The K-THEORY OF p-COMPACT GROUPS

A. Jeanneret & A. Osse^{∗}

Introduction

Let G be a compact Lie group and BG its classifying space. Atiyah and Segal have shown that the complex K-theory ring K^{∗}(BG; ZZ) is isomorphic to the I-adic completion of the complex representation ring R(G). Assume G is connected and fix a maximal torus T ⊂ G with Weyl group W. The preceding result is equivalent to the isomorphism

K^{∗}(BG; ZZ) ≅ K^{∗}(BT; ZZ)^{W} ,  (∗)

where the last term stands for the ring of invariants of the natural W -action on K^{∗}(BT; ZZ).

In [12] Dwyer and Wilkerson introduce the concept of p-compact group, where p stands for a prime. Their original results and subsequent works ([13], [19], [20]) show that these objects constitute a natural homotopy theoretic generalization of compact Lie groups. For instance, p-compact groups have maximal tori, Weyl groups, etc. We refer to section 1 for the precise definitions. In this introduction, we would like to emphasize the fact that all the structure of a p-compact group is concentrated at the single prime p. Therefore it is natural to consider p-adic K-theory rather than ordinary K-theory. In this framework, our main result is the following generalization of the isomorphism (∗):

Theorem Let X be a connected p-compact group, i : T ⟶ X a maximal torus and W the corresponding Weyl group. The classifying map Bi : BT ⟶ BX induces a ring isomorphism

K^{∗}(BX; ZZp̂) ≅ K^{∗}(BT; ZZp̂)^{W} .

∗Supported by grant No 20-43215.95 of the Swiss National Fund for Scientific Research

As a first application we study the ordinary complex K-theory of finite loop spaces. More precisely if L is a connected finite loop space, we show that K^{1}(BL; ZZ) = 0 and K^{0}(BL; ZZ) is torsion free and without zero divisor. Our results are much more complete for finite loop spaces with maximal tori. In the case of a compact connected Lie group, we obtain a non-analytical proof of the isomorphism (∗). Moreover we have the following generalization of a result due to Notbohm and Smith ([23, Theorem 5.1]):

Theorem Let L be a connected finite loop space. Then L admits a maximal torus if and only if there exists a compact connected Lie group G such that K^{∗}(BL; ZZ) is λ-isomorphic to K^{∗}(BG; ZZ).

In Section 1, we recall the basic definitions of the theory of p-compact groups. In the same section we use a theorem of Kane and Lin to deduce that the p-adic K-theory of a 1-connected p-compact group is an exterior algebra. As the reader will see, our arguments depend heavily on this result. Section 2 contains the crucial step of the proof of our main result, namely the map Bi : BT ⟶ BX above induces a finite ring homomorphism in mod p K-theory. To achieve this goal, we appeal to Dwyer’s transfer and we show that any p-compact toral group P “embeds” into some unitary group U(N). The technical Section 3 deals with the reduction of the general situation to the 1-connected case. In Section 4 we combine Dwyer’s transfer with Kane and Lin’s theorem and the finiteness result of Section 2 to conclude. Some consequences of our main result are given in Section 5. The last section is devoted to the announced application to finite loop spaces.

Notations. Throughout the paper, p is a fixed prime number, ZZp̂ the ring of p-adic integers and Qp̂ = Q ⊗ ZZp̂ the fraction field of ZZp̂. For any space Y the symbol HQp̂^{∗}(Y) stands for Q ⊗ H^{∗}(Y; ZZp̂) and Yp̂ for the Bousfield-Kan

p-completion of Y.

1 Backgrounds

The purpose of this section is to fix the notation and to recall the definitions and the results we are going to use.

A loop space X is a triple (X, BX; e), where X is a space, BX a connected pointed space and e : X ⟶ ΩBX a homotopy equivalence; BX is called the classifying space of X. Such a loop space will be called a p-compact group if the following additionnal conditions are satisfied:

1. X is IF_{p}-finite, i.e, H^{∗}(X; IF_{p}) is a finite dimensional IF_{p}-vector space; 2. π_{0}(X) is a finite p-group and π_{n}(X) is a finitely generated ZZp̂-module

for any n ≥ 1.

A morphism f : X ⟶ Y between two p-compact groups is a pointed map Bf : BX ⟶ BY. The morphism f is a monomorphism (respectively, an epimorphism) if the homotopy fiber Y/X of Bf is IF_{p}-finite (respectively, the classifying space of a p-compact group). A short exact sequence X ⟶^{f} Y ⟶^{g} Z of p-compact groups is a sequence such that BX ⟶^{Bf} BY ⟶^{Bg} BZ is a fibration up to homotopy. Two morphisms f, g : X ⟶ Y are conjugate if the maps Bf and Bg are freely homotopic. The centralizer of a morphism f : X ⟶ Y of p-compact groups is the loop space

C_{Y} (f(X)) := (ΩMap(BX, BY)_{Bf}, Map(BX, BY)_{Bf}; id) .

The morphism f will be called central if the basepoint evaluation map ev : Map(BX, BY)_{Bf} ⟶ BY is a homotopy equivalence.

A p-compact torus (of rank n) is a p-compact group T such that BT ≃ K(ZZp̂^{n}, 2). A p-compact toral group P is a p-compact group fitting into a short exact sequence T ⟶ P ⟶ π, where T is a p-compact torus and π a finite p-group. A maximal torus for a p-compact group X is a monomorphism i : T ⟶ X whose centralizer is a p-compact toral group. One of the fundamental results of [12] says that any p-compact group admits a maximal torus, unique up to conjugacy.

Let i : T ⟶ X be a maximal torus for a p-compact group X. We replace the map Bi : BT ⟶ BX by an equivalent fibration BT′ ⟶ BX; the Weyl space W_{T} (X) is defined as the space of self-maps of BT′ over BX. In Proposition 9.5 of [12], it is shown that the space W_{T} (X) is homotopically discrete and W_{T} (X) := π_{0}(W_{T} (X)) is a finite group under composition; W_{T} (X) (or simply W_{X}) is called the Weyl group of X (with respect to the maximal torus i). By construction the Weyl space W_{T} (X) acts on BT′; the Borel construction of this action gives a loop space called the normalizer of T and denoted N (T). Thus we have a homotopy fibration sequence BT ⟶ BN (T) ⟶ BW_{T} (X). In general, the loop space N (T) is not a p-compact group. However one obtains a p-compact group N_{p}(T) by performing the Borel construction for the action of the submonoid W_{T} (X)_{p} given by the union of components of W_{T} (X) which project onto a p-Sylow subgroup of W_{T} (X); N_{p}(T) is called the p-normalizer of T.

For future use, some other important results of [12] are recorded in the following Theorem 1.1 Let i : T ⟶ X be a maximal torus for a connected p-compact group X, with Weyl group W_{X}.

1. The homotopy action of W_{X} on BT induces a faithful representation of W_{X} as a reflection group in the Qp̂-vector space HQp̂^{2}(BT).

2. The map Bi induces a ring isomorphism

HQp̂^{∗}(BX) ≅ HQp̂^{∗}(BT)^{W_{X}} .

Throughout the paper we will be dealing with ZZ/2-graded K-theories. Following Sullivan, we define the K-theory with p-adic coefficients as the inverse limit

K^{∗}(−; ZZp̂) = lim_{←} K^{∗}(−; ZZ/p^{r}) ,

where K^{∗}(−; ZZ/p^{r}) denotes the K-theory with ZZ/p^{r}-coefficients. It can be shown that K^{0}(−; ZZp̂) is represented by ZZp̂ × BUp̂, while K^{1}(−; ZZp̂) is represented by Up̂; as usual U stands for the infinite unitary group.

The K-theory of 1-connected mod p finite H-spaces has been computed by Kane and Lin (see [14, §44-1]). For the p-compact groups, their result implies

Theorem 1.2 Let X be a 1-connected p-compact group.

