UNSTABLE SPLITTINGS OF CLASSIFYING
SPACES OF pCOMPACT GROUPS
D. Notbohm
Abstract. Dwyer and Wilkerson gave a definition of a pcompact group, whi*
*ch
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
pcompact group has a maximal torus, a normalizer of the maximal torus an*
*d a Weyl
group. The believe or hope that pcompact groups enjoys most properties o*
*f compact
Lie groups establishes a program for the classification of these objects.*
* Following
the classification of compact connected Lie groups, one step in this prog*
*ram is to
show that every simply connected pcompact group splits into a product of*
* simply
connected simple pcompact groups. The proof of this splitting theorem is*
* based on
the fact that every classifying space of a pcompact 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 : BX ! X), where X and BX are
toplogical spaces, where BX is pointed, and where e is a homotopy equivalence
between the loop space BX of BX and X. Such a triple is called a pcompact
group, if X is Fpfinite, i.e H*(X; Fp) is finite, and if BX is a pcomplete co*
*nnected
space. This notion was introduced by Dwyer and Wilkerson in [7]. Examples of
pcompact 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; BG^p ' G^p) is a
pcompact group.
In recent work of Dwyer and Wilkerson [7, 8] and of Moller and the author [1*
*2]
it turned out that pcompact groups are a good homotopy theoretic generalisation
of compact Lie groups. Inparticular, it was shown that pcompact groups enjoy
quite a lot of the properties of compact Lie groups; e.g there exist always max*
*imal
tori, normalizer of the maximal tori and Weyl groups [7]. The maximal torus
of a pcompact group X is a map BTX ! BX of an EilenbergMacLane space
BTX ' K((Z^p)n; 2) into the classifying space BX of X with certain properties.
The Weyl group, denoted by WX , is a finite group, and the normalizer of the
maximal torus is a map BN(TX )! BX, where BN(TX ) fits into a fibration
BTX ! BN(TX )! BWX such that the triangle
BTX G________________BNX//
GGG ww
GGG www
## ww
BX
______________
1980 Mathematics Subject Classification (1985 Revision). 55 R 35, 55 P 35, 2*
*2 E 20.
Key words and phrases. loop space, classifying space, pcompact group, compa*
*ct Lie group.
Typeset by AM ST*
*EX
1
2
commutes up to homotopy. For exact definitions of these notions see Section 2.
In general, the normalizer, also denoted by NX , does not give a pcompact grou*
*p,
because the space BNX is not pcomplete (the fundamental group ss1(BNX ) might
not be a finite pgroup). Nevertheless, the space BNX establishes a finite loop*
* space
NX := (NX ; BNX ; BNX ' NX ), where NX is Fpfinite. This tripel behaves
similar as pcompact groups. There exists a pcompact subgroup of PX NX ,
which we construct by restricting the fibration to the classifying space of the*
* p
Sylow subgroup of WX . The classifying space BPX is pcomplete, because ss1(BPX*
* )
is a finite pgroup and because BTX is pcomplete. This follows from [4, II 5*
*.1,
5.2]. The map BPX ! BX is called the ptoral Sylow subgroup of X and plays
the same role as the ptoral Sylow subgroup of a compact Lie group.
One of the main question about pcompact groups asks for a classification of
these objects. The naive approach to believe that the analogy between pcompact
groups and compact Lie groups is as good as possible produces a lot of `theorem*
*s'
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 me*
*ans
to express everything in terms of classifying spaces, e.g. a homomorphism X! Y
between pcompact groups is a pointed map BX! BY between the classifying
spaces, and a pcompact group X ~= X1 x X2 splits into a product of pcompact
groups if the there exists a homotopy equivalence BX ' BX1 x BX2. The task of
the naive approach consists of finding `new' proofs (in terms of homotopy theor*
*y)
for `old' results, which also work for the larger class of pcompact groups.
The classification of compact connected Lie groups says that, for every comp*
*act
connected Lie group, there exists a finite covering, which is a product of simp*
*le
simply connected Lie groups and a torus. In [12] was shown that the first part
of this result is true for connected pcompact groups, namely every connected
pcompact 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 pcompact group is a product of simple simply connected pcompact
groups. A `new' proof of this second step is the main purpose of this paper.
We call a pcompact group X simply connected if the space X is simply con
nected. 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 pcompact
group. Then, X ~= X1 x :::Xn splits into a product of simple simply connected
pcompact groups.
There exists also a notion of a center of a pcompact group[8] [12], which i*
*s the
generalization of the group or Lie group theoretic center. The center is alway*
*s a
pcompact group. Again, for details see Section 2. A pcompact 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 pcompact
group. Then, X ~=X1 x :::Xn splits into a product of simple centerfree connected
pcompact groups.
The proofs of both theorem are based on a general splitting criteria for pc*
*ompact
groups.
3
1.3 Theorem. A pcompact group X splits into product X ~= X1 x X2 of p
compact groups if and only if the normalizer NX of the maximal torus TX ! X
splits into a product NX ~=N1 x N2.
For the proof of Theorem 1.3 we have to study maps from classifying spaces i*
*nto
almost BZ=plocal spaces. A space A is called BZ=plocal if the map
A! map(BZ=p; A) is an equivalence and called almost BZ=plocal if the map
A! map(BZ=p; A)const into the component of the constant map is an equiva
lence. Examples of BZ=plocal and almost BZ=plocal spaces are provided by
Fpfinite spaces. A Fpfinite space K is BZ=plocal. This follows from the Su*
*l
livan conjecture [10]. By [15], the space BHE(K) is almost BZ=plocal. Here,
