ON THE FUNCTOR ‘CLASSIFYING

SPACE’ FOR COMPACT LIE GROUPS

by Dietrich Notbohm

Abstract. Two compact connected Lie groups G and H are isomorphic as Lie groups if and only if the associated classifying spaces BG and BH are homotopy equivalent. The proof is based on the following fact: The functor ‘classifying space’ induces a map from group extension of the form G ⟶ H ⟶ Γ to fibrations of the form BG ⟶ X ⟶ BΓ which turns out to be a bijection for Γ a finite and G a compact connected Lie group.

1. Introduction.

In [R], Rector introduced the idea of studying compact connected Lie groups from the homotopy point of view, i.e in terms of classifying spaces. As a Lie group, G has a very rich structure. How much of this structure is already contained in the classifying space BG ? This question has a long history and was again recently raised by G.Mislin at the Göttingen conference on classifying spaces in August 91 [Sm].

For G a finite group, or more generally for a discrete group, G ≅ π_{1}(BG). Two compact connected simple Lie groups G and H are isomorphic as Lie groups iff G and H are homotopy equivalent iff BG and BH are homotopy equivalent. The first equivalence is proved in [B–B], the second is a consequence of the first, because ΩBG ≃ G. For semi simple Lie groups the first equivalence is not true. Counterexamples may be found in [B–B]. But by a result of [C–W–W], the isomorphy type of a semi simple Lie group G depends only on the isomorphy type of the normalizer N(T_{G}) of the maximal torus T_{G}. In this paper, we ask for the relation between G, N(T_{G}), BN(T_{G}), and BG. W_{G} denotes always the Weyl group of G.

Theorem A. For two compact connected Lie groups G and H, the following statements are equivalent.

(1) G and H are isomorphic as Lie groups.

(2) N(T_{G}) and N(T_{H}) are isomorphic as Lie groups.

(3) BN(T_{G}) and BN(T_{H}) are homotopy equivalent.

(4) BG and BH are homotopy equivalent.

In the case of compact Lie groups we have the following result.

Theorem B. Two compact Lie groups G and H are isomorphic as Lie groups iff BG and BH are homotopy equivalent.

The proofs are based on a comparison between group extensions of the form G ⟶ H ⟶ Γ and fibrations of the form BG ⟶ BH ⟶ BΓ, where G, H, and Γ are compact Lie groups, G is connected, and Γ is finite.

Typeset by AMS-TEX

For two CW –complexes F and B, we denote by F ib(B, F) the set of equivalence classes of fibrations

F ⟶ E ⟶ B .

The equivalence relation is given by fiber homotopy equivalences of the form

F ⟶ E_{1} ⟶ B
f↓  ^{g}↓  ∥
F ⟶ E_{2} ⟶ B .

where f is homotopic to the identity on F. This is an equivalence relation, because, for an inverse of (g, f)

F ⟶ E_{2} ⟶ B
f^{−1}↓ ^{g^{−1}}↓  ∥
F ⟶ E_{1} ⟶ B ,

f^{−1} is again homotopic to the identity.

For two groups G and Γ, we denote by Ext(Γ, G) the set of equivalence classes of group extensions G ⟶ H ⟶ Γ. The equivalence relation is the usual one.

There is a natural map

B : Ext(Γ, G) ⟶ F ib(BΓ, BG) ,

which turns out to be a bijection as the next statement says.

Theorem C. For a finite group Γ and a compact connected Lie group G, the map

B : Ext(Γ, G) ⟶ F ib(BΓ, BG)

is a bijection.

The paper is organized as follows: Section 2 contains some results about maps between classifying spaces. In section 3 we discuss the relation between group extensions of compact Lie groups and the associated fibrations. In particuliar, we prove theorem C. Some facts about compact connected Lie groups we recall in section 4. The final section is devoted to the proofs of the theorems A and B.

When we deal with p–adic completion, we do this in the sense of Bousfield and Kan [B–K].

It is pleasure to thank G.Mislin for a helpful discussion on this subject.

2. Maps between classifying spaces.

Let α : P ⟶ H be a homomorphism between compact Lie groups. The centralizer of the image of α in H is denoted by C_{H}(α). The obvious homomorphism

C_{H}(α) × P ⟶ H

passes to a map between classifying spaces. The adjoint is given by

ad : BC_{H}(α) ⟶ map(BP, BH)_{Bα} .

map(BP, BH)_{Bα} denotes the component of Bα of the mapping space map(BP, BH). This construction was used to show that, for P a p–toral group, ad is a mod–p equivalence [D–Z] [No 1]. Using a result of [B–N], there is a slightly stronger statement, namely

ad : BC_{H}(α)_{p}^{∧} ⟶ map(BP, BH_{p}^{∧})_{Bα}

is an equivalence (see also [J–M–O]). Here we are interested in a global result in a particular case.

