ℤ_{p})⊗ℚ is not in general isomorphic to H^{∗}(X, ℚ_{p}). For instance if X = Bℤ/p^{∞}, then H_{ℚ_{p}}^{∗}(X) is a polynomial ring over the p-adic rationals on a generator in degree 2, while H^{∗}(X, ℚ_{p}) is trivial. The use of H_{ℚ_{p}}^{∗} as the appropriate cohomology theory for our purpose goes back to Dwyer and Wilkerson [DW], in their first paper on p-compact groups.

2. The rational cohomology

Two preparatory lemmas are needed before we prove our main claim.

Lemma 2.1. Let P be a discrete p-toral group with maximal torus P_{0} ≤ P. Then H_{ℚ_{p}}^{∗}(BP) ≅ H_{ℚ_{p}}^{∗}(BP_{0})^{P/P_{0}}.

Proof. This is of course a particular case of a much more general statement. Up to homotopy, BP_{0} is a covering space of BP with group P/P_{0}, and so one has the usual transfer map

Tr: H^{∗}(BP_{0}, ℤ_{p}^{∧}) → H^{∗}(BP, ℤ_{p}^{∧}),

where Tr∘Res is multiplication by |P/P_{0}|. Hence after tensoring with ℚ this composite is an isomorphism. On the other hand, the composition the other way Res ∘ Tris norm map for the action of P/P_{0} on H_{ℚ_{p}}^{∗}(BP_{0}), and hence the image of restriction is the subgroup of invariants H_{ℚ_{p}}^{∗}(BP_{0})^{P/P_{0}}. □

To prove the theorem, we will use the subgroup decomposition for p-local compact groups [BLO3, Proposition 4.6]. Hence the following lemma is an essential ingredient. In order to state it, we need to recall some notation and terminology.

For a fusion system F over a discrete p-toral group S, we denote by O(F) the orbit category associated to F, i.e., the category with the same objects and with morphisms Mor_{O(F)}(P, Q) = Rep_{F}(P, Q) =^{def} Hom_{F}(P, Q)/Inn(Q). For P, Q ∈ S, let N_{S}(P, Q) denote the transporter set consisting of all elements of S which conjugate P into Q. If F′ is a full subcategory of F, we denote by O(F′) the full subcategory of O(F) whose objects are those of F′. If Γ is a finite group, we denote by O_{p}(Γ) the category whose objects are the p-subgroups of Γ and whose morphisms are Mor(P, Q) = C_{Γ}(P)\N_{Γ}(P, Q)/Inn(Q).

Lemma 2.2. Let F be any saturated fusion system over a discrete p-toral group S. Define

F^{∗}: O(F^{c})^{op} ⟶ ℚ-mod

on objects by setting F^{∗}(P) = H_{ℚ_{p}}^{∗}(BP). On morphisms, F^{∗} sends the class of φ ∈ Hom_{F}(P, P′) to the homomorphism induced by Bφ. Then F^{∗} is acyclic, namely \underleftarrow{lim}^{i}_{O(F^{c})}(F^{∗}) = 0 for all i > 0.

Proof. Set Q = C_{S}(S_{0}) ◁ S, and Γ = Out_{F}(Q). Then Q is F-centric, and is weakly closed in F since S_{0} is. Let F_{≥Q} denote the full subcategory of F whose objects are those P ≤ S which contain Q, and let

Θ: O(F_{≥Q}) ⟶ O_{p}(Γ)

be the functor which sends an object P to Out_{P} (Q) ≤ Γ, and a morphism φ ∈ Rep_{F}(P, P′) to the class of φ|_{Q} ∈ N_{Γ}(Θ(P), Θ(P′)) (see [BLO3, Lemma 5.7]). For each p-subgroup Π ≤ Γ, regarded as a group of automorphisms of S_{0}, define

Φ^{∗}(Π) = H_{ℚ_{p}}^{∗}(BS_{0})^{Π}.

This defines a graded functor Φ^{∗}: O_{p}(Γ)^{op} → ℚ-mod. Furthermore, for each P ≤ S which contains Q,

F^{∗}(P) = H_{ℚ_{p}}^{∗}(BQ)^{P/Q} = Φ^{∗}(Θ(P)).