HE(K) denotes the monoid of self equivalences of K. In general, this is not an
Fpfinite space. Moreover, if K is pcomplete and a loop space, the classifying
space BHE(K) is also pcomplete [15]. These are the main example we apply the
next theorem to.
1.4 Theorem. Let P! X be a ptoral Sylow subgroup of a pcompact group
X, and let f : BX! A be a map into a connected pcomplete almost BZ=plocal
space.
(1) The restriction fBP is nullhomotopic if and only if f is nullhomotopi*
*c.
(2) The map A! map(BX; A)constis an equivalence.
Remark. In [11] is proved a similar result for maps between classifying spaces *
*of
pcompact 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 dimensio*
*nal
cohomology groups of pseudo reflection groups. The proof of Theorem 1.4 is carr*
*ied
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 Wil*
*k
erson also got proofs for similar results.
2. Background.
In this section we recall the basic notions about pcompact 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 pcompact groups or loop spaces is an isomorphism if
f g
Bf : BY! BX is an equivalence. A sequence X! Y! Z of pcompact groups
Bf Bg
is short exact if the associated sequence BX ! BY ! BZ is a fibration up
to homotopy. A monomorphism of pcompact groups is a map BX! BY whose
homotopy fiber, denoted by Y=X, is Fpfinite. A subgroup Y! X of pcompact
4
group X is a monomorphism of pcompact groups.
2.2 pcompact tori, pcompact toral groups and finite extensions of
pcompact tori : A pcompact torus is a pcompact group(T; BT; BT ' T ),
where BT ' K((Z^p)n; 2) is an EilenbergMacLane space of degree 2.
A finite group or a finite loop space is a tripel (K; BK; BK ' K) such that
BK is an EilenbergMacLane space of a finite group of degree 1.
A finite extension of a padic torus T is a tripel (N; BN; BN ' N) which fits
into a short exact sequence of loop spaces T! N! W =: N=T , where W is a
finite loop space. A pcompact toral group P is a finite extension of a pcomp*
*act
torus T such that the quotient P=T is a finite pgroup. Inparticular, every *
*p
compact toral group is a pcompact group.
2.3 Conjugation and subconjugation : Two homomorphisms f; g : Y! X
of pcompact groups are called conjugate if the induced maps Bf; Bg : BY! BX
are homotopic.
For a homomorphism f : Y ! X of pcompact groups and for a pcompact
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 CX (f(Y )) to be the loop space given*
* by
the triple
CX (f(Y )) := (map(BY; BX)Bf ; map(BY; BX)Bf ; id) :
The evaluation at the basepoint ev : map(BY; BX)Bf! BX establishes a homo
morphism CX (f(Y ))! X of loop spaces. If Y is a pcompact toral group the ce*
*n
tralizer CX (f(Y )) is again a pcompact group and the evaluation CX (f(Y ))! *
* X
is a monomorphism [7, 5.1, 5.2 and 6.1].
A subgroup Z! X of a pcompact group X is called central if the monomor
phism CX (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 pcompact group X,
the centralizer CX (X) is a pcompact group and Z(X) := CX (X)! X is the
center of X. For every pcompact group X there exists a short exact sequence
Z(X)! X! X=Z(X) =: P X of pcompact groups, and, if X is connected, the
quotient P X has a trivial center [12, 4.7].
We call a pcompact group X centerfree if Z(X) is the trivial group.
2.5 Maximal tori : The maximal torus of a pcompact group X is a monomor
phism TX ! X of a pcompact torus into X such that the centralizer CX (TX ) i*
*s a
pcompact toral group, whose component of the unit is given by TX .
2.6 Theorem [7, 8.11, 8.13 and 9.1]. Let X be a pcompact group.
(1) The pcompact group X has a maximal torus TX ! X.
(2) Any subtorus T! X of X is subconjugated to the maximal torus TX !
X.
5
(3) Any two maximal tori of X are conjugated.
(4) If X is connected then TX ! CX (TX ) is an isomorphism.
(5) If X is connected every finite cyclic subgroup Z=pn! X of X is subcon
jugate to TX .
2.7 Weyl spaces and Weyl groups: Let TX ! X be a maximal torus of
a pcompact group. We think of BTX ! BX as being a fibration. The Weyl
space WT (X) is defined to be the mapping space of all fiber maps over the iden
tity on BX. Then each component of WT (X) is contractible and the Weyl group
WT (X) := ss0(WT (X)) is a finite group under composition [DW, 9.5].
2.8 Theorem [7, 9.5 and 9.7]. Let TX ! X be the maximal torus of a connected
pcompact group X.
(1) The action of WX on BTX induces representations
WX ! Aut(H2(BTX ; Z^p) Q) ~=Gl(n; Q^p)
and
WX ! Aut(H2(BTX ; Z^p) Q) ~=Gl(n; Q^p)
which are monomorphisms whose images are generated by pseudoreflec
tions.
(2) The map H*Q^p(BX)! H*Q^p(BTX )WX is an isomorphism.
2.9 Normalizers, pnormalizers of maximal tori and ptoral Sylow sub
groups : Let i : TX ! X be a maximal torus of a pcompact group X. Again
we think of BTX ! BX as being a fibration. The Weyl space WX acts on BTX
via fiber maps This establishes a monoid homomorphism WX ! HE(BTX ) where
HE(BTX ) denotes the monoid of all self equivalences of BTX . Passing to class*
*i
fying spaces establishes a map BWX ! BHE(BTX ) which can be thought of as
being a classifying map of the fibration BTX ! BN(TX )! BWX . The total
space gives the classifying space of the normalizer N(TX ) of TX . This is alwa*
*ys a
finite extension of the pcompact torus TX .
Let Wp be the union of those components of WX corresponding to a pSylow
subgroup Wp of WX . The restriction of the above construction to Wp gives the
classifying space of the pnormalizer Np(TX ).
Since the action of WX respects the map BTX ! BX, the monomorphism
TX ! X extends to a loop map N(TX )! X. The pnormalizer fits into an
exact sequence TX ! Np(TX )! Wp and is therefore a pcompact toral group.
The restriction Np(TX )! X is a monomorphism [7, 9.9] and is a ptoral Sylow
subgroup of X, i.e. every pcompact toral subgroup P! X of X is subconjugate
to Np(TX ).
6