2.1 Proposition. Let ρ : H⟶N be a homomorphism from a compact connected Lie group H into a finite extension of a torus. Then,

BC_{N}(ρ) ⟶ map(BH, BN)_{Bρ}

is a homotopy equivalence.

Proof. BN fits into the fibration

BT ⟶ BN ⟶ BW .

Here T is the toral part of N, respectively the component of the unit, and W := N/T, which acts on T. Because H is connected, the image of ρ is contained in T. Let Iso(ρ) ⊂ W be the group of those elements which act trivially on the image of ρ. The above construction induces a commutative diagram

∐_{w∈W/Iso(ρ)} BT  ⟶  BC_{N}(ρ)  ⟶  BW
↓  ↓  ↓
∐_{w∈W/Iso(ρ)} map(BH, BT)_{wBρ} ⟶ map(BH, BN)_{Bρ} ⟶ map(BH, BW)_{c} ,

where c denotes the constant map. Since the components of C_{N}(ρ) are given by Iso(ρ), the upper row is a fibration. The bottom row is a fibration, because every map BH⟶BT is induced by a homomorphism [L-M-S]. The left and right columns are equivalences. For the right column this follows from a short calculation, and for the left column the same kind of calculation works, observing that BT is an H– space and that therefore all components of map(BH, BT) are homotopy equivalent. Therefore, the middle column is an equivalence, which proves the statement. □

Let K ⟶ G ⟶ H be an exact sequence of compact Lie groups, and let X be a space. Maps BG ⟶ X can be considered as extensions of maps BK ⟶ X in the following way (see [D–Z]). The space B̃K := EG/K carries a free H–action, is homotopy equivalent to BK, H–equivariant homotopy equivalent to B̃K × EH, and B̃K/H = BG. We get

map(BG, X) = map(B̃K/H, X)
= map_{H}(B̃K, X)
≃ map_{H}(EH × B̃K, X)
≃ map_{H}(EH, map(B̃K, X))
= map(B̃K, X)^{hH} .

The last equation can be used as the definition of the homopy fixed point set. The group H acts trivially on X, and map_{H}( , ) denotes the mapping space of equivariant maps. Because the homotopy fixed points can be interpreted as sections in the bundle EH ×_{H} map(B̃K, X), this description leads to the following lemma (see [No 3]):

2.2 Lemma. Let f_{K} : BK⟶Xbe a map and map(BK, X)_{f_{K}} be simple.

(1) The obstructions for extending f_{K} to a map f_{G} : BG⟶X are contained in the groups H^{∗+1}(BH, π_{∗}(map(B̃K, X)_{f_{K}}) (twisted coefficients).

(2) The homotopy classes of extensions BG⟶X are classified by obstructions in H^{∗}(BH, π_{∗}(map(B̃K, X)_{f_{K}}) (twisted coeficients).

3. Fibrations of classifying spaces and group extensions.

The purpose of this section is to prove theorem C. This is done in several steps. First we consider a finite group Γ, acting on a torus T via a homomorphism ρ : Γ⟶Aut(T). Here, Aut(T) denotes the group of automorphisms. Fibrations of the form

BT ⟶ Y ⟶ BΓ

can be classified by homotopy classes of maps

BΓ⟶BHE(BT) ,

where HE(BT) is the monoid of self equivalences of BT [St]. We denote by SHE(BT) the component of the identity. π_{0}(HE(BT)) ≅ Aut(T) is the group of the components. Hence, we have a fibration

BSHE(BT) ⟶ BHE(BT) ⟶ BAut(T) .

For every fibration BT ⟶Y ⟶BΓ, the composition

BΓ⟶BHE(BT)⟶BAut(T)

is homotopic to a map Bρ, where ρ : Γ⟶Aut(T) describes the action of π_{1}(BΓ) on π_{2}(BT). Also, every group extension T ⟶N⟶Γ establishes a homorphism ρ : Γ⟶Aut(T). By Ext_{ρ}(Γ, T) and F ib_{ρ}(BΓ, BT) we denote the set of group extensions or fibrations associated to a fixed homomorphism ρ : Γ⟶Aut(T). Then, for every ρ, we get a map

B : Ext_{ρ}(Γ, T) ⟶ F ib_{ρ}(BΓ, BT) .

The canonical map (see section 2)

BT ⟶ SHE(BT)

is a homotopy equivalence and a homomorphism of monoids. Moreover, this map is Γ–equivariant, because Γ acts via homomorphisms. We get an equivalence

BBT ⟶BSHE(BT) ,

which is also Γ–equivariant.

Let ϕ, ψ : BΓ⟶BHE(BT) be two classifying maps associated to the same action ρ : Γ ⟶ Aut(T), i.e. the compositions \overline{ϕ}, \overline{ψ} : BΓ⟶BHE(BT)⟶BAut(T) are homotopic. The obstructions for lifting this homotopy lie in the groups H^{∗}(BΓ; π_{∗}(BSHE(BT))) which vanish for ∗ ≠ 3. Therefore, the difference between ϕ ans ψ can be measured by the obstruction class

d(ϕ, ψ) ∈ H^{3}(BΓ; π_{3}(BSHE(BT))) ≅ H^{3}(BΓ; π_{2}(BT)) .

