FIBRATIONS OF CLASSIFYING SPACES

by Kenshi Ishiguro and Dietrich Notbohm

Abstract. We investigate fibrations of the form Z ⟶ Y ⟶ X, where two of the three spaces are classifying spaces of compact connected Lie groups. We obtain certain finiteness conditions on the third space which make it also a classifying space. Our results allow to express some of the basic notions in group theory in terms of homotopy theory, i.e. in terms of classifying spaces. As an application we prove that every retract of the classifying space of a compact connected Lie group is again a classifying space.

1. Introduction.

In group theory one has the notions of a subgroup, i.e. a monomorphism, of a normal subgroup, of a kernel, and of an epimorphism. Following the idea of Rector [24] and Rector–Stasheff [26] to describe group theory or Lie group theory from the homotopy point of view, one may look for definitions of these notions in terms of classifying spaces. For a monomorphism, results of Quillen [23], Dwyer and Wilkerson [6] and the second author [20] give a strong hint to say that a map f : BH⟶BG between the classifying spaces of compact Lie groups is a monomorphism if H^{∗}(BH; 𝔽_{p}) is a finitely generated module over the ring H^{∗}(BG; 𝔽_{p}) for any prime p. A similiar definition is already given in [24]. The notion of a kernel is discussed in [20]. In order to find homotopy theoretical definitions for a normal subgroup and an epimorphism, we are led to study fibrations of classifying spaces.

Suppose two of the three spaces of a fibration are homotopy equivalent to classifying spaces of compact connected Lie groups (at p). One might ask if the third space is homotopy equivalent to the classifying space of a compact Lie group (at p). There are 3 types of fibrations to consider:

(1) BH ⟶ BG ⟶ X

(2) BH ⟶ Y ⟶ BK

(3) Z ⟶ BG ⟶ BK We see a negative answer to each of the three types. For type (1) and type (3), it is easy to find counter-examples; for instance, BS^{3} ⟶ BSO(3) ⟶ K(ℤ/2, 2) and S^{2} ⟶ BS^{1} ⟶ BS^{3}, respectively. For type (2), J. Møller pointed out to us that there is a fibration BS^{3} ⟶ Y ⟶ BS^{3} such that Y ∈ Genus(BS^{3} × BS^{3}), but Y ≄ BS^{3} × BS^{3}.

If the loop spaces ΩX and ΩZ satisfies certain finiteness conditions, however, Theorem 1 and Theorem 3 stated below give the affirmative answers and provide homotopy theoretical definitions for a normal subgroup and an epimorphism. The

Typeset by AMS-TEX

finiteness conditions are automatically satisfied if the map BH ⟶ BG is induced by a monomorphism, and if the map BG ⟶ BK is induced by an epimorphism. Theorem 2 will show that we can find a compact connected Lie group G such that Y ∈ Genus(BG); i.e. the p–completed space Y_{p}^{∧} is homotopy equivalent to BG_{p}^{∧} for any prime p.

Theorem 1. Suppose H and G are compact connected Lie groups.

(1) If BH_{p}^{∧} ⟶^{f} BG_{p}^{∧} ⟶ X is a fibration for a space X and if H^{∗}(ΩX; 𝔽_{p}) is finite, there is a compact connected Lie group K such that X ≃ BK_{p}^{∧}. In particuliar, if f is of form (Bi)_{p}^{∧} for a monomorphism i : H ⟶ G, then

H is a normal subgroup of G, and hence the given fibration BH_{p}^{∧} ⟶^{(Bi)_{p}^{∧}} BG_{p}^{∧} ⟶ X is equivalent to BH_{p}^{∧} ⟶ BG_{p}^{∧} ⟶ B(G/H)_{p}^{∧}.

(2) If BH ⟶ BG ⟶ X is a fibration for a space X whose loop space is homotopy equivalent to a finite CW–complex, there is a compact connected Lie group K such that X ≃ BK.

Theorem 2. Suppose H and K are compact connected Lie groups.

(1) If BH_{p}^{∧} ⟶ Y ⟶ BK_{p}^{∧} is a fibration for some space Y, there is a compact connected Lie group G such that Y ≃ BG_{p}^{∧}.

(2) If BH ⟶ Y ⟶ BK is a fibration for some space Y, there is a compact connected Lie group G such that Y ∈ Genus(BG) .

To state a result in the third case we need to consider unstable Adams operations ψ^{p^{k}} : BG_{p}^{∧}⟶BG_{p}^{∧}, where G is a compact connected Lie group. We denote the homotopy fiber of ψ^{p^{k}} by F (G, p^{k}) and call it an Adams fiber of p–type. If G is simply–connected and simple, we call F (G; p^{k}) a special Adams fiber. As shown in Proposition 3.2, the Adams fiber is always homotopy equivalent to a CW –complex of finite type and p–complete. When G is a torus, it is clear that the Adams fiber is the classifying space of a finite abelian p–group. However, a special Adams fiber is never the completion of a classifying space of a compact Lie group (Proposition 3.2).

Theorem 3. Suppose G and K are compact connected Lie groups.

(1) Let Z ⟶ BG_{p}^{∧} ⟶^{f} BK_{p}^{∧} be a fibration. If H^{∗}(ΩZ; ℤ_{p}^{∧}) ⊗ ℚ is a finite dimensional ℚ_{p}^{∧}–vector space, there is a compact Lie group Γ and special Adams fibers F_{1}, ⋯, F_{r} of p–type such that Z ≃ BΓ_{p}^{∧} × ∏_{i=1}^{r} F_{i}. If, in addition, H^{∗}(ΩZ; 𝔽_{p}) is finite, then Z ≃ BΓ_{p}^{∧}. In particuliar, if f ≃ (Bρ)_{p}^{∧} for a homomorphism ρ : G ⟶ K, then ρ is an epimorphism.

(2) Let Z ⟶ BG ⟶ BK be a fibration. If H^{∗}(ΩZ; ℚ) is a finite dimensional ℚ–vector space, there exists a compact Lie group Γ and special Adams fibers F_{i}, 1 ≤ i ≤ s, such that Z ≃ BΓ × ∏_{i=1}^{s} F_{i}. If ΩZ is homotopy equivalent to a finite CW –complex, then Z ≃ BΓ.

According to these results we suggest the following definitions: A map BH ⟶ BG or BH_{p}^{∧} ⟶ BG_{p}^{∧} is called normal if there exists a fibration BH ⟶ BG ⟶ X or BH_{p}^{∧} ⟶ BG_{p}^{∧} ⟶ X such that X satisfies the finiteness conditions of Theorem 1. A map BG ⟶ BK or BG_{p}^{∧} ⟶ BK_{p}^{∧} is called a weak epimorphism if there exists a fibration Z ⟶ BG ⟶ BK or Z ⟶ BG_{p}^{∧} ⟶ BK_{p}^{∧} such that H^{∗}(ΩZ; ℚ) is a finite dimensional ℚ–module or that H^{∗}(ΩZ; ℤ_{p}^{∧}) ⊗ ℚ is a finite dimensional ℚ_{p}^{∧}– module, respectively. If ΩZ is equivalent to a finite CW –complex or if in addition H^{∗}(ΩZ; 𝔽_{p}) is finite then the map is called an epimorphism.

Our key result to prove the Theorems 1, 2, and 3 comes from observation of the following fibrations. Suppose π is a finite p–subgroup of a compact Lie group G

such that π_{0}(G) is a p–group. We consider the fibrations Bπ ⟶^{(Bi)_{p}^{∧}} BG_{p}^{∧} ⟶ X where i : π ⟶ G is the inclusion and X is a 1–connected space. The space X is 1–connected if and only if the quotient space G/π is connected. If π is a normal subgroup of G, then Bπ ⟶ BG ⟶ B(G/π) is a fibration. Moreover, if this fibration is nilpotent, it is preserved by p–completion. The following shows that this is the only case.

Theorem 4. Suppose π is a finite p–subgroup of a compact Lie group G such that

π_{0}(G) is a p–group. Let i : π ⟶ G be the inclusion. If Bπ ⟶^{(Bi)_{p}^{∧}} BG_{p}^{∧} ⟶ X is a fibration for a simply–connected space X, then π is a normal subgroup of G. In particular, if G is connected, then π is a central subgroup of G.

The idea of the proofs of our main results is the following: We have to consider fibrations whose fibers are given by the classifying space BH of a compact connected Lie group H. Using Stasheff’s result [27] fibrations of this type are classified by maps into BHE(BH), which is the classifying space of the monoid of self equivalences of BH. The space HE(BH_{p}^{∧}) is calculated by Jackowski, McClure and Oliver [11] [12]. There exists a complete description of the components, and after completion they prove that for any self–equivalence f : BH ⟶ BH the mapping space map(BH, BH)_{f} is homoytopy equivalent to BZ(H) , where Z(H) denotes the center of H. This allows to get our hands on fibrations with fiber BH.

It is well known that for a compact connected Lie group G, there is a covering α_{G} ⟶ G̃ ⟶ G such that G̃ = G_{s} × T is a product of a simply-connected Lie group G_{s} and a torus T, and that α_{G} is a finite central subgroup of G̃. We call α_{G} ⟶ G̃ ⟶ G a universal finite covering of G. If G is semi-simple, then it is the universal covering of G in the usual sense.

Given a fibration, we can find another fibration involved with a universal finite covering. We will first discuss this kind of fibration and then deduce to the original fibration using Theorem 4. For example, in our proof of Theorem 1, we will consider the following homotopy commutative diagram:

BH′ ⟶ BG̃ ⟶ X̃
↓  ↓  ↓
BH ⟶ BG ⟶ X

First we find K̃ such that X̃ ≃ BK̃. Theorem 4 then provides K with X ≃ BK.

We have another application of Theorem 4. Suppose X is a retract of the p– completed space BG_{p}^{∧}, where G is a compact connected Lie group. The first author showed [10] that there is a compact Lie group K such that X ≃ BK_{p}^{∧}. Our result is the un–completed version of this result.

Theorem 5. Suppose G is a compact connected Lie group. A retract of BG is homotopy equivalent to a classifying space BK for some compact connected Lie group K.

The group K in this theorem, however, need not be a subgroup of G. We will give such an example in §6.

