UNSTABLE SPLITTINGS OF CLASSIFYING

SPACES OF p––COMPACT GROUPS

D. Notbohm

Abstract. Dwyer and Wilkerson gave a definition of a p–compact group, which is a loop space with certain properties and a good generalisation of the notion of compact Lie groups in terms of classifying spaces and homotopy theory; e.g. every p–compact group has a maximal torus, a normalizer of the maximal torus and a Weyl group. The believe or hope that p–compact groups enjoys most properties of compact Lie groups establishes a program for the classification of these objects. Following the classification of compact connected Lie groups, one step in this program is to show that every simply connected p–compact group splits into a product of simply connected simple p–compact groups. The proof of this splitting theorem is based on the fact that every classifying space of a p–compact group splits into a product if the normalizer of the maximal torus does.

1. Introduction.

A loop space is a triple X = (X, BX, e : ΩBX ⟶^{≃} X), where X and BX are toplogical spaces, where BX is pointed, and where e is a homotopy equivalence between the loop space ΩBX of BX and X. Such a triple is called a p–compact group, if X is 𝔽_{p}–finite, i.e H^{∗}(X; 𝔽_{p}) is finite, and if BX is a p–complete connected space. This notion was introduced by Dwyer and Wilkerson in [7]. Examples of p–compact groups are given by the completion of a compact connected Lie group. For a compact connected Lie group G, the tripel (G_{p}^{∧}, BG_{p}^{∧}, ΩBG_{p}^{∧} ≃ G_{p}^{∧}) is a p–compact group.

In recent work of Dwyer and Wilkerson [7, 8] and of Møller and the author [12] it turned out that p–compact groups are a good homotopy theoretic generalisation of compact Lie groups. Inparticular, it was shown that p–compact groups enjoy quite a lot of the properties of compact Lie groups; e.g there exist always maximal tori, normalizer of the maximal tori and Weyl groups [7]. The maximal torus of a p–compact group X is a map BT_{X} ⟶ BX of an Eilenberg–MacLane space BT_{X} ≃ K((ℤ_{p}^{∧})^{n}, 2) into the classifying space BX of X with certain properties. The Weyl group, denoted by W_{X}, is a finite group, and the normalizer of the maximal torus is a map BN(T_{X}) ⟶ BX, where BN(T_{X}) fits into a fibration BT_{X} ⟶ BN(T_{X}) ⟶ BW_{X} such that the triangle

[figure: \begin{aligned}BT_{X} & & BN_{X} \\ & BX \end{aligned}]

Typeset by AMS-TEX

1980 Mathematics Subject Classification (1985 Revision). 55 R 35, 55 P 35, 22 E 20.

Key words and phrases. loop space, classifying space, p–compact group, compact Lie group.

commutes up to homotopy. For exact definitions of these notions see Section 2. In general, the normalizer, also denoted by N_{X}, does not give a p–compact group, because the space BN_{X} is not p–complete (the fundamental group π_{1}(BN_{X}) might not be a finite p–group). Nevertheless, the space BN_{X} establishes a finite loop space N_{X} := (N_{X}, BN_{X}, ΩBN_{X} ≃ N_{X}), where N_{X} is 𝔽_{p}–finite. This tripel behaves similar as p–compact groups. There exists a p–compact subgroup of P_{X} ⊂ N_{X} , which we construct by restricting the fibration to the classifying space of the p– Sylow subgroup of W_{X}. The classifying space BP_{X} is p–complete, because π_{1}(BP_{X}) is a finite p–group and because BT_{X} is p–complete. This follows from [4, II 5.1, 5.2]. The map BP_{X} ⟶ BX is called the p–toral Sylow subgroup of X and plays the same role as the p–toral Sylow subgroup of a compact Lie group.

One of the main question about p–compact groups asks for a classification of these objects. The naive approach to believe that the analogy between p–compact groups and compact Lie groups is as good as possible produces a lot of ‘theorems’ and conjectures. Because of the lack of an equivalent for the Lie algebra, one cannot translate all the proofs of statements about compact Lie groups. Translation means to express everything in terms of classifying spaces, e.g. a homomorphism X ⟶ Y between p–compact groups is a pointed map BX ⟶ BY between the classifying spaces, and a p–compact group X ≅ X_{1} × X_{2} splits into a product of p–compact groups if the there exists a homotopy equivalence BX ≃ BX_{1} × BX_{2}. The task of the naive approach consists of finding ‘new’ proofs (in terms of homotopy theory) for ‘old’ results, which also work for the larger class of p–compact groups.

The classification of compact connected Lie groups says that, for every compact connected Lie group, there exists a finite covering, which is a product of simple simply connected Lie groups and a torus. In [12] was shown that the first part of this result is true for connected p–compact groups, namely every connected p–compact group has a finite covering, which is a product of a simply connected p– compact group and a torus. For the second step one has to show that every simply connected p–compact group is a product of simple simply connected p–compact groups. A ‘new’ proof of this second step is the main purpose of this paper.

We call a p–compact group X simply connected if the space X is simply connected. The definition of simple has to wait until Section 2.

1.1 Theorem. Let p be an odd prime. Let X be a simply connected p–compact group. Then, X ≅ X_{1} × ...X_{n} splits into a product of simple simply connected p–compact groups.

There exists also a notion of a center of a p–compact group[8] [12], which is the generalization of the group or Lie group theoretic center. The center is always a p–compact group. Again, for details see Section 2. A p–compact group X is called centerfree if the center Z(X) of X is the trivial group, i.e. the classifying space BZ(X) is contractible.

1.2 Theorem. Let p be an odd prime. Let X be a centerfree connected p–compact group. Then, X ≅ X_{1} × ...X_{n} splits into a product of simple centerfree connected p–compact groups.

The proofs of both theorem are based on a general splitting criteria for p–compact groups.

1.3 Theorem. A p–compact group X splits into product X ≅ X_{1} × X_{2} of p– compact groups if and only if the normalizer N_{X} of the maximal torus T_{X} ⟶ X splits into a product N_{X} ≅ N_{1} × N_{2}.

For the proof of Theorem 1.3 we have to study maps from classifying spaces into almost Bℤ/p–local spaces. A space A is called Bℤ/p–local if the map A ⟶ map(Bℤ/p, A) is an equivalence and called almost Bℤ/p–local if the map A ⟶ map(Bℤ/p, A)_{const} into the component of the constant map is an equivalence. Examples of Bℤ/p–local and almost Bℤ/p–local spaces are provided by 𝔽_{p}–finite spaces. A 𝔽_{p}–finite space K is Bℤ/p–local. This follows from the Sullivan conjecture [10]. By [15], the space BHE(K) is almost Bℤ/p–local. Here, HE(K) denotes the monoid of self equivalences of K. In general, this is not an 𝔽_{p}–finite space. Moreover, if K is p–complete and a loop space, the classifying space BHE(K) is also p–complete [15]. These are the main example we apply the next theorem to.

1.4 Theorem. Let P ⟶ X be a p–toral Sylow subgroup of a p–compact group X, and let f : BX ⟶ A be a map into a connected p–complete almost Bℤ/p–local space.

(1) The restriction f|_{BP} is nullhomotopic if and only if f is nullhomotopic.

(2) The map A ⟶ map(BX, A)_{const} is an equivalence.

Remark. In [11] is proved a similar result for maps between classifying spaces of p–compact groups.

The paper is organized as follows: In Section 2 we recall material about p– compact groups, mostly from [7], and prove some auxiliary lemmas necessary for the proof of Theorem 1.3. Section 3 contains a calculation of some low dimensional cohomology groups of pseudo reflection groups. The proof of Theorem 1.4 is carried out in Section 4, and the proof of Theorem 1.3 in Section 5. The final section is devoted to the proofs of Theorem 1.1 and Theorem 1.2.

One remark about references. There is some overlap between the papers [7] and [12]. For most citation referring to one of these papers one could also use the other one. We used the one which was first at hand.

Finally, we would like to point out that, independently of us, Dwyer and Wilkerson also got proofs for similar results.

2. Background.

In this section we recall the basic notions about p–compact groups from [7]. Most of the notions are motivated by classical Lie group theory and by passing to classifying spaces. For keeping things short and because the analogy to compact Lie groups is discussed in [7, 8] and [12], we omit motivations.

2.1 Isomorphisms, monomorphisms, subgroups and exact sequences : A homomorphism Y ⟶ X of p–compact groups or loop spaces is an isomorphism if

Bf : BY ⟶ BX is an equivalence. A sequence X ⟶^{f} Y ⟶^{g} Z of p–compact groups

is short exact if the associated sequence BX ⟶^{Bf} BY ⟶^{Bg} BZ is a fibration up to homotopy. A monomorphism of p–compact groups is a map BX ⟶ BY whose homotopy fiber, denoted by Y/X, is 𝔽_{p}–finite. A subgroup Y ⟶ X of p–compact group X is a monomorphism of p–compact groups.

2.2 p-compact tori, p–compact toral groups and finite extensions of p–compact tori : A p–compact torus is a p–compact group(T, BT, ΩBT ≃ T), where BT ≃ K((ℤ_{p}^{∧})^{n}, 2) is an Eilenberg–MacLane space of degree 2.

A finite group or a finite loop space is a tripel (K, BK, ΩBK ≃ K) such that BK is an Eilenberg–MacLane space of a finite group of degree 1.

A finite extension of a p–adic torus T is a tripel (N, BN, ΩBN ≃ N) which fits into a short exact sequence of loop spaces T ⟶ N ⟶ W =: N/T, where W is a finite loop space. A p–compact toral group P is a finite extension of a p-compact torus T such that the quotient P/T is a finite p–group. Inparticular, every p– compact toral group is a p–compact group.