For a fixed ϕ, this induces a one to one correspondence between the homotopy classes of lifts of \overline{ϕ} and the group H^{3}(BΓ; π_{2}(BT).

Let s : BΓ⟶BHE(BT) be the classifying map of the fibration BT ⟶B(T ⋊ Γ)⟶BΓ, where T ⋊ Γ denotes the semi direct product. The above considerations prove the following lemma:

3.1 Lemma. For a fixed action ρ : Γ⟶Aut(T), the map

\begin{matrix}F ib_{ρ}(BΓ, BT) & ⟶ & H^{3}(Γ; π_{2}(BT)) \\ (ϕ : BΓ→BHE(BT)) & ↦ & d(s, ϕ) \end{matrix}

is a bijection.

Now we are prepared to analyse the map B in this special case.

3.2 Proposition. The canonical map

\begin{aligned}B : & Ext(Γ, T) & ⟶ & F ib(BΓ, BT) \\ & T →N→Γ & ↦ & BT →BN→BΓ \end{aligned}

is a bijection.

Proof. We only have to prove that, for a fixed homorphism ρ : Γ ⟶ Aut(T), the map B : Ext_{ρ}(Γ, T)⟶F ib_{ρ}(BΓ, BT) is a bijection. Elements of Ext_{ρ}(Γ, T) are described by cohomology classes in H^{2}(Γ; T). The canonical map

H^{2}(Γ; T) ⟶ H^{3}(Γ; π_{2}(BT))

is a bijection. We get a diagram

Ext_{ρ}(Γ, T) ⟶^{B} F ib_{ρ}(BΓ, BT)
≅↓  ≅↓
H^{2}(Γ; T) ⟶^{≅} H^{3}(BΓ; π_{2}(BT)) ,

which commutes. This can be shown by comparing the constructions of the obstruction classes which induce the vertical bijections. Hence, B is also a bijection. □

Now, let A be a compact abelian Lie group, i.e A ≅ A′ × T is a product of a finite abelian group A′ and a torus T. Let Γ be a finite group acting on A. The exact sequence

T ⟶ A ⟶ A/T ≅ A′

is a sequence of Γ–modules. For any group extension A⟶G⟶Γ, the subgroup T is normal in G. This leads to a diagram of exact sequences

T  T ⟶ 1
↓  ↓  ↓
A ⟶ G ⟶ Γ
↓  ↓  ↓
A/T ⟶ G/T ⟶ Γ .

3.3 Proposition. Let A = A′ × T be a product of a finite abelian group A′ and a torus T, and let Γ be a finite group. Then,

Ext(Γ, A) ⟶ F ib(BΓ, BA)

is a bijection.

Poof. Let A⟶G⟶Γ and A⟶H⟶Γ be two exact sequences, such that the associated fibrations are equivalent, i.e there exists an equivalence

BA ⟶ BG ⟶ BΓ
ϕ_{A}↓  ^{ϕ}↓  ∥
BA ⟶ BH ⟶ BΓ ,

where ϕ_{A} ≃ id. In particuliar, the composition BT ⟶^{Bi} BA ⟶^{ϕ_{A}} BA lifts to a map ϕ_{T} : BT ⟶ BT such that Biϕ_{T} ≃ ϕ_{A}Bi, and ϕ_{T} is homotopic to the identity on BT. Because T is normal in G and H, we get a diagram

BT ⟶ BG ⟶ B(G/T)
ϕ_{T} ↓  ^{ϕ}↓  \overline{ϕ}↓
BT ⟶ BH ⟶ B(H/T) .

We have to show that the map \overline{ϕ} exists making the diagram commutative. Because H/T is a finite group, map(BT, B(H/T))_{c} ≃ B(H/T), where c is the constant map. This follows, because, for the loops of the mapping space of pointed maps, Ωmap_{∗}(BT, BH/T)_{c} ≃ map_{∗}(BT, H/T) is contractible. We can apply [No 2; Lemma 2.1 and the preceeding remarks] which tells us that [B(G/T), B(H/T)] ⟶ [BG, B(H/T)] is an injection and that the image consists of those maps BG ⟶ B(H/T), whose restriction to BT is homotopically trivial. Hence, the map \overline{ϕ} exists as desired and is a homotopy equivalence. Because G/T and H/T are finite groups, the map \overline{ϕ} is given by an isomorphism G/T ≅ H/T of groups.

Now we can think of G and H as elements in Ext(G/T, T). The induced fibrations are equivalent. By proposition 3.2, we have G ≅ H as elements in Ext(G/T, T). Because H⟶H/T ≅ G/T ⟶Γ is the projection H⟶Γ, this proves the injectivity of the map.