2.10 Proposition. Let Np(TX )! X be the ptoral Sylow subgroup of a p
compact group X. Then, the following holds:
(1) Every pcompact torus subgoup P 0! X of X is subconjugated to Np(TX ).
(2) The induced map H*(BX; Fp)! H*(BNp(TX ); Fp) is a monomorphism.
Proof. The Euler characteristic of X=Np(TX ) is coprime to p [7, proof of Theor*
*em
2.3]. Therefore, part (1) follows from [7, 2.14] and part (2) from [7, 9.8].
We also call Np(TX ) the pnormalizer of TX .
2.11 Simply connected and simple pcompact groups : A pcompact
group X is called simply connected, if X is simply connected, and X is called s*
*im
ple, if the associated representation WX ! Aut(H2(BTX ; Z^p) Q) is irreducib*
*le.
By [5], this is equivalent to the fact that the associated complex representati*
*on is
irreducible. The notion of simple is motivated by the classical situation. For *
*a com
pact connected Lie group G, the representation WG! H2(BTG ; C) is irreducible
if and only if G is simple.
2.12 Elementary abelian subgroups2.13 Let X be a pcompact group, and
let i : P! X be the ptoral 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 j0 : E! P . Such a subconjugation is called special if
CP (j0(E))! CX (j(E)) is a ptoral Sylow subgroup.
2.13 Lemma.
(1) For every monomorphism j : E! X, there exists a special subconjugation
j0 : E! P .
(2) Let
E ______ff______E1//@@
@@ """
j @@ OO""""j1"
X
be a diagram commuting up to conjugation. and let j0 : E! P be a
special subconjugation of j. Then, there exists a special subconjugati*
*on
j01: E1! P such that
E ______ff______E1//??
?? """
j0 ??OO"""j01""
P
commutes up to conjugation.
Proof. Let P 0! CX (j(E)) be the ptoral Sylow subgroup. Because j(E) is centr*
*al
in CX (j(E)), it is also contained in P 0[12, 4.]. By Proposition 2.10, there e*
*xists a
j k
subconjugation k : P 0! P . The composition j0 : E! P 0! P is a subconjugat*
*ion
of j into P and CP (j0(E)) ~=P 0. Hence, j0 is special. This proves part (1).
7
Let E2 E1 be a complement of E, i.e. E1 ~= E E2. Taking the adjoint of
j1 : E E2! X establishes map j2 : E2! CX (j(E)). Because j0 was special,
P 0:= CP (j0(E))! CX (j(E)) is a ptoral Sylow subgroup. Applying part (1)
yields a special subconjugation j02: E2! P 0, and passing back to maps into P*
* and
X proves part (2).
The elemantary abelian subgroups of a pcompact group X build a category
Ap(X), which we will call the Quillen category of X. An object of Ap(X) is a
iE
monomorphism E ! X, where E is a nontrivial elementary abelian group, and a
morphism is a triangle
E1 ______________/E2/A
AA "
AA ""
A """""
X
which is commutative up to conjugation. For the functor
OE : Ap(X)! T op : (E! X) 7! BCX (E)
there exists a map
: hocolim OE ! BX :
!
Ap(X)
Let * : Ap(X)! Ab be the functor given by *(E! X) := H*(BCX (E); Fp).
The following theorem is a collection of results of [8, 8.1 and the proof, 8.2].
2.14 Theorem [8]. Let X be pcompact group.
(1) The category Ap(X) is modp acyclic, i.e for the constant functor FZ=p
taking as value Z=p, we have
ae0 fori > 0
limi FZ=p =
Ap(X) Z=p fori = 0
(2) The map : hocolim (OE)! BX is a homotopy equivalence.
! Ap(X)
(3) We have ae
0 fori > 0
lim i * =
Ap(X) Z=p fori = 0
We finish this section with a result, necessary for later purpose.
2.15 Proposition. Let X be a pcompact group and i : P! X a ptoral 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 maxim*
*al
torus of X. This inclusion factors over i : T! P . (Confusing notations we de*
*note
all inclusions by i.) If we think a of the map BT! BP as a fibration respecti*
*vely
8
__
a covering, then we get a free action of P := P=T on BT . By_[9, 5.1], there ex*
*ists a
natural transformation of functors map(BP; )! map(BT; )hP , 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 CX (T ) is subconjugate to*
* P
(Proposition 2.10). Moreover,_this map is P_equivariant,_and induces therefore*
* an
equivalence map(BT; BP )hPBi! map(BT; BX)hPBi. 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 m*
*oti
vation 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 padic rationals. A faithful representation ae : W! Gl(*
*U) of
a finite group W is called a pseudo reflection group, if the image ae(W ) is ge*
*nerated
by pseudo reflections. The representation is called a honest real reflection gr*
*oup, if
ae(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 repre*
*sen
tation making W to a pseudo reflection group. A pseudo reflection group is call*
*ed
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 H1(W; Z^p), H1(W ; Fp), H2(W; Z^p),
H2(W; Fp), H1(W; Z^p), H1(W ; Fp), H2(W; Z^p), H2(W; Fp), H3(W ; Z^p) all vanis*
*h.
Proof. Because W is a finite group, it is sufficient to look at H1(W ; Z^p) a*
*nd
H2(W ; Z=p). Then, for the other groups, the statement follows by universal co
efficient theorems. Because every pseudo reflection group splits into a produc*
*t of
irreducible pseudo reflection groups and because of the K"unnethformula we also
can assume that W is irreducible.
The group H1(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 o*
*f order
coprime to p. Thus, H1(W ; Z^p) = 0.
The cohomology group H2(W ; Fp) 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 deno*
*te
the set of reflections of W and RB R a minimal set of generators of W Let
oe1; :::oen denote the elements of RB . Then there exists integers mi;jsuch t*
*hat
W ~=< oe1; :::; oen > =; < (oeioej)mi;j : 1 i; j n > is the quotient of the f*
*ree group
generated by the elements of RB dividing out only relations of the form (oeioe*
*j)mi;j
[3, chap. 5]. Let ^s: W! N be a set theoretic section. Then we define s : RB!*
* W
by soe := ^s(oe)p. Because the extension is central, the map s does not depend *
*on ^s.
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(oei)s(oej))mi;j= 1 for all 1 i;*
* j n.
9
Let W 0 W be the subgroup generated by oe1 and oe2, and let N0 N be the