This paper consists of 6 more sections. Each of the next five sections contains the proof of one of the above theorems. Section 2 is devoted to the discussion of normal p–subgroups of compact Lie groups; i.e. for a finite p–subgroup π of a connected Lie group G there exists a fibration Bπ ⟶ BG ⟶ X. Inparticuliar Theorem 4 is proved in this section. Theorem 2 is proved in §3 and Theorem 1 in §4. In §5 the study of Adams fibers leads to a proof of Theorem 3. Retracts of classifying spaces are discussed in §6. Some results about mapping spaces of classifying spaces of connected Lie groups are contained in the appendix §7. These results are used in the proofs of our main theorems.

The work in this paper was done when the first author visited SFB 170 in Göttingen in 1991-92. He wishes to express his gratitude for their hospitality.

2. Normal p–subgroups of connected Lie groups.

In this section we will prove Theorem 4 and some other results about fibrations of type Bπ ⟶ BG ⟶ X. To prove Theorem 4 we need the following lemma.

Lemma 2.1. Suppose Z is a subgroup of N and N is a subgroup of a group G. If Z is central in G and N/Z is a normal subgroup of the quotient group G/Z, then N is a normal subgroup of G.

Proof. Suppose x ∈ N and xZ ∈ N/Z. For any g ∈ G we see that g(xZ)g^{−1} = yZ for some y ∈ N, since N/Z is a normal subgroup of G/Z. Since Z is central in G, it follows that g(xZ)g^{−1} = gxg^{−1}Z. Consequently gxg^{−1} ∈ yZ ⊂ N. Therefore N is a normal subgroup of G. □

Proof of Theorem 4. Let Z(π) denote the center of the finite p–group π and let j : Z(π) ⟶ π be the inclusion. Since π_{0}(G) is a p–group, G/π_{p}^{∧} ⟶ Bπ_{p}^{∧} ⟶ BG_{p}^{∧} is, as the completion of a fibration, again a fibration [4]. The loop space ΩX is mod–p equivalent to the finite connected CW–complex G/π. Thus, by the Sullivan conjecture [14], the evaluation induces an equivalence map(BZ(π), X)_{0} ≃ X between the component of the constant map and X. We have the homotopy commutative diagrams of fibrations:

map_{∗}(BZ(π), Bπ)_{Bj} ⟶ map(BZ(π), Bπ)_{Bj} ⟶^{ev} Bπ
↓  ↓  ↓
map_{∗}(BZ(π), BG_{p}^{∧})_{Bi} ⟶ map(BZ(π), BG_{p}^{∧})_{Bi} ⟶^{ev} BG_{p}^{∧}
↓  ↓  ↓
map_{∗}(BZ(π), X)_{0} ⟶ map(BZ(π), X)_{0}  ⟶^{ev} X

Here ev denotes an evaluation map and induces a homotopy equivalence in the top and bottom rows. Let C_{G}(Z(π)) denote the centralizer of Z(π) in G. Since the map BC_{G}(Z(π))_{p}^{∧} ⟶ BG_{p}^{∧} is a homotopy equivalence and π_{0}(G) is a p–group, a result of Mislin–Thevenaz [15] shows Z(π) is central in G. Thus we obtain the homotopy commutative diagrams of fibrations:

BZ(π) ⟶ BZ(π) ⟶ ∗
↓  ↓  ↓
Bπ  ⟶  BG_{p}^{∧}  ⟶ X
↓  ↓  ↓
B(π/Z(π)) ⟶ B(G/Z(π))_{p}^{∧} ⟶ X

Because the composition BZ(π) ⟶ BG ⟶ X is nullhomotopic, the right arrow in the bottom row exists [9]. Notice that the bottom fibration B(π/Z(π)) ⟶ (B(G/Z(π))_{p}^{∧} ⟶ X satisfies the assumption of this theorem. Repeating the same procedure, we obtain a sequence of fibrations:

Bπ  ⟶  BG_{p}^{∧}  ⟶ X
↓  ↓  ↓
B(π/Z(π)) ⟶ B(G/Z(π))_{p}^{∧} ⟶ X
↓  ↓  ↓
⋮  ⟶  ⋮  ⟶ ⋮
↓  ↓  ↓
∗  ⟶  B\overline{G}_{p}^{∧}  ⟶ X

where \overline{G} is a suitable quotient group. Since a finite p–group has a non-trivial center, the fibre of the bottom fibration is contractible. Using Lemma 2.1, an inductive argument shows that π is a normal subgroup of G. □

Corollary 2.2. Suppose π is a finite subgroup of a compact connected Lie group G. If Bπ ⟶^{Bi} BG ⟶ X is a fibration, then π is central in G.

Proof. The exact sequence of homotopy groups implies that π is abelian and X is 1–connected. Since the fibration is nilpotent, it is preserved by p–completion. Notice that (Bπ)_{p}^{∧} = Bπ_{p}, where π_{p} is a p–Sylow subgroup of π, and that X is p–equivalent to the finite complex G/π. Theorem 4 shows that for any p the p– Sylow subgroup of π is normal in G. Since G is connected, π_{p} is central in G. Consequently, π is central in G. □

Remark. If there is a fibration Bπ ⟶^{Bi} BG ⟶ X, then the loop space ΩX is homotopy equivalent to the quotient space G/π. This means that G/π admits a loop space structure that makes the quotient map G ⟶ G/π deloopable. However we point out that G/π can have a loop structure even if π is not central. In fact, there is a compact Lie group which is homotopy equivalent to G/π such that π is not central in G. Such an example is given by the following. Let i : ℤ/k ⟶ U(n) be the monomorphism which sends x ∈ ℤ/k to the diagonal unitary n × n matrix diag(x, 1, …, 1). Then U(n)/ℤ/k is homeomorphic to SU(n) × S^{1}, and ℤ/k is not a central subgroup of U(n) for n ≥ 2.

We will discuss the mod p version of Corollary 2.2. In this case, the finite subgroup π need not be abelian. Our result will show, however, that π must be p– nilpotent. Casacuberta shows [5] that π is p–nilpotent if and only if (Bπ)_{p}^{∧} ≃ Bπ_{p}, where π_{p} is a p–Sylow subgroup of π.

In [9] the kernel ker(f) of a map f : BG ⟶ X is defined. This is the set of all elements g ∈ N_{p}T_{G} in the p–toral Sylow subgroup N_{p}T_{G} of G, which generates a finite p–group ⟨g⟩, such that f|_{B⟨g⟩} is nullhomotopic. Under certain finiteness conditions on the space X, ker(f) is a normal subgroup of N_{p}T_{G} [20]. In particuliar, if (p, |W_{G}|) = 1, ker(f) is an abelian group.

The following observation, formulated as a lemma, is needed for later purpose. The proof is obvious.

Lemma 2.3. Suppose the following diagram of groups is commutative:

H̃ \underrightarrow{ψ̃} G̃
q_{H} ↓  ↓^{q_{G}}
H ⟶^{ψ} G

where the vertical homomorphisms are surjective. If the image ψ̃(H̃) is normal in G̃, then ψ(H) is normal in G. (This result holds when ”normal” is replaced by ”central”.)

Theorem 2.4. Suppose π is a finite subgroup of a compact connected Lie group

G. If (Bπ)_{p}^{∧} ⟶^{(Bi)_{p}^{∧}} (BG)_{p}^{∧} ⟶^{f} X is a fibration, the following hold:

(1) The finite group π is p–nilpotent.

(2) Its p–Sylow subgroup π_{p} is central in G.

Proof. Notice that if π_{p} is a p–Sylow subgroup of π, then π_{p} is a subgroup of Kerf [20]. We need to consider four cases.

\underline{Case 1.} G is simple, |W (G)| ≡ 0 mod p except that G ≠ G_{2} and p = 3 In this case, according to results of [9, 10], we see π_{p} is a central subgroup of G. Suppose N(π_{p}) denotes the normalizer in π. Using Swan’s theorem [28] we can show that

H^{∗}(Bπ; 𝔽_{p}) ≅ H^{∗}(BN(π_{p}); 𝔽_{p}) ≅ H^{∗}(Bπ_{p}; 𝔽_{p})^{N(π_{p})/π_{p}} ≅ H^{∗}(Bπ_{p}; 𝔽_{p}) . Therefore (Bπ)_{p}^{∧} ≃ Bπ_{p}.

\underline{Case 2.} |W (G)| ≢ 0 mod p Notice that π_{p} is abelian. We claim that π_{p} is trivial. If π_{p} was non-trivial, we would find a nontrivial subgroup Z/p ⊂ π_{p}. We consider the diagram associated with map(Bℤ/p, −).

map(Bℤ/p, (Bπ)_{p}^{∧})_{Bj} ⟶ (Bπ)_{p}^{∧}
↓  ↓
map(Bℤ/p, BG_{p}^{∧})_{Bi} ⟶ BG_{p}^{∧}
↓  ↓
map(Bℤ/p, X)_{0} ⟶ X

By Proposition 7.4, map(Bℤ/p, (Bπ)_{p}^{∧})_{Bj} ≃ BC_{π}(ℤ/p)_{p}^{∧}, therefore the top arrow is rationally homotopy equivalent. The bottom arrow is a homotopy equivalence. Because (p, |W_{G}|) = 1, the subgroup ℤ/p ⟶^{i} G is not central, and BC_{G}(i) ⟶ BG is rationally not a homotopy equivalence. This is a contradiction and implies the desired result.

\underline{Case 3.} G = G_{2} and p = 3 From [9], we see that π_{3} is a subgroup of ℤ/3. Note that map(Bℤ/3, (BG_{2})_{3}^{∧})_{Bi} ≃ BSU(3)_{3}^{∧}. If π_{3} = ℤ/3, a diagram analoguous to the above produces a contradiction. Consequently, π_{3} is trivial.