2.3 Conjugation and subconjugation : Two homomorphisms f, g : Y ⟶ X of p–compact groups are called conjugate if the induced maps Bf, Bg : BY ⟶ BX are homotopic.

For a homomorphism f : Y ⟶ X of p–compact groups and for a p–compact toral subgroup i : P ⟶ X we say that P is subconjugate to Y if there exists a homomorphism j : P ⟶ Y such that fj and i are conjugate.

2.4 Centralizers and centers : For a homorphism f : Y ⟶ X between p– compact groups, we define the centralizer C_{X}(f(Y)) to be the loop space given by the triple

C_{X}(f(Y)) := (Ωmap(BY, BX)_{Bf}, map(BY, BX)_{Bf}, id) .

The evaluation at the basepoint ev : map(BY, BX)_{Bf} ⟶ BX establishes a homomorphism C_{X}(f(Y)) ⟶ X of loop spaces. If Y is a p–compact toral group the centralizer C_{X}(f(Y)) is again a p–compact group and the evaluation C_{X}(f(Y)) ⟶ X is a monomorphism [7, 5.1, 5.2 and 6.1].

A subgroup Z ⟶ X of a p–compact group X is called central if the monomorphism C_{X}(Z) ⟶ X is an isomorphism. The center Z(X) of X is the maximal central subgroup of X [8, 1.2] [12, 4.3, 4.4]. To give an explicit definition we use a result of Dwyer and Wilkerson [8, 1.3]. For every p–compact group X, the centralizer C_{X}(X) is a p–compact group and Z(X) := C_{X}(X) ⟶ X is the center of X. For every p–compact group X there exists a short exact sequence Z(X) ⟶ X ⟶ X/Z(X) =: P X of p–compact groups, and, if X is connected, the quotient P X has a trivial center [12, 4.7].

We call a p–compact group X centerfree if Z(X) is the trivial group.

2.5 Maximal tori : The maximal torus of a p–compact group X is a monomorphism T_{X} ⟶ X of a p-compact torus into X such that the centralizer C_{X}(T_{X}) is a p-compact toral group, whose component of the unit is given by T_{X}.

2.6 Theorem [7, 8.11, 8.13 and 9.1]. Let X be a p–compact group.

(1) The p–compact group X has a maximal torus T_{X} ⟶ X.

(2) Any subtorus T ⟶ X of X is subconjugated to the maximal torus T_{X} ⟶ X.

(3) Any two maximal tori of X are conjugated.

(4) If X is connected then T_{X} ⟶ C_{X}(T_{X}) is an isomorphism.

(5) If X is connected every finite cyclic subgroup ℤ/p^{n} ⟶ X of X is subconjugate to T_{X}.

2.7 Weyl spaces and Weyl groups: Let T_{X} ⟶ X be a maximal torus of a p–compact group. We think of BT_{X} ⟶ BX as being a fibration. The Weyl space W_{T} (X) is defined to be the mapping space of all fiber maps over the identity on BX. Then each component of W_{T} (X) is contractible and the Weyl group W_{T} (X) := π_{0}(W_{T} (X)) is a finite group under composition [D-W, 9.5].

2.8 Theorem [7, 9.5 and 9.7]. Let T_{X} ⟶ X be the maximal torus of a connected p–compact group X.

(1) The action of W_{X} on BT_{X} induces representations

W_{X} ⟶ Aut(H^{2}(BT_{X}; ℤ_{p}^{∧}) ⊗ ℚ) ≅ Gl(n, ℚ_{p}^{∧})

and

W_{X} ⟶ Aut(H_{2}(BT_{X}; ℤ_{p}^{∧}) ⊗ ℚ) ≅ Gl(n, ℚ_{p}^{∧})

which are monomorphisms whose images are generated by pseudoreflections.

(2) The map H_{ℚ_{p}^{∧}}^{∗}(BX) ⟶ H_{ℚ_{p}^{∧}}^{∗}(BT_{X})^{W_{X}} is an isomorphism.

2.9 Normalizers, p–normalizers of maximal tori and p–toral Sylow subgroups : Let i : T_{X} ⟶ X be a maximal torus of a p–compact group X. Again we think of BT_{X} ⟶ BX as being a fibration. The Weyl space W_{X} acts on BT_{X} via fiber maps This establishes a monoid homomorphism W_{X} ⟶ HE(BT_{X}) where HE(BT_{X}) denotes the monoid of all self equivalences of BT_{X}. Passing to classifying spaces establishes a map BW_{X} ⟶ BHE(BT_{X}) which can be thought of as being a classifying map of the fibration BT_{X} ⟶ BN(T_{X}) ⟶ BW_{X}. The total space gives the classifying space of the normalizer N(T_{X}) of T_{X}. This is always a finite extension of the p-compact torus T_{X}.

Let W_{p} be the union of those components of W_{X} corresponding to a p–Sylow subgroup W_{p} of W_{X}. The restriction of the above construction to W_{p} gives the classifying space of the p–normalizer N_{p}(T_{X}).

Since the action of W_{X} respects the map BT_{X} ⟶ BX, the monomorphism T_{X} ⟶ X extends to a loop map N(T_{X}) ⟶ X. The p–normalizer fits into an exact sequence T_{X} ⟶ N_{p}(T_{X}) ⟶ W_{p} and is therefore a p-compact toral group. The restriction N_{p}(T_{X}) ⟶ X is a monomorphism [7, 9.9] and is a p–toral Sylow subgroup of X, i.e. every p–compact toral subgroup P ⟶ X of X is subconjugate to N_{p}(T_{X}).

2.10 Proposition. Let N_{p}(T_{X}) ⟶ X be the p–toral Sylow subgroup of a p– compact group X. Then, the following holds:

(1) Every p-compact torus subgoup P′ ⟶ X of X is subconjugated to N_{p}(T_{X}).

(2) The induced map H^{∗}(BX, 𝔽_{p}) ⟶ H^{∗}(BN_{p}(T_{X}); 𝔽_{p}) is a monomorphism.

Proof. The Euler characteristic of X/N_{p}(T_{X}) is coprime to p [7, proof of Theorem 2.3]. Therefore, part (1) follows from [7, 2.14] and part (2) from [7, 9.8]. □

We also call N_{p}(T_{X}) the p–normalizer of T_{X}.

2.11 Simply connected and simple p–compact groups : A p–compact group X is called simply connected, if X is simply connected, and X is called simple, if the associated representation W_{X} ⟶ Aut(H_{2}(BT_{X}; ℤ_{p}^{∧}) ⊗ ℚ) is irreducible. By [5], this is equivalent to the fact that the associated complex representation is irreducible. The notion of simple is motivated by the classical situation. For a compact connected Lie group G, the representation W_{G} ⟶ H_{2}(BT_{G}; ℂ) is irreducible if and only if G is simple.

2.12 Elementary abelian subgroups2.13 Let X be a p–compact group, and let i : P ⟶ X be the p–toral Sylow subgroup. Let j : E ⟶ X a monomorphism of an elementary abelian group E into X. By 2.10, the subgroup E is subconjugte to P via a homomorphism j′ : E ⟶ P. Such a subconjugation is called special if C_{P} (j′(E)) ⟶ C_{X}(j(E)) is a p–toral Sylow subgroup.

2.13 Lemma.

(1) For every monomorphism j : E ⟶ X, there exists a special subconjugation j′ : E ⟶ P.

(2) Let

[figure: \begin{aligned}E & & E_{1} \\ & X \end{aligned}; α; j; j_{1}]

be a diagram commuting up to conjugation. and let j′ : E ⟶ P be a special subconjugation of j. Then, there exists a special subconjugation j′_{1} : E_{1} ⟶ P such that

[figure: \begin{aligned}E & & E_{1} \\ & P \end{aligned}; α; j′; j′_{1}]

commutes up to conjugation.

Proof. Let P′ ⟶ C_{X}(j(E)) be the p–toral Sylow subgroup. Because j(E) is central in C_{X}(j(E)), it is also contained in P′ [12, 4.]. By Proposition 2.10, there exists a

subconjugation k : P′ ⟶ P. The composition j′ : E ⟶^{j} P′ ⟶^{k} P is a subconjugation of j into P and C_{P} (j′(E)) ≅ P′. Hence, j′ is special. This proves part (1).

Let E_{2} ⊂ E_{1} be a complement of E, i.e. E_{1} ≅ E ⊕ E_{2}. Taking the adjoint of j_{1} : E ⊕ E_{2} ⟶ X establishes map j_{2} : E_{2} ⟶ C_{X}(j(E)). Because j′ was special, P′ := C_{P} (j′(E)) ⟶ C_{X}(j(E)) is a p–toral Sylow subgroup. Applying part (1) yields a special subconjugation j′_{2} : E_{2} ⟶ P′, and passing back to maps into P and X proves part (2). □

The elemantary abelian subgroups of a p–compact group X build a category A_{p}(X), which we will call the Quillen category of X. An object of A_{p}(X) is a

monomorphism E ⟶^{i_{E}} X, where E is a nontrivial elementary abelian group, and a morphism is a triangle

[figure: \begin{aligned}E_{1} & & E_{2} \\ & X \end{aligned}]

which is commutative up to conjugation. For the functor

ϕ : A_{p}(X) ⟶ T op : (E ⟶ X) ↦ BC_{X}(E)

there exists a map

Φ : \underrightarrow{hocolim}_{A_{p}(X)}(ϕ) ⟶ BX .