subgroup defined by the pull back diagram
Z=pO _____N0_//____W_0//
OOOO
OOOOO  
fflffl fflffl
Z=p ______N//_____/W/ :
By the classification list of pseudo reflection groups [5] there exist only thr*
*ee re
flection groups generated by two honest reflections, namely Z=2 x Z=2, 3 and
the dieder group D12 of 12 elements. A short calculation shows that in all cas*
*es
H2(W 0; Z=p) = 0. For Z=2 x Z=2 this is obvious, for 3 one uses the fact that
3=Z=3 ~=Z=2, and for D12 one observes that 3 D12 is a subgroup of index 2.
Therefore there exists a group thoeretic section s0: W 0! N0. By the independe*
*nce
of s, for i = 1; 2, we have s(oei) = s0(oei)p = s0(oepi) = s0(oei). This shows*
* that s
satisfies the relation for oe1 and oe2, and analogously, all relation. Hence, t*
*he map s
can be extended to a group theoretic section, the sequence Z=p! N! W splits,
and H2(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.
no._ grouporder__ p_rime_ subgroup_ index_
2a r . mn1 . n! p n n r . mn1
12 48 p = 3 3 8
29 28 . 3 . 5 p = 5 5 25
31 64 . 6! p = 5 5 3 . 27
34 108 . 9! p = 7 7 25 . 35*
* :
In no. 2a, the number m divides p  1 and r divides m. For no. 2a the informa
tion about the subgroup might be found in [17] and for all other numbers in [1].
Therefore, in all these cases, H2(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 Wi! Gl(Ui) be a
pseudo reflection group and Li Ui be a [Wi]sublattice.
(1) The map
H3(W1; L1)! H3(W1; L1 x L2)
is an isomorphism.
(2) The map
H3(W1; L1) H3(W2; L2)! H3(W1 x W2; L1 x L2)
is an isomorphism.
10
Proof. Part (2) is a consequence of (1). By Proposition 3.1 we have
H2(W1; H1(W2; L1)) = H1(W1; H2(W2; L1)) = H0(W1; H3(W2; L1)) = 0 :
Hence, part (1) follows from the HochschildSerre spectral sequence for calcula*
*ting
H3(W1 x W2; L1).
4. Maps from pcompact groups into almost BZ=plocal spaces.
In [8] is set up a induction principle for proving statements about pcompact
group, which we will use in this and the next section and which we recall here.
The cohomolgical dimesion of a Fpfinite space is given by the maximal degree
of the nonvanishing modp cohomology groups. For two pcompact groups X and
Y , we say that X < Y if the cohomological dimension of X is smaller than the o*
*ne
of Y or if both have the same cohomological dimension but ss0(Y ) has a smaller
order than ss0(X).
4.1 Definition. A class Cl of pcompact groups is called saturated if it satis*
*fies
the following 5 conditions:
(1) If X 2 Cl and Y ~= X, then Y 2 Cl, i.e. Cl is closed under equivalences.
(2) The trivial pcompact group belongs to Cl.
(3) If the identity component X0 of X is in Cl and if any pcompact 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 2 Cl for all pcompact groups s*
*uch
that Y < X, then X 2 Cl.
4.2 Theorem [8, 9.2]. Any saturated class of pcompact groups contains all p
compact groups.
4.3 Remark. Our definition of a saturated class is not exactly the same as Dwy*
*er
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 prin
ciple. That is we have to show that the class Cl of all pcompact groups satisf*
*ying
(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 pcompact group into an almost BZ=plocal space. If f is
null homotopic, the restriction fBP is nullhomotopic.
If X and Y are isomorphic pcompact groups then they have isomorphic ptoral
Sylow subgroups PX and PY fitting into a homotopy commutative diagram
BPX ______BX//
' '
fflffl fflffl
BPY ______BY// :
Hence, X satisfied the statement if and only if Y does. This is Step (1).
11
The trivial group obviously satisfies the statement, which is Step 2. Going *
*a little
aside trip, we want to prove the statement for pcompact toral groups. The fir*
*st
part is obvious. Let P be a pcompact toral group. The loop space A is BZ=p
local [15, x1]. The proof of the Sullivan conjecture for pcompact groups [8, *
*9.3]
does not only apply to Fpfinite spaces, but also to BZ=plocal spaces without *
*any
change of the arguments. Therefore, for the mapping space map*(BP; A)const of
pointed maps, we have map(BP; A)const' map*(BP; A) is contractible. Hence,
the map A! map(BP; A)constis an equivalence.
Now let P! X be the ptoral Sylow subgroup of a pcompact group X. The
composition P ! X! ss0(X) is an epimorphism [12, 3.8]. This establishes a
commutaive diagram of fibrations
BP0 ______BP//______Bss0(X)//OO
  OOOO
  OOOO
fflffl fflffl
BX0 ______BX//______Bss0(X)// ;
and P0! X0 is a ptoral Sylow subgroup of X0 [12, 3.9]. Let
f : BX! A be a map. If fBP ' const and if X0 2 Cl, then fBX0 ' const.
Thinking of BX0! BX as a fibration, the space BX0 carries a free ss := ss0(X)
action. Taking the trivial action on A, the map A! map(BX0; A)const is an
equivalence, ssequivariant and establishes equivalences
(*) map(Bss; A) ' Ahss! (map(BX0; A)const)hss' map(BX; A)f0BX0 'const:
Here,_Ahssdenotes the homotopy fixedpoint set of the ssaction on A (see [9]).
Let f : Bss! A be the map corresponding to f :_BX!_ A. Applying_the_same
trick to map(BP; A) allows to calculate the map f_. In this case, f correspon*
*ds
to th constant map const : BP! A, and hence, 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 2 Cl. This is the third step in t*
*he
induction process.
For any connected pcompact group X with ptoral Sylow subgroup P! X,
there exists a diagram of fibrations
__
BZOO______BP//O______BP//
OOOO  
OOOOO  
fflffl ___fflff;l
BZ ______BX// ______BX//
__ ___
where Z = Z(X) is the center of X and where_P ! X := X=Z is a ptoral
Sylow subgroup of the centerfree group X . Actually, both fibrations are princi*
*pal
bundles, because Z! P and Z! X are central [2, 7.2]. Let f : BX! A be a
map such that fBP ' const. Then fBZ ' const and A! map(BZ; A)const. In
this situation we can apply a lemma of Zabrodsky [19] (see also [10]), which te*
*lls
12
us that in the diagram
map(BP; A)f0BZO'constOoo'___map(B__P;OA)O
 