\underline{Case 4.} G is any compact connected Lie group. Let α_{G} ⟶ G̃ ⟶ G be a universal finite covering of G. The group extension α_{G} ⟶ π̃ ⟶ π induced by the monomorphism i : π ⟶ G is central. Consequently the fibration Bα_{G} ⟶ Bπ̃ ⟶ Bπ is nilpotent, and is preserved by p–completion. We have the homotopy commutative diagram:

(Bα_{G})_{p}^{∧} ⟶ (Bα_{G})_{p}^{∧} ⟶ ∗
↓  ↓  ↓
(Bπ̃)_{p}^{∧} ⟶ (BG̃)_{p}^{∧} ⟶ X
↓  ↓  ↓
(Bπ)_{p}^{∧} ⟶ (BG)_{p}^{∧} ⟶ X

Suppose G̃ = ∏_{i} G̃_{i} where each G̃_{i} is either a simply-connected simple Lie group or a torus. Let G̃_{a} = ∏_{j} G̃_{j} with |W (G̃_{j})| ≢ 0 mod p or G̃_{j} = G_{2} and p = 3. If q : G̃ ⟶ G̃_{a} is the projection and π̃_{p} is a p–Sylow subgroup of π̃, an argument analogous to the ones used in Case 2 and Case 3 shows that q(π̃_{p}) = 1. Hence π̃_{p} is central. This implies that (Bπ̃)_{p}^{∧} ≃ Bπ̃_{p}. Since α_{G} is abelian, we see (Bα_{G})_{p}^{∧} ≃ B(α_{G})_{p}. Therefore (Bπ)_{p}^{∧} ≃ Bπ_{p}. Finally we see π_{p} is central by lemma 2.3. □

Theorem 2.4 is also true if the total space of the fibration is BO(n). We note that the orthogonal group O(n) is disconnected and that the center of O(n) is isomorphic to the finite group ℤ/2.

Corollary 2.5. Suppose π is a finite subgroup of an orthogonal group O(n). If Bπ ⟶^{Bi} BO(n) ⟶ X is a nilpotent fibration, then π is central in O(n).

Proof. Notice that (Bπ)_{p}^{∧} ⟶ BO(n)_{p}^{∧} ⟶ X_{p}^{∧} is a fibration for any p. We recall that if n = 2m or 2m + 1 and p is odd, then BO(n)_{p}^{∧} = BSO(2m + 1)_{p}^{∧} [3]. Hence the p–Sylow subgroup π_{p} is trivial by Theorem 2.4. According to Theorem 4, we see that π = π_{2} is normal in O(n), and hence central in O(n). □

3. Fibrations of type BH ⟶ Y ⟶ BK.

Proof of Theorem 2. (1) Let α_{K} ⟶ K̃ ⟶ K be a universal finite covering of the compact connected Lie group K. We consider the diagram:

∗ ⟶ Bα_{K} _{p}^{∧} ⟶ Bα_{K} _{p}^{∧}
↓  ↓  ↓
BH_{p}^{∧} ⟶ Ỹ ⟶ BK̃_{p}^{∧}
↓  ↓  ↓
BH_{p}^{∧} ⟶ Y ⟶ BK_{p}^{∧}

where Ỹ is the pullback. The middle horizontal fibration is classified by a map BK̃_{p}^{∧} ⟶ B(HE(BH_{p}^{∧})) [27]. This map lifts to B(SHE(BH_{p}^{∧})). Since H is connected, we see B(SHE(BH_{p}^{∧})) ≃ B^{2}(Z(H))_{p}^{∧}, [12]. Taking a self covering of K̃ , if necessary, we may assume the classifying map of the fibration is trivial. Consequently Ỹ ≃ B(H × K̃)_{p}^{∧} and Y ≃ (B(H × K̃)/α_{K})_{p}^{∧} by Theorem 4.

(2) We have the following diagram analogous to (1):

∗ ⟶ Bα_{K} ⟶ Bα_{K}
↓  ↓  ↓
BH ⟶ Ỹ ⟶ BK̃
↓  ↓  ↓
BH ⟶ Y ⟶ BK

Since (B^{2}Z(H))_{p}^{∧} is contractible for large prime p, taking a suitable finite selfcovering of K̃, we can assume that the p–completion of the middle horizontal fibration BH_{p}^{∧} ⟶ Ỹ_{p}^{∧} ⟶ BK̃_{p}^{∧} is trivial at any prime p. Consequently Ỹ_{p}^{∧} ≃ B(H ×K̃)_{p}^{∧} at any p. Thus Y_{p}^{∧} ≃ BG_{p}^{∧} for any p if G = (H × K̃)/α_{K}. □

4. Fibrations of the type BH ⟶ BG ⟶ X.

This chapter is devoted to the proof of theorem 1:

Proof of Theorem 1. (1) First we assume the compact connected Lie group G is semi–simple. We claim that H is semi–simple. If not, the Serre spectral sequence of the fibration G_{p}^{∧} ⟶ ΩX ⟶ BH_{p}^{∧} with coefficients ℚ would tell us that H^{1}(ΩX; ℚ) = 0 and H^{2}(ΩX; ℚ) ≠ 0. This is a contradiction, since ΩX is a mod–p finite H–space and H^{∗}(ΩX; ℚ) ≅ ⋀_{ℚ_{p}^{∧}}(x_{2k_{1}+1}, ..., x_{2k_{n}+1}) an exterior algebra generated by odd degree elements.

Next we will show that there is a compact connected semi–simple Lie group K such that X ≃ BK_{p}^{∧}. Let α be the kernel of the homomorphism π_{2}(BG_{p}^{∧}) ⟶ π_{2}(X). If β := π_{2}(BG_{p}^{∧}) and γ := π_{2}(X), we obtain the homotopy commutative diagram:

Bα ⟶ Bβ ⟶ Bγ
↓  ↓  ↓
F ⟶ BG̃_{p}^{∧} ⟶ X̃
↓  ↓  ↓
BH_{p}^{∧} ⟶ BG_{p}^{∧} ⟶ X

where G̃ is a universal covering of G, where X̃ is the homotopy fibre of the map X ⟶ K(π_{2}X, 2) whose induced homomorphism on the second homotopy group is isomorphic, and where F is the homotopy fibre of the induced map BG̃_{p}^{∧} ⟶ X̃. We claim here that F ≃ (BH′)_{p}^{∧} for some semi–simple Lie group H′. If δ = π_{2}(BH_{p}^{∧}), then we have the homotopy commutative diagram:

∗ ⟶ Bδ ⟶ Bδ
↓  ↓  ↓
Bα ⟶ F̃ ⟶ BH̃_{p}^{∧}
↓  ↓  ↓
Bα ⟶ F ⟶ BH_{p}^{∧}

where H̃ is a universal covering of H, and F̃ is the pullback. The middle horizontal fibration Bα ⟶ F̃ ⟶ BH̃_{p}^{∧} is characterized, according to Stasheff’s theorem [27], by a map BH̃_{p}^{∧} ⟶ B(HE(Bα)). Since H is semi–simple, the space BH̃_{p}^{∧} is 3– connected. Hence the characteristic map is trivial, and we see F̃ ≃ B(α × H̃)_{p}^{∧}. Thus F ≃ B((α × H̃)/δ)_{p}^{∧}. Let H′ = (α × H̃)/δ. Since F is 1–connected, the Lie group H′ is connected. We next claim that the fibration F ≃ (BH′)_{p}^{∧} ⟶ BG̃_{p}^{∧} ⟶ X̃ is trivial. The fibration is characterized by a map X̃ ⟶ BHE(BH′_{p}^{∧}) [27], which lifts to BSHE(BH′_{p}^{∧}) ≃ B^{2}Z(H′)_{p}^{∧} [12] and is therefore nullhomotopic, since X̃ is 2–connected. Here SHE( ) denotes the component of the identity of HE( ). It follows that X̃ is a retract of BG̃_{p}^{∧}. A result of [10] implies X̃ ≃ BK̃_{p}^{∧} for some subgroup K̃ of G̃, where K̃ = ∏_{i} K̃_{i}, a product group of 1–connected simple Lie groups. Since H^{∗}(ΩX; 𝔽_{p}) is finite, using the Sullivan conjecture we can show that the map Bγ ⟶ BK̃_{p}^{∧} is induced by a monomorphism γ ⟶ K̃. If K is the quotient group K̃/γ, Theorem 4 implies X ≃ BK_{p}^{∧}.

Suppose f = (Bi)_{p}^{∧} for a monomorphism i : H ⟶ G. We will show i(H)◁G using some results in §7. Notice that we now have the following homotopy commutative diagram:

BH̃_{p}^{∧} ⟶^{(Bĩ)_{p}^{∧}} BG̃_{p}^{∧} ⟶^{r} BK̃_{p}^{∧}
↓  ↓  ↓
BH_{p}^{∧} ⟶^{(Bi)_{p}^{∧}} BG_{p}^{∧} ⟶ BK_{p}^{∧}

where ĩ is a lifted homomorphsim of Lie groups. We need a general result about maps between classifying spaces of simple Lie groups. Suppose G′ and K′ are 1– connected compact simple Lie groups. If both homotopy sets [(BG′)_{p}^{∧}, B(K′)_{p}^{∧}] and [(BK′)_{p}^{∧}, (BG′)_{p}^{∧}] are non-trivial, then (BG′)_{p}^{∧} is homotopy equivalent to (BK′)_{p}^{∧}. This is a consequence of the following two results: (1) [(BG′)_{p}^{∧}, B(K′)_{p}^{∧}] ≠ 0 implies that rank(G′) ≤ rank(K′) and that the Weyl group W (G′) is a subgroup of W (K′), [1]. (2) [BSp(n)_{2}^{∧}, BSpin(2n + 1)_{p}^{∧}] = 0 for n ≥ 3, [9]. Since BK̃_{p}^{∧} is a retract of BG̃_{p}^{∧}, each (BK̃_{i})_{p}^{∧} is also a retract of BG̃_{p}^{∧}. By Proposition 6.1, for each i ,we can find a factor subgroup G̃_{j_{i}} of G̃ such that r|_{(BG̃_{j_{i}})_{p}^{∧}} : (BG̃_{j_{i}})_{p}^{∧} ⟶ (BK̃_{i})_{p}^{∧} is a homotopy equivalence. Let G̃_{K} = ∏_{i} G̃_{j_{i}} , the product of such subgroups of G̃, and let G̃_{H} be the product of the rest of factor subgroups of G̃ so that G̃ = G̃_{H} × G̃_{K}. Since the restricted map r|_{(BG̃_{K})_{p}^{∧}} is a homotopy equivalence, we see ĩ(H̃) = G̃_{H}.