1. The K-theory of X is an exterior algebra

K^{∗}(X; ZZp̂) ≅ E_{ZZp̂}(η_{1}, ...η_{r}) ,

where the generators η_{1}, ...η_{r} are in K^{1}(X; ZZp̂).

2. The K-theory of BX is a power series ring

K^{∗}(BX; ZZp̂) ≅ ZZp̂[[ξ_{1}, ...ξ_{r}]] ,

where the generators ξ_{1}, ...ξ_{r} are in K^{0}(BX; ZZp̂).

Proof. By Theorem 1.1 above and Theorem 3.1 in [4], there is a CWcomplex W of finite type with BX ≃ Wp̂. Since BX is 2-connected, the construction in [4] also shows that W is 1-connected. Given this fact, Proposition VI.6.5 of [9] implies that (ΩW)p̂ ≃ Ω (Wp̂) ≃ X. Hence standard arguments show that H_{∗}(ΩW; ZZ_{(p)}) is finitely generated over ZZ_{(p)}. As ΩW is an associative H-space, we can use Corollary 10.4 of [7] to show that K^{∗}(X; ZZp̂) ≅ K^{∗}(ΩW; ZZp̂) is an exterior algebra over ZZp̂. The second assertion follows from a Rothenberg-Steenrod spectral sequence argument and the fact that there are no phantoms in p-adic K-theory. □

We write HQp̂^{∗∗}(−) for the direct product ∏_{n≥0} HQp̂^{n}(−); it is a ZZ/2-

graded Qp̂-algebra, graded by:

HQp̂^{even}(−) = ∏_{n≥0} HQp̂^{2n}(−), HQp̂^{odd}(−) = ∏_{n≥0} HQp̂^{2n+1}(−) .

The Chern character

ch : K^{∗}(−; ZZp̂) ⟶ HQp̂^{∗∗}(−)

is a ZZ/2-graded ring homomorphism whose definition is as in [15, page 282].

Proposition 1.1 Let X be a 1-connected p-compact group.

1. The rationalization of the Chern character

ch ⨂ Qp̂ : K^{∗}(X; ZZp̂) ⨂ Qp̂ ⟶ HQp̂^{∗∗}(X)

is an isomorphism.

2. The Chern character

ch : K^{∗}(BX; ZZp̂) ⟶ HQp̂^{∗∗}(BX)

is a monomorphism.

Note. For trivial algebraic reasons, the rationalization of the Chern character for BX is not necessarily surjective. One has to use completed tensor product in order to obtain an isomorphism.

Proof. Let us first consider the case of the space X. Proceeding as above, there is a 1-connected CW-complex of finite type V whose p-completion is homotopy equivalent to X. Take any finite subcomplex Ṽ of V such that Ṽ ↪ V induces an isomorphism in cohomology with ZZp̂-coefficients and observe that the composition

Ṽ ↪ V → Vp̂ ≃ X

induces an isomorphism in p-adic K-theory. The problem is now reduced to the case of a finite complex where the assertion is true.

To treat the case of BX, let us fix the exterior generators η_{1}, ...η_{r} of K^{∗}(X; ZZp̂). By the first part of the proof, we have HQp̂^{∗}(X) ≅ E_{Qp̂}(y_{1}, ...y_{r}),

where y_{i} = ch(η_{i}) for i = 1, …, r. The map α : ΣX ⟶ BX, adjoint to the homotopy equivalence e : X ⟶ ΩBX, induces a commutative diagram

K^{∗}(BX; ZZp̂) \overline{α^{∗}_{}}K^{∗}(ΣX; ZZp̂)

[figure: HQp̂^{∗∗}(BX); ch; α^{∗∗}; HQp̂^{∗∗}(ΣX); ch]

Now observe that the generators ξ_{1}, ...ξ_{r} of K^{∗}(BX; ZZp̂) may be chosen so that α^{∗}(ξ_{i}) = σ(η_{i}), with σ the suspension isomorphism. Similarly, it is possible to choose generators x_{1}, ..., x_{r} of HQp̂^{∗∗}(BX) such that α^{∗∗}(x_{i}) = σ(y_{i}).

The commutative diagram above implies that ch(ξ_{i}) is congruent, modulo decomposables, to x_{i}. We now invoke Theorem 1.2 to conclude the proof. □

2 A finiteness result

Let R and S be graded commutative rings; a ring homomorphism ϕ : R ⟶ S is called finite if S is a finitely generated ϕ(R)-module. The present section is devoted to the proof of

Theorem 2.1 Let X be a p-compact group and i : T ⟶ X a maximal torus. Then the map Bi induces a finite ring homomorphism

Bi^{∗} : K^{∗}(BX; IF_{p}) ⟶ K^{∗}(BT; IF_{p}) .

This result will follow from a sequence of five propositions. The principal ingredients are the ”main theorem” of [12] and Dwyer’s transfer ([11]).

Proposition 2.1 If X is a p-compact group, then there are homogeneous classes b_{1}, ..., b_{r} ∈ H^{∗}(BX; IF_{p}) which generate a polynomial subalgebra R = IF_{p}[b_{1}, ..., b_{r}] in H^{∗}(BX; IF_{p}) and such that the inclusion R ⊂ H^{∗}(BX; IF_{p}) is finite.

Proof. By Theorem 2.4 of [12] (alias the ”main theorem”), H^{∗}(BX; IF_{p}) is a finitely generated graded algebra over IF_{p}. It suffices then to invoke the graded version of Noether normalization theorem (as stated for example in [5, Theorem 2.2.7]). □

Before going farther, a well-known observation is in order. Suppose that there exists a monomorphism h : X ⟶ U(n)p̂, where U(n)p̂ denotes the p-compact group obtained by p-completing the unitary group U(n). Let j : T (n) ⟶ U(n)p̂ be a maximal torus; then there exists a monomorphism g : T ⟶ T (n) making the following diagram commutative:

BT (n)
[figure: Bg]
BT ⟶^{Bi} BX ⟶^{Bh}BU(n)p̂^{^{Bj}}

The map Bg induces a surjective (hence finite!) homomorphism in mod p K-theory. As well-known Bj^{∗} : K^{∗}(BU(n)p̂; IF_{p}) ⟶ K^{∗}(BT (n); IF_{p}) is also finite. It follows that Bi^{∗} : K^{∗}(BX; IF_{p}) ⟶ K^{∗}(BT; IF_{p}) is finite.

Unfortunately we don’t know yet if every p-compact group admit a monomorphism into some U(n)p̂. But for the present purpose, the following proposition will be sufficient.

Proposition 2.2 Let P be a p-compact toral group. Then there exists a monomorphism ϕ : P ⟶ U(n)p̂ for some integer n.

Proof. If P is finite the answer is well-known; thus we may assume that P fits into a short exact sequence T ⟶ P ⟶ π, where T is a p-compact torus of rank r ≥ 1 and π a finite p-group. The discrete approximation of this sequence gives rise to a short exact sequence of discrete groups (see Proposition 3.7 in [13])

{1} ⟶ T̆ ⟶ P̆ ⟶ π ⟶ {1} ,

with T̆ = (ZZ/p^{∞})^{r}. We write l for the order of π and embed the latter into the symmetric group Σ_{l}, via the regular representation. By a theorem of Kaloujnine and Krasner (Theorem 7.37 in [25]), there exists an embedding of P̆ into the wreath product T̆ ≀ Σ_{l} ≅ (ZZ/p^{∞} ≀ Σ_{l})^{r}. Our candidate for the monomorphism ϕ is the p-completion of the composite

P̆ ↪^{α} (ZZ/p^{∞} ≀ Σ_{l})^{r} ↪^{β} (S^{1} ≀ Σ_{l})^{r} ↪^{γ} U(l)^{r} ⊂ U(lr) .