 
 
map(BX; A)f0BZ 'const oo'___map(B___X; A)
__ ___
the horizontal lines are homotopy equivalences._Let f : BX ! A be the map cor*
*re
sponding_to f. The top row implies that fB__P' const. Thus, by the assumptions,
f is also nullhomotopic as well as f. This finishes the proof of the fourth st*
*ep in
the induction.
Now let X be a centerfree connected pcompact group. Then, for an object
j : E! X, the centralizer CX (j(E)) is smaller than X. We choose a special
subconjugation j0 : E! P of j (Lemma 2.13). Then, CP (j0(E))! CX (j(E)) is
a ptoral Sylow subgroup and CP (j0(E)) is subconjugate to P . Let f : BX! A
be a map such that fBP ' const. Then fBCP (j0(E))and fBCX (j(E))are also nu*
*ll
homotopic and A! map(BCX (j(E)); A)constis an equvalence (by induction hy
pothesis). Moreover, these maps are compatible with all morphisms in the Quillen
category Ap(X). In the sequence
holim A ! holim map(BCX (E); A)const
 
Ap(X) Ap(X)
! map(hocolim BCX (E) ; A)F! map(BX; A)F
!
Ap(X)
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 f0 : BX! A such that for every ob
ject E ! X of Ap(X) the restriction f0BCX (E) is null homotopic. Using the
BousfieldKan spectral sequence for calculating the first expression, Theorem 2*
*.14
shows that holimAp(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 pcompact group with ptoral Sylow subgroup P! X.
Let F! E! BX be a fibration, such that F is Fpfinite, pcomplete 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 retrict*
*ion
to BP is null homotopic. The space BHE(F ) is connected, pcomplete and almost
BZ=plocal. Thus, the statement follows from Theorem 1.4.
A similar statement as the last corollary is also true if the fiber is the c*
*lassifying
space of a pcompact torus.
13
4.5 Proposition. Let T be a pcompact torus, let X be a pcompact group with
ptoral 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 fibrati*
*on 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 pcompact torus, and let X be a pcompact group with p
toral Sylow subgroup P! X. If, for a map f : BX! BHE(BT ), the restriction
fBP is null homotopic, then f is null homotopic.
Proof. Let P0! X0 be the ptoral Sylow subgroup of the component of the unit.
In the diagram
BSHE(BT ) ' B2T