Thus Φ^{∗} ∘ Θ ≅ F^{∗}|_{O(F_{≥Q})}.

For each P ≤ S, Out_{Q}(P) acts trivially on F^{∗}(P) since Q centralises P_{0}, and F^{∗}(P) is a subring of H_{ℚ_{p}}^{∗}(BP_{0}). So by [BLO3, Lemma 5.7],

\underleftarrow{lim}^{∗}_{O(F^{c})}(F^{∗}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(Γ)}(Φ^{∗}).

The functor Φ^{∗} is a Mackey functor on O_{p}(Γ), and hence is acyclic (see [JM, Proposition 5.14] or [JMO, Proposition 5.2]). □

We are now ready to prove our main theorem.

Theorem 2.3. Let G = (S, F, L) be a p-local compact group. Then

H_{ℚ_{p}}^{∗}(BG) ≅ H_{ℚ_{p}}^{∗}(BS_{0})^{W (G)}.

Proof. Let π: L → O(F^{c}) be the projection, and let B̃: O(F^{c}) → Top denote the left homotopy Kan extension of the constant functor on L along π. Then there is a homotopy equivalence

hocolim_{O(F^{c})} B̃ → |L|,

and for each object P ∈ O(F^{c}), B̃(P) ≃ BP [BLO3, Proposition 4.6]. Consider the Bousfield-Kan spectral sequence [BK] for cohomology of the homotopy colimit, with coefficients in the p-adic integers ℤ_{p}. Since ℚ is flat as a ℤ-module, one can tensor the spectral sequence with ℚ to get a spectral sequence for p-adic rational cohomology

E_{2}^{p,q} = \underleftarrow{lim}^{p}_{O(F^{c})} H_{ℚ_{p}}^{q}(B̃(−)) ⟹ H_{ℚ_{p}}^{p+q}(|L|).

By Lemma 2.2, the higher limits all vanish and we obtain the formula

H_{ℚ_{p}}^{∗}(|L|) ≅ \underleftarrow{lim}_{O(F^{c})} H_{ℚ_{p}}^{∗}(B̃(−)).  (1)

For each F-centric subgroup P ≤ S, let ι_{P} : P → S denote the inclusion. The inverse limit in (1) consists of all elements x ∈ H_{ℚ_{p}}^{∗}(BS) such that φ^{∗} ∘ ι_{Q}^{∗}(x) = ι_{P}^{∗} (x) for all F-centric subgroups P, Q ≤ S, and all morphisms φ ∈ Hom_{F}(P, Q).

Let φ: P → Q be any morphism in F, where P and Q are F-centric. Then by [BLO3, Lemma 2.4] the restriction φ|_{P_{0}} coincides with the restriction to P_{0} of some automorphism σ ∈ W (G). Let x ∈ H_{ℚ_{p}}^{∗}(BS_{0})^{W (G)} ≤ H_{ℚ_{p}}^{∗}(BS) be any element. Then ι_{P}^{∗} (x) ∈ H_{ℚ_{p}}^{∗}(BP) ≤ H_{ℚ_{p}}^{∗}(P_{0}), and ι_{Q}^{∗}(x) ∈ H_{ℚ_{p}}^{∗}(BQ) ≤ H_{ℚ_{p}}^{∗}(BQ_{0}), and

φ^{∗}(ι_{Q}^{∗}(x)) = σ^{∗}(ι_{Q}^{∗}(x)) = ι_{P}^{∗} σ^{∗}(x) = ι_{P}^{∗} (x).

Hence

H_{ℚ_{p}}^{∗}(BS_{0})^{W (G)} ≤ \underleftarrow{lim}_{O(F^{c})} H_{ℚ_{p}}^{∗}(B̃(−)).