Let Φ^{∗} : A_{p}(X) ⟶ Ab be the functor given by Φ^{∗}(E ⟶ X) := H^{∗}(BC_{X}(E); 𝔽_{p}). The following theorem is a collection of results of [8, 8.1 and the proof, 8.2].

2.14 Theorem [8]. Let X be p–compact group.

(1) The category A_{p}(X) is mod–p acyclic, i.e for the constant functor F_{ℤ/p} taking as value ℤ/p, we have

\underleftarrow{lim}_{A_{p}(X)}^{i} F_{ℤ/p} = \{ \begin{matrix}0 for i > 0 \\ ℤ/p for i = 0 \end{matrix}

(2) The map Φ : \underrightarrow{hocolim}_{A_{p}(X)}(ϕ) ⟶ BX is a homotopy equivalence.

(3) We have

\underleftarrow{lim}_{A_{p}(X)}^{i} Φ^{∗} = \{ \begin{matrix}0 for i > 0 \\ ℤ/p for i = 0 \end{matrix}

We finish this section with a result, necessary for later purpose.

2.15 Proposition. Let X be a p–compact group and i : P ⟶ X a p–toral Sylow subgroup. Then BZ(P) ⟶ map(BP, BP)_{id} ⟶ map(BP, BX)_{Bi} is a sequence of homotopy equivalences.

Proof. The first equivalence follows from [8, 1.2]. Let i : T ⟶ X be a maximal torus of X. This inclusion factors over i : T ⟶ P. (Confusing notations we denote all inclusions by i.) If we think a of the map BT ⟶ BP as a fibration respectively a covering, then we get a free action of \overline{P} := P/T on BT. By [9, 5.1], there exists a natural transformation of functors map(BP, ) ⟶ map(BT, )^{h\overline{P}}, which, for every target, is an equivalence. The map map(BT, BP)_{Bi} ⟶ map(BT, BX)_{Bi} is an equivalence because T is a maximal torus and because C_{X}(T) is subconjugate to P (Proposition 2.10). Moreover, this map is \overline{P} –equivariant, and induces therefore an equivalence map(BT, BP)_{Bi}^{h\overline{P}} ⟶ map(BT, BX)_{Bi}^{h\overline{P}}. The component of id : BP ⟶ BP is obviuosly mapped onto the component Bi : BP ⟶ BX, which finishes the proof of the statement. □

For further definitions, background and explanations, inparticular for the motivation of these notions, we refer the reader to [7,8] and [12].

3. Low dimensional cohomology groups of pseudo reflection groups. The main result of this chapter states vanishing results for some cohomology

groups of pseudo reflection groups at odd primes. Let U be a finite dimensional vector space over the p–adic rationals. A faithful representation ρ : W ⟶ Gl(U) of a finite group W is called a pseudo reflection group, if the image ρ(W) is generated by pseudo reflections. The representation is called a honest real reflection group, if ρ(W) is generated by honest reflections and if the representation is already defined over ℚ. A finite group W is a pseudo reflection group, if there exists a representation making W to a pseudo reflection group. A pseudo reflection group is called irreducible, if the associated representation is irreducible.

3.1 Proposition. Let W be a pseudo reflection group. Then, for an odd prime p, the homology and cohomology groups H_{1}(W, ℤ_{p}^{∧}), H_{1}(W; 𝔽_{p}), H_{2}(W, ℤ_{p}^{∧}), H_{2}(W, 𝔽_{p}), H^{1}(W, ℤ_{p}^{∧}), H^{1}(W; 𝔽_{p}), H^{2}(W, ℤ_{p}^{∧}), H^{2}(W, 𝔽_{p}), H^{3}(W; ℤ_{p}^{∧}) all vanish.

Proof. Because W is a finite group, it is sufficient to look at H_{1}(W; ℤ_{p}^{∧}) and H^{2}(W; ℤ/p). Then, for the other groups, the statement follows by universal coefficient theorems. Because every pseudo reflection group splits into a product of irreducible pseudo reflection groups and because of the Künneth–formula we also can assume that W is irreducible.

The group H_{1}(W, ℤ) is isomorphic to the abelinization of W. Because W is generated by elements of order dividing p − 1, this is a finite abelian group of order coprime to p. Thus, H_{1}(W; ℤ_{p}^{∧}) = 0.

The cohomology group H^{2}(W; 𝔽_{p}) classifies central extensions of the form ℤ/p ⟶ N ⟶ W. We want to show that every central extension of this form splits.

First we consider the case of an honest real reflection group. Let R ⊂ W denote the set of reflections of W and R_{B} ⊂ R a minimal set of generators of W Let σ_{1}, ...σ_{n} denote the elements of R_{B}. Then there exists integers m_{i,j} such that W ≅< σ_{1}, ..., σ_{n} > /, < (σ_{i}σ_{j})^{m_{i,j}} : 1 ≤ i, j ≤ n > is the quotient of the free group generated by the elements of R_{B} dividing out only relations of the form (σ_{i}σ_{j})^{m_{i,j}} [3, chap. 5]. Let ŝ: W ⟶ N be a set theoretic section. Then we define s : R_{B} ⟶ W by sσ := ŝ(σ)^{p}. Because the extension is central, the map s does not depend on ŝ. We want to show that s can be extended to group theoretic section W ⟶ N. That is to say that s has to satify the relation (s(σ_{i})s(σ_{j}))^{m_{i,j}} = 1 for all 1 ≤ i, j ≤ n.

Let W′ ⊂ W be the subgroup generated by σ_{1} and σ_{2}, and let N′ ⊂ N be the subgroup defined by the pull back diagram

[figure: \begin{matrix}ℤ/p & N_{′} & W_{′} \\ ℤ/p & N & W & . \end{matrix}]

By the classification list of pseudo reflection groups [5] there exist only three reflection groups generated by two honest reflections, namely ℤ/2 × ℤ/2, Σ_{3} and the dieder group D_{12} of 12 elements. A short calculation shows that in all cases H^{2}(W′, ℤ/p) = 0. For ℤ/2 × ℤ/2 this is obvious, for Σ_{3} one uses the fact that Σ_{3}/ℤ/3 ≅ ℤ/2, and for D_{12} one observes that Σ_{3} ⊂ D_{12} is a subgroup of index 2. Therefore there exists a group thoeretic section s′ : W′ ⟶ N′. By the independence of s, for i = 1, 2, we have s(σ_{i}) = s′(σ_{i})^{p} = s′(σ_{i}^{p}) = s′(σ_{i}). This shows that s satisfies the relation for σ_{1} and σ_{2}, and analogously, all relation. Hence, the map s can be extended to a group theoretic section, the sequence ℤ/p ⟶ N ⟶ W splits, and H^{2}(W; ℤ/p) = 0 for honest real reflection groups.

If W is a pseudo reflection group, then by the classification list [5], the order of W is coprime to p or W is one of the groups of number 1, 2b for p = 3, 28, 35, 36 or 37, which all describe honest real reflection groups or belongs to one of the numbers 2a, 12, 29, 31 or 34. We refere here to the numbering of [5]. In all the latter cases we only have to consider one prime. The following table indicates this prime and denotes a subgroup of index coprime to p. Moroever, the subgroup is a honest real reflection group.

\underline{no.}  \underline{grouporder}  \underline{prime}  \underline{subgroup}  \underline{index}
2a  r · m^{n−1} · n!  p ≤ n  Σ_{n}  r · m^{n−1}
12  48  p = 3  Σ_{3}  8
29  2^{8} · 3 · 5  p = 5  Σ_{5}  2^{5}
31  64 · 6!  p = 5  Σ_{5}  3 · 2^{7}
34  108 · 9!  p = 7  Σ_{7}  2^{5} · 3^{5} .

In no. 2a, the number m divides p − 1 and r divides m. For no. 2a the information about the subgroup might be found in [17] and for all other numbers in [1]. Therefore, in all these cases, H^{2}(W; ℤ/p) also vanishes. □

The following proposition is needed in Section 6.

3.2 Proposition. Let p be an odd prime. For i = 1, 2 let W_{i} ⟶ Gl(U_{i}) be a pseudo reflection group and L_{i} ⊂ U_{i} be a [W_{i}]–sublattice.

(1) The map

H^{3}(W_{1}; L_{1}) ⟶ H^{3}(W_{1}; L_{1} × L_{2})

is an isomorphism.

(2) The map

H^{3}(W_{1}; L_{1}) ⊕ H^{3}(W_{2}; L_{2}) ⟶ H^{3}(W_{1} × W_{2}; L_{1} × L_{2})

is an isomorphism.

Proof. Part (2) is a consequence of (1). By Proposition 3.1 we have

H^{2}(W_{1}; H^{1}(W_{2}; L_{1})) = H^{1}(W_{1}; H^{2}(W_{2}; L_{1})) = H^{0}(W_{1}; H^{3}(W_{2}; L_{1})) = 0 .

Hence, part (1) follows from the Hochschild–Serre spectral sequence for calculating H^{3}(W_{1} × W_{2}; L_{1}). □

4. Maps from p–compact groups into almost Bℤ/p–local spaces. In [8] is set up a induction principle for proving statements about p–compact

group, which we will use in this and the next section and which we recall here.

The cohomolgical dimesion of a 𝔽_{p}–finite space is given by the maximal degree of the nonvanishing mod–p cohomology groups. For two p–compact groups X and Y, we say that X < Y if the cohomological dimension of X is smaller than the one of Y or if both have the same cohomological dimension but π_{0}(Y) has a smaller order than π_{0}(X).

4.1 Definition. A class Cl of p–compact groups is called saturated if it satisfies the following 5 conditions:

(1) If X ∈ Cl and Y ≅ X, then Y ∈ Cl, i.e. Cl is closed under equivalences.