f fflffl
BP __________BX//_____________/BHE(BT/)
  
  
fflffl fflffl __f fflffl
BP=P0 ___'__Bss0(BX)//________/Bss0(HE(BT/))
__
the map f is null homotopic. This follows because
ss := ss0(HE(BT ) ~= Gl(H2(BT ; Z^p)) is a discrete group, because therefore Bss
is almost BZ=plocal, 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 compositi*
*on
BP! Bss0(X)! Bss and BX! Bss0(X)! Bss to BHE(BT ) are classified
by the obstruction groups H3(BP; ss3(BT )) or H3(BX; ss3(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 pcompact groups.
In this chapter we want to prove the main "technical" theorem which allows
several interesting applications to pcompact groups. Let X be a pcompact group
with normalizer NX of the maximal torus TX ! X of X. Let NX ~=N1 x N2 be
a splitting into two factors. For i = 1; 2, we want to construct a subgroup Yi!*
* X
of X such that this inclusion induces an isomorphism NYi ~=Ni in a way which we
will make precise working out the construction.
For any pcompact group X there exists a fibration
BX0! BX! Bss
where X0 denotes the connected component of the unit and ss the group of the
components. The composition BNX ! BX! Bss factors over BWX ! Bss,
which is induced by a homomorphism WX ! ss. The kernel is given by the Weyl
group WX0 [12, 3.8], which is a pseudo reflection group. If NX ~=N1 x N2, the
14
Weyl group WX ~=W1xW2 also splits into two factors as well as the maximal torus
TX ~=T1 x T2. Every pseudo reflection oe 2 WX0 is contained either in W1 or *
*in
W2. Let Wi0 WX0 be the subgroup generated by all pseudo reflection contained
in Wi, and let N0ibe the counterimage of Wi0in Ni. Then, we have splittings
WX0 ~= W10x W20, NX0 ~= N01x N02and ss ~=ss1 x ss2. Now we define Y 0=: CX (T2).
This is a subgroup of maxiaml rank. For later use we note the following lemma:
5.1 Lemma. NY 0= CNX (T2) = N1 x CN2 (T2).
Proof. The Weylgroup WY 0is given by all elements of WX acting trivially on T2
[8, 7.6]. Hence, WY 0= W1 x ^W2, where W^2 = WY 0\ W2 W2. Let q : NX ! WX
be the projection. Then, N0Y = q1 (WY 0) = CNX (T2) = N1 x CN2 (T2). For the
last equivalence, there is a remark in order. The loops spaces NX and N2 are n*
*ot
pcompact groups. But the types of centralizers in question are studied in [14,
3.7]. That result implies that CNX (T2) as well as CN2 (T2) are finite extensio*
*ns of
a pcompact tori.
By the above lemma, N1 NY 0and ss1 ss0(Y 0). Let Y 00be defined by the pull
back diagram
BY 00 _________BY/0/
 
 
fflffl fflffl
Bss1______/Bss1/x Bss02 :
By construction, we have NY 00= N1 x T2. Applying the functor map(BT2; ) shows
that T2 Y 00is central, and dividing out this central subgroup gives a diagram*
* of
fibrations
BT2O______BN1/x/BT2O ____________BN1//
OOO
OOOOO  
OO fflffl fflffl__
BT2 _________/BY/00_________B(Y/00=T2)/=: BY :
Both lines_are principal fibration [2, 7.2]. Thus, the lower fibration_is class*
*ified by a
map BY ! B2T2, which is determined by a cohomology class in H3(BY ; ss3(B2T2)*
*).
The restriction_of this map to BN1 is homotopically trivial, and the induced
map H3(BY ; ss3(B2T2))! H3((BN1; ss3(B2T2)) is a monomorphism (Proposition
2.10). Hence, the lower row is equivalent to the trivial fibration and we can *
*split
BY 00' BY x BT2 in such a way that we get a commuative diagram
BN1 x BT2M
nnn MMMM
nnnn MMM
wwnnnn M&&
BY x BT2 _______________________BY_00// :
__
The composition BY! BY 00! BY is an equivalence. Hence, N1 ~=NY Y is
the normalizer of the maximal torus. The pcompact subgroup Y Y 00 X is the
one which we associate with N1.
15
5.2 Lemma. Let X be a connected pcompact group. Let NX ~=N1 x N2 split
into two factors, and let Y1 X be the subgroup associated to N1.
(1) We have Y1 x T2 ~=CX (T2).
(2) There exists an isomorphism Z(X) ~=Z(Y1) x Z(Y2) making the diagram
Z(X) ______~=Z(Y1)_x/Z(Y2)/
 
 
fflffl fflffl
(TX )WX _____(T1)W1_x_(T2)W2___
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 pcompact group such that NX ~= N1 x N2. For
ji
i = 1; 2, let Yi ! X be the subgroup associated to Ni. Then there exists a
homotopy equvalence j : Y1 x Y2! X making the following diagram commutative
up to homotopy for i = 1; 2
N1 x N2 ______NX//
 
 
fflfflj fflffl
Y1 xOY2_______/X/Ou::
 jiuuu
 uuu
 uu
Yi :
Proof. The proof is carried out in several steps and based on the induction pri*
*nciple
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 conditi*
*on
(S). Let Cl be the class of all pcompact groups satifying the statement. We wa*
*nt
to show that Cl is a saturated class.
~=
Step 1: Let X be a pcompact group such that NX ~=N1 x N2. Let Y ! X be
an isomorphism of pcompact groups. Then, the normalizer NY ~= NX ~=N1 x N2
also splits. Let Yi! Y be the subgroup associated to Ni. If Y satifies the th*
*eorem,
the composition Y1 x Y2 ~= Y ~=X establishes the desired splitting of X. Henc*
*e,
the class Cl is closed under equivalences.
Step 2 : Theorem 5.3 is satified by ptoral groups. This is obvious.
Step 3: Let us assume that the theorem is true for connected pcompact group*
*s.
We want to show that it is true in general.
16
Let X be a pcompact group satifying condition (S), and let X0 be the compon*
*ent
of the unit. If Ti! Yi is a central subgroup for i = 1; 2, the Weyl group WX *
* acts
trivially on TX , and the inclusion CX (TX )! X is a sugroup of maximal rank *
*and
induces an isomorphism between the Weyl groups. Moreover, both spaces fit into
a diagram
C0 := CX0 (TX )______C/:=/CX (TX ) ____________ss0(C)//
  