The morphism γ is just the inclusion of the normalizer of the standard maximal torus of U(l)^{r}. By a result of Quillen (see [24]), Bγ and the classifying map of the last inclusion induce finite homomorphisms on mod p cohomology. The morphism β is given by the natural inclusion ZZ/p^{∞} ⊂ S^{1}; therefore Bβ induces an isomorphism on mod p cohomology. By Proposition 9.11 of [12], we will be done if we can prove that the composite

B(β ∘ α) : BP̆ ⟶ B(S^{1} ≀ Σ_{l})^{r}

induces a finite morphism on mod p cohomology.

In the proof of Proposition 12.1 of [12], it is shown that there exists a finite subgroup inclusion ν : P_{m} ↪ P̆ with Bν^{∗} : H^{∗}(BP̆; IF_{p}) ⟶ H^{∗}(BP_{m}; IF_{p}) injective and finite. Consider now the diagram

P_{m}
[figure: β∘α∘ν]
[figure: P̆; ν; ∩; ⊂; β∘α; (S^{1} ≀ Σ_{l})^{r}]

The ring homomorphism B(β∘α∘ν)^{∗} is finite, by Quillen’s result ([24]). Since H^{∗}(B(S^{1} ≀ Σ_{l})^{r}; IF_{p}) is noetherian and Bν^{∗} injective, the ring homomorphism B(β ∘ α)^{∗} is finite; it follows that Bϕ^{∗} is also finite. □

Proposition 2.3 Let X be a p-compact group and i : T ⟶ X a maximal torus. If the Atiyah-Hirzebruch spectral sequence

E_{2}^{∗,∗}(BX) = H^{∗}(BX; K^{∗}(pt; IF_{p})) ⟹ K^{∗}(BX; IF_{p})

degenerates (i.e, there exists an integer n_{0} such that E_{n_{0}}(BX) ≅ E_{∞}(BX)), then the ring homomorphism Bi^{∗} : K^{∗}(BX; IF_{p}) ⟶ K^{∗}(BT; IF_{p}) is finite.

Proof. We write Z_{n}(BX) ⊂ E_{2}(BX) for the pre-image of the submodule of cycles in E_{n}(BX). We clearly have a chain of inclusions

E_{2}(BX) ⊃ Z_{2}(BX) ⊃ ... ⊃ Z_{n}(BX) ⊃ ... ⊃ Z_{∞}(BX) = ⋂_{n≥2} Z_{n}(BX) .

For n ≥ 2, set

R_{n} = IF_{p}[b_{1}^{p^{n−1}}, ..., b_{r}^{p^{n−1}}] ⊗ K^{∗}(pt; IF_{p}) ⊂ E_{2}(BX) ,

where b_{1}, ..., b_{r} are as in Proposition 2.1. For all n ≥ 2, the inclusion R_{n} ⊂ E_{2}(BX) is finite and (by construction) R_{n} is contained in Z_{n}(BX). By hypothesis there is an integer n_{0} such that Z_{n_{0}}(BX) = Z_{∞}(BX), thus R_{n_{0}} ⊂ Z_{∞}(BX). Since i : T ⟶ X is a monomorphism, Bi induces a finite ring homomorphism in mod p cohomology; it follows that the composite

R_{n_{0}} ↪ E_{2}(BX) ⟶^{Bi^{∗}} E_{2}(BT)

is also finite. At the E_{∞}-level, we obtain a diagram

R_{n_{0}} ⟶ E_{∞}(BX) ⟶^{E_{∞}(Bi^{∗})} E_{∞}(BT) ,

whose composite coincides with the preceding one (since E_{2}(BT) = E_{∞}(BT)). In particular the morphism R_{n_{0}} ⟶ E_{∞}(BT) is finite; this imply the finiteness of the map E_{∞}(Bi^{∗}) : E_{∞}(BX) ⟶ E_{∞}(BT). We now invoke Corollary 1 (page 41) of [6] to conclude that Bi^{∗} : K^{∗}(BX; IF_{p}) ⟶ K^{∗}(BT; IF_{p}) is finite as claimed. □

Proposition 2.4 Let X be a p-compact group and i : T ⟶ X a maximal torus. Let N_{p} denote the p-normalizer of this maximal torus. If the Atiyah-Hirzebruch spectral sequence degenerates for BN_{p}, then it also degenerates for BX.

Proof. Let j : N_{p} ⟶ X be the canonical monomorphism and

τ : (BX_{+})p̂ ⟶ ((BN_{p})_{+})p̂

be the transfer associated to the map Bj (see [11, Example 1.1]). We consider the diagram

\begin{aligned}E_{2}(BX) & ⊃ Z_{2}(BX) ⊃ ... ⊃ Z_{n}(BX) ⊃ & ... & ⊃ & Z_{∞}(BX) \\ E_{2}(BN_{p}) & [figure: ^{τ∗}; ^{Bj∗}] & ⊃ Z_{2}(BN_{p}) ⊃ ... ⊃ Z_{n}(BN_{p}) ⊃ & ... & ⊃ & Z_{∞}(BN_{p}) \end{aligned}

Since τ^{∗} is induced by a stable map, we have τ^{∗}(Z_{n}(BN_{p})) ⊂ Z_{n}(BX) for all n ≥ 2. By hypothesis there exists an integer n_{0} such that Z_{n_{0}}(BN_{p}) = Z_{∞}(BN_{p}). Let x ∈ Z_{n_{0}}(BX), then Bj^{∗}(x) ∈ Z_{n_{0}}(BN_{p}) = Z_{∞}(BN_{p}) and

x = χ^{−1} (τ^{∗} ∘ Bj^{∗})(x) ∈ Z_{∞}(BX) ,

where χ is the Euler characteristic of the space X/N_{p}. Dwyer and Wilkerson have shown that χ is invertible mod p ([12]). Thus we have proved that Z_{n_{0}}(BX) = Z_{∞}(BX). □

Proposition 2.5 For any p-compact toral group P, the Atiyah-Hirzebruch spectral sequence for BP degenerates.

Proof. By Proposition 2.2 there exists a monomorphism ϕ : P ⟶ U(n)p̂. Hence the induced homomorphism Bϕ^{∗} : H^{∗}(BU(n); IF_{p}) ⟶ H^{∗}(BP; IF_{p}) is finite. Since H^{∗}(BU(n); IF_{p}) is concentrated in even degrees, the naturality of the spectral sequence implies that R = Im(Bϕ^{∗}) ⊗ K^{∗}(pt; IF_{p}) consists of permanent cycles. We can now invoke Proposition 4.1 of [3] to conclude. □

To complete the proof of Theorem 2.1, it suffices to recall that the pnormalizer N_{p} is p-toral. We close the section by our main application of this theorem.

Theorem 2.2 Let X be a 1-connected p-compact group and i : T ⟶ X a maximal torus. The ring homomorphism

Bi^{∗} : K^{∗}(BX; ZZp̂) ⟶ K^{∗}(BT; ZZp̂)

makes K^{∗}(BT; ZZp̂) into a free and finitely generated K^{∗}(BX; ZZp̂)-module.

Proof. For simplicity, we set

S_{X} := K^{∗}(BX; ZZp̂) and S_{T} := K^{∗}(BT; ZZp̂) .

It is well-known that S_{T} is a power series ring over ZZp̂. By Theorem 1.2, the same is true for S_{X}. Consequently we have

K^{∗}(BX; IF_{p}) = S_{X} ⊗_{ZZp̂} IF_{p} and K^{∗}(BT; IF_{p}) = S_{T} ⊗_{ZZp̂} IF_{p} .