Conversely, let y ∈ H_{ℚ_{p}}^{∗}(BS) ≤ H_{ℚ_{p}}^{∗}(BS_{0}) be an element which is stable under each morphism in F between centric subgroups, and let σ ∈ W (G). By Alperin’s fusion theorem, σ can be decomposed into a sequence σ = σ_{1} ∘ σ_{2} ∘ ⋯ σ_{n}, where each σ_{i} ∈ W (G) can be extended to an automorphism of some F-centric subgroup P_{i} ≤ S. But since y is stable under each of the σ_{i}^{∗}, it is also stable under σ^{∗}. This shows that

\underleftarrow{lim}_{O(F^{c})} H_{ℚ_{p}}^{∗}(B̃(−)) ≅ H_{ℚ_{p}}^{∗}(BS_{0})^{W (G)}

and thus completes the proof of our claim. □

3. Applications

For a compact Lie group G, one defines an unstable Adams operation of degree ζ to be a selfmap of the classifying space inducing multiplication by ζ^{i} on rational cohomology in dimension 2i, where ζ is an integer. An analogous definition is made for p-compact groups, except ζ is required to be a p-adic unit, and rational cohomology is replaced by p-adic rational cohomology. Unstable Adams operations are a very important concept in the homotopy theory of classifying spaces of compact Lie groups and p-compact groups.

In [JLL], it is shown that p-local compact groups also admit unstable Adams operations. Let G = (S, F, L) be a p-local compact group and let ζ be a p-adic unit. A normal Adams automorphism of degree ζ on S is an automorphism ϕ ∈ Aut(S) which restricts to the ζ-power map on S_{0}, and induces the identity on the group of components S/S_{0}. A geometric unstable Adams operation of degree ζ on G is a selfmap Ψ of BG, such that there exist a normal Adams automorphism ϕ of degree ζ on S, with the property that ι ∘ Bϕ ≃ Ψ ∘ ι. Here ι: BS → BG is the canonical inclusion. (See [JLL, Definitions 2.3, 3.4]) Theorem A allows us to define geometric unstable Adams operations of p-local compact groups, along the lines of the classical cohomological definition.

The following lemma is an analogue of a theorem of Notbohm [N, Proposition 4.1].

Lemma 3.1. Let G = (S, F, L) be a p-local compact group, and let T be a discrete p-torus. Then there is an isomorphism

Hom(T, S_{0})/W (G) ⟶^{≅} [BT, BG],

where W (G) acts by left translation. Also, two maps f, h: BT → BG are homotopic if and only if they induce the same homomorphism on H_{ℚ_{p}}^{∗}(−).

Proof. By [BLO3, Theorem 6.3 (a)] there is an isomorphism of sets

Rep(T, L) ⟶^{≅} [BT, BG],

where Rep(T, L) =^{def} Hom(T, S)/∼, with α ∼ β if and only if there is some φ ∈ Hom_{F}(α(T), β(T)) such that φ ∘ α = β. Since T is a discrete p-torus, the image of every homomorphism from it to S is contained in S_{0}, and by [BLO3, Lemma 2.4 (b)], every homomorphism in F between subgroups of S_{0} is the restriction of some element in W (G). Thus

Rep(T, L) ≅ Hom(T, S_{0})/W (G),

as claimed.

It remains to prove the last statement. Two maps f, h: BT → BG that are homotopic clearly induce the same map on cohomology. Conversely, assume that f, h: BT → BG are two maps such that f^{∗} = g^{∗}: H_{ℚ_{p}}^{∗}(BG) ⟶ H_{ℚ_{p}}^{∗}(BT). Let α, β: T ⟶ S_{0} be homomorphisms such that f = ι ∘ Bα and g = ι ∘ Bβ, where ι: BS_{0} ⟶ BG is the inclusion of the maximal torus.

We will show that there is w ∈ W (G) such that w ∘ α = β, following the argument used by Adams and Mahmud to prove [AM, Theorem 1.7]: an argument based on the uniqueness of factorisation in the polynomial ring H_{ℚ_{p}}^{∗}(BS_{0}). For simplicity, write V = H_{ℚ_{p}}^{2}(BS_{0}) and V′ = H_{ℚ_{p}}^{2}(BT). For each w ∈ W (G), define

V (w) = { x ∈ V | Bβ^{∗}(x) = B(w ∘ α)^{∗}(x) } = Ker((Bβ^{∗} − B(w ∘ α)^{∗})|_{V} ) .

For each x ∈ V, set

x̂ = ∏_{w∈W (G)} Bw^{∗}(1 + x) ∈ S(V) ≅ H_{ℚ_{p}}^{∗}(BS_{0})

where S(V) denotes the symmetric algebra on the ℚ_{p}-vector space V. Since x̂ is W (G)- invariant, Theorem 2.3 implies that x̂ ∈ Im(ι^{∗}), and hence that Bα^{∗}(x̂) = Bβ^{∗}(x̂). In other words,

∏_{w∈W (G)}(1 + Bα^{∗}Bw^{∗}x) = ∏_{w∈W (G)}(1 + Bβ^{∗}Bw^{∗}x) ∈ S(V′) .