  
fflffl fflffl fflffl
X0 __________________X//__________/WX/=WX0 ~= ss0(X) :
Because C0 is connected [12, ???], the top row is an exact sequence of pcompact
groups and therefore C0 is the connected component of the unit of C. The left
column desribes a subgroup of maximal rank and WC ~= WX . By [12, 3.11] follows
that the left column is an isomorphism of pcompact groups. The isomomorphism
in the lower right corner follows from [12, 3.8], which also implies that ss0(C*
*) ~=
WC =WC0 . Thus the right column is an isomorphism of a pcompact groups as well
as the middle vertical arrow. By 2.5, the pcompact group C is a pcompact toral
group and so is X. This case we already considered in Step 2. Hence, we can
assume that T1! Y1 is not a central subgroup, that CY1(T1) is smaller than Y1,
that the Weyl group of CX (T1) is a proper subgroup of WX and that CX (T1) is
smaller than X (if both have the same cohomological dimension, then CX (T1) has
less components than X).
As explained at the beginning of this section, the normalizer NX0 ~=N01x N02
also splits as well as the group ss := ss0(X) ~=ss1 x ss2 of the components. Fo*
*r the
associated subgroups we have fibrations
BYi;0! BYi! Bssi :
Moreover, by the hypothesis, we have a commutative diagram
N1;0x N2;0 ______NX0//
 
 
fflfflj fflffl
Y1;0xOY2;0O_______X0//r99
 jirrr
 rr r
 rr
Yi;0 :
Here, Yi;0denote the component of the unit of Yi and Ni;0the normalizer of the
maximal torus of Yi;0. We want to show that there exists an extension Y1xY2! X
of the lower horizontal arrow.
Looking at homogeneous spaces we have a commutative diagram of fibrations
Y2;0 _________Y2//______ss2//OO
OOOO
'  OOOO
fflffl fflffl OO
X0=Y1;0 _____X=Y1_//_____ss2// ;
17
which implies that the vertical middle arrow is a homotopy equivalence. Moreove*
*r,
this map is given by the composition Y2! X! X=Y1.
ki
Let Pi ! Yi be the the ptoral Sylow subgroup of Yi. Taking pullbacks induc*
*es
a diagram of fibrations
Y2 _______Y2_________Y2____________
  
  
fflffl fflffl fflffl
(*) EP ________E//______BY1//
  
  
fflffl fflffl fflffl
BP1 _____BY1_//_____BX// :
The left column establishes a proxy action of P1 on Y2. For the definition of a
proxy action we refer the reader to [7, x10]. Obviously, the induced fibrations*
* have
id
sections sP : BP1! EP and s : BY1! E which come from the lift BY1 ! BY1.
Applying the functor map(BP; ) to the left and right fibration yields a pull ba*
*ck
diagram
(Y2hP1)sP__________________(Y2hP1)s______________
 
 
fflffl fflffl
(**) map(BP1; EP )sP ________/map(BP1;/BY1)Bki
 
 
fflffl fflffl
map(BP1; BY1)Bk1 ______map(BP1;/BX)Bj1Bk1/ ;
where Y2hP1 denotes the homotopy fixed point set of the proxy action of P1 on
Y2 (again for definitions see [7, x10]. In the fibers we only take those compon*
*ents
of the homotopy fixedpoint sets which belong to the associated sections. The
map_BZ(P1)! map(BP1; BY1)Bki is an equivalence (Proposition 2.15). Now, let
P 1 denote the group of components of P1. Replacing BT1 by BgT1 to transform_
BT1! BP1 into a fibration, i.e into a covering, we get an action_of P 1on gBT*
*1. By
[9], we have an equivalence of functors map(BP1; ) ' map(BgT1; )hP 1. This allo*
*ws
to calculate the mapping space in the lower right corner of the above pull back
diagram. By Lemma 5.1, we have NCX (T1)= N^1x NY2, where N^1:= CNY1 (T1) ~=
CY1(T1) is a pcompact group. The isomorphism follows because T1! Y1 is a
maximal torus. Moreover, the centralizer CX (T1) is smaller than X. Hence we can
apply the induction hypothesis to establish a commutative diagram
BN^1 x BNY2 _'____BNCX/(T1)/
 
 
fflffl ' fflffl
BN^1 x BY2 ______BCX_(T1)// :
18
The inclusions Y1 x Y2 N^1x Y2 ~=CX (T1)! X induce equivalences
'
M2 := map(BgT1; BN^1 x BY2)Bi ! M1 := map(BgT1; BY1 x BY2)Bi
'
M2 ! M3 := map(BgT1; BX)Bi
__
which are also P 1equivariant. Therefore, these equivalences_induce equivalen*
*ces
between the homoytopy fixedpoint sets. The component of MhP11_related to the
inclusion P1! Y1! Y2 corresponds to the component of MhP31 related to the
inclusion P1! Y1! X. Therefore, we get an equivalence
BZ(P1) x BY2 ' map(BP1; BY1 x BY2)Bi ' map(BP1; BX)Bi :
The pull back digram (**) translates now to
__ ____________ __
(Y2hP 1)sP____________ (Y2hP 1)s
 