(2) The trivial p–compact group belongs to Cl.

(3) If the identity component X_{0} of X is in Cl and if any p–compact group Y, such that Y < X, is in Cl then X also belongs to Cl.

(4) If X is connected, and if X/Z(X) is in Cl, then X is in Cl.

(5) If X is connected and centerfree, and Y ∈ Cl for all p–compact groups such that Y < X, then X ∈ Cl.

4.2 Theorem [8, 9.2]. Any saturated class of p–compact groups contains all p– compact groups.

4.3 Remark. Our definition of a saturated class is not exactly the same as Dwyer and Wilkerson give (there is minor difference in (3)), but their argument also works in our situation to prove Theorem 4.2.

Proof of Theorem 1.4. We want to prove the statement by the induction principle. That is we have to show that the class Cl of all p–compact groups satisfying (1) and (2) is a saturated class.

One direction of part (1) is obvious. Let f : BX ⟶ A be a map from the classifying space of a p–compact group into an almost Bℤ/p–local space. If f is null homotopic, the restriction f|_{BP} is nullhomotopic.

If X and Y are isomorphic p–compact groups then they have isomorphic p–toral Sylow subgroups P_{X} and P_{Y} fitting into a homotopy commutative diagram

[figure: \begin{matrix}BP_{X} & BX \\ BP_{Y} & BY & . \end{matrix}; ≃; ≃]

Hence, X satisfied the statement if and only if Y does. This is Step (1).

The trivial group obviously satisfies the statement, which is Step 2. Going a little aside trip, we want to prove the statement for p-compact toral groups. The first part is obvious. Let P be a p-compact toral group. The loop space ΩA is Bℤ/p– local [15, §1]. The proof of the Sullivan conjecture for p–compact groups [8, 9.3] does not only apply to 𝔽_{p}–finite spaces, but also to Bℤ/p–local spaces without any change of the arguments. Therefore, for the mapping space map_{∗}(BP, A)_{const} of pointed maps, we have Ωmap(BP, A)_{const} ≃ map_{∗}(BP, ΩA) is contractible. Hence, the map A ⟶ map(BP, A)_{const} is an equivalence.

Now let P ⟶ X be the p-toral Sylow subgroup of a p–compact group X. The composition P ⟶ X ⟶ π_{0}(X) is an epimorphism [12, 3.8]. This establishes a commutaive diagram of fibrations

[figure: \begin{matrix}BP_{0} & BP & Bπ_{0}(X) \\ BX_{0} & BX & Bπ_{0}(X) & , \end{matrix}]

and P_{0} ⟶ X_{0} is a p–toral Sylow subgroup of X_{0} [12, 3.9]. Let f : BX ⟶ A be a map. If f|_{BP} ≃ const and if X_{0} ∈ Cl, then f|_{BX_{0}} ≃ const. Thinking of BX_{0} ⟶ BX as a fibration, the space BX_{0} carries a free π := π_{0}(X) action. Taking the trivial action on A, the map A ⟶ map(BX_{0}, A)_{const} is an equivalence, π–equivariant and establishes equivalences

(*) map(Bπ, A) ≃ A^{hπ} ⟶ (map(BX_{0}, A)_{const})^{hπ} ≃ map(BX, A)_{f′|_{BX_{0}} ≃const} .

Here, A^{hπ} denotes the homotopy fixed–point set of the π–action on A (see [9]). Let \overline{f} : Bπ ⟶ A be the map corresponding to f : BX ⟶ A. Applying the same trick to map(BP, A) allows to calculate the map \overline{f}. In this case, \overline{f} corresponds to th constant map const : BP ⟶ A, and hence, \overline{f} is homotopic to the constant map. This shows that f ≃ const, which is part (1). The second part follows from the equivalences in (*). Therefore, we have X ∈ Cl. This is the third step in the induction process.

For any connected p–compact group X with p–toral Sylow subgroup P ⟶ X, there exists a diagram of fibrations

[figure: \begin{matrix}BZ & BP & B\overline{P} \\ BZ & BX & B\overline{X} & , \end{matrix}]

where Z = Z(X) is the center of X and where \overline{P} ⟶ \overline{X} := X/Z is a p–toral Sylow subgroup of the centerfree group \overline{X}. Actually, both fibrations are principal bundles, because Z ⟶ P and Z ⟶ X are central [2, 7.2]. Let f : BX ⟶ A be a map such that f|_{BP} ≃ const. Then f|_{BZ} ≃ const and A ⟶ map(BZ, A)_{const}. In this situation we can apply a lemma of Zabrodsky [19] (see also [10]), which tells us that in the diagram

[figure: \begin{matrix}map(BP, A)_{f′|_{BZ}≃const} & map(B\overline{P} , A) \\ map(BX, A)_{f′|_{BZ}≃const} & map(B\overline{X}, A) \end{matrix}; ≃; ≃]

the horizontal lines are homotopy equivalences. Let \overline{f} : B\overline{X} ⟶ A be the map corresponding to f. The top row implies that \overline{f}|_{B\overline{P}} ≃ const. Thus, by the assumptions, \overline{f} is also nullhomotopic as well as f. This finishes the proof of the fourth step in the induction.

Now let X be a centerfree connected p–compact group. Then, for an object j : E ⟶ X, the centralizer C_{X}(j(E)) is smaller than X. We choose a special subconjugation j′ : E ⟶ P of j (Lemma 2.13). Then, C_{P} (j′(E)) ⟶ C_{X}(j(E)) is a p–toral Sylow subgroup and C_{P} (j′(E)) is subconjugate to P. Let f : BX ⟶ A be a map such that f|_{BP} ≃ const. Then f|_{BC_{P} (j′(E))} and f|_{BC_{X}(j(E))} are also null homotopic and A ⟶ map(BC_{X}(j(E)), A)_{const} is an equvalence (by induction hypothesis). Moreover, these maps are compatible with all morphisms in the Quillen category A_{p}(X). In the sequence

\underleftarrow{holim}_{A_{p}(X)}(A) ⟶ \underleftarrow{holim}_{A_{p}(X)}(map(BC_{X}(E), A)_{const})

⟶ map(\underrightarrow{hocolim}_{A_{p}(X)}(BC_{X}(E)), A)_{F} ⟶ map(BX, A)_{F}

all maps are equivalences; the first because of the above argument, the second by general nonsense and the third because of Theorem 2.14. By F we denoted the set of all homotopy classes of maps f′ : BX ⟶ A such that for every object E ⟶ X of A_{p}(X) the restriction f′|_{BC_{X}(E)} is null homotopic. Using the Bousfield–Kan spectral sequence for calculating the first expression, Theorem 2.14 shows that holim_{A_{p}(X)}A ≃ A, that F = {const}, that f ≃ const and that A ⟶ map(BX, A)_{const} is a homotopy equivalence. This finishes the proof of the last induction step as well as the proof of the theorem. □

4.4 Corollary. Let X be a p–compact group with p–toral Sylow subgroup P ⟶ X. Let F ⟶ E ⟶ BX be a fibration, such that F is 𝔽_{p}–finite, p–complete and a loop space. If the restriction to BP induces a fibration, fiber homotopy equivalent to the trivial fibration, then the fibration itself is fiber homotopically trivial.

Proof. By [18], the fibration is classified by a map BX ⟶ BHE(F). The retriction to BP is null homotopic. The space BHE(F) is connected, p–complete and almost Bℤ/p–local. Thus, the statement follows from Theorem 1.4. □

A similar statement as the last corollary is also true if the fiber is the classifying space of a p-compact torus.

4.5 Proposition. Let T be a p–compact torus, let X be a p–compact group with p–toral Sylow subgroup P ⟶ X, and let BT ⟶ E ⟶ BX be a fibration. If the restriction to BP induces a fiber homotopic trivial fibration, then the fibration itself is fiber homotopic trivial.

The proof is analogously as for Corollary 4.4, but based on the lemma

4.6 Lemma. Let T be a p–compact torus, and let X be a p–compact group with p– toral Sylow subgroup P ⟶ X. If, for a map f : BX ⟶ BHE(BT), the restriction f|_{BP} is null homotopic, then f is null homotopic.

Proof. Let P_{0} ⟶ X_{0} be the p–toral Sylow subgroup of the component of the unit. In the diagram

[figure: \begin{aligned}& & BSHE(BT) ≃ B^{2}T \\ BP & BX & BHE(BT) \\ BP/P_{0} & Bπ_{0}(BX) & Bπ_{0}(HE(BT)) \end{aligned}; f; ≃; \overline{f}]

the map \overline{f} is null homotopic. This follows because π := π_{0}(HE(BT) ≅ Gl(H_{2}(BT; ℤ_{p}^{∧})) is a discrete group, because therefore Bπ is almost Bℤ/p–local, and because we can apply the Zabrodsky lemma in this case as in the proof of Theorem 1.4. Homotopy classes of lifts of the composition BP ⟶ Bπ_{0}(X) ⟶ Bπ and BX ⟶ Bπ_{0}(X) ⟶ Bπ to BHE(BT) are classified by the obstruction groups H^{3}(BP, π_{3}(BT)) or H^{3}(BX, π_{3}(BT)). The existence of a transfer as a stable map [6] shows that the inclusion BP ⟶ BX induces a monomorphism H^{∗}(BP, M) ⟶ H^{∗}(BX, M) for any systems of coefficients. Hence, the map f is also nullhomotopic, which proves the statement. □

5. Splittings of p–compact groups.

In this chapter we want to prove the main ”technical” theorem which allows several interesting applications to p–compact groups. Let X be a p–compact group with normalizer N_{X} of the maximal torus T_{X} ⟶ X of X. Let N_{X} ≅ N_{1} × N_{2} be a splitting into two factors. For i = 1, 2, we want to construct a subgroup Y_{i} ⟶ X of X such that this inclusion induces an isomorphism N_{Y_{i}} ≅ N_{i} in a way which we will make precise working out the construction.