Consequently ĩ(H̃) is a normal subgroup of G̃. Lemma 2.3 implies that H ◁ G is a normal subgroup. This finishes the proof of part (1) for G semi simple.

Now we prove (1) for the general case. For a compact connected Lie group G, we can find a group extension \overline{G}_{s} ⟶ G ⟶ \overline{T}_{g}, where \overline{G}_{s} is semi-simple and \overline{T}_{g} is a torus (for details see for example [19]). Note that H^{2}(B\overline{T}_{g}; ℤ) = H^{2}(BG; ℤ) and hence \overline{T}_{g} = K(H^{2}(BG; ℤ), 1). If H′ is the kernel of the composite homomorphsim H ⟶ G ⟶ \overline{T}_{g}, we obtain the homotopy commutative diagram:

(BH′)_{p}^{∧} ⟶  (B\overline{G}_{s})_{p}^{∧}  ⟶  X̃
↓  ↓  ↓
BH_{p}^{∧} ⟶  BG_{p}^{∧}  ⟶  X
↓  ↓  ↓
B(H/H′)_{p}^{∧} ⟶ K(H^{2}(BG; ℤ), 2)_{p}^{∧} ⟶ K(H^{2}(X; ℤ), 2)

Since \overline{G}_{s} is semi-simple, we see X̃ ≃ (BK′)_{p}^{∧} for some compact connected Lie group K′. Since H/H′ is a torus, we see that K(H^{2}(X; ℤ), 2) is the p–completion of the classifying space of some torus. Theorem 2 shows X ≃ BK_{p}^{∧} for some compact connected Lie group K.

Suppose f = (Bi)_{p}^{∧}. Let α_{G} ⟶ G̃ ⟶ G be a universal finite covering, where G̃ = G_{s} × T_{g} is a product of a simply–connected Lie group G_{s} and a torus T_{g}. (Warning: G_{s} and T_{g} can be different from \overline{G}_{g} and \overline{T}_{g} respectively.) For suitable universal finite coverings H_{s} × T_{h} ⟶ H and G_{s} × T_{g} ⟶ G, Lie theory provides a homomorphism ĩ : H_{s} × T_{h} ⟶ G_{s} × T_{g} such that the following diagram of Lie groups commutes:

H_{s} × T_{h} \underrightarrow{ĩ} G_{s} × T_{g}
↓  ↓
H ⟶^{i} G

Let H̃ = H_{s} × T_{h}. The above diagram induces the following homotopy commutative diagram:

BH̃_{p}^{∧} ⟶^{(Bĩ)_{p}^{∧}} BG̃_{p}^{∧} ⟶ BK′_{p}^{∧}
↓  ↓  ↓
BH_{p}^{∧} ⟶^{(Bi)_{p}^{∧}} BG_{p}^{∧} ⟶ BK_{p}^{∧}

for a suitable compact connected Lie group K′. By Proposition 7.1 we see that ĩ must be a product homomorphism: ĩ = ĩ_{s} × ĩ_{t} : H_{s} × T_{h} ⟶ G_{s} × T_{g}. We can show that ĩ_{s} : H_{s} ⟶ G_{s} is a monomorphism, and ĩ_{s}(H_{s}) ◁ G_{s}, since G_{s} is semi–simple. Consequently ĩ(H_{s} × T_{h}) ◁ G_{s} × T_{g}. Lemma 2.3 implies that i(H) is a normal subgroup of G, which completes the proof of (1) in the general case.

(2) We will prove the un-completed version of (1). The argument is analoguous to the p–adic case. Again we first assume G is semi–simple. Likewise to the p–adic case, we obtain the following diagram

Bπ ⟶ B(π_{1}G) ⟶ B(π_{2}X)
↓  ↓  ↓
BH′ ⟶ BG̃ ⟶ X̃
↓  ↓  ↓
BH ⟶ BG ⟶ X

where π is a suitable finite abelian group. The fibration BH′ ⟶ BG̃ ⟶ X̃ is trivial at any prime p. Thus, for each prime p, there is a subgroup K̃(p) of G̃ such that X ≃ BK̃(p)_{p}^{∧}. If T_{H′} and TG̃ are maximal tori, there is an induced map B(TG̃/T_{H′}) ⟶ X̃, since the composition BT_{H′} ⟶ BTG̃ ⟶ BG̃ ⟶ X̃ is nullhomotopic [9]. For each p, the induced map provides a maximal torus after p–adic completion, because X̃_{p}^{∧} ≃ BK(p)_{p}^{∧}. Therefore, the homotopy fiber of the map B(TG̃/T_{H′}) ⟶ X̃ is simply-connected and the mod–p cohomology and rational cohomology look like the cohomology of a finite complex. Moreover, X and B(TG̃/T_{H′}) have the same rank. This means that BTG̃/T_{H′} ⟶ X̃ is integrally a maximal torus in the sense of [21].

Next we will show that the space X̃ has a genus type of one of the BK̃(p)’s. In fact, we can take p = 2 ; i.e. X ∈ Genus(BK̃(2)). Suppose α_{p} : X̃_{p}^{∧} ⟶ BK̃(p) is the homotopy equivalence. We see the following equivalences:

H^{∗}(X̃; ℚ^{∧}) ≅ H^{∗}(BK̃(p); ℚ^{∧}) ≅ H^{∗}(BT̃(p); ℚ^{∧})^{W (K̃(p))}

for any p. Here T̃(p) is a maximal torus of the Lie group K̃(p), and W (K̃(p)) is the Weyl group. If TX̃ is a maximal torus of X̃, there is a map α′_{p} : BTX̃ _{p}^{∧} ⟶ BT̃(p)_{p}^{∧} such that the following diagram commutes up to homotopy:

X̃_{p}^{∧} ⟶^{α_{p}} BK̃(p)_{p}^{∧}
↑  ↑
BTX̃ _{p}^{∧} ⟶^{α′_{p}} BT̃(p)_{p}^{∧}

[21]. Consequently all of the ℚ^{∧}–representations of the Weyl groups W (K̃(p)) are equivalent each other. Here we need the following fact. Suppose K_{1} and K_{2} are products of a torus and simply-connected simple Lie groups. If the rational representaion of W (K_{1}) is equivalent to that of W (K_{2}), then the classifing space BK_{1} is p–equivalent to BK_{2} for any odd prime p. One can show this by using the classification of simply-connected simple Lie groups, since W (K_{1}) ≅ W (K_{2}) implies K_{1} = K_{2} up to Sp(n) and Spin(2n + 1). Consequently X_{p}^{∧} ≃ BK̃(p)_{p}^{∧} ≃ BK̃(2)_{p}^{∧} for p odd. Since X̃ has a maximal torus, a result of Notbohm-Smith [22] shows that X̃ ≃ BK̃(2). Therefore X ≃ BK if K = K̃(2)/π_{2}X.

In the general case, using a group extention \overline{G}_{s} ⟶ G ⟶ \overline{T}_{g}, we obtain the homotopy commutaive diagram

BH′ ⟶ B\overline{G}_{s} ⟶  BK̃
↓  ↓  ↓
BH ⟶ BG ⟶  X
↓  ↓  ↓
B(H/H′) ⟶ B\overline{T}_{g} ⟶ K(H^{2}(X; ℤ), 2)

We can show that X has a maximal torus. By Theorem 2, there exists a compact connected Lie group K′ such that X ∈ Genus(BK′) and hence, X ≃ BK for a suitable compact connected Lie group K [18]. □

5. Fibrations of type Z ⟶ BG ⟶ BK.

In this section we prove Theorem 3. First we have to investigate Adams fibers, which were defined, in the introduction, as the homotopy fibers of unstable Adams operations. Let G be a compact connected Lie group and α_{G} ⟶ G̃ ⟶^{q_{G}} G a universal finite cover of G, where G̃ = G_{s} × T is a product of a simply connected Lie group G_{s} and a torus T. Note that G_{s} = ∏_{i} G_{i} a product of simple simply connected Lie groups G_{i}. An unstable Adams operation ψ^{r} : BG_{p}^{∧} ⟶ BG_{p}^{∧} of degree r ∈ ℤ_{p}^{∧} gives rise to a diagram

(*)  \begin{matrix}BG̃_{p}^{∧} & ⟶^{ψ^{r}} & BG̃_{p}^{∧} \\ ^{Bq_{G}}↓ & ^{Bq_{G}}↓ \\ BG_{p}^{∧} & ⟶^{ψ^{r}} & BG_{p}^{∧} & . \end{matrix}

The map ψ^{r} exists if and only if r is a p–adic unit or p is coprime to the order of the Weyl group W_{G} [29] [30] [8]. The homotopy fiber F (G, r) of ψ^{r} is nontrivial if r = p^{k}u, where u is a unit and k ≥ 1. Thus, F (G, r) ≃ F (G, p^{k}). If k ≥ 1, then (p, |W_{G}|) = 1 and Bq_{G} : BG̃_{p}^{∧} ⟶ BG_{p}^{∧} is an equivalence. This means that both Adams fibers F (G̃, p^{k}) and F (G, p^{k}) are equivalent. Moreover, F (G̃, p^{k}) ≃ ∏_{i} F (G_{i}, p^{k}) × B(ℤ/p^{k})^{n}, where n is the dimension of T. This proves the following lemma:

Lemma 5.1. Adams fibers of p–type are homotopy equivalent to a finite product of special Adams fibers of p–type and of the classifying space of a finite abelian p–group.