Our Theorem 2.1 and Theorem 8.4 (page 58) of [17], imply that the homomorphism Bi^{∗} : S_{X} ⟶ S_{T} is finite. We now observe that both rings are noetherian, local, regular and of the same dimension. And so we can apply Proposition 22 (page IV-37) of [26] to conclude that S_{T} is free over S_{X}. □

3 Reduction to the one-connected case

The following results of [20] and [19] will play an important role in this section.

Theorem 3.1 Let X be a connected p-compact group and π the torsion subgroup of π_{1}(X). Set Y = X < 1 > ×S, where X < 1 > denotes the 1connected cover of X and S a p-compact torus of rank dim_{Qp̂}(π_{1}(X) ⊗ Qp̂).

1. There is a short exact sequence of p-compact groups

{1} ⟶ π ⟶^{f} Y ⟶^{g} X ⟶ {1} ,

where the monomorphism f : π → Y is central.

2. Let i_{1} : T_{1} ⟶ X < 1 > and i : T_{X} ⟶ X be maximal tori for the respective p-compact groups. Set also T_{Y} = T_{1} × S, then

j = i_{1} × id : T_{Y} ⟶ Y

is a maximal torus and there exists a short exact sequence

{1} ⟶ π ⟶^{φ} T_{Y} ⟶^{γ} T_{X} ⟶ {1}

which makes the following diagram commutative (up to homotopy):

Bπ = Bπ

[figure: BT_{Y}; Bφ; Bj; BY; Bf]

[figure: BT_{X}; Bγ; Bi; BX; Bg]

3. Let W_{Y} (respectively W_{X}) denote the Weyl group of the maximal torus j (respectively i); note that, by construction, the group W_{Y} acts trivially on BS. There exists an isomorphism Φ : W_{Y} → W_{X} such that, for any w in the Weyl group W_{Y}, the diagram

[figure: BT_{Y}; w; BT_{Y}; BT_{X}; Bγ; Φ(w); BT_{X}; Bγ]

commutes (up to homotopy).

Proof. The first part is proved in [20, Theorem 5.4], while the last two are contained in [19, Theorem 2.5]. □

In order to exploit fully the theorem above, we must study some properties of central monomorphisms. The main tool of this study is an observation of Dwyer and Wilkerson:

Lemma 3.1 ([13], Lemma 5.3) Let

\begin{aligned}& & Z_{1} \\ Z_{2} & \overline{^{f_{2}}} & Z_{0} & ^{^{f_{1}}} \end{aligned}

be a diagram of p-compact groups and suppose that the morphism f_{2} is central. Then there is up to conjugacy a unique morphism of p-compact groups μ(f_{1}, f_{2}) : Z_{1} × Z_{2} → Z_{0} which is conjugate to f_{1} on Z_{1} × {1} and to f_{2} on {1} × Z_{2}.

Until further notice we will keep the data and notation of Theorem 3.1. The classifying space Bπ possesses a multiplication

Bμ : Bπ × Bπ → Bπ ;

the unit for Bμ will be chosen to be the basepoint of Bπ. Let R denote either a ring ZZ/p^{r} (r ≥ 1) or the p-adic ring ZZp̂. By a well-known result of Atiyah, K^{∗}(Bπ; R) is free and finitely generated over R. This fact implies (see [2, Lemma 1.4]) that K^{∗}(Bπ × BZ; R) is naturally isomorphic to K^{∗}(Bπ; R) ⊗_{R} K^{∗}(BZ; R) for any p-compact group Z. Thus the multiplication of Bπ induces a Hopf algebra structure on K^{∗}(Bπ; R) (see [2]).

By Lemma 3.1 there is up to conjugacy a unique morphism k : π×Y → Y which is conjugate to f on π × {1} and to the identity on {1} × Y. It easily checked that the homomorphism

Bk^{∗} : K^{∗}(BY; R) ⟶ K^{∗}(Bπ; R) ⊗_{R} K^{∗}(BY; R)

defines a K^{∗}(Bπ; R)-comodule structure on K^{∗}(BY; R).

Let us now consider the maximal torus j : T_{Y} → Y, with Weyl group W_{Y}. By Lemma 6.5 of [13], φ is up to conjugacy the unique morphism making the following diagram commutative

BT_{Y}
[figure: Bφ]
[figure: Bπ; Bf; BY; Bj]

This uniqueness implies that the morphism φ is W_{Y} -invariant, i.e, for any w in the Weyl group W_{Y}, the morphisms φ and w ∘ φ are conjugate.

We proceed as above to obtain a morphism κ : π × T_{Y} → T_{Y} (unique up to conjugacy) such that

Bκ^{∗} : K^{∗}(BT_{Y}; R) ⟶ K^{∗}(Bπ; R) ⊗_{R} K^{∗}(BT_{Y}; R)

defines a K^{∗}(Bπ; R)-comodule structure on K^{∗}(BT_{Y}; R). The uniqueness of κ and the W_{Y} -invariance of φ imply that, for all w ∈ W_{Y}, the induced map w^{∗} : K^{∗}(BT_{Y}; R) ⟶ K^{∗}(BT_{Y}; R) is a K^{∗}(Bπ; R)-comodule homomorphism. Once again Lemma 3.1 implies that Bj^{∗} : K^{∗}(BY; R) ⟶ K^{∗}(BT_{Y}; R) is a K^{∗}(Bπ; R)-comodule homomorphism.

Since BX is 1-connected, the fibration

Bπ ⟶^{Bf} BY ⟶^{Bg} BX

is equivalent to a principal one; more precisely there is a principal fibration

Bπ↪B̃X ⟶^{p} BX

and a homotopy equivalence h : BY → B̃X, such that the diagram

Bπ = Bπ

[figure: BY; Bf; h; B̃X; ∩]

[figure: BX; Bg; =; BX; p]

commutes (up to homotopy).

Finally we observe that the action of Bπ on B̃X defines a K^{∗}(Bπ; R)- comodule structure on K^{∗}(B̃X; R). As above, the uniqueness of Bk implies that the map h induces a K^{∗}(Bπ; R)-comodule isomorphism between the rings K^{∗}(B̃X; R) and K^{∗}(BY; R).

For future reference we record some of the assertions above in

Lemma 3.2 1. The map h : BY ⟶ B̃X induces an isomorphism of K^{∗}(Bπ; R)-comodules between K^{∗}(B̃X; R) and K^{∗}(BY; R).

2. For all w ∈ W, w^{∗} : K^{∗}(BT_{Y}; R) ⟶ K^{∗}(BT_{Y}; R) is a K^{∗}(Bπ; R)- comodule homomorphism.

3. Bj^{∗} : K^{∗}(BY; R) ⟶ K^{∗}(BT_{Y}; R) is a K^{∗}(Bπ; R)-comodule homomorphism.

We write A for K^{∗}(Bπ; ZZp̂) and A_{r} for K^{∗}(Bπ; ZZ/p^{r}). The universal coefficient formula implies K^{∗}(Bπ; ZZ/p^{r}) ≅ K^{∗}(Bπ; ZZp̂) ⨂ ZZ/p^{r}, that is, A_{r} ≅ A ⨂ ZZ/p^{r}.

The E_{2}-term of the the Rothenberg-Steenrod spectral sequence of the principal fibration

[figure: Bπ; ⊂; B̃X; BX; p]

is given by

E_{2}^{∗,∗} = Cotor_{A_{r}}^{∗}(K^{∗}(B̃X; ZZ/p^{r}); ZZ/p^{r}) .

Recall that Cotor_{A_{r}}^{n}(−; −) is the n-th derived functor of the cotensor product −□_{A_{r}}− (we refer to [2] for more details).

The next proposition is the main ingredient for the study of this spectral sequence.

Proposition 3.1 ([2], Prop. 3.5) For all A_{r}-comodules B and all n > 0,

Cotor_{A_{r}}^{n}(B; ZZ/p^{r}) = 0 .