Since S(V′) is a unique factorization domain, there is w ∈ W (G) such that (1+Bβ^{∗}x) = λ(1 + Bα^{∗}Bw^{∗}x), for some λ ∈ ℚ_{p}^{×}. Then λ = 1 and hence Bβ^{∗}x = Bα^{∗}Bw^{∗}x. In particular, x ∈ V (w).

This proves that V = ⋃_{w∈W (G)} V (w). Since ℚ_{p} is infinite, V finite dimensional, and W (G) finite, there is w ∈ W (G) such that V = V (w) (cf. [AM, Lemma 3.1]). Hence Bβ^{∗} = B(w ∘ α)^{∗}. Since Hom(T, S_{0}) injects into Hom(H_{ℚ_{p}}^{∗}(BS_{0}), H_{ℚ_{p}}^{∗}(T)), it now follows that w ∘ α = β ∈ Hom(T, S_{0}), and hence that f ≃ g as maps BT ⟶ BG. □

Proposition 3.2. Let G = (S, F, L) be a p-local compact group, and let ζ be a p-adic unit. Then any geometric unstable Adams operation Ψ of degree ζ induces multiplication by ζ^{i} on H_{ℚ_{p}}^{2i}(BG) . If S_{0} is self centralising in S, then any self equivalence Ψ of BG which induces multiplication by ζ^{i} on H_{ℚ_{p}}^{2i}(BG) for each i is a geometric unstable Adams operation on F.

Proof. Let ι: BS → BG be the canonical inclusion (induced by the distinguished monomorphism δ_{S}: S → Aut_{L}(S)), and set ι_{0} = ι|_{S_{0}}. If ψ is a geometric unstable Adams operation on G of degree ζ, then by definition, there exists a normal Adams automorphism ϕ of S such that Ψ ∘ ι ≃ ι ∘ Bϕ, and hence Ψ ∘ ι_{0} ≃ ι_{0} ∘ B(ϕ|_{S_{0}}). For each i ≥ 0, ϕ|_{S_{0}} induces multiplication by ζ^{i} on H_{ℚ_{p}}^{2i}(BS_{0}), and hence Ψ does the same on H_{ℚ_{p}}^{2i}(BG).

Assume now that S_{0} is self centralising in S. Let Ψ: BG → BG be a self equivalence such that Ψ^{∗} is multiplication by ζ^{i} on H_{ℚ_{p}}^{2i}(BG). By [BLO3, Theorem 6.3(a)] and Lemma 3.1, the natural maps

End(S)/ Aut_{F}(S) ⟶^{≅} [BS, BG] and End(S_{0})/W (G) ⟶^{≅} [BS_{0}, BG]  (2)

are bijections. Hence there is φ ∈ End(S) such that ι ∘ Bφ ≃ Ψ ∘ ι, and φ ∈ Aut(S) since Ψ is a homotopy equivalence. Let φ_{0} ∈ Aut(S_{0}) be the restriction of φ to S_{0}, let ζ̂ denote the ζ-power map on S_{0}, and set ρ = ζ̂ ∘ φ_{0}^{−1} ∈ Aut(S_{0}). Then

Bφ_{0}^{∗} ∘ ι_{0}^{∗} = Bζ̂^{∗} ∘ ι_{0}^{∗}: H_{ℚ_{p}}^{∗}(BG) ⟶ H_{ℚ_{p}}^{∗}(BS_{0}),

and by Lemma 3.1, there is w ∈ W (G) such that w ∘ φ_{0} = ζ̂ ∈ Aut(S_{0}).