For any p–compact group X there exists a fibration

BX_{0} ⟶ BX ⟶ Bπ

where X_{0} denotes the connected component of the unit and π the group of the components. The composition BN_{X} ⟶ BX ⟶ Bπ factors over BW_{X} ⟶ Bπ, which is induced by a homomorphism W_{X} ⟶ π. The kernel is given by the Weyl group W_{X_{0}} [12, 3.8], which is a pseudo reflection group. If N_{X} ≅ N_{1} × N_{2}, the Weyl group W_{X} ≅ W_{1} ×W_{2} also splits into two factors as well as the maximal torus T_{X} ≅ T_{1} × T_{2}. Every pseudo reflection σ ∈ W_{X_{0}} is contained either in W_{1} or in W_{2}. Let W′_{i} ⊂ W_{X_{0}} be the subgroup generated by all pseudo reflection contained in W_{i}, and let N′_{i} be the counterimage of W′_{i} in N_{i}. Then, we have splittings W_{X_{0}} ≅ W′_{1} × W′_{2}, N_{X_{0}} ≅ N′_{1} × N′_{2} and π ≅ π_{1} × π_{2}. Now we define Y′ =: C_{X}(T_{2}). This is a subgroup of maxiaml rank. For later use we note the following lemma:

5.1 Lemma. N_{Y′} = C_{N_{X}} (T_{2}) = N_{1} × C_{N_{2}}(T_{2}).

Proof. The Weylgroup W_{Y′} is given by all elements of W_{X} acting trivially on T_{2} [8, 7.6]. Hence, W_{Y′} = W_{1} × Ŵ_{2}, where Ŵ_{2} = W_{Y′} ∩ W_{2} ⊂ W_{2}. Let q : N_{X} ⟶ W_{X} be the projection. Then, N′_{Y} = q^{−1}(W_{Y′}) = C_{N_{X}} (T_{2}) = N_{1} × C_{N_{2}}(T_{2}). For the last equivalence, there is a remark in order. The loops spaces N_{X} and N_{2} are not p–compact groups. But the types of centralizers in question are studied in [14, 3.7]. That result implies that C_{N_{X}} (T_{2}) as well as C_{N_{2}}(T_{2}) are finite extensions of a p–compact tori. □

By the above lemma, N_{1} ⊂ N_{Y′} and π_{1} ⊂ π_{0}(Y′). Let Y′′ be defined by the pull back diagram

[figure: \begin{matrix}BY_{′}_{′} & BY_{′} \\ Bπ_{1} & Bπ_{1} × Bπ′_{2} & . \end{matrix}]

By construction, we have N_{Y′′} = N_{1} ×T_{2}. Applying the functor map(BT_{2}, ) shows that T_{2} ⊂ Y′′ is central, and dividing out this central subgroup gives a diagram of fibrations

[figure: \begin{matrix}BT_{2} & BN_{1} × BT_{2} & BN_{1} \\ BT_{2} & BY_{′}_{′} & B(Y′′/T_{2}) =: B\overline{Y} & . \end{matrix}]

Both lines are principal fibration [2, 7.2]. Thus, the lower fibration is classified by a map B\overline{Y} ⟶ B^{2}T_{2}, which is determined by a cohomology class in H^{3}(B\overline{Y}; π_{3}(B^{2}T_{2})). The restriction of this map to BN_{1} is homotopically trivial, and the induced map H^{3}(B\overline{Y}; π_{3}(B^{2}T_{2})) ⟶ H^{3}((BN_{1}; π_{3}(B^{2}T_{2})) is a monomorphism (Proposition 2.10). Hence, the lower row is equivalent to the trivial fibration and we can split BY′′ ≃ BY × BT_{2} in such a way that we get a commuative diagram

[figure: \begin{aligned}& BN_{1} × BT_{2} \\ BY × BT_{2} & & BY_{′}_{′} & . \end{aligned}]

The composition BY ⟶ BY′′ ⟶ B\overline{Y} is an equivalence. Hence, N_{1} ≅ N_{Y} ⊂ Y is the normalizer of the maximal torus. The p–compact subgroup Y ⊂ Y′′ ⊂ X is the one which we associate with N_{1}.

5.2 Lemma. Let X be a connected p–compact group. Let N_{X} ≅ N_{1} × N_{2} split into two factors, and let Y_{1} ⊂ X be the subgroup associated to N_{1}.

(1) We have Y_{1} × T_{2} ≅ C_{X}(T_{2}).

(2) There exists an isomorphism Z(X) ≅ Z(Y_{1}) × Z(Y_{2}) making the diagram

[figure: \begin{matrix}Z(X) & Z(Y_{1}) × Z(Y_{2}) \\ (T_{X})^{W_{X}} & (T_{1})^{W_{1}} × (T_{2})^{W_{2}} \end{matrix}; ≅]

commutative.

Proof. The first part follows directly from the above construction. By [8, 7.5] the center can be calculated, only knowing the normalizer of the maximal torus. This proves part (2). □

Theorem 1.3 is now contained in the following statement.

5.3 Theorem. Let X be a p–compact group such that N_{X} ≅ N_{1} × N_{2}. For

i = 1, 2, let Y_{i} ⟶^{j_{i}} X be the subgroup associated to N_{i}. Then there exists a homotopy equvalence j : Y_{1} × Y_{2} ⟶ X making the following diagram commutative up to homotopy for i = 1, 2

[figure: \begin{matrix}N_{1} × N_{2} & N_{X} \\ Y_{1} × Y_{2} & X \\ Y_{i} & . \end{matrix}; j; j_{i}]

Proof. The proof is carried out in several steps and based on the induction principle of Dwyer and Wilkerson explained in Section 4. The proof takes the rest of this chapter. For abbreviation, we collect the assumptions of the theorem in condition (S). Let Cl be the class of all p–compact groups satifying the statement. We want to show that Cl is a saturated class.

Step 1: Let X be a p–compact group such that N_{X} ≅ N_{1} × N_{2}. Let Y ⟶^{≅} X be an isomorphism of p–compact groups. Then, the normalizer N_{Y} ≅ N_{X} ≅ N_{1} × N_{2} also splits. Let Y_{i} ⟶ Y be the subgroup associated to N_{i}. If Y satifies the theorem, the composition Y_{1} × Y_{2} ≅ Y ≅ X establishes the desired splitting of X. Hence, the class Cl is closed under equivalences.

Step 2 : Theorem 5.3 is satified by p–toral groups. This is obvious.

Step 3: Let us assume that the theorem is true for connected p–compact groups. We want to show that it is true in general.

Let X be a p–compact group satifying condition (S), and let X_{0} be the component of the unit. If T_{i} ⟶ Y_{i} is a central subgroup for i = 1, 2, the Weyl group W_{X} acts trivially on T_{X}, and the inclusion C_{X}(T_{X}) ⟶ X is a sugroup of maximal rank and induces an isomorphism between the Weyl groups. Moreover, both spaces fit into a diagram

[figure: \begin{matrix}C_{0} := C_{X_{0}}(T_{X}) & C := C_{X}(T_{X}) & π_{0}(C) \\ X_{0} & X & W_{X}/W_{X_{0}} ≅ π_{0}(X) & . \end{matrix}]

Because C_{0} is connected [12, ???], the top row is an exact sequence of p–compact groups and therefore C_{0} is the connected component of the unit of C. The left column desribes a subgroup of maximal rank and W_{C} ≅ W_{X}. By [12, 3.11] follows that the left column is an isomorphism of p–compact groups. The isomomorphism in the lower right corner follows from [12, 3.8], which also implies that π_{0}(C) ≅ W_{C}/W_{C_{0}}. Thus the right column is an isomorphism of a p–compact groups as well as the middle vertical arrow. By 2.5, the p–compact group C is a p-compact toral group and so is X. This case we already considered in Step 2. Hence, we can assume that T_{1} ⟶ Y_{1} is not a central subgroup, that C_{Y_{1}}(T_{1}) is smaller than Y_{1}, that the Weyl group of C_{X}(T_{1}) is a proper subgroup of W_{X} and that C_{X}(T_{1}) is smaller than X (if both have the same cohomological dimension, then C_{X}(T_{1}) has less components than X).

As explained at the beginning of this section, the normalizer N_{X_{0}} ≅ N′_{1} × N′_{2} also splits as well as the group π := π_{0}(X) ≅ π_{1} × π_{2} of the components. For the associated subgroups we have fibrations

BY_{i,0} ⟶ BY_{i} ⟶ Bπ_{i} .

Moreover, by the hypothesis, we have a commutative diagram

[figure: \begin{matrix}N_{1,0} × N_{2,0} & N_{X_{0}} \\ Y_{1,0} × Y_{2,0} & X_{0} \\ Y_{i,0} & . \end{matrix}; j; j_{i}]

Here, Y_{i,0} denote the component of the unit of Y_{i} and N_{i,0} the normalizer of the maximal torus of Y_{i,0}. We want to show that there exists an extension Y_{1} ×Y_{2} ⟶ X of the lower horizontal arrow.

Looking at homogeneous spaces we have a commutative diagram of fibrations

[figure: \begin{matrix}Y_{2,0} & Y_{2} & π_{2} \\ X_{0}/Y_{1,0} & X/Y_{1} & π_{2} & , \end{matrix}; ≃]

which implies that the vertical middle arrow is a homotopy equivalence. Moreover, this map is given by the composition Y_{2} ⟶ X ⟶ X/Y_{1}.