 
fflffl fflffl
map(BP1; EP )sP _________BZ(P1)//
 
 
fflffl fflffl
BZ(P1) __________/BZ(P1)/x BY2 :
This shows that __ __
(Y2hP 1)sP ' (Y2hP 1)s ' Y2 :
Taking the adjoint we can construct a map Y2 x BP1! EP which is a homotopy
equivalence and gives a trivialization of the fibration Y2! EP! BP1. By The*
*o
rem 1.4, the fibration Y2! E! BY1 is also trivial, i.e. we have a fiber hom*
*otopy
equivalence Y2 x BY1! E.
Applying the functor map(BY1; ) to the middle and right columns in the diagr*
*am
(*) establishes another pull back diagram
map(BY1; E)s ' BZ1 x Y2 ______map(BY1;/BY1)id/' BZ1
 
 
fflffl fflffl
map(BY1; BY1)id ' BZ1 __________map(BY1;/BX)Bji/ ;
where Z1 := Z(Y1) denotes the center of Y1 The equivalences follow from the abo*
*ve
fiber homotopy equivalence and from [8, 1.3]. Passing to loop spaces shows that
map(BY1; BX)Bj1) =: BY20is a pcompact group.
The pull back diagram
BZ1 ' map(BY1; BY1)id _________/BZ(P1)/' map(BP1; BY1)Bi
 
 
fflffl fflffl
BY20' map(BY1; BX)Bi ______BZ(P1)/x/BY2 ' map(BP1; BX)Bi
19
shows that BZ1! BY20has a section BY20! BZ1 and that the homotopy fiber is
equivalent to BY2 (the right column allows a section). Moreover, the diagram al*
*so
shows that the composition BY2! BY20! BX equals the inclusion BY2! BX.
Passing to adjoints shows the existence of a map BY1 x BY2! BX, which also
extends the map BN1 x BN2! BX. This completes the proof of the third step.
5.4 Remark. The arguments for calculating map(BP1; BX)Bi ' BZ(P1) x BY2
always work when X satifies the condition (S). We only have to assume the in
duction hypothesis, and that T1! Y1 is not central. The ptoral Sylow subgroup
P = P1 x P2 splits into a product. Hence, we always get a map BP1 x BY2! BX,
extending the composition BP1 x BP2! BN1 x BN2! BX. For the calcula
tion of the mapping space map(BY1; BX)Bi, we only needed in addition that the
composition Y2! X! X=Y1 is a homotopy equivalence.
Step 4: We have to show that the theorem is true for a connected pcompact
group if it is true for the associated centerfree pcompact group. Let_X_ be a
connected pcompact group, let Z := Z(X) be the center of X, and let X :=
X=Z(X) be the associated centerfree group. Let NX ~=N1 x N2, let Y1; Y2! X
be the associated subgroups, let Pi Ni Yi be a ptoral Sylow subgroup, and
let Zi := Z(Yi) denote the center of Yi. The center Zi is a subgroup of Pi [12,*
* 4.3].
Because Wi acts trivially on Zi, the inclusion Zi Ni is central [14, 3.7] By [*
*2,
7.2], we can construct a diagram of principal bundles
___ ___
BZ1 x BZ2OO ______BN1/x/BN2 ______BN/1/x BN 2
OOO
OOOO  
OOO  
fflffl ___fflffl
BZOO______________BX//______________BX//OOOO
  
  
  __
BZi _____________/BYi/______________BY/i/ :
__ ___ ___
By the construction, the subgroup Y i! X_ is associated_to N i. Thus, by the
hypothesis, there exists an equivalence BY 1 x BY 2! BX extending the map on
the normalizers. The diagram of the classifying maps of the principle bundles
__ __
BY 1 x BY 2 ______B2Z1/x/B2Z2OO
OO
' OOOOOO
___fflffl________//
BX B2Z
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
BY1 x BY2! BX. This finishes the proof of Step 4.
Step 5: We have to show that the statement is true for a centerfree connected
pcompact group, if it is satified by all smaller pcompact groups. The proof *
*of
20
this step is even more complicate than the one of Step 2 and divided into sev
eral claims. The outline is as follows: The first major step is the constructi*
*on of
an equivalence between the Quillen categories of X and Y1 x Y2 (Claim 3). Let
FX : Ap(X)! Ab be the functor given by FX (E! X) := H*(BC(E); Fp). Anal
ogously, we define a functor FY : Ap(Y1 x Y2)! Ab. Using the equivalence of
the Quillen categories, we then show that there exists a natural transformation
FX ! FY , which is an isomorphism on the objects (Claim 6). The modp de
composition theorem of pcompact groups (Theorem 2.14) establishes an isomor
phism H*(BX; Fp) ~=H*(BY1 x BY2; Fp) which is compatible with the equivalence
BNX ' BN1 x BN2 (Claim 7). An EilenbergMoore spectral sequence argument
allows to calculate the fiber of BY1! BX which turns out to be equivalent to *
*Y2
(Claim 8). Now similiar arguments as in the proof of Step 2 are applicable and
'
establish the desired equivalence BY1 x BY2 ! BX (Claim 9).
Claim 1: The spaces Yi are connected.
Proof. This follows from Lemma 5.2 and from [12, 3.11], which says that, for co*
*n
nected pcompact groups, centralizers of tori are always connected.
Let iX : P! NX ! X be the ptoral Sylow subgroup. Then P = P1 x P2
splits and iY : P1 x P2! N1 x N2! Y1 x Y2 is also the ptoral Sylow subgroup
of Y1 x Y2.
Claim 2: Let j1; j2 : E! P be two monomorphisms of an elementary abelian
subgroup E into P . Then BiY Bj1 and BiY Bj2 are homotopic if and only if
BiX Bj1 ' BiX