Let BA ⟶ E ⟶ BΓ be a fibration. Then, the long exact sequence of the homotopy groups shows that Γ′ := π_{1}(E) is a finite group and that π_{2}(BA) ≅ π_{2}(E). Therefore there exists a fibration BT ⟶ E ⟶ BΓ′. By Proposition 3.2 this implies that there exists a connected Lie group G such that E ≃ BG. By construction, the kernel of the composition G ⟶ Γ′ ⟶ Γ is given by A. This proves the surjectivity of the map. □

Finally, we consider the case of a compact connected Lie group G. To every exact sequence

G ⟶ H ⟶ Γ

is associated a homomorphism ρ : Γ⟶Out(G), where Out(G) := Aut(G)/Inn(G) is the set of outer automorphisms of G, i.e automorphisms modulo conjugations. Again, this can be detected from the fibration BG⟶BH⟶BΓ. The composition BΓ⟶BHE(BG)⟶Bπ_{0}(HE(BG)) ≃ BOut(G) is given by Bρ. The first map is the classifying map of the fibration. The isomorphism π_{0}(HE(BG) ≅ Out(G) follows from [JMO 2].

It is well known that, for a comact connected Lie group, the projection Aut(G)⟶Out(G) has a section s : Out(G)⟶Aut(G). This section is not unique in general. We fix a choosed section s for the following. For any homomorphism ρ : Γ⟶Out(G), the group extension G⟶G ⋊ Γ⟶Γ realizes ρ, where Γ acts on G via ρ and the section s. The set Ext_{ρ}(Γ, G) is nonempty.

The canonical map Out(G)⟶Aut(Z(G)) gives rise to an action ρ : Γ⟶Aut(Z(G)), also denoted by ρ. The semi direct products fit into

Z(G) ⟶ Z(G) ⋊ Γ ⟶ Γ
↓  ↓  ∥
G ⟶ G ⋊ Γ ⟶ Γ .

Fixing these elements in Ext_{ρ}(G) and Ext_{ρ}(Z(G)), the cohomology group H^{2}(Γ; Z(G)) acts transitively and freely on both extension sets [MacL]. This proves the following lemma:

3.4 Lemma.

(1) There exists an isomorphism

Φ : Ext_{ρ}(Γ, Z(G)) ⟶ Ext_{ρ}(Γ, G) .

(2) If

ζ : Z(G) ⟶ N ⟶ Γ
↓  ↓  ∥
ξ : G ⟶ H ⟶ Γ

is a commutative diagram, then Φ(ζ) = ξ.

Next we will construct the analogous map on the level of fibrations, i.e. a map F ib(BΓ, BZ(G))⟶F ib(BΓ, BG). We will use the classification theorem of Stasheff [S]. We have to find a way to pass from BHE(BZ(G)) to BHE(BG). Because Z(G) is a product of a finite abelian group and a torus, the canonical map BZ(G)⟶SHE(BZ(G)) is an equivalence. But BZ(G)⟶SHE(BG) is only an equivalence after completion [J–M–O 1,2] [No 2]. Therefore, completion comes into the play. We define X^{∧} := ∏_{p} X_{p}^{∧} to be the product of all p–adic completions.

3.5 Lemma.

(1)

BHE(BG) ≃ BSHE(BG) ×_{Out(G)} EOut(G)

and

BSHE(BG)^{∧} ×_{Out(G)} EOut(G) ≃ BBZ(G)^{∧} ×_{Out(G)} EOut(G) .

(2)

BHE(BZ(G)) ≃ BSHE(BZ(G)) ×_{Aut(Z(G))} EAut(Z(G))
≃ BBZ(G) ×_{Aut(Z(G))} EAut(Z(G)) .

Proof. Out(G) acts on SHE(BG) via the section s : Out(G)⟶Aut(G) and conjugation. This establishes a homomorphism of monoids

SHE(BG) ⋊ Out(G)⟶HE(BG) ,

which, passing to classifying spaces, fits into

BSHE(BG) ⟶ BSHE(BG) ×_{Out(G)} EOut(G) ⟶ BOut(G)
∥  ↓  ∥
BSHE(BG) ⟶  BHE(BG)  ⟶ BOut(G) .

Obviously, the middle arrow is an equivalence.

As already mentioned, the map BZ(G)⟶SHE(BG) induces an equivalence after completion [J–M–O 2] or [No 2]. Moreover, it is a homorphism of monoids and is equivariant with respect to the action of Out(G). This establishes the equivalence

BSHE(BG)^{∧} ×_{Out(G)} EOut(G) ≃ BZ(G)^{∧} ×_{Out(G)} EOut(G)

and finishes the proof of the first part.

The proof of (2) is analogous. But we don’t have to deal with completions. Z(G) is a product of a finite abelian group and a torus. This implies that BZ(G)⟶SHE(BZ(G)) is an integral equivalence. □

Now we are prepared to prove the analogous of lemma 3.4 for fibrations of the aasociated classifying spaces. Let F ib_{ρ}(BΓ, BG) denote the set of fibrations, associated to ρ : Γ⟶Out(G) in the above sense.