Let P_{i} ⟶^{k_{i}} Y_{i} be the the p–toral Sylow subgroup of Y_{i}. Taking pullbacks induces a diagram of fibrations

(*)  [figure: \begin{matrix}Y_{2} & Y_{2} & Y_{2} \\ E_{P} & E & BY_{1} \\ BP_{1} & BY_{1} & BX & . \end{matrix}]

The left column establishes a proxy action of P_{1} on Y_{2}. For the definition of a proxy action we refer the reader to [7, §10]. Obviously, the induced fibrations have

sections s_{P} : BP_{1} ⟶ E_{P} and s : BY_{1} ⟶ E which come from the lift BY_{1} ⟶^{id} BY_{1}. Applying the functor map(BP, ) to the left and right fibration yields a pull back diagram

(**)  [figure: \begin{matrix}(Y_{2}^{hP_{1}})_{s_{P}} & (Y_{2}^{hP_{1}})_{s} \\ map(BP_{1}, E_{P})_{s_{P}} & map(BP_{1}, BY_{1})_{Bk_{i}} \\ map(BP_{1}, BY_{1})_{Bk_{1}} & map(BP_{1}, BX)_{Bj_{1}Bk_{1}} & , \end{matrix}]

where Y_{2}^{hP_{1}} denotes the homotopy fixed point set of the proxy action of P_{1} on Y_{2} (again for definitions see [7, §10]. In the fibers we only take those components of the homotopy fixed–point sets which belong to the associated sections. The map BZ(P_{1}) ⟶ map(BP_{1}, BY_{1})_{Bk_{i}} is an equivalence (Proposition 2.15). Now, let \overline{P}_{1} denote the group of components of P_{1}. Replacing BT_{1} by B̃T_{1} to transform BT_{1} ⟶ BP_{1} into a fibration, i.e into a covering, we get an action of \overline{P}_{1} on B̃T_{1}. By [9], we have an equivalence of functors map(BP_{1}, ) ≃ map(B̃T_{1}, )^{h\overline{P}_{1}}. This allows to calculate the mapping space in the lower right corner of the above pull back diagram. By Lemma 5.1, we have N_{C_{X}(T_{1})} = N̂_{1} × N_{Y_{2}}, where N̂_{1} := C_{N_{Y_{1}}} (T_{1}) ≅ C_{Y_{1}}(T_{1}) is a p–compact group. The isomorphism follows because T_{1} ⟶ Y_{1} is a maximal torus. Moreover, the centralizer C_{X}(T_{1}) is smaller than X. Hence we can apply the induction hypothesis to establish a commutative diagram

[figure: \begin{matrix}BN̂_{1} × BN_{Y_{2}} & BN_{C_{X}(T_{1})} \\ BN̂_{1} × BY_{2} & BC_{X}(T_{1}) & . \end{matrix}; ≃; ≃]

The inclusions Y_{1} × Y_{2} ⟵ N̂_{1} × Y_{2} ≅ C_{X}(T_{1}) ⟶ X induce equivalences

M_{2} := map(B̃T_{1}, BN̂_{1} × BY_{2})_{Bi} ⟶^{≃} M_{1} := map(B̃T_{1}, BY_{1} × BY_{2})_{Bi}
M_{2} ⟶^{≃} M_{3} := map(B̃T_{1}, BX)_{Bi}

which are also \overline{P}_{1}–equivariant. Therefore, these equivalences induce equivalences between the homoytopy fixed–point sets. The component of M_{1}^{h\overline{P}_{1}} related to the inclusion P_{1} ⟶ Y_{1} ⟶ Y_{2} corresponds to the component of M_{3}^{h\overline{P}_{1}} related to the inclusion P_{1} ⟶ Y_{1} ⟶ X. Therefore, we get an equivalence

BZ(P_{1}) × BY_{2} ≃ map(BP_{1}, BY_{1} × BY_{2})_{Bi} ≃ map(BP_{1}, BX)_{Bi} .

The pull back digram (∗∗) translates now to

[figure: \begin{matrix}(Y_{2}^{h\overline{P}_{1}})_{s_{P}} & (Y_{2}^{h\overline{P}_{1}})_{s} \\ map(BP_{1}, E_{P})_{s_{P}} & BZ(P_{1}) \\ BZ(P_{1}) & BZ(P_{1}) × BY_{2} & . \end{matrix}]

This shows that

(Y_{2}^{h\overline{P}_{1}})_{s_{P}} ≃ (Y_{2}^{h\overline{P}_{1}})_{s} ≃ Y_{2} .

Taking the adjoint we can construct a map Y_{2} × BP_{1} ⟶ E_{P} which is a homotopy equivalence and gives a trivialization of the fibration Y_{2} ⟶ E_{P} ⟶ BP_{1}. By Theorem 1.4, the fibration Y_{2} ⟶ E ⟶ BY_{1} is also trivial, i.e. we have a fiber homotopy equivalence Y_{2} × BY_{1} ⟶ E.

Applying the functor map(BY_{1}, ) to the middle and right columns in the diagram (∗) establishes another pull back diagram

[figure: \begin{matrix}map(BY_{1}, E)_{s} ≃ BZ_{1} × Y_{2} & map(BY_{1}, BY_{1})_{id} ≃ BZ_{1} \\ map(BY_{1}, BY_{1})_{id} ≃ BZ_{1} & map(BY_{1}, BX)_{Bj_{i}} & , \end{matrix}]

where Z_{1} := Z(Y_{1}) denotes the center of Y_{1} The equivalences follow from the above fiber homotopy equivalence and from [8, 1.3]. Passing to loop spaces shows that map(BY_{1}, BX)_{Bj_{1}}) =: BY′_{2} is a p–compact group.

The pull back diagram

[figure: \begin{matrix}BZ_{1} ≃ map(BY_{1}, BY_{1})_{id} & BZ(P_{1}) ≃ map(BP_{1}, BY_{1})_{Bi} \\ BY′_{2} ≃ map(BY_{1}, BX)_{Bi} & BZ(P_{1}) × BY_{2} ≃ map(BP_{1}, BX)_{Bi} \end{matrix}]

shows that BZ_{1} ⟶ BY′_{2} has a section BY′_{2} ⟶ BZ_{1} and that the homotopy fiber is equivalent to BY_{2} (the right column allows a section). Moreover, the diagram also shows that the composition BY_{2} ⟶ BY′_{2} ⟶ BX equals the inclusion BY_{2} ⟶ BX. Passing to adjoints shows the existence of a map BY_{1} × BY_{2} ⟶ BX, which also extends the map BN_{1} × BN_{2} ⟶ BX. This completes the proof of the third step.

5.4 Remark. The arguments for calculating map(BP_{1}, BX)_{Bi} ≃ BZ(P_{1}) × BY_{2} always work when X satifies the condition (S). We only have to assume the induction hypothesis, and that T_{1} ⟶ Y_{1} is not central. The p–toral Sylow subgroup P = P_{1} × P_{2} splits into a product. Hence, we always get a map BP_{1} × BY_{2} ⟶ BX, extending the composition BP_{1} × BP_{2} ⟶ BN_{1} × BN_{2} ⟶ BX. For the calculation of the mapping space map(BY_{1}, BX)_{Bi}, we only needed in addition that the composition Y_{2} ⟶ X ⟶ X/Y_{1} is a homotopy equivalence.

Step 4: We have to show that the theorem is true for a connected p–compact group if it is true for the associated centerfree p–compact group. Let X be a connected p–compact group, let Z := Z(X) be the center of X, and let \overline{X} := X/Z(X) be the associated centerfree group. Let N_{X} ≅ N_{1} × N_{2}, let Y_{1}, Y_{2} ⟶ X be the associated subgroups, let P_{i} ⊂ N_{i} ⊂ Y_{i} be a p–toral Sylow subgroup, and let Z_{i} := Z(Y_{i}) denote the center of Y_{i}. The center Z_{i} is a subgroup of P_{i} [12, 4.3]. Because W_{i} acts trivially on Z_{i}, the inclusion Z_{i} ⊂ N_{i} is central [14, 3.7] By [2, 7.2], we can construct a diagram of principal bundles

[figure: \begin{matrix}BZ_{1} × BZ_{2} & BN_{1} × BN_{2} & B\overline{N}_{1} × B\overline{N}_{2} \\ BZ & BX & B\overline{X} \\ BZ_{i} & BY_{i} & B\overline{Y}_{i} & . \end{matrix}]

By the construction, the subgroup \overline{Y}_{i} ⟶ \overline{X} is associated to \overline{N}_{i}. Thus, by the hypothesis, there exists an equivalence B\overline{Y}_{1} × B\overline{Y}_{2} ⟶ BX extending the map on the normalizers. The diagram of the classifying maps of the principle bundles

[figure: \begin{matrix}B\overline{Y}_{1} × B\overline{Y}_{2} & B^{2}Z_{1} × B^{2}Z_{2} \\ B\overline{X} & B_{2}Z \end{matrix}; ≃]

commutes up to homotopy, because the homotopy classes of the horizontal arrows are determined by cohomology classes and because the restrictions of both maps to the normalizers are equal. Taking fibers establishes the desired equivalence BY_{1} × BY_{2} ⟶ BX. This finishes the proof of Step 4.