Fix a morphism ι̂ ∈ Mor_{L}(S_{0}, S) such that π(ι̂) is the inclusion, and regard this as the inclusion of S_{0} in S in the category L. By [BLO3, Lemma 4.3(a)], for each g ∈ S, there is a unique restriction δ_{S_{0}}(g) ∈ Aut_{L}(S_{0}) of δ_{S}(g) ∈ Aut_{L}(S); i.e., a unique morphism such that ι̂∘ δ_{S_{0}}(g) = δ_{S}(g) ∘ ι̂. Identify S and S_{0} with their images in Aut_{L}(S_{0}). Let α ∈ Aut_{L}(S_{0}) be a lift of w, i.e., π(α) = w. By Axiom (C), for each

t ∈ S_{0}, α ∘ δ_{S_{0}}(t) = δ_{S_{0}}(w(t)) ∘ α. Hence, c_{α}|_{S_{0}} = w, and so χ =^{def} c_{α}|_{S} ∘ φ: S → Aut_{L}(S_{0})

restricts to w ∘ φ_{0} = ζ̂ on S_{0}.

Now, for each g ∈ S, χ ∘ c_{g} = c_{χ(g)} ∘ χ as automorphisms of S_{0}, and since χ|_{S_{0}} = ζ̂ is central in Aut(S_{0}), c_{g}|_{S_{0}} = c_{χ(g)}|_{S_{0}}. Since S_{0} is self centralising in S, it follows that for each g ∈ S, g ≡ χ(g) (mod S_{0}). In particular, χ(S) = S, and χ induces the identity on S/S_{0}. Thus χ is a normal Adams automorphism of S of degree ζ. Also,

ι ∘ Bχ ≃ ι ∘ B(c_{α}|_{S}) ∘ Bφ ≃ ι ∘ Bφ ≃ Ψ ∘ ι ,

and thus Ψ is a geometric unstable Adams operation on G as claimed. □

If G = (S, F, L) is a p-local compact group, and P ≤ S is a subgroup satisfying a certain mild condition (fully normalised), then one can define the normaliser fusion system, N_{F}(P), which is shown in [BLO6, Theorem 2.3] to be a saturated fusion system. The normaliser linking system N_{L}(P) can be defined in exactly the same way as in [BLO2, Definition 6.1], and the proof of [BLO2, Lemma 6.2] applies verbatim to show that N_{L}(P) is a centric linking system associated to N_{F}(P). Thus in this case N_{G}(P) = (N_{S}(P), N_{F}(P), N_{L}(P)) is a p-local compact subgroup of G.

In particular, the maximal torus S_{0}, is fully normalised, since it is unique in its F-conjugacy class, and we may consider the inclusion

N_{G}(S_{0}) ⟶ G .  (3)

Then, S_{0} is the maximal torus in both G and N_{G}(S_{0}), and from the definition of morphisms in the normaliser fusion system [BLO6, Definition 2.1],

W (G) = Aut_{F}(S_{0}) = Aut_{N_{F} (S_{0})}(S_{0}) = W (N_{G}(S_{0})).

Thus one obtains as an immediate corollary of Theorem A, that the inclusion (3) induces an isomorphism in p-adic rational cohomology. This is analogous to the corresponding statements for compact Lie groups and p-compact groups.

References

[AM] J.F. Adams & Z. Mahmud, Maps between classifying spaces, Inv. Math. 35 (1976), 1–41 [BK] P. Bousfield & D. Kan, Homotopy limits, completions, and localizations, Lecture notes in

math. 304, Springer-Verlag (1972)

[BLO2] C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, Journal Amer. Math. Soc. 16 (2003), 779–856

[BLO3] C. Broto, R. Levi, & B. Oliver, Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups, Geom. & Topol. 11 (2007), 315427.

[BLO6] C. Broto, R. Levi, & B. Oliver, In Preparation.

[DW] W. G. Dwyer, & C. W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces. Ann. of Math. (2) 139 (1994), no. 2, 395–442.

[JLL] F. Junod, R. Levi and A. Libman, Unstable Adams operations of p-local compact groups, Alg. and Geom. Topol. 12 (2012) 4974

[JM] Stefan Jackowski, J. McClure, Homotopy decomposition of classifying spaces via elementary abelian subgroups. Topology 31 (1992), no. 1, 113132.

[JMO] S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of self-maps of BG via Gactions, Annals of Math. 135 (1992), 184–270

[N] D. Notbohm, Abbildungen zwischen klassifizierenden Räume, Dissertation, Göttingen, 1988

Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain

  E-mail address: broto@mat.uab.es

Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, U.K.

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

LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

  E-mail address: bob@math.univ-paris13.fr