3.6 Lemma.

(1) There exists a bijection

Ψ : F ib_{ρ}(BΓ, BZ(G)) ⟶ F ib_{ρ}(BΓ, BG) .

(2) If

ϵ :BZ(G) ⟶ X ⟶ BΓ
↓  ↓  ∥
η : BG ⟶ Y ⟶ BΓ

is a commutative diagram of fibrtaions, then Ψ(ϵ) = η.

Proof. By [J–M–O 2] and [B–N], we have BZ(G)^{∧} ≃ SHE(BG)^{∧} ≃ SHE(BG^{∧}) and BZ(G)^{∧} ≃ SHE(BZ(G))^{∧} ≃ SHE(BZ(G)^{∧})). Because Γ is finite,

H^{i}(Γ; π_{j}(BSHE(BG))) ≅ H^{i}(Γ; π_{j}(BSHE(BG)) ⊗ ℤ^{∧})
≅ H^{i}(Γ; π_{j}(BSHE(BG^{∧}))) .

These are the obstruction groups for lifting maps and homotopies in the fibrations BHE(BG)⟶BOut(G) and BSHE(BG^{∧})×_{Out(G)} EOut(G)⟶BOut(G). Therefore, completion induces bijections

F ib_{ρ}(BΓ, BG) ⟶^{≅} F ib_{ρ}(BΓ, BG^{∧}) ,

and, analogously,

F ib_{ρ}(BΓ, BZ(G)) ⟶^{≅} F ib_{ρ}(BΓ, BZ(G)^{∧}) .

Let E be the pull back of BHE(BZ(G))⟶BAut(Z(G)) along the map BOut(G)⟶BAut(Z(G)), i.e. E = BSHE(BZ(G)) ×_{Out(G)} EOut(G). Because ρ maps Γ into Out(G), every element in F ib_{ρ}(BΓ, BZ(G)) is classified by a map BΓ⟶E. Lemma 3.5 implies that there exists an diagram

BSHE(BZ(G)) ⟶  E  ⟶ BOut(G)
↓  ↓  ∥
BSHE(BG) ⟶ BHE(BG) ⟶ BOut(G) ,

which becomes an equivalence of fibrations after fiberwise completion (for fiberwise completion see [B–K]). Therefore, any element in F ib_{ρ}(BΓ, BZ(G)^{∧} is classified by a map BΓ⟶BSHE(BZ(G)^{∧}) ×_{Out(G)} EOut(G). Therefore, in the commutative diagram

F ib_{ρ}(BΓ, BZ(G)) ⟶^{Ψ} F ib_{ρ}(BΓ, BG)
↓  ↓
F ib_{ρ}(BΓ, BZ(G)^{∧}) ⟶ F ib_{ρ}(BΓ, BG^{∧})

the lower horizontal map is a bijection. The vertical arrows are also bijections which proves (1). The second part is obvious.

□

Finally, we can prove theorem C.

Proof of theorem C. We choose a homorphism ρ : Γ⟶Out(G). Then, by lemma 3.4 and lemma 3.6, the diagram

Ext_{ρ}(Γ, Z(G)) ⟶^{B} F ib_{ρ}(BΓ, BZ(G))
Φ↓  ^{Ψ}↓
Ext_{ρ}(Γ, G) ⟶^{B} F ib_{ρ}(BΓ, BG)

commutes, and Φ and Ψ are bijections. The upper horizontal map is also a bijection by proposition 3.3. □

4. Compact connected Lie Groups.

For any compact connected Lie group G there exists a finite covering

1 ⟶ K ⟶ G̃ = G_{s} × T ⟶ G ⟶ 1

of compact Lie groups, where G_{s} is simply connected and T a torus (e.g. see [BtD]). Such coverings we call universal finite. G̃ is unique up to isomorphism, but the homomorphism G̃ ⟶ G is not unique. We can choose a finite self covering T ⟶ T and can take the composition G_{s} × T ⟶ G_{s} × T ⟶ G. Among the universal finite coverings of G, there is a minimal one, characterized by the condition that K ⟶ G_{s} × T ⟶G_{s} is an injection. The minimal universal finite covering is a quotient of all the others.

Every universal finite covering establishes the following commutative diagrams of exact sequences of connected Lie groups:

(*)  \begin{matrix}K_{s} & ⟶ & G_{s} & ⟶ & G_{s}/K_{s} := \overline{G}_{s} \\ ↓ & & ↓ & & ↓ \\ K & ⟶ & G_{s} × T & ⟶ & G \\ ↓ & & ↓ & & ↓ \\ \overline{K} = K/K_{s} & ⟶ & T & ⟶ & T/\overline{K} := \overline{T} & ′ \end{matrix}

where K_{s} := K ∩ (G_{s} × \{0\}). We call \overline{G}_{s} the semi simple part of G. Because H^{1}(\overline{G}_{s}; ℤ) = 0, the sequence \overline{G}_{s}⟶G⟶\overline{T} induces isomorphisms H^{1}(\overline{T}; ℤ) ≅ H^{1}(G; ℤ) and H^{2}(B\overline{T}; ℤ) ≅ H^{2}(BG; ℤ). These isomorphisms determine the maps G⟶\overline{T} and BG⟶B\overline{T}, and therefore, the fibration B\overline{G}_{s}⟶BG⟶B\overline{T}.