The problem with the above fibration is that K^{∗}(B̃X; ZZ/p^{r}) is not a finite ZZ/p^{r}-module, hence the spectral sequence may not converge. This difficulty can be solved by replacing BX by the p-completion Zp̂ of a CW-complex of finite type Z ([4]). For the m-th skeleton Z^{(m)} of Z, consider the induced principal fibration

[figure: Bπ; ⊂; p^{−1}(Z^{(m)}); Z^{(m)}; p]

By Proposition 3.1, the Rothenberg-Steenrod spectral sequence of this fibration collapses. It follows that

K^{∗}(BX; ZZ/p^{r}) ≅ lim_{⟵} K^{∗}(Z^{(m)}; ZZ/p^{r})
≅ lim_{⟵} K^{∗}(p^{−1}(Z^{(m)}); ZZ/p^{r})□_{A_{r}}ZZ/p^{r}
≅ K^{∗}(B̃X; ZZ/p^{r})□_{A_{r}}ZZ/p^{r} .

Note that the third isomorphism is due to the fact that A_{r} is a free ZZ/p^{r}- module. By invoking Lemma 3.2, we conclude that the map Bg induces an isomorphism

K^{∗}(BX; ZZ/p^{r}) ≅ K^{∗}(BY; ZZ/p^{r})□_{A_{r}}ZZ/p^{r} .

Theorem 3.2 Let X be a connected p-compact group, π the torsion subgroup of π_{1}(X) and A = K^{∗}(Bπ; ZZp̂). Let Bπ ⟶^{Bf} BY ⟶^{Bg} BX be the fibration of Theorem 3.1. The map Bg induces an isomorphism

Bg^{∗} : K^{∗}(BX; ZZp̂) ⟶^{≅} K^{∗}(BY; ZZp̂)□_{A}ZZp̂ .  (1)

Note. The isomorphism (1) is equivalent to the exactness of the sequence

0 ⟶ K^{∗}(BX; ZZp̂) ⟶^{Bg^{∗}} K^{∗}(BY; ZZp̂) ⟶^{Ψ} K^{∗}(BY; ZZp̂) ⊗ A ,

where Ψ is the defining morphism for the cotensor product.

Proof. We give only the main steps. To begin with, we observe the following:

1. A ≅ lim_{⟵} A_{r} .

2. K^{∗}(BY; ZZp̂) ≅ lim_{⟵} K^{∗}(BY; ZZ/p^{r}) .

3. The modules A and A_{r} are free over their respective ground rings.

We apply the left exact functor lim_{⟵} (−) to the inverse system of exact sequences

0 ⟶ K^{∗}(BX; ZZ/p^{r}) ⟶^{Bg^{∗}} K^{∗}(BY; ZZ/p^{r}) ⟶^{Ψ} K^{∗}(BY; ZZ/p^{r}) ⊗ A_{r} .

The observations stated above lead to the result. □

Our next step is to investigate the relationship between the K-theory of BX and the K-theory of its maximal torus BT_{X}. Of course Theorem 3.2 applies to the fibration Bπ ⟶^{Bφ} BT_{Y} ⟶^{Bγ} BT_{X}. Combining this with Lemma 3.2 and Theorem 3.2 we obtain that, for any w in the Weyl group W_{X}, the following diagram (with exact rows) commutes:

[figure: 0 →; K^{∗}(BT_{X}; ZZp̂); Bγ^{∗}; K^{∗}(BT_{Y}; ZZp̂); ψ; K^{∗}(BT_{Y}; ZZp̂) ⊗ A]

[figure: 0 →; K^{∗}(BT_{X}; ZZp̂); w^{∗}; Bγ^{∗}; K^{∗}(BT_{Y}; ZZp̂); Φ(w)^{∗}; ψ; K^{∗}(BT_{Y}; ZZp̂) ⊗ A; Φ(w)^{∗}⊗id]

It follows that the projection Bγ : BT_{Y} ⟶ BT_{X} induces an isomorphism

K^{∗}(BT_{X}; ZZp̂)^{W_{X}} ≅ K^{∗}(BT_{Y}; ZZp̂)^{W_{Y}} □_{A}ZZp̂ .  (2)

The isomorphisms (1) and (2) fit together in the commutative diagram:

K^{∗}(BX; ZZp̂) ≅ K^{∗}(BY; ZZp̂)□_{A}ZZp̂

[figure: K^{∗}(BT_{X}; ZZp̂)^{W_{X}}; Bi^{∗}; ≅; K^{∗}(BT_{Y}; ZZp̂)^{W_{Y}} □_{A}ZZp̂; Bj^{∗}]

The theorem we want to prove asserts that the ring homomorphism Bi^{∗} : K^{∗}(BX; ZZp̂) ⟶ K^{∗}(BT_{X}; ZZp̂)^{W_{X}} is an isomorphism. Obviously this will be achieved if we can prove that Bj^{∗} is an isomorphism onto the ring of invariants K^{∗}(BT_{Y}; ZZp̂)^{W_{Y}}. We recall here that

BY = BX < 1 >×BS, BT_{Y} = BT_{1} × BS and Bj = Bi_{1} × id .

As the Weyl group W_{Y} acts trivially on S, it suffices to show that Bi_{1} induces an isomorphism onto the ring of invariants. In other words it is enough to deal with the 1-connected case. This is settled in the next section.

4 The one-connected case

In this section X is a 1-connected p-compact group, i : T ⟶ X a maximal torus and j : N ⟶ X the normalizer of i. Thus we have a commutative diagram

BT
[figure: Bi]
[figure: BN; Bh; Bj; BX]

Let τ : (BX_{+})p̂ ⟶ (BN_{+})p̂ be the transfer associated to the map Bj ([11, Example 1.1]). Since the Euler characteristic of X/N is equal to 1, the composite

(BX_{+})p̂ ⟶^{τ} (BN_{+})p̂ ⟶^{(Bi)p̂} (BX_{+})p̂

is a homotopy equivalence; its inverse will be denote ψ. We introduce further notations:

i) τ̃ is the composite

(BX_{+})p̂ ⟶^{ψ} (BX_{+})p̂ ⟶^{τ} (BN_{+})p̂ ;

ii) F is defined by the cofibre sequence

(BX_{+})p̂ ⟶^{τ̃} (BN_{+})p̂ ⟶^{ν} F ;

iii) θ denotes the composite

(BN_{+})p̂ ⟶^{∇} (BN_{+})p̂ ∨ (BN_{+})p̂ ⟶^{ν∨(Bj)p̂} F ∨ (BX_{+})p̂ .

Proposition 4.1 1. For E^{∗}(−) = H^{∗}(−; ZZp̂) or K^{∗}(−; ZZp̂), the map

E^{∗}(θ) : E^{∗}(F) ⨁ E^{∗}(BX_{+}) ⟶ E^{∗}(BN_{+}) , (y, z) ↦ ν^{∗}(y)+(Bj)^{∗}(z)

is an isomorphism.

2. For any y ∈ K^{∗}(F; ZZp̂), (Bh^{∗} ∘ ν^{∗})(y) = 0.

Proof. Let us first notice that E^{∗}(BX_{+}) ≅ E^{∗}((BX_{+})p̂), and similarly for BN_{+}. By the definition of τ̃, we have

(Bj)p̂ ∘ τ̃ ∼ id_{(BX_{+})p̂} .  (3)

Granted this fact, the long exact sequence in E^{∗}(−) of the cofibration

(BX_{+})p̂ ⟶^{τ̃} (BN_{+})p̂ ⟶^{ν} F

yields an inverse for E^{∗}(θ).