Let E be the homotopy fiber of the completion map BG ⟶ BG_{p}^{∧}. The homotopy groups of E are uniquely p–divisible. An Adams operation ψ^{p^{k}} induces an equivalence ψ^{p^{k}} : E ⟶^{≃} E. Therefore, in the commutative diagram of fibrations

(**)  \begin{matrix}∗ & ⟶ & E & ⟶ & E \\ ↓ & & ↓ & & ↓ \\ F & ⟶ & BG & ⟶^{ψ^{p^{k}}} & BG \\ ^{≃}↓ & & ↓ & & ↓ \\ F (G, p^{k}) & ⟶ & BG_{p}^{∧} & ⟶^{ψ^{p^{k}}} & BG_{p}^{∧} \end{matrix}

the map F ⟶ F (G, p^{k}) is an equivalence. We use this equivalence to calculate the cohomology of Adams fibers. If (p, |W_{G}|) = 1, the mod–p cohomology and the p–adic cohomology of BG are polynomial, denoted by H^{∗}(BG; 𝔽_{p}) ≅ 𝔽_{p}[x_{1}, ⋯, x_{n}] and H^{∗}(BG; ℤ_{p}^{∧}) ≅ ℤ_{p}^{∧}[x_{1}, ⋯, x_{n}], where x_{i} has degree 2r_{i}. It follows that H^{∗}(G; 𝔽_{p}) ≅ Λ_{𝔽_{p}}(y_{1}, ⋯, y_{n}) and H^{∗}(G; ℤ_{p}^{∧}) ≅ Λ_{ℤ_{p}^{∧}}(y_{1}, ⋯, y_{n}), which are exterior algebras with generators y_{i} of degree 2r_{i} −1. We use the same notation for the generators.

Proposition 5.2. Let k ≥ 1 and (p, |W_{G}|) = 1.

(1) F (G, p^{k}) is a p–complete CW –complex of finite type.

(2)

H^{∗}(F (G, p^{k}); ℤ) ≅ ℤ[x_{1}, ⋯, x_{n}]/(p^{kr_{i}}x_{i} : 1 ≤ i ≤ n) ,
where (p^{kr_{i}}x_{i} : 1 ≤ i ≤ n) is the ideal generated by the elements p^{kr_{i}}x_{i}.

(3)

H^{∗}(F (G, p^{k}); 𝔽_{p}) ≅ H^{∗}(G; 𝔽_{p}) ⊗ H^{∗}(BG; 𝔽_{p})
≅ Λ_{𝔽_{p}}(y_{1}, ⋯, y_{n}) ⊗ ℤ/p[x_{1}, ⋯, x_{n}] .

as algebras. The Steenrod operations P^{j}, j ≥ 1, act trivially on y_{i} and in the obvious way on H^{∗}(BG; 𝔽_{p}). For the r_{i}-th order Bockstein β_{r_{i}} we have β_{r_{i}}(y_{i}) = x_{i}.

(4)

H^{∗}(ΩF (G, p^{k}); 𝔽_{p}) ≅ H^{∗}(ΩG; 𝔽_{p}) ⊗ H^{∗}(G; 𝔽_{p})

as algebras. In particuliar, H^{∗}(ΩF (G, p^{k}); 𝔽_{p}) is not finite and ΩF (G, p^{k}) is not equivalent to a finite CW –complex.

(5) If F (G, p^{k}) is a special Adams fiber, i.e G is simple simply-connected , then F (G, p^{k}) is not homotopy equivalent to the p–adic completion of the classifying space of a compact Lie group .

Proof. For G a torus of dimension n, F (G, p^{k}) ≃ B(ℤ/p^{k})^{n}. Some straightforward and easy calculations prove everything in this case. Therefore, we only have to prove the statements for simply connected Lie groups. For abbreviation, we define F := F (G, p^{k}).

Part (1) follows from the diagram (**). In order to calculate the cohomology we consider the diagram

G ⟶ F ⟶ BG
∥  ↓  _{ψ^{p^{k}}} ↓
G ⟶ ∗ ⟶ BG .

For both p–adic and mod–p cohomology, in the Serre spectral sequence of the bottom fibration the generators y_{i} are mapped on x_{i} by the transgression. In mod–p cohomolgy, ψ^{p^{k}} induces the trivial map and the spectral sequences of the top row collapses. In p–adic cohomology, ψ^{p^{k}}^{∗}(x_{i}) = p^{kr_{i}}x_{i} and the generators of H^{∗}(G; 𝔽_{p}) are mapped on p^{kr_{i}}x_{i} by the transgression in the spectral sequence of the top row. This shows that the isomorphism in (2) and (3) are isomorphisms of algebras. Because the reduced cohomology is pure torsion, we can deal with ℤ_{p}^{∧} instead of ℤ as coefficients.

In integral cohomology the class x_{i} generates a cyclic group of order p^{kr_{i}}. We see that β_{r_{i}}(y_{i}) = x_{i}. To calculate the action of the Steenrod powers we look at the mod–p Serre spectral sequence of the fibration F ⟶ BG ⟶ BG. The elements y_{i} ∈ H^{∗}(F; 𝔽_{p}) are transgressive and mapped onto x_{i} + decomposables. Let us assume that P^{j}(y_{i}) ≠ 0. Then P^{j}(y_{i}) is transgressive. Because (p, |W_{G}|) = 1, we have p > r_{i} for all i. In particuliar P^{j}(y_{i}) is a sum of products of elements. Each summand contains at least one y_{l} as factor. Hence, some differential of degree less than max(2r_{i}) acts nontriviallly on P^{j}(y_{i}), which is a contradiction.

Statement (4) follows by similiar considerations. H^{∗}(ΩG; 𝔽_{p}) ≅ ⨂_{i} D[z_{i}] is the tensor algebra of division algebras generated by classes z_{i} in degree 2r_{i} − 2. We also have a diagram of fibrations

ΩG ⟶ ΩF ⟶ G
∥  ↓  _{ψ^{p^{k}}} ↓
ΩG ⟶ ∗ ⟶ G .

In the Serre spectral sequence for mod–p cohomology of the bottom row all generators z_{i} are transgressive. Hence the spectral sequence for the top row collapses. This proves the first part of (4). The second part of (4) follows from the fact that H^{∗}(ΩG; 𝔽_{p}) is not finite.

Finally, let Γ be a compact Lie group , such that F (G, p^{k}) ≃ BΓ_{p}^{∧}. Then, Γ contains a nontrivial element of order p^{l}, l ≥ 1, which gives a subgroup ℤ/p^{l} ⊂ Γ. The composition Bℤ/p^{l} ⟶ BΓ_{p}^{∧} ≃ F (G; p^{k}) ⟶ BG_{p}^{∧} lifts to BG and is induced by a homomorphism ρ : ℤ/p^{l} ⟶ G [7]. By [23], H^{∗}(Bℤ/p^{l}; 𝔽_{p}) is a finitely generated H^{∗}(BΓ; 𝔽_{p})–module, and, by (3), a finitely generated H^{∗}(BG; 𝔽_{p})–module. Thus, ρ is a monomorphism [23]. Because ψ^{p^{k}} ∘Bρ is nullhomotopic, l ≤ k. In particuliar, Γ is a finite group. Otherwise we would find elements of order p^{l} for every l ∈ ℕ. Now we choose l = 1, i.e. we restrict to the subgroup ℤ/p ⊂ ℤ/p^{l} We get a diagram of fibrations

map(Bℤ/p, BΓ_{p}^{∧})_{Bi} ⟶ map(Bℤ/p, BG_{p}^{∧})_{Bρ} ⟶^{(ψ^{p^{k}})^{∗}} map(Bℤ/p, BG_{p}^{∧})_{const}
ev↓  ^{ev}↓  ^{ev}↓
BΓ_{p}^{∧}  ⟶  BG_{p}^{∧}  ⟶^{ψ^{p^{k}}}  BG_{p}^{∧}  .

By [7] and Proposition 7.4, map(Bℤ/p, BΓ_{p}^{∧})_{Bi} ≃ BC_{Γ}(ℤ/p)_{p}^{∧} and map(Bℤ/p, BG_{p}^{∧})_{Bρ} ≃ BC_{G}(ρ)_{p}^{∧}. The right vertical arrow is a homotopy equivalence. The left vertical arrow is rationally a homotopy equivalence, because Γ is a finite group. The homotopy fiber of (ψ^{p^{k}})^{∗} is connected and given by map(Bℤ/p, BΓ_{p}^{∧})_{Bi}. If G is simple and simply-connected, the subgroup ℤ/p^{l} ⊂ G is not central, since (p, |W_{G}|) = 1. The centralizer C_{G}(ℤ/p) is connected, of maximal rank, W_{C_{G}(ℤ/p)} ⊂ W_{G} is a proper subgroup [19]. Hence, BC_{G}(ρ) ⟶ BG is not a rational homotopy equivalence. This is a contradiction and finishes the proof. □

Let G and K be compact connected Lie groups and let Z ⟶ BG_{p}^{∧} ⟶^{f} BK_{p}^{∧} be a fibration such that H^{∗}(ΩZ; ℤ_{p}^{∧}) ⊗ ℚ is a finite ℚ_{p}^{∧}–module. As a finitely generated connected Hopf algebra, H^{∗}(ΩZ; ℤ_{p}^{∧}) ⊗ ℚ ≅ Λ_{ℚ_{p}^{∧}}(y_{1}, ⋯, y_{r}) is an exterior algebras generated by classes y_{i} of degree 2l_{i} − 1. The functor ⊗_{ℤ}ℚ is exact. The spectral sequence for p–adic cohomology of a fibration gives a spectral sequence converging to H^{∗}( ; ℤ_{p}^{∧}) ⊗ ℚ. Therefore, H^{∗}(Z; ℤ_{p}^{∧}) ⊗ ℚ ≅ ℚ_{p}^{∧}[x_{1}, ⋯, x_{r}] is a polynomial algebra generated by the classes x_{i} of degree 2l_{i}. Because BG and BK are rationally homotopy equivalent to a product of even dimensional Eilenberg–MacLane spaces, the spectral sequence for H^{∗}( ; ℤ_{p}^{∧}) ⊗ ℚ of the fibration Z ⟶ BG_{p}^{∧} ⟶ BK_{p}^{∧} collapses and H^{∗}(BG; ℤ/p)⊗ℚ ≅ H^{∗}(Z; ℤ_{p}^{∧})⊗H^{∗}(BK; ℤ_{p}^{∧})⊗ℚ. This implies that H^{∗}(BK_{p}^{∧}; ℤ_{p}^{∧}) ⊗ ℚ ⟶ H^{∗}(BG_{p}^{∧}; ℤ_{p}^{∧}) ⊗ ℚ is a monomorphism and has a left inverse as map of algebras.