By the above diagram (∗), the map T_{G}^{W_{G}}⟶\overline{T} is an epimorphism. Restricting the projection G⟶\overline{T} to N(T_{G}) or T_{G}^{W_{G}} and passing to the classifying spaces yields a commutative diagram of fibrations

B(T \overline{G}_{s}^{W_{G}}) ⟶ B(T_{G}^{W_{G}}) ⟶ B\overline{T}
↓  ↓  ∥
BN(T_{\overline{G}_{s}}) ⟶ BN(T_{G}) ⟶ B\overline{T}
↓  ↓  ∥
B\overline{G}_{s} ⟶ BG ⟶ B\overline{T} .

4.1 Lemma. The canonical inclusion Z(\overline{G}_{s})⟶T_{\overline{G}_{s}}^{W_{Gs}} induces an injection

F ib(BT_{p}^{∧}, B\overline{G}_{s}_{p}^{∧}) ⟶ F ib(BT_{p}^{∧}, B(T_{\overline{G}_{s}}^{W_{Gs}})_{p}^{∧}) .

Proof. Elements in F ib(BT_{p}^{∧}, B\overline{G}_{s}_{p}^{∧}) are classified by homotopy classes of maps BT_{p}^{∧}⟶BHE(B\overline{G}_{s}_{p}^{∧}) [St]. Because BT_{p}^{∧} is simply connected, every homotopy class lifts to a homotopy class BT_{p}^{∧}⟶BSHE(B\overline{G}_{s}_{p}^{∧}).

The map BZ(\overline{G}_{s})⟶SHE(B\overline{G}_{s}) is an equivalence after completion [J–M–O 2] [No 2] and induces an equivalence BZ(\overline{G}_{s})_{p}^{∧}⟶SHE(B\overline{G}_{s}_{p}^{∧}) [B–N]. Moreover, it is a homomorphism of monoids and passes to a map between the classifying spaces. The composition

BSHE(B\overline{G}_{s}_{p}^{∧}) ⟵^{≃} BBZ(\overline{G}_{s})_{p}^{∧} ⟶ BB(T_{\overline{G}_{s}}^{W_{G}})_{p}^{∧} ⟶^{≃} BSHE(B(T_{\overline{G}_{s}}^{W_{G}}))_{p}^{∧}

establishes the desired map.

Since \overline{G}_{s} is semi simple, Z(\overline{G}_{s}) and T_{\overline{G}_{s}}^{W_{G}} are finite abelian groups. Therefore,

[BT_{p}^{∧}, BBZ(\overline{G}_{s})_{p}^{∧}] ≅ H^{2}(BT; S_{p}Z(\overline{G}_{s}))
⟶ H^{2}(BT; S_{p}(T_{\overline{G}_{s}}^{W_{G}})) ≅ [BT_{p}^{∧}, BB(T_{\overline{G}_{s}}^{W_{G}})]

is an injection. Here, S_{p}Z(G_{s}) denotes the p–Sylow subgroup. □

5. The proof of theorem A and B.

For the proof of theorem A, we need a result about self maps of the classifying space of a compact connected Lie group.

5.1 Lemma. Let f : BG_{s} ×BT ⟶ BG_{s} ×BT be a self map, where G_{s} is a simply connected Lie group and where T is a torus. If f is rationally an equivalence, there exist a map f_{s} : BG_{s} ⟶ BG_{s} and a homomorphism α : T ⟶ T, such that f ≃ f_{s} × Bα. Both, f_{s} and Bα are rationally equivalences.

Proof. For a rational equivalence f : BG_{s} × BT ⟶ BG_{s} × BT, there exist homorphisms β : T_{G_{s}} × T ⟶ T_{G_{s}} × T and γ : W_{G_{s}} ⟶ W_{G_{s}}, such that β is W_{G_{s}}–equivariant with respect to γ, i.e. βw = γ(w)β for w ∈ W_{G_{s}}, and such that

BT_{G_{s}} × BT ⟶^{Bβ} BT_{G_{s}} × BT
↓  ↓
BG_{s} × BT ⟶^{f} BG_{s} × BT

commutes up to homotopy. This all follows from [A–M] and [No 1]. Moreover Bβ is a rational equivalence and γ an isomorphism.

We can describe β by a matrix

A = ( \begin{matrix}β_{s,s} & β_{T,s} \\ β_{s,T} & β_{T,T} \end{matrix} ) ,

where β_{s,T} : T_{G_{s}} ⟶ T_{G_{s}} × T ⟶^{β} T_{G_{s}} × T ⟶ T. All the other entries are given by analogous compositions, and all entries are W_{G_{s}}–equivariant. Because, up to homotopy, every map BG_{s} ⟶ BT is trivial, the homomorphism β_{s,T} is constant. Because W_{G_{s}} acts trivially on T, we get β_{T,s} = β_{T,s}w = γ(w)β_{T,s} for all w ∈ W_{G_{s}}. This shows that the image of β_{T,s} is contained in the finite group T_{G_{s}}^{W_{Gs}}, and hence, that β_{T,s} is constant.

Now we define f_{s} to be the composition

BG_{s} ⟶ BG_{s} × BT ⟶^{f} BG_{s} × BT ⟶ BG_{s}

and set α := β_{T,T}. Then, f_{s} × Bα and f induce the same self map of the maximal torus. Thus, both maps are homotopic [J–M–O 2] [No 2]. □

Proof of theorem A. To get more insight in the relation between the different conditions in theorem A, we will prove more than just a circle of implications. The conclusions from (1) to (2) and (4), and from (2) to (3) are obvious. (3) implies (2) by proposition 3.2. Moreover, we show that (3) and (4) are equivalent and that (4) implies (1).

Let us assume that condition (3) is satisfied, namely BN(T_{G}) ≃ BN(T_{H}). We will show that BG ≃ BH, which is condition (4). By (2), W_{G} ≅ W_{H}, and T_{G} ≅ T_{H} as W_{G}–modules. Thus,

H^{2}(BG; ℤ) ≅ H^{2}(BT_{G}; ℤ)^{W_{G}} ≅ H^{2}(BT_{H}; ℤ)^{W_{H}} ≅ H^{2}(BH; ℤ) .

Because H^{2}(BG, ℤ) and H^{2}(BH; ℤ) determine the semi simple parts of G and H (see section 4), we yield a diagram of fibrations

(**)  \begin{aligned}B\overline{G}_{s} & ⟵ & BN(T_{\overline{G}_{s}}) & ≃ & BN(T_{\overline{H}_{s}}) & ⟶ & B\overline{H}_{s} \\ ↓ & & ↓ & & ↓ & & ↓ \\ BG & ⟵ & BN(T_{G}) & ≃ & BN(T_{H}) & ⟶ & BH \\ ↓ & & ↓ & & ↓ & & ↓ \\ BT & & BT & = & BT & & BT & . \end{aligned}

\overline{G}_{s} and \overline{H}_{s} are semi simple, and, by (2), they have isomorphic normalizers. Therefore, \overline{G}_{s} ≅ \overline{H}_{s} as Lie groups [C–W–W]. Moreover, the two fibrations

B(T_{\overline{G}_{s}}^{W_{G}}) ⟶ B(T_{G}^{W_{G}}) ⟶ BT
≃↓  ^{≃}↓  ∥
B(T_{\overline{H}_{s}}^{W_{H}}) ⟶ B(T_{H}^{W_{H}}) ⟶ BT

are also isomorphic. Because BT is simply connected, localization and completion preserve the fibrations. By lemma 4.1, the left and right columns in (∗∗) are equivalent as fibrations after completion. Hence, there exists an equivalence f_{p}^{∧} : BG_{p}^{∧} ⟶ BH_{p}^{∧}, which fits into

BT_{G}_{p}^{∧} ⟶^{≃} BT_{H} _{p}^{∧}
↓  ↓
BG_{p}^{∧} ⟶^{f_{p}^{∧}} BH_{p}^{∧} .

The top horizontal arrow is induced by a homotopy equivalence f_{T} : BT_{G} ⟶ BT_{H} which comes from the equivalence BN(T_{G}) ≃ BN(T_{H}). Since BG and BN(T_{G}) are rationally equivalent, we also have an equivalence f_{ℚ} : BG_{ℚ} ⟶ BH_{ℚ}, which is compatible with the map f_{T} : BT_{G} ⟶BT_{H}. Therefore, the coherence conditions for glueing together all these maps via the arithmetic square are satisfied. This yields an equivalence BG ≃ BH.

Next we will conclude (3) from (4); i.e. we assume that BG ≃ BH. Because BG and BH have the same rational cohomology, G and H must have the same rank. The composition BT_{G}⟶BG ≃ BH can be realized by a homomorphism ρ : T_{G}⟶H [No 1] which, up to conjugation, splits over an isomorphism ρ : T_{G}⟶T_{H}. The commutativity of

BT_{G} ⟶_{≃}^{Bρ} BT_{H}
↓  ↓
BG ⟶^{≃} BH

shows that W_{G} ≅ W_{H} and that the two representations of W_{G}, given by the action on BT_{G} and BT_{H} are conjugate.

Because map(BT_{G}, BN(T_{H}))_{Bρ} ≃ BT_{H} (proposition 2.1), there is only one obstruction γ(f_{N}) for the existence of an extension

f_{N} : BN(T_{G}) ⟶ BN(T_{H})

of BT_{G}⟶BT_{H}⟶BN(T_{H}) (lemma 2.2). This obstruction class lies in

H^{3}(W_{G}; π_{2}(map({ \atop BT_{G}}, BN(T_{H}))_{Bρ})) ≅ H^{3}(W_{G}; π_{2}(BT_{H}))

≅ ∏_{p} H^{3}(W_{G}; π_{2}(BT_{H} _{p}^{∧}))

≅ ∏_{p} H^{3}(W_{G}; π_{2}(map(BT_{G}, BH_{p}^{∧})_{Bρ})) .

Here, all cohomology groups are taken with twisted coefficients. The first isomorphism follows from proposition 2.1, the second because W_{G} is a finite group, and the third because, by [No 1] and [B–N], the canonical map BT_{H} _{p}^{∧} = BC_{H}(ρ) ⟶ map(BT_{G}, BH_{p}^{∧})_{Bρ} is a homotopy equivalence (see also section 2). By these isomorphisms, the obstruction γ(f_{N}) is mapped to ∏_{p} γ(f_{G}_{p}^{∧}), the obstruction for extending

BT_{G}⟶ ∏_{p} BH_{p}^{∧}

to BN(T_{G}). This extension obviously exists. Hence, the obstruction γ(f_{N}) vanishes, and the extension

f_{N} : BN(T_{G})⟶BN(T_{H})

exists. By analogous arguments, we can also construct the extension

g_{N} : BN(T_{H})⟶BN(T_{G})

of BT_{H} ⟶^{Bρ^{−1}} BT_{G} ⟶ BN(T_{G}). Both are equivalences, because the action of W_{G} ≅ π_{1}(BN(T_{G})) on π_{2}(BT_{G}) is faithful as well as the action of W_{H} on π_{2}(BT_{H}). This is statement (3).

To finish the proof of the equivalence of the first four conditions, we have to show that (4) implies (1). If BG ≃ BH, G and H are locally isomorphic [Sch]; i.e. the universal finite covers of G and H are isomorphic as Lie groups. Let G_{s} × T be this universal finite cover.

The equivalence f : BG ⟶^{≃} BH lifts to a map f̃: BG_{s} × BT ⟶BG_{s} × BT [No 2]. We yield a diagram

BK_{G} ⟶ BG_{s} × BT ⟶ BG
↓  f̃↓  ^{f}↓^{≃}
BK_{H} ⟶ BG_{s} × BT ⟶ BH .

The universal finite cover of H can be chosen to be minimal, i.e. the homomorphism K_{H}⟶G_{s} × T ⟶G_{s} is an injection (see section 4). Up to homotopy,

f̃ = f_{s} × Bα : BG_{s} × BT ⟶ BG_{s} × BT

is a product (lemma 5.1). The restriction f_{s}|_{BK_{G}} is induced by a homomorphism. Hence, the kernel of K_{G} ⟶ G_{s} is contained in the kernel of K_{H} ⟶ G_{s} and therefore trivial. That is that the universal finite cover G_{s}×T ⟶ G is also minimal. Applying the same considerations to the homotopy inverse f^{−1} : BH ⟶^{≃} BG shows, that f̃ = f_{s} × Bα is a homotopy equivalence and that K_{G} ≅ K_{H}.

Up to homotopy, f_{s} is given by outer automorphisms and Adams operations with positive exponents [J–M–O 2] [Mø] [No 2]. The map f_{s} is an equivalence. Hence, all Adams operations have exponent +1, and f_{s} is homotopic to Bβ for a suitable outer automorphism β of G_{s}. The isomorphism

β × α : G_{s} × T ⟶ G_{s} × T

maps K_{G} onto K_{H} and, therefore, G ≅ H as Lie groups which is condition (1). This finishes the proof of theorem A. □

Proof of theorem B. Let G and H be compact Lie groups and f : BG ⟶ BH a homotopy equivalence. Because f induces an isomorphism on the fundamental groups, we have π_{0}(G) ≅ π_{0}(H) =: π_{0}. Moreover, the universal covers BG_{e} and BH_{e} are equivalent. G_{e} is the component of the unit of G. Thus, f induces an commutative diagram of fibrations

BG_{e} ⟶ BG ⟶ Bπ_{0}
≃↓  ↓  ∥
BH_{e} ⟶ BH ⟶ Bπ_{0} .

Theorem A implies that G_{e} ≅ H_{e}. Hence, the two groups G and H can be thought of as being elements in Ext(Bπ_{0}, BG_{e}). Proposition 3.7 shows that G ≅ H as Lie groups. □

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Dietrich Notbohm, Mathematisches Institut, Bunssenstraße 3-5, 3400 Göttingen, Germany.