By the second part of Theorem 1.1, Bj^{∗} : HQp̂^{∗}(BX) ⟶ HQp̂^{∗}(BN) is

an isomorphism; hence (by what we have just proved), H̃Qp̂^{∗}(F) = 0. The

second claim follows from a diagram chase in

[figure: K^{∗}(F; ZZp̂) ⊕ K^{∗}(BX; ZZp̂); ≅; K^{∗}(BN; ZZp̂); Bh^{∗}; K^{∗}(BT; ZZp̂)]

[figure: HQp̂^{∗∗}(BX); ch; ≅; HQp̂^{∗∗}(BN); ch; Bh^{∗∗}; HQp̂^{∗∗}(BT); ch; ∩]
□

Proposition 4.2 The maps τ̃∘Bi and Bh induce the same homomorphism in p-adic K-theory, hence

Im(Bi^{∗}) = Im(Bh^{∗}) ⊂ K^{∗}(BT; ZZp̂) .

Proof. Let x ∈ K̃^{∗}(BN; ZZp̂). By Proposition 4.1, there exist y ∈ K̃^{∗}(F; ZZp̂) and z ∈ K̃^{∗}(BX; ZZp̂) such that x = ν^{∗}(y) + Bj^{∗}(z). On the one hand, Proposition 4.1 implies

Bh^{∗}(x) = Bh^{∗}(ν^{∗}(y) + Bj^{∗}(z))
= (Bh^{∗} ∘ ν^{∗})(y) + (Bj ∘ Bh)^{∗}(z)
= 0 + Bi^{∗}(z) .

On the other hand, (3) implies

(Bi^{∗} ∘ τ̃^{∗})(x) = (Bi^{∗} ∘ τ̃^{∗})(ν^{∗}(y) + Bj^{∗}(z))
= (Bi^{∗} ∘ τ̃^{∗} ∘ ν^{∗})(y) + Bi^{∗}((Bj ∘ τ̃)^{∗})(z)
= 0 + Bi^{∗}(z) .

And we have obtained Bi^{∗} ∘ τ̃^{∗} = Bh^{∗}. □

The preceding proposition reduces the determination of the image of Bi^{∗} to that of Bh^{∗}. A partial description of Im(Bh^{∗}), which is sufficient for our purpose, is provided by

Proposition 4.3 Let |W | denote the order of the Weyl group W = π_{0}(N), then

Im(Bh^{∗}) ⊗ ZZp̂[1/|W |] = K^{∗}(BT; ZZp̂)^{W} ⊗ ZZp̂[1/|W |] .

Proof. As easily checked, there is (up to homotopy) a fibre square

[figure: BT × W; p; BT; BT; pr_{1}; Bh; BN; Bh]

where p(x, w) = w · x and pr_{1}(x, w) = x, for any x ∈ BT and w ∈ W. Let tr be the transfer associated to the map Bh. Using the naturality and the product formulae for the transfer (see Theorem 2.6 and Theorem 2.8 in [11]) and applying p-adic K-theory, we obtain, for any ξ ∈ K^{∗}(BT; ZZp̂):

(Bh^{∗} ∘ tr^{∗})(ξ) = ∑_{w∈W} w^{∗}(ξ) .  (4)

The proposition is an immediate consequence of this equation. □

Note. As the reader may have noticed, the equality (4) is the well known double coset formula for finite coverings. If one could prove such a formula for the fibration Bi : BT ⟶ BX, then most of the arguments in this section would drastically simplify.

We are now in position to complete the proof of our main result:

Proposition 4.4 Let X be a 1-connected p-compact group, i : T ⟶ X a maximal torus and W the Weyl group. Then the classifying map Bi induces an isomorphism

K^{∗}(BX; ZZp̂) ≅ K^{∗}(BT; ZZp̂)^{W} .

Proof. The injectivity of Bi^{∗} follows from Proposition 1.1 and the commutativity of the diagram

[figure: K^{∗}(BX; ZZp̂); Bi^{∗}; K^{∗}(BT; ZZp̂); HQp̂^{∗∗}(BX); ch; ∩; ⊂; Bi^{∗∗}; HQp̂^{∗∗}(BT); ∩; ch]

To show that the image of Bi^{∗} is the ring of invariants, we consider the diagram

K^{∗}(BT; ZZp̂)^{W} ^{⊂} F rac(K^{∗}(BT; ZZp̂)^{W})

[figure: K^{∗}(BX; ZZp̂); Bi^{∗}; ∪; ⊂; F rac(K^{∗}(BX; ZZp̂)) ,]

where F rac(−) denotes the field of fractions of the corresponding integral domain. By Propositions 4.2 and 4.3, the right vertical map is an isomorphism. In Theorem 2.2, we showed that Bi^{∗} : K^{∗}(BX; ZZp̂) ↪ K^{∗}(BT; ZZp̂) is finite, in particular K^{∗}(BX; ZZp̂) ↪ K^{∗}(BT; ZZp̂)^{W} is integral. As K^{∗}(BX; ZZp̂) is integrally closed (it is a power series ring over ZZp̂), we obtain that the left vertical map is also an isomorphism. □

5 Applications to p-compact groups

Having completed the proof of our main result, we can safely turn to some of its consequences. The first one generalizes Proposition 1.2 in [23]:

Theorem 5.1 Let T be a p-compact torus and X a connected p-compact group. The natural map

[BT, BX] ⟶ Hom_{λ}(K^{∗}(BX; ZZp̂), K^{∗}(BT; ZZp̂))

is a bijection, where Hom_{λ}(−, −) stands for the set of λ-ring homomorphisms.

Proof Let i_{X} : T_{X} ⟶ X be a maximal torus for X and W_{X} the corresponding Weyl group. It is not difficult to check (or see Proposition 1.2 of [23]) the bijectivity of the natural map

[BT, BT_{X}] ⟶^{≅} Hom_{λ}(K^{∗}(BT_{X}; ZZp̂), K^{∗}(BT; ZZp̂)) .

The Weyl group W_{X} acts on the domain and codomain of this map, and the latter is equivariant with respect to these actions. As easily seen, left composition with Bi_{X} induces a commutative diagram

[BT, BT_{X}]/W_{X} \overline{≅_{}}Hom_{λ}(K^{∗}(BT_{X}; ZZp̂), K^{∗}(BT; ZZp̂))/W_{X}

[figure: [BT, BX]; Hom_{λ}(K^{∗}(BX; ZZp̂), K^{∗}(BT; ZZp̂))]

Proposition 8.11 of [12] and Proposition 4.1 of [21] imply that the left vertical arrow of the diagram is bijective. Next we combine our main result with Theorem 4.1 of [27] to obtain the surjectivity of the right vertical arrow. To show its injectivity, we first observe that a nonzero vector space over an infinite field cannot be a union of a finite number of its proper subspaces (see [16, page 78]). It follows that we can adapt the proof of [22, Lemma 7.1] to obtain our injectivity result. This completes the proof of our theorem. □

Our second application extends Proposition 1.1 to all connected p-compact groups.

Theorem 5.2 Let X be a connected p-compact group.

1. The Chern character

ch : K^{∗}(BX; ZZp̂) ⟶ HQp̂^{∗∗}(BX)

is a monomorphism.

2. For all x ∈ HQp̂^{∗}(BX), there exists ξ ∈ K^{∗}(BX; ZZp̂) such that

ch(ξ) = Mx + higher terms ,

where M denotes the order of the Weyl group of X.

Proof Let i_{X} : T_{X} ⟶ X be a maximal torus and W_{X} the corresponding Weyl group. The naturality of the Chern character and of the action of W_{X} implies the commutativity of the diagram

[figure: K^{∗}(BX; ZZp̂); Bi_{X}^{∗}; K^{∗}(BT; ZZp̂)^{W_{X}}; HQp̂^{∗∗}(BX); ch; Bi_{X}^{∗∗}; HQp̂^{∗∗}(BT)^{W_{X}}; ch]

To complete the proof, we invoke our main result and the fact that the claims are true for tori. □

The third application will be useful in the sequel:

Proposition 5.1 Let X be a connected p-compact group and let

E_{2}^{s,t} = H^{s}(BX; K^{t}(pt; ZZp̂) ⟹ K^{s+t}(BX; ZZp̂)

be the p-adic Atiyah-Hirzebruch spectral sequence for BX. Given s, t. there exists an integer r = r(s, t) such that E_{r}^{s,t} = E_{∞}^{s,t}.