By [2] or [21], the map f : BG_{p}^{∧} ⟶ BK_{p}^{∧} gives rise to a commutative diagram

BT_{G}_{p}^{∧} ⟶^{f_{T}} BT_{K} _{p}^{∧}
↓  ↓
BG_{p}^{∧} ⟶^{f} BK_{p}^{∧} .

Passing to cohomology we obtain

H^{∗}(BT_{G}_{p}^{∧}; ℤ_{p}^{∧}) ⊗ ℚ ⟵^{f_{T}^{∗}} H^{∗}(BT_{K} _{p}^{∧}; ℤ_{p}^{∧}) ⊗ ℚ
↑  ↑
H^{∗}(BG_{p}^{∧}; ℤ_{p}^{∧}) ⊗ ℚ ⟵^{f^{∗}} H^{∗}(BK_{p}^{∧}; ℤ_{p}^{∧}) ⊗ ℚ .

By [1], there exists a homomorphism W_{G} ⟶ W_{K} such that f_{T}^{∗} is W_{G}–equivariant.

Now we assume that both G = G_{s} × T and K = K_{s} × S are products of a simply-connected Lie group G_{s} or K_{s} and a torus T or S. Moreover, G_{s} = ∏_{i} G_{i} and K_{s} = ∏_{j} K_{j} are products of simple simply connected Lie groups G_{i} or K_{j}. By the proof of Proposition 7.1, G_{s} = G_{1} × G_{2} and T = T_{1} × T_{2} such that f factors over BG_{2}_{p}^{∧} × BT_{2}_{p}^{∧}. This establishes the diagram

BG_{p}^{∧} ≃ BG_{1}_{p}^{∧} × BG_{2}_{p}^{∧} × BT_{1}_{p}^{∧} × BT_{2}_{p}^{∧} ⟶^{f} BK_{p}^{∧}
↓  ∥
BG_{2}_{p}^{∧} × BT_{2}_{p}^{∧}  ⟶^{\overline{f}} BK_{p}^{∧} .

The map \overline{f} : BG_{2}_{p}^{∧} × BT_{2}_{p}^{∧} ⟶ BK_{p}^{∧} induces an isomorphism in H^{∗}( ; ℤ_{p}^{∧}) ⊗ ℚ– cohomology. Therefore, T_{2} ≅ S and G_{2} and K_{s} are of the same rational type, i.e the classifying spaces are rationally equivalent. Spin(2n + 1) and Sp(n), n ≥ 3 are the only nonisomorphic simple simply connected Lie groups of the same rational type.

Actually, BSpin(2n + 1)_{p}^{∧} ≃ BSp(n)_{p}^{∧} for odd primes, and there are no essential maps BSpin(2n + 1) ⟶ BSp(n) and vice versa with n ≥ 3 [9]. Thus, if we replace Spin(2n + 1) by Sp(n) for odd primes, we can assume that G_{2} ≅ K_{s} as Lie groups.

Self maps of BK ≃ BK_{s} × BS, K_{s} = ∏_{j} K_{j}, are analysed in [12]. The result implies that

\overline{f} ≃ σ ∘ (∏_{j} Bα_{j} ∘ e_{j}^{d_{j}} ∘ ψ^{r_{j}}) × A_{T} .

σ permutes equivalent factors of K_{s}, α_{j} is an outer autumorphism, e is an exceptional isogeny, which is always a mod–p equivalence if it occurs [1], d_{j} ∈ {0, 1}, ψ^{r_{j}} is an unstable Adams operation of degree r_{j} ∈ ℤ_{p}^{∧}, and A_{T} is a self map of BS_{p}^{∧}. Therefore, the homotopy fiber hofib(\overline{f}) of \overline{f} is equivalent to a product ∏_{i} F_{i} × Bπ of special Adams fibers F_{i} of p–type and the classifying space Bπ of a finite abelian p–group. F_{i} is the homotopy fiber of ψ^{r_{i}} and Bπ the homotopy fiber of A_{T}.

We summarize these considerations in the following proposition, which is a special case of Theorem 3. Two compact connected Lie groups G and H are said to have the same p–adic type if BG_{p}^{∧} ≃ BH_{p}^{∧}.

Proposition 5.3. Let G and K be product groups of a simply connected Lie group and a torus.

(1) Let Z ⟶ BG_{p}^{∧} ⟶^{f} BK_{p}^{∧} be a fibration such that H^{∗}(ΩZ; ℤ_{p}^{∧}) ⊗ ℚ is a finite dimensional ℚ_{p}^{∧}–vector space. Then, G ≅ G′ × G′′, such that G′′ and K have the same p–adic type. Moreover, there exist a finite number of special Adams fibers F_{i} of p–type and a finite abelian p–group π such that

Z ≃ BG′_{p}^{∧} × ∏_{i} F_{i} × Bπ .

(2) Let Z ⟶ BG ⟶^{f} BK be a fibration such that H^{∗}(ΩZ; ℚ) is a finite dimensional ℚ–vector space. Then G ≅ G′ × G′′ and G′′ ≅ K. Moreover, there exist a finite number of special Adams fibers F_{i} and a finite abelian group π such that

Z ≃ BG′ × ∏_{i} F_{i} × Bπ

Proof. There is something left to prove. The above considerations show that G ≅ G′ × G′′ such that G′′ and K have the same p–adic type. Moreover, the map f splits over BG′′_{p}^{∧}. This yields a diagram

BG′_{p}^{∧}  BG′_{p}^{∧} ⟶ ∗
↓  ↓  ↓
Z ⟶ BG_{p}^{∧} ⟶^{f} BK_{p}^{∧}
↓  ↓  ∥
F ⟶ BG′′_{p}^{∧} ⟶ BK_{p}^{∧} ,

where F ≃ ∏_{i} F_{i} × Bπ. Here, F_{i} is a special Adams fiber of p–type and π a finite abelian p–group. The middle vertical fibration is trivial and establishes an equivalence Z ≃ BG′_{p}^{∧} × F, which proves (1).

The above considerations are also applicable to the uncompleted version. This completes the proof. □

Now we are prepared for the proof of Theorem 3.

Proof of Theorem 3. Let Z ⟶ BG_{p}^{∧} ⟶^{f} BK_{p}^{∧} be a fibration, such that H^{∗}(ΩZ; ℤ_{p}^{∧}) ⊗ ℚ is a finite dimensional ℚ_{p}^{∧}–vector space. We can choose universal finite coverings α_{G} ⟶ G̃ ⟶^{q_{G}} G and α_{K} ⟶ K̃ ⟶^{q_{K}} K such that f lifts to a map f̃: BG̃ ⟶ BK̃. We get a diagram

Z̃ ⟶ BG̃_{p}^{∧} \underrightarrow{f̃} BK̃_{p}^{∧}
↓  ^{Bq_{G}}↓  ^{Bq_{K}} ↓
Z ⟶ BG_{p}^{∧} ⟶^{f} BK_{p}^{∧}

Notice that both q_{G} and q_{K} are rationally equivalences. Thus we get

H^{∗}(ΩZ; ℤ_{p}^{∧}) ⊗ ℚ ≅ H^{∗}(ΩZ̃; ℤ_{p}^{∧}) ⊗ ℚ .

Because all spaces are p–completed, we can assume that α_{G} and α_{K} are finite abelian p–groups.

We need a refinement of the splitting in Proposition 5.3. By that proposition, G̃ ≅ G′ × G′′, such that G′′ ≅ G′′_{s} × T, a product of a simply connected Lie group G′′_{s} and a torus T, has the same p–type as K̃. The extension map f̃ factors over f̃′′ : BG′′_{s}_{p}^{∧}×BT_{p}^{∧} ⟶ BK̃_{p}^{∧}. Up to permutations and equivalences f̃′′ ≃ ∏ ψ^{r_{i}} ×A_{T}, r_{i} ∈ ℤ_{p}^{∧}. We split G′′_{s} = G_{2} × G_{3} into a product. G_{2} collects those factors of G′′_{s} where r_{i} is a p–adic unit, and G_{3} the others, i.e. G̃ ≅ G′ × G_{2} × G_{3} × T. An Adams operation ψ^{r} only exists on BH_{p}^{∧}, H a compact connected Lie group, if r is a p–adic unit or if p is coprime to the order of W_{H} [29] [30] [8]. Therefore, the composition α_{G} ⟶ G̃ ⟶ G_{3} is constant.

Because K̃ has the same p–type as G′′, we get an analogous splitting K̃ = K_{2} × K_{3} × T, and α_{K} ⟶ K̃ ⟶ K_{3} is also constant. This yields a commutative diagram

BG̃_{p}^{∧}  ⟶^{Bq_{G}}  BG_{p}^{∧}
∥  ∥
BG′_{p}^{∧} × BG_{2}_{p}^{∧} × BG_{3}_{p}^{∧} × BT_{p}^{∧} ⟶^{Bq_{G}} B(G′ × G_{2} × T/α_{G})_{p}^{∧} × BG_{3}_{p}^{∧}
Bp̃r↓  ^{Bpr}↓
BG_{2}_{p}^{∧} × BG_{3}_{p}^{∧} × BT_{p}^{∧}  ⟶^{B\overline{q}_{G}} B(G_{2} × T/α_{G})_{p}^{∧} × BG_{3}_{p}^{∧}
f̃′′↓  ^{f′′}↓
BK̃_{p}^{∧} = BK_{2}_{p}^{∧} × BK_{3}_{p}^{∧} × BT_{p}^{∧} ⟶^{Bq_{K}} B(K_{2} × T/α_{K})_{p}^{∧} × BK_{3}_{p}^{∧} .

The map f̃′′ ≃ f̃′′′ × f_{3}, where

f̃′′′ ≃ Bρ_{p}^{∧} : B(G_{2} × T)_{p}^{∧} ⟶ B(K_{2} × T)_{p}^{∧}

is induced by a homomorphism ρ̃: G_{2} × T ⟶ K_{2} × T, and where f_{3} is a product of Adams operations. The map f′′ exists according to a result of [20],. We have an anologous splitting of f′′, namely f′′ ≃ f′′′ × f_{3}, where

f′′′ ≃ Bρ_{p}^{∧} : B((G_{2} × T)/α_{G})_{p}^{∧} ⟶ B((K_{2} × T)/α_{K})_{p}^{∧}

is induced by a homomorphism ρ : (G_{2} × T)/α_{G} ⟶ (K_{2} × T)/α_{K}. This implies that

Z̃ ≃ Bker(ρ̃∘ p̃r)_{p}^{∧} × F ,

(proposition 5.3) and therefore that

Z ≃ Bker(ρ ∘ \overline{pr})_{p}^{∧} × F .