Step 5: We have to show that the statement is true for a centerfree connected p–compact group, if it is satified by all smaller p–compact groups. The proof of this step is even more complicate than the one of Step 2 and divided into several claims. The outline is as follows: The first major step is the construction of an equivalence between the Quillen categories of X and Y_{1} × Y_{2} (Claim 3). Let F_{X} : A_{p}(X) ⟶ Ab be the functor given by F_{X}(E ⟶ X) := H^{∗}(BC(E); 𝔽_{p}). Analogously, we define a functor F_{Y} : A_{p}(Y_{1} × Y_{2}) ⟶ Ab. Using the equivalence of the Quillen categories, we then show that there exists a natural transformation F_{X} ⟶ F_{Y}, which is an isomorphism on the objects (Claim 6). The mod-p decomposition theorem of p–compact groups (Theorem 2.14) establishes an isomorphism H^{∗}(BX; 𝔽_{p}) ≅ H^{∗}(BY_{1} × BY_{2}; 𝔽_{p}) which is compatible with the equivalence BN_{X} ≃ BN_{1} × BN_{2} (Claim 7). An Eilenberg–Moore spectral sequence argument allows to calculate the fiber of BY_{1} ⟶ BX which turns out to be equivalent to Y_{2} (Claim 8). Now similiar arguments as in the proof of Step 2 are applicable and

establish the desired equivalence BY_{1} × BY_{2} ⟶^{≃} BX (Claim 9).

Claim 1: The spaces Y_{i} are connected.

Proof. This follows from Lemma 5.2 and from [12, 3.11], which says that, for connected p–compact groups, centralizers of tori are always connected. □

Let i_{X} : P ⟶ N_{X} ⟶ X be the p–toral Sylow subgroup. Then P = P_{1} × P_{2} splits and i_{Y} : P_{1} × P_{2} ⟶ N_{1} × N_{2} ⟶ Y_{1} × Y_{2} is also the p–toral Sylow subgroup of Y_{1} × Y_{2}.

Claim 2: Let j_{1}, j_{2} : E ⟶ P be two monomorphisms of an elementary abelian subgroup E into P. Then Bi_{Y} Bj_{1} and Bi_{Y} Bj_{2} are homotopic if and only if Bi_{X}Bj_{1} ≃ Bi_{X}Bj_{2}.

Proof. We prove the statement via an induction over the rank of E. Let us assume that E = ℤ/p. Because X is centerfree, the actions of W_{1} on T_{1} and W_{2} on T_{2} are nontrivial. Inparticular, T_{i} ⟶ Y_{i} is not a central subgroup. The composition Bi_{Y} Bj_{i} as well as Bi_{X}Bj_{i} factor over BP_{1} × BY_{2}. This follows from Remark 5.4. Because Y_{2} is connected (Claim 1), the second coordinate, as a cyclic subgroup, of both maps is subconjugate to T_{2} (Theorem 2.6). Hence, we can assume that the second corodinate of j_{i} : E ⟶ P_{1} × P_{2} takes image in T_{2}. Analogously, we can assume that the first cordinate also takes image in T_{1}. That is to say that j_{i} : E ⟶ P_{1} × P_{2} represents E as a subgroup of the torus. By [13, 4.2], the two maps Bi_{Y} Bj_{1} and Bi_{Y} Bj_{2} are homotopic if and only if Bj_{1} and Bj_{2} differ by a Weyl group element if and only if Bi_{X}Bj_{1} ≃ Bi_{X}Bj_{2}. This proves Claim 2 for E = ℤ/p.

Now let us assume that the rank of E is bigger than 1. Let E_{1} ⊂ E denote the first coordinate of E and E_{2} ⊂ E a complement of E_{1}, i.e. E ≅ E_{1} ⊕ E_{2}. By what we already proved we can assume that, for i = 1, 2, the restriction j_{1}|_{E_{1}} = j_{2}|_{E_{1}} : E_{1} ⟶ P_{1} × P_{2} is a subgroup of the maximal torus. Let j′_{i} : E_{2} ⟶ C_{P_{1}×P_{2}}(E_{1}) be the adjoint of j_{i}. The centralizers X′ := C_{X}(E_{1}) and C_{Y_{1}×Y_{2}}(E_{1}) are subgroups of maximal rank, smaller than the centerfree group X, and the normalizers of the maximal tori are given by C_{N_{X}} (E_{1}) = N′_{1} × N′_{2} which again splits as well as C_{P_{1}×P_{2}}(E_{1}) = P′_{1} × P′_{2}. The latter centralizer is a p–toral Sylow subgroup of X′ and Y′_{1} × Y′_{2}. Moreover, Y′_{i} := C_{Y_{i}}(i_{Y_{i}}j_{i}(E_{1})) ⟶ Y_{i} is the subgroup associated to N′_{i}. Therefore, by induction hypothesis, there exist an equivalence Y′_{1} × Y′_{2} ⟶^{≃} X′ making the diagram

[figure: \begin{aligned}& & BY′_{1} × BY′_{2} \\ BP′_{1} × BP′_{2} & BN′_{1} × BN′_{2} \\ & & BX′ \end{aligned}]

commutative up to homotopy.

The two maps Bi_{X}Bj_{1}, Bi_{X}Bj_{2} : BE_{1} × BE_{2} ⟶ BX are homotopic if the adjoints BE_{2} ⟶ BX′ are homotopic. The same is true for Y_{1} × Y_{2}. The above homotopy commutative diagram shows that the claim is true for the adjoints. This finishes the induction and the proof of Claim 2.

Claim 3: The Quillen categories of X and Y_{1} × Y_{2} are isomorphic, i.e. there exist a functor A_{p}(Y_{1} × Y_{2}) ⟶ A_{p}(X), which is an isomorphism of categories.

Proof. By Proposition 2.10, every elementary abelian subgroup is subconjugate to P_{1} × P_{2}. Hence the statement follows from Claim 2.

Claim 4: Let j : E ⟶ P := P_{1} × P_{2} be a monomorphism. Then, the map j is a special subconjugation of i_{Y} j if and only of it is a special subconjugation for i_{X}j.

Proof. Let E ⟶^{j} P be a special subconjugation of i_{Y} j and E ⟶^{j′} P a special subconjugation of i_{X}j. By Claim 2, the composition i_{Y} j′ is conjugate to i_{Y} j. Hence, the centralizer P_{X} = C_{P} (j′(E)), which is the p–toral Sylow subgroup of C_{X}(i_{X}j(E)), is subconjugated to the p–toral Sylow subgroup P_{Y} = C_{P} (j(E)) of C_{Y} (i_{Y} j(E)) and vice versa. Therefore P_{Y} and P_{X} are isomorphic. This proves the statement. □

Claim 5: Let E ⟶^{j} P be a monomorphism, which is a special subconjugation of i_{X}j and i_{Y} j. Then, there exists an equivalence f_{E} : BC_{Y} (E) ⟶^{≃} BC_{X}(E) making the diagram

[figure: \begin{aligned}& BC_{P} (E) \\ BC_{Y} (E) & & BC_{X}(E) \end{aligned}; ≃]

commutative up to homotopy.

Proof. Let E_{0} ≅ ℤ/p ⊂ E denote the first coordinate of E. By Claim 2 and Lemma 2.13 we can assume that E_{0} ⟶ P is a toral subgroup and a special subconjugation of E_{0} ⟶ P ⟶ X and E_{0} ⟶ P ⟶ Y. As shown in the proof of Claim 2 (using the induction hypothesis), there exists a homotopy commutative diagram

[figure: \begin{aligned}& BC_{P} (E_{0}) \\ BC_{Y} (E_{0}) & & BC_{X}(E_{0}) & . \end{aligned}; ≃]

Passing to adjoints and taking centralizers finishes the proof of Claim 5. □

Claim 6: There exists a natural transformation F_{Y} ⟶ F_{X}, which is an isomorphism on objects.

Proof. Let E ⟶^{j} P be a special subconjugation of j_{Y} := i_{Y} j and j_{X} := i_{X}j. Then we define

ϕ_{E} := H^{∗}(f_{E}, 𝔽_{p}) : H^{∗}(BC_{Y} (E); 𝔽_{p}) ⟶ H^{∗}(BC_{X}(E); 𝔽_{p}) .

Let E_{1} ⟶ E_{2} be a morphism of A_{p}(Y) ≅ A_{p}(X). By Lemma 2.13, for i = 1, 2, there exist monomorphisms j_{i} : E_{i} ⟶ P, which are special lifts of j_{i,Y} and j_{i,X} such that the diagram

[figure: \begin{aligned}BE_{1} & & BE_{2} \\ & BP \end{aligned}]

commutes up to homotopy. Passing to centralizers gives another diagram, namely

[figure: \begin{aligned}& BC_{Y} (E_{2}) & & BC_{Y} (E_{1}) \\ BC_{P} (E_{2})′ & & BC_{P} (E_{1}) \\ & BC_{X}(E_{2}) & & BC_{X}(E_{1}) & . \end{aligned}; f_{E_{2}}; f_{E_{1}}]

The two triangle commute up to homotopy (Claim 5) as well as the upper and the lower parallelogram. For any p–compact group U with p–toral Sylow subgroup P_{U} ⟶ U, the map H^{∗}(BU; 𝔽_{p}) ⟶ H^{∗}(BP_{U}; 𝔽_{p}) is an injection (Proposition 2.10). The diagonal arrows induce injections in mod-p cohomology, because we always chose special subconjugation. Therefore, the square commutes at least in mod-p cohomology, and the maps Φ_{E} establish a natural transformation F_{Y} ⟶ F_{X}, which is an isomorphism on the objects. □

Claim 7: There exists an isomorphism

ϕ : H^{∗}(BX; 𝔽_{p}) ⟶^{≅} H^{∗}(BY; 𝔽_{p})

of algebras over the Steenrod algebra such that the composition H^{∗}(BX; 𝔽_{p}) ⟶^{ϕ} H^{∗}(BY; 𝔽_{p}) ⟶ H^{∗}(BY_{i}, 𝔽_{p} equals the map H^{∗}(BX; 𝔽_{p}) ⟶ H^{∗}(BY_{i}; 𝔽_{p}) induced by the inclusion BY_{1} ⟶ BX.