Proof. Recall that there are no phantoms in p-adic K-theory. Thus one can adapt the “proof of necessity” of Theorem 3.3 in [10], but their reference to Theorem 3.2 of that paper has to be replaced by our Theorem 5.2. This change in their argument is needed because of the following example: the work of Anderson and Hodgkin ([2]) imply that K̃^{∗}(K(ZZ, 3); ZZp̂) = 0, but

HQp̂^{3}(K(ZZ, 3)) = Qp̂. □

6 Applications to finite loop spaces

In this section we study the K-theory of finite loop spaces; let us first recall the basic definitions.

A loop space (L, BL; e) is said finite if H_{∗}(L; ZZ) is a finitely generated ZZ-module. By a well-known result of Hopf, H^{∗}(L; Q) is an exterior algebra whose number of generators is called the rank of the finite loop space. A connected finite loop space L admits a maximal torus if there is a pointed map Bi : BT ⟶ BL, with BT = K(ZZ^{n}; 2), satisfying the two conditions:

1. The homotopy fiber L/T of Bi is ZZ-finite, that is, H_{∗}(L/T; ZZ) is a finitely generated ZZ-module.

2. The ranks of T and L are equal.

The Weyl group of a maximal torus i : T ⟶ L is defined exactly as in Section 1. The relationship between finite loop spaces and p-compact groups is given by

Proposition 6.1 ([21], Prop. 1.8) Let L be a connected finite loop space with maximal torus i : T ⟶ L . Then for any prime p:

1. The triple (Lp̂, BLp̂; ep̂) is a p-compact group and the p-completed map ip̂ : Tp̂ ⟶ Lp̂ is a maximal torus.

2. Let W (respectively, Wp̂) be the Weyl group of the maximal torus i (respectively, ip̂). The p-completion induces an isomorphism Wp̂ ≅ W.

Let us begin with a general result on the K-theory of finite loop spaces:

Theorem 6.1 Let L be a connected finite loop space. Then

1. The Chern character

ch : K^{0}(BL; ZZ) ⟶ H^{even}(BL; Q)

is injective. Consequently the ring K^{0}(BL; ZZ) is torsion free and has no zero divisor.

2. K^{1}(BL; ZZ) = 0.

Proof. For the first point, consider the commutative diagram

[figure: K^{0}(BL; ZZ); ê; ∏_{p} K^{0}(BL; ZZp̂); H^{even}(BL; Q); ch; ∏_{p} HQp̂^{even}(BL); (ch_{p})]

By Theorem 5.2, the right vertical map is injective. Since

H_{∗}(BL; π_{∗+1}(BU) ⊗ Q) = 0 ,

Lemma 2 in [18] implies that the map ê is also injective.

Let us deal with the second point. For any sufficiently large prime number p, H^{∗}(BL; ZZp̂) is concentrated in even degrees; hence the p-adic Atiyah-Hirzebruch spectral sequence is trivial for such p. This observation, our Proposition 5.1, Lemma 4.1 in [1] and Theorem 3.3 in [10] imply that:

K^{1}(BL; ZZ) ≅ lim_{←} K^{1}(BL^{(n)}; ZZ) ,

where the inverse limit is taken over the skeleta of BL. Now Lemma 2 in [18] implies that K^{1}(BL; ZZ) injects into ∏ K^{1}(BL; ZZp̂) and this product is zero by our main result. □

For finite loop spaces with maximal tori, we have an integral version of our main result:

Theorem 6.2 Let L be a connected finite loop space with maximal torus i : T ⟶ L and W the corresponding Weyl group. Then the map Bi induces a ring isomorphism:

K^{∗}(BL; ZZ) ≅ K^{∗}(BT; ZZ)^{W} .

Proof. The version of Sullivan’s arithmetic square given by W. Meier (in [18]) applies in our case. It gives rise to the following commutative diagram, with exact rows:

[figure: 0 →; K^{0}(BL; ZZ); K^{0}(BL; Q) ⊕ K^{0}(BL; ẐZ); K^{0}(BL; A_{f})]

[figure: 0 →; K^{0}(BT; ZZ); Bi^{∗}; K^{0}(BT; Q) ⊕ K^{0}(BT; ẐZ); Bi^{∗}; K^{0}(BT; A_{f}); Bi^{∗}]

(5)

where ẐZ is the profinite completion of ZZ and A_{f} = ẐZ ⊗ Q the ring of finite adeles.

Recall that if E is a Q-algebra (in particular, E = Q or E = A_{f}), then

K^{0}(−; E) ≅ ∏_{n≥0} H^{2n}(−; E) .

In our case, Theorem 1.2 of [21] implies that K^{0}(BL; E) ≅ K^{0}(BT; E)^{W}. On the one hand the map Bi : BT ⟶ BL, p-completion and Proposition 6.2 provide a commutative square

K^{0}(BLp̂; ZZp̂) \overline{≅_{}}K^{0}(BL; ZZp̂)

[figure: K^{0}(BTp̂; ZZp̂)^{Wp̂}; ≅; ≅; K^{0}(BT; ZZp̂)^{W}]

On the other hand K^{0}(−; ẐZ) ≅ ∏ K^{0}(−; ZZp̂), where the product is taken over all the prime numbers. With all these observations, an easy diagram chase in (5) yields K^{0}(BL; ZZ) ≅ K^{0}(BT; ZZ)^{W}. □

In [21, Theorem 1.2] Møller and Notbohm showed that the Weyl group of a connected finite loop space with maximal torus is crystallographic. This combines with the preceding theorem to imply that a connected finite loop space with a maximal torus has the K-theory of the classifying space of some compact connected Lie group. We will now show that this condition characterizes the finite loop spaces which have maximal tori.

Theorem 6.3 Let L be a connected finite loop space. Then L admits a maximal torus if and only if there exists a compact connected Lie group G such that K^{∗}(BL; ZZ) is λ-isomorphic to K^{∗}(BG; ZZ).

This result can be viewed as a generalization of a theorem of Notbohm-Smith ([23, Theorem 5.1]). In fact our proof will follow the same pattern as theirs. We deal first with some preliminaries.

Let L be a connected finite loop space. Consider the two filtrations of K^{∗}(BL; ZZ) defined by setting, for n = 0, 1, …:

i) S_{n}(L) = Ker(K^{∗}(BL; ZZ) ⟶ K^{∗}(BL^{(n−1)}; ZZ)), where BL^{(n)} is the n-th skeleton of BL.

ii) C_{n}(L) = {ξ ∈ K^{∗}(BL; ZZ) s.t ch_{r}(ξ) = 0 , for r = 0, 1, …, n − 1}, where ch_{r}(−) denotes the r-th component of the Chern character.

Then one easily checks that:

1. For any n, S_{n}(L) and C_{n}(L) are λ-ideals of K^{∗}(BL; ZZ) and S_{n}(L) ⊂ C_{n}(L). Moreover if L is a torus T, then S_{n}(T) = C_{n}(T) = I^{n} where I is the augmentation ideal.

2. For any prime p,

K^{∗}(BL; ZZp̂) ≅ lim_{←}(K^{∗}(BL; ZZ)/S_{n}(L) ⊗ ZZp̂) .

3. For any prime p,

K^{∗}(BL; Qp̂) ≅ lim_{←}(K^{∗}(BL; ZZ)/S_{n}(L) ⊗ Qp̂)
≅ lim_{←}(K^{∗}(BL; ZZ)/C_{n}(L) ⊗ Qp̂) .

Next we consider the integral version of Theorem 5.1.

Proposition 6.2 Let L be a connected finite loop space and T a torus. Then the natural map

α : [BT, BL] ⟶ Hom_{λ}(K^{∗}(BL; ZZ), K^{∗}(BT; ZZ))

is a bijection.

Proof Let φ : K^{∗}(BL; ZZ) ⟶ K^{∗}(BT; ZZ) be a λ-homomorphism. Consider the composite

φ̄_{n} : K^{∗}(BL; ZZ)/S_{n}(L) ⟶ K^{∗}(BL; ZZ)/C_{n}(L) ⟶ K^{∗}(BT; ZZ)/I^{n} ,

where the first arrow is the canonical surjection and the second arrow is induced by φ (the notation is as above). Then define

φp̂ : K^{∗}(BL; ZZp̂) ⟶ K^{∗}(BT; ZZp̂)

as the inverse limit of the φ̄_{n} ⊗ ZZp̂’s. Since φp̂ is a λ-homomorphism, it is induced by a map fp̂ : BTp̂ ⟶ BLp̂ (see Theorem 5.1). On the rational side, the λ-homomorphism φ induces (via the Chern character) a graded homomorphism

φ_{Q} : H^{∗}(BL; Q) ⟶ H^{∗}(BT; Q) .

As BL and BT are rationally products of Eilenberg-MacLane spaces, φ_{Q} is induced by a map f_{Q} : BT_{Q} ⟶ BL_{Q}. To obtain a map f : BT ⟶ BL out of the fp̂’s and f_{Q}, we will use Sullivan’s arithmetic square (as presented in [18, Theorem 4]). Thus it suffices to check that, for any prime p, the two composites

BT ⟶ BTp̂ ⟶^{fp̂} BLp̂ ⟶ (BLp̂)_{Q} ,

BT ⟶ BT_{Q} ⟶^{f_{Q}} BL_{Q} ⟶ (BLp̂)_{Q}

are homotopic. By choosing a a rational equivalence BL ⟶ ∏_{i=1}^{k} K(ZZ, 2n_{i}) and observing that the two composites represent elements of the product ∏_{i=1}^{k} H^{2n_{i}}(BT; Qp̂), it is sufficient to show the commutativity of the following diagram

HQp̂^{∗}(BLp̂) \overline{^{f ∗ ; p̂}} HQp̂^{∗}(BTp̂)
[figure: c_{p}(L); ≅; c_{p}(T); ≅]
H^{∗}(BL; Qp̂)  H^{∗}(BT; Qp̂)
[figure: c_{0}(L); ≅; c_{0}(T); ≅]
H^{∗}(BL_{Q}; Qp̂) \overline{^{f_{Q}^{∗}}} H^{∗}(BT_{Q}; Qp̂)

This is equivalent to showing that the two homomorphisms

ψ_{1} = c_{p}(T) ∘ f ∗ ; p̂ ∘ c_{p}(L)^{−1} and ψ_{2} = c_{0}(T) ∘ f_{Q}^{∗} ∘ c_{0}(L)^{−1}

are equal. Since ψ_{1} and ψ_{2} are Qp̂-linear, it is sufficient to check the equality on the subring H^{∗}(BL; Q) ⊂ H^{∗}(BL; Qp̂). Assume that we are given x ∈ H^{k}(BL; Q). As K^{1}(BL; ZZ) = 0, there exists η ∈ K^{0}(BL; ZZ) and an integer M such that ch(η) = Mx + higher terms (see [10, Theorem 3.2]). By construction the following diagram is commutative

H^{∗∗}(BL; Qp̂)←^{ch}K^{∗}(BL; ZZp̂) ←^{i_{p}(L)}K^{∗}(BL; ZZ)→^{ch}H^{∗∗}(BL; Q) →^{i_{0}(L)}H^{∗∗}(BL; Qp̂)

[figure: H^{∗∗}(BT; Qp̂); ψ_{1}; ←^{ch}; K^{∗}(BT; ZZp̂); φp̂; ←^{i_{p}(T)}; K^{∗}(BT; ZZ); φ; →^{ch}; H^{∗∗}(BT; Q); φ_{Q}; →^{i_{0}(T)}; H^{∗∗}(BT; Qp̂); ψ_{1}]

By Theorem 4 in [18], we have ch ∘ i_{p}(L) = i_{0}(L) ∘ ch on the upper horizontal line and ch ∘ i_{p}(T) = i_{0}(T) ∘ ch on the lower one. This implies the desired equality; and we have shown that α is surjective. Its injectivity is a consequence of Theorem 2 and Lemma 2 in [18] and of our Theorem 5.1. □

Proof (of Theorem 6.3) One of the implication has been established in Theorem 6.2; thus we are left with the second one. By hypothesis, there is a torus T and a finite group W of λ-automorphisms of K^{∗}(BT; ZZ) such that

K^{∗}(BL; ZZ) ≅^{λ} K^{∗}(BT; ZZ)^{W} .

Let φ denote the composite K^{∗}(BL; ZZ) ≅ K^{∗}(BT; ZZ)^{W} ⊂ K^{∗}(BT; ZZ) ; the preceding theorem provides us with a map f : BT ⟶ BL inducing φ in (integral) K-theory. Clearly the ranks of T and L are equal. Thus we will be done if we can show that the homotopy fiber V of f is ZZ-finite. We have the following properties:

1. V is homotopy equivalent to a CW-complex of finite type;

2. For any prime p, V is IF_{p}-good and π_{1}(Vp̂) ≅ π_{1}(V) ⊗ ZZp̂ (see section 7 in [8]).

Let us now fix a prime p and consider the p-completion fp̂ : BTp̂ ⟶ BLp̂. Let also Bi_{L} : BT_{L} ⟶ BLp̂ be a maximal p-torus. By Proposition 8.11 in [12], there exists a map Bψ : BTp̂ ⟶ BT_{L} such that Bi_{L} ∘ Bψ = fp̂. In p-adic K-theory this yields the following commutative diagram

K^{∗}(BT_{L}; ZZp̂)
[figure: Bψ^{∗}]
[figure: K^{∗}(BLp̂; ZZp̂); Bi_{L}^{∗}; f ∗ ; p̂; K^{∗}(BTp̂; ZZp̂)]

Since K^{∗}(BL; ZZp̂) ≅ K^{∗}(BT; ZZp̂)^{W} (via f ∗ ; p̂), Theorem 4.1 in [27] provides us with a λ-ring homomorphism Φ : K^{∗}(BTp̂; ZZp̂) ⟶ K^{∗}(BT_{L}; ZZp̂) such that Φ ∘ f ∗ ; p̂ = Bi_{L}^{∗}. Passing to fields of fractions and using some Galois theory, one checks that the composites Φ ∘ Bψ^{∗} and Bψ^{∗} ∘ Φ are λ-isomorphisms. In other words, Bψ induces an isomorphism in p-adic K-theory. It follows that Bψ is a homotopy equivalence, so that fp̂ : BTp̂ ⟶ BLp̂ is also a maximal torus. Since the homotopy fiber of fp̂ is Vp̂ (see [8, Proposition 4.2]), we obtain that H_{∗}(Vp̂; IF_{p}) ≅ H_{∗}(V; IF_{p}) is a finite dimensional IF_{p}-vector space. Property 1) above now combines with the universal coefficient theorem to imply that H_{∗}(V; ZZ) is a finitely generated abelian group. □

Aknowledgements. We would like to thanks W.G. Dwyer, U. Suter and U. Würgler for helpful discussions. Part of this work was done while the second author was visiting the Department of Mathematics of the Ohio State University (Columbus). He would like to thank the members of this institution, especially H. Glover and G. Mislin, for their warm hospitality.

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