Here, ker( ) denotes the kernel of a homomorphism, and F is a product of special Adams fibers of p–type, in particuliar the homotopy fiber of f_{3}.

For later purpose, we observe that, for a homomorphism α : Γ ⟶ Γ′ between two compact connected Lie groups, ker(α) fits into an exact sequence

ker_{e}(α) ⟶ ker(α) ⟶ π_{0}(ker(α)) ,

where ker_{e}( ) denotes the component of the unit. The group π_{0}(ker(α)) is abelian and acts trivially on ker_{e}(α). This is obviously true for products of a simply connected Lie group and a torus, and therefore, using universal finite coverings, also true in the general case. Hence, Bker_{e}(α) and Bker(α) are rationally homotopy equivalent. In the above situation, π_{0}(ker(ρ∘pr)) is an abelian p–group and ker_{e}(ρ∘ pr) ≅ G′/(α_{G} ∩ G′).

If H^{∗}(ΩZ; 𝔽_{p}) is finite, Adams fibers of p–type can’t occur (Proposition 5.2). This finishes the proof of part (1).

To prove the second statement, that is the uncompleted version, we also pass to universal finite coverings, yielding a diagram

Z̃ ⟶ BG̃ \underrightarrow{f̃} BK̃
↓ ^{Bq_{G}}↓  ^{Bq_{K}} ↓
Z ⟶ BG ⟶^{f} BK .

We also have a splitting G̃ = G′ × G′′, where f̃|_{BG′} is nullhomotopic, and where f factors over a rational equivalence f̃′′ : BG′′ ⟶ BK̃. In fact, f̃′′ is a mod–p equivalence for almost all primes. Because BK and BK̃ are simply-connected , completion maintains the fibrations [4]. By the proof of (1), the rationalisation Z_{ℚ} is equivalent to BG′/(α_{G} ∩ G′)_{ℚ}, and Z_{p}^{∧} ≃ BΓ(p) × F (p), where F (p) a product of special Adams fibers of p–type, and where Γ(p) is a compact Lie group. Moreover, there exists a finite abelian p–group π(p) and an exact sequence G′/(S_{p}α_{G} ∩ G′) ⟶ Γ(p) ⟶ π(p), where S_{p}α_{G} denote the p–Sylow subgroup. Using the arithmetique square and the fact that the rationalisations of Bπ(p) and F (p) are trivial, one can show that Z ≃ Z′ × F. Here F is a product of special Adams fibers and Z′ is the total space of a fibration B(G′/(α_{G} ∩ G′)) ⟶ Z′ ⟶ Bπ, where π is a finite abelian group. By [18], this implies that Z′ is the classifying space of a compact Lie group.

If ΩZ is equivalent to a finite CW –complex, Adams fibers can’t occur (Proposition 5.2 (4)). This finishes the proof. □

6. Retracts of classifying spaces.

In this section we prove Theorem 5. As a remark, it will be shown that the Lie group K in Theorem 5 need not be a subgroup of G.

Proof of Theorem 5. Let α_{G} ⟶ G̃ ⟶ G be a universal finite covering of the compact connected Lie group G. If X is a retract of BG, an argument similar to the one used in the proof of the main theorm of [10] shows that one can find a retract X̃

of BG̃ : Bα_{X} ⟶ Bα_{G} ⟶ Bα_{X}

↓  ↓  ↓
X̃ ⟶^{i} BG̃ ⟶^{r} X̃
↓  ↓  ↓
X ⟶ BG ⟶ X

where the composite r·i is homotopy equivalent to the identity map. Along the line of the proof again, one can show that for any prime p there is a simply connected Lie group K̃(p) which is a product of some factor subgroups of G̃ such that X̃_{p}^{∧} ≃ BK̃(p)_{p}^{∧}.

First we claim that X̃ has a maximal torus in the sense of [25] or [21]. The composite map i · r|_{BT_{G}} : BTG̃ ⟶ BG̃ is of form Bρ for a homomorphsim ρ : TG̃ ⟶ G̃ [16]. We can show that the homotopy fibre of the map r|_{Bρ(T_{G})} ⟶ X̃ is a

retract of the finite CW –complex G̃/TG̃ and hence homotopy equivalent to a finite CW –complex. Since ρ(TG̃) is a retract of TG̃, the monomorphism H^{∗}(X; ℚ) ⟶ H^{∗}(Bρ(TG̃); ℚ) is finite. Hence rank(X) = rank(ρ(TG̃)).

Next we claim that the space X̃ has a genus type of BK̃(2) . We see that

H^{∗}(X̃; ℚ^{∧}) ≅ H^{∗}(BK̃(p); ℚ^{∧}) ≅ H^{∗}(BT̃(p); ℚ^{∧})^{W (K̃(p))}

for any p. Here T̃(p) is a maximal torus of K̃(p). An argument similar to the one used in the proof of Theorem 1 (2) implies

X_{p}^{∧} ≃ BK̃(p)_{p}^{∧} ≃ BK̃(2)_{p}^{∧}

for p odd. If K̃ = K̃(2), then X̃ ≃ BK̃. We now have a fibration Bα_{X} ⟶ BK̃ ⟶ X. Let f be the map Bα_{X} ⟶ BK̃, and let S_{p}α_{X} be the p–Sylow subgroup of the finite abelian group α_{X}. Notice that BS_{p}α_{X} ⟶ BK̃_{p}^{∧} ⟶ X_{p}^{∧} is a fibration. Since S_{p}α_{X} is a p–group, there is a monomorphism i_{p} : S_{p}α_{X} ⟶ K̃ such that f_{p}^{∧} ≃ (Bi_{p})_{p}^{∧}. Theorem 4 shows i_{p}(S_{p}α_{X}) is central in K̃. Consequently, the map f : Bα_{X} ⟶ BK̃ factorizes over an homotopy equivalence X ≃ BK if K = K̃/α_{X}. □

Remark. There are compact connected semi-simple Lie groups K and G which satisfy the following:

(1) BK is a retract of BG.

(2) K is not a subgroup of G.

The following is such an example. Let K̃ = SU(n) and G̃ = SU(n) × SU(n). The center of the special unitary group SU(n) is isomorphic to ℤ/n, and consists of the scalar matrices

( \begin{matrix}ζ \\ & . & . & . \\ & & & & ζ \end{matrix} ) with ζ^{n} = 1

For a positive integer k, let η_{k} : ℤ/k ⟶ G̃ be the monomorphism as follows:

η_{k}(ζ) = ( \begin{matrix}ζ \\ & . & . & . \\ & & & & ζ \end{matrix} ) × ( \begin{matrix}ζ^{k} \\ & . & . & . \\ & & & & ζ^{k} \end{matrix} )

Now we define G = G̃/η_{k}(ℤ/n) and K = K̃/ℤ/n = P SU(n). When k is prime to n!, we have the commutative diagram:

BK̃ ⟶^{id×ψ^{k}} BG̃
↓  ↓
BK ⟶^{f} BG

where ψ^{k} is an unstable Adams operation and f is the induced map. If q : BG̃ ⟶ BK̃ is the projection to the first factor subgroup of G̃, then q ·(id×ψ^{k}) ≃ id. Thus BK is a retract of BG. However, K is not a subgroup of G if k ≠ 1.

7. Appendix.

This section contains some results about mapping spaces of classifying spaces of compact Lie groups needed in the proofs of the main results.

For R = ℤ or R = ℤ_{p}^{∧} we denote by X_{R} the space X itself or the p–adic completion of X respectively.

Proposition 7.1. Let R = ℤ or R = ℤ_{p}^{∧}. Let G = G_{s} × T_{1} and K = K_{s} × T_{2} be products of a a simply–connected compact Lie group and a torus. Let f : BG_{R} ⟶ BK_{R} be a map such that f^{∗} : H^{∗}(BK_{R}; R) ⊗ ℚ ⟶ H^{∗}(BG_{R}; R) ⊗ ℚ is a monomorphsim. Then we can find two subgroups G′ and G′′ of G satisfying the following:

(1) G = G′ × G′′

(2) G′and G′′ are also products of a simply connected Lie group and a torus.

(3) The restriction f|_{BG′′} is nullhomotopic so that f factors over f′ : BG′_{R} ⟶ BK_{R}

(4) f′^{∗} : H^{∗}(BK_{R}; R) ⊗ ℚ ⟶ H^{∗}(BG′_{R}; R) ⊗ ℚ is a monomorphism

(5) G′ and K have the same rank.

If f^{∗} has a left inverse as a map between algebras, then f′^{∗} is an isomorphism.

Corollary 7.2. Let G be a product of a simply–connected compact Lie group and a torus, and let H and K be (arbitrary) connected compact Lie groups.

(1) Any fibration of the form (BH)_{p}^{∧} ⟶ (BG)_{p}^{∧} ⟶ (BK)_{p}^{∧} is fiber homotopically trivial.

(2) Any integral fibration of the form BH ⟶ BG ⟶ BK is fiber homotopically trivial.

Proof. The Serre spectral sequence for rational cohomology of the fibration (completed or un-completed) collapses since everything is concentrated in even degrees. By the last proposition, this shows, using the notation from above, that H^{∗}(BK_{R}; R)ℚ ⊗ H^{∗}(BH_{R}; R) ⊗ ℚ ≅ H^{∗}(BG_{R}; R) ⊗ ℚ. Hence, the maps BH_{R} ⟶ BG′′_{R} as well as BG′_{R} ⟶ BK_{R} are rationally equivalences. Both map can only have a finite kernel. The first map makes H^{∗}(BH; ℤ/p) into a finitely generated H^{∗}(BG′′; ℤ/p) module. By [23] this implies that BH_{R} ⟶ BG′′_{R} has a trivial kernel and is therefore an equivalence. This shows that the fibration is trivial.