Proof. We have the following sequence of isomorphisms:

H^{∗}(BX; 𝔽_{p}) ≅ lim_{\overleftarrow{A_{p}(X)}} F_{X} ≅ lim_{\overleftarrow{A_{p}(Y)}} F_{Y} ≅ H^{∗}(BY; 𝔽_{p}) .

The first and last isomorphism follow from Theorem 2.14 and the middle isomorphism from Claim 6. The second part of the statement follows from the construction of the natural transformation F_{Y} ⟶ F_{X}. □

Claim 8: The composition Y_{2} ⟶ X ⟶ X/Y_{1} is a homotopy equivalence.

Proof. In the pull back diagram

[figure: \begin{matrix}E & BY_{1} \\ BY_{2} & BX \end{matrix}]

we want to calculate the mod–p cohomology of E via the Eilenberg–Moore spectral sequence. The E_{2} is given by

\begin{matrix}E_{2} = & T or_{H^{∗}(BX;𝔽_{p})}(H^{∗}(BY_{1}; 𝔽_{p}), H^{∗}(BY_{2}; 𝔽_{p})) \\ ≅ & T or_{H^{∗}(BY_{1};𝔽_{p})}(H^{∗}(BY_{1}; 𝔽_{p}), 𝔽_{p}) \\ ≅ & T or_{𝔽_{p}}(𝔽_{p}, 𝔽_{p}) . \end{matrix}

The functor ⊗_{𝔽_{p}}H^{∗}(BY_{i}; 𝔽_{p}) is exact. This implies the second isomorphism and also, together with Claim 7, the first isomorphism. Therefore, E is mod–p acyclic. Moreover E is p–complete, because all the other involved spaces are p–complete, Hence, E ≃ ∗. The map between the fibers is obviously given by the composition Y_{2} ⟶ X ⟶ X/Y_{1}. This proves the statement. □

Claim 9: There exists an equivalence BY_{1}×BY_{2} ⟶^{≃} BX, such that the diagram

[figure: \begin{matrix}BN_{1} × BN_{2} & BN_{X} \\ BY_{1} × BY_{2} & BX \end{matrix}; ≃]

commutes up to homotopy.

Proof. Using Claim 8, the same arguments as in the proof of Step 2 are applicable (see Remark 5.4). This establishes an equivalence BY_{1} × BY_{2} ⟶ BX with the desired properties. In this case the proof even simplifies a little, because BY_{1} and BY_{2} are centerfree connected groups, which is to say that, for i = 1, 2, the mapping spaces map(BY_{i}, BY_{i})_{id} ≃ BZ(Y_{i}) are contractible. □.

Claim 9 finishes the induction for the proof of Step 5, which was the last step to complete the proof of Theorem 5.3. □

6. Simply connected and centerfree connected p–compact groups. In this section we want to prove Theorem 1.2 and Theorem 1.3. We call a

finite extension N of a p-compact torus T simple if the associated representation N/T ⟶ Gl(H_{2}(BT; ℤ_{p}^{∧}) ⊗ ℚ is irreducible.

6.1 Lemma. Let N be a finite extension of a p-compact torus T, such that N/T acts as a pseudo reflection group on H_{2}(BT; ℤ_{p}^{∧}) ⊗ ℚ. Then, up to order, there exists at most one splitting N ≅ N_{1} × ... × N_{n} into simple finite extensions of p–compact tori.

Proof. The existence of a splitting into simple factors shows that we have splittings W := N/T ≅ W_{1} ×...×W_{n} for the quotient, L_{T} := H_{2}(BT; ℤ_{p}^{∧})⊗ℚ ≅ L_{1} ⊕...⊕L_{n} for the 2–dimensional homology of BT and T ≅ T_{1} × ... × T_{n} for the torus T itself. By standard representation theory of pseudo reflection groups, these splittings are unique up to order. Thus, the splitting of N is also unique up to order. □

Theorem 1.2 is contained in the following statement.

6.2 Theorem. Let p be an odd prime. Let X be a centerfree connected p–compact group. Then, the following holds:

(1) The normalizer N_{X} ≅ N_{1} × ... × N_{n} splits, up to order, uniquely into a product of simple factors. For each factor, the qoutient W_{i} := N_{i}/T_{i} acts on H_{2}(BT_{i}; ℤ_{p}^{∧}) as a pseudo reflection group.

(2) The p–compact group X ≅ X_{1} × ... × X_{n} splits into a product of simple centerfree connected p–compact groups.

(3) For a splitting X ≅ X_{1} × ... × X_{n} into simple centerfree connected p– compact groups, the normalizers of the maximal tori of the factors are determined by the factors of N_{X}, i.e , after reordering the factors, there exist isomorphisms N_{X_{i}} ≅ N_{i} such that the diagrams

[figure: \begin{matrix}BN_{X_{i}} ≃ BN_{i} & BN_{X} \\ BX_{i} & BX \end{matrix}]

commute up to homotopy.

Proof. The action of W_{X} on L := H_{2}(BT_{X}; ℤ_{p}^{∧}) represents W as a pseudo reflection group, and the lattice L is centerfree. Therefore, by [16, 1.3], the lattice L ≅ L_{1} ⊕ ... ⊕ L_{n} splits into a direct sum of sublattices, such that W_{i} acts trivially on L_{j} for i ≠ j. Let T_{i} := T_{L_{i}} be the p-adic torus associated to L_{i}. Moreover, T ≅ T_{L} ≅ T_{1} ⊕ ... ⊕ T_{n} is the torus associated to L.

The fibration BT_{X} ⟶ BN_{X} ⟶ BW_{X} is classified by a map c_{N} : BW_{X} ≃ ∏_{i} BW_{i} ⟶ ∏ BHE(BT_{i}). Let c_{0} : BW_{X} ⟶ ∏_{i} BHE(BT_{i}) be the classifying map of the product of the fibrations BT_{i} ⟶ BN′ ⟶ BW_{i} given by the semi direct product. Since BSHE(BT_{i}) ≃ B^{2}T_{i}, the difference between c_{N} and c_{0} is measured by a cohomology class in

H^{3}(BW_{X}; ∏_{i} π_{3}(B^{2}T_{i})) ≅ H^{3}(BW_{X}; ∏ L_{i}) ≅ ∏_{i} H^{3}(BW_{i}; L_{i}) .

The last isomorphism follws from Lemma 2.2. This cohomology class decribes also a product of fibrations of the desired form which is fiber homotopy equivalent to BT_{X} ⟶ BN_{X} ⟶ BW_{X}. This establishes a splitting into simple factors. The uniqueness follows from Lemma 6.1, which proves (1).

Now we can apply Theoerem 1.3, which proves the existence of a splitting. By Lemma 5.2, each factor is centerfree. This is part (2).

Every splitting of X establishes a splitting of N_{X} into simple factors, which are uniquely determined up to order (Lemma 6.1). This proves part (3). □

Theorem 1.1 is contained in the following result.

6.3 Theorem. Let p be an odd prime. Let X be a simply connected p–compact group. Then, the following holds:

(1) The normalizer N_{X} ≅ N_{1} × ... × N_{n} splits, up to order, uniquely into a product of simple factors. For each factor, the qoutient W_{i} := N_{i}/T_{i} acts on H_{2}(BT_{i}; ℤ_{p}^{∧}) as a pseudo reflection group.

(2) The p–compact group X ≅ X_{1} × ... × X_{n} splits into a product of simple simply connected p–compact groups.

(3) For a splitting X ≅ X_{1} × ... × X_{n} into simple simply connected p–compact groups, the normalizers of the maximal tori of the factors are determined by the factors of N_{X}, i.e , after reordering the factors, there exist isomorphisms N_{X_{i}} ≅ N_{i} such that the diagrams

[figure: \begin{matrix}BN_{X_{i}} ≃ BN_{i} & BN_{X} \\ BX_{i} & BX \end{matrix}]

commute up to homotopy.

Proof. Let \overline{X} := X/Z(X) be the associated centerfree connected p–compact group. The center Z := Z(X) of X is a finite group [12, 5.3]. All these spaces fit into a principal fibration

BZ ⟶ BX ⟶ B\overline{X} .

with classifying map B\overline{X} ⟶ B^{2}Z [2, 7.2]. Hence, there exists an isomorphism π_{2}(B\overline{X}) ≅ π_{1}(BZ) ≅ Z.

By Theorem 6.2, there exist splittings N\overline{X} ≅ ∏_{i} \overline{N}_{i} of N\overline{X} and \overline{X} ≅ ∏_{i} \overline{X}_{i} of X into simple centerfree factors. Let Z ≅ ∏_{i} Z_{i} be the associated splitting. The above classifying map also splits into a product of maps B\overline{X}_{i} ⟶ B^{2}Z_{i}. The product of the fibers X_{i} gives a splitting of X ≅ ∏_{i} X_{i} into simple simply connected p–compact groups. This also establishes a splitting of N_{X} ≅ ∏_{i} N_{X_{i}} into simple factors. The uniqueness properties for these splittings follow on the one hand from Lemma 6.1, on the other hand analogously as in the proof of Theroem 6.2. □

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Mathematisches Institut, Bunsenstr. 3-5, D–3400 Göttingen, Germany, e-mail: notbohm@cfgauss.uni-math.gwdg.de.