The proof of Proposition 7.1 is based on the following lemma:

Lemma 7.3. Let 𝔽 be a field of characteristic zero, let W = ∏_{i} W_{i} be a finite product of finite groups W_{i}, 1 ≤ i ≤ n, and let M_{i} be an irreducible 𝔽[W_{i}]–module. Then, M := ⨁_{i} M_{i} is a direct sum of irreducible 𝔽[W]–modules and, for every 𝔽[W]–submodule N ⊂ M, there exist a subset I ⊂ {1, ..., n} such that N maps onto ⨁_{i∈I} M_{i} and such that the projection N ⊂ M ⟶ M_{i} is trivial for i ∉ I. In particuliar, N ≅ ⨁_{i∈I} M_{i}.

Proof. Obviously, each M_{i} is an irreducible W –module. Notice that N splits into irreducible 𝔽[W]–modules N_{j} i.e. N ≅ ⨁_{j} N_{j}. The composition N ⊂ M ⟶ M_{i} is either trivial or an epimorphism so that each summand N_{j} is isomorphic to a summand M_{i_{j}}. Without loss of generality we can assume that all compositions are epimorphisms. The action of W on M can be described by diagonal matrices D(w_{1}, ⋯, w_{n}) for w_{i} ∈ W_{i}, and the map ⨁_{j} N_{j} ⟶ ⨁_{j} M_{i_{j}} by a matrix Φ = (ϕ_{j,i}). At least one entry in each column and each row is nontrivial and hence an isomorphism. Let N_{j} be a summand on which all factors W_{i}, i ≠ i_{0}, act trivially. Then,

( \begin{matrix}w_{1} \\ & w_{2} \\ & & . & . & . \\ & & & & & w_{n} \end{matrix} ) · ( ϕ_{j,1} ; ϕ_{j,2} ; ⋮ ; ϕ_{j,n} ) = ( ϕ_{j,1} ; ϕ_{j,2} ; ⋮ ; ϕ_{j,n} ) · w_{i_{o}} .

If, for k ≠ i_{o}, ϕ_{j,k} is nontrivial, then w_{i_{o}} = ϕ_{j,k}^{−1} · w_{k} · ϕ_{j,k} for all w_{i_{0}} ∈ W_{i_{0}} and w_{k} ∈ W_{k}, which is a contradiction. Therefore, ϕ_{j,k} = 0 for k ≠ i_{0} and ϕ_{j,i_{0}} is an isomorphism. Moreover Φ is a square matrix. In particuliar, it is a monomial matrix, i.e every row and every column has exactly one nontrivial entry, which is an isomorphism. This finishes the proof. □

Proof of Proposition 7.1. First we assume that G = G_{s} and K = K_{s}, i.e. G and K are simply-connected. Let G = ∏ G_{i} and K = ∏ K_{j} be the splitting into simple simply connected Lie group. By [16] or [21], the map f lifts to a map f_{T} : (BT_{G})_{R} ⟶ (BT_{K})_{R} between the classifyng spaces of the maximal tori. Moreover, by [1] there exists a homorphism W_{G} ⟶ W_{K} such that the induced map

f_{T}^{∗} : H^{∗}((BT_{K})_{R}; R) ⊗ ℚ ⟶ H^{∗}((BT_{G})_{R}; R) ⊗ ℚ

is W_{G}–equivariant. The isomorphism

H^{2}((BT_{G})_{R}; R) ≅ ⨁_{i} H^{2}((BT_{G_{i}})_{R}; R) ⊗ ℚ

is a splitting into irreducible ℚ_{R}[W_{G}]–modules of the form as assumed in the previous lemma. Thus there exists a splitting G = G′ × G′′ such that

H^{2}((BT_{K})_{R}; R) ⊗ ℚ ≅ H^{2}((BT_{G′})_{R}; R) ⊗ ℚ ,

i.e. G′ and K have the same rank, and f|_{BG′′_{R}} is trivial in H^{∗}( ; R)⊗ℚ–cohomology and therefore nullhomotopic [12]. By a standard trick we can divide out BG′′ ( e.g. see [9]), i.e. f factors over a map f′ : BG′_{R} ⟶ BK_{R}. The induced map in H^{∗}( ; R)⊗ ℚ–cohomology is still a monomorphism and has a left inverse as algebra map if f^{∗} does. Because both polynomial rings H^{∗}(BG′_{R}; R) ⊗ ℚ and H^{∗}(BH_{R}; R) ⊗ ℚ have the same transcendence degree over ℚ_{R}, f′^{∗} is an isomorphism.

Now we consider the general case, i.e G = G_{s} × T_{1} and K = K_{s} × T_{2}. Then f ≃ f_{s}×f_{T}, where f_{s} : BG_{s}_{R} ⟶ BK_{s}_{R} and f_{T} : BT_{1}_{R} ⟶ BT_{2}_{R}. This follows from the proof of [19, Theorem 16.2], which is the uncompleted version of proposition 7.1 for self maps. The same arguments also apply in our case. H^{∗}(f_{s}; R) ⊗ ℚ and H^{∗}(f_{T}; R) ⊗ ℚ are monomorphisms. Then, G_{s} = G′_{s} × G′′_{s} splits into two groups with the desired properties. By similiar arguments, T_{1} splits into a product of two tori with the same properties. This finishes the proof. □

The following proposition as well as the proof was pointed out to the second author by G.Mislin.

Proposition 7.4. Let V be an elementary abelian p–group and G a compact Lie group. Then, for any homomorphism ρ : V ⟶ G,

(map(BV, BG)_{Bρ})_{p}^{∧} ≃ map(BV, BG_{p}^{∧})_{Bρ} .

Proof. By [13, 3.4.5] or [7], BC_{G}(ρ)_{p}^{∧} ⟶ (map(BV, BG)_{Bρ})_{p}^{∧} is a homotopy equivalence. The classifying space BG is p–good in the sense of [4]. By [13, 3.3.2 and 3.4.3], the mapping space map(BV, BG_{p}^{∧})_{Bρ} is p–complete, and BC_{G}(ρ)_{p}^{∧} ⟶ map(BV, BG_{p}^{∧})_{Bρ} is a homotopy equivalence. □

References

1. J.F. ADAMS AND Z.MAHMUD, Maps between classifying spaces, Inventiones Math. 35 (1976), 1–41.

2. J.F. ADAMS and Z. WOJTKOWIAK, Maps between p–completed classifying spaces, Proc. Royal Soc. Edingburgh 112a (1989), 231-325.

3. A. BOREL, Topology of Lie groups and characteristic classes, Bull. AMS 61 (1955), 397-432.

4. A. BOUSFIELD and D. KAN, Homotopy limits, completions and localisations, SLNM 304 (1972).

5. C. CASACUBERTA, The behaviour of homology in the localization of finite groups, Canad. Math. Bull. 34(3) (1991), 311-320.

6. W.G. DWYER and C.W. WILKERSON, Mapping spaces of nullhomotopic maps, Astérisque 191 (1990), 97-108.

7. W.G. DWYER and A. ZABRODSKY, Maps between classifying spaces, Proc. 1986 Barcelona conference, SLNM 1298, 106-119.

8. K.ISHIGURO, Unstable Adams operations on classifying spaces, Math. Proc. Camb. Phil. Soc. 102 (71) (1987), 71-75.

9. K. ISHIGURO, Classifying spaces of compact simple Lie groups and p–tori, Proc. of 1990 Barcelona Conf., SLNM 1509, 210-226.

10. K. ISHIGURO, Retracts of classifying spaces, Adams memorial symposium on algegraic topology volume 1, London Math. Soc. Lecture Notes 175, Cambridge Univ. Press (1992), 271–280.

11. S. JACKOWSKI, J.E. McCLURE AND B. OLIVER, Homotopy classification of self-maps of BG via G–actions Part I and Part II, Ann. of Math. 135 (1992), 183–226, 227–270.

12. S. JACKOWSKI, J.E. McCLURE AND B. OLIVER, Self homotopy equivalences of BG, Preprint.

13. J. LANNES, Sur les espaces fonctionelles dont la source est la classifiant d’un p–groupe abélien élemémentaires,, Publ. Math. IHES 75 (1992), 135-244.

14. H. MILLER, The Sullivan conjecture on maps from classifying spaces, Ann. Math. 120 (1984), 39-87.

15. G. MISLIN AND J. THÉVENAZ, The Z^{∗}–Theorem for compact Lie groups, Math. Ann. 291 (1991), 103–111.

16. D. NOTBOHM, Maps between classifying spaces, Math. Z. 207 (1991), 153-168.

17. D. NOTBOHM, Maps between classifying spaces and applications, to appear in JPPA.

18. D. NOTBOHM, Fake Lie groups and maximal tori IV, Math. Ann. 294 (1992), 109-116.

19. D. NOTBOHM, Homotopy uniqueness of classifying spaces of compact connected Lie groups at primes dividing the order of the Weyl group,, to appear in Topology Göttingen 1991.

20. D. NOTBOHM, Kernels of maps between classifying spaces,, Math. Gott. Heft 17 (1992).

21. D. NOTBOHM and L. SMITH, Fake Lie groups and maximal tori I, Math. Ann. 288 (1990), 637-661.

22. D. NOTBOHM and L. SMITH, Fake Lie groups and maximal tori III, Math. Ann. 290 (1991), 629–642.

23. D. QUILLEN, The spectrum of an equivariant cohomology ring I, Ann. Math. 94 (1971), 549-572.

24. D. RECTOR, Subgroups of finite dimensional topological groups, JPPA 1 (1971), 253-273.

25. D. RECTOR, Loop structures on the homotopy type of S^{3}, SLNM 418 (1974), 121–138.

26. D. RECTOR and J. STASHEFF, Lie groups from a homotopy point of view, in Localisations in group theory and homotopy theory, Proc. of the Seattle symposium 1974, SLNM 418 (1974), 121-131.

27. J. STASHEFF, A classification theorem for fibre spaces, Topology 2 (1963), 239–246.

28. R.G. SWAN, The p–period of a finite group, Illinois J. of Math. 4 (1972), 341–346.

29. D.SULLIVAN, Geometric topology part I, localisations periodicity, and Galois symmetry, MIT-notes (1970).

30. C.W. WILKERSON, Self-maps of classifying spaces, LNM 418 (1974), 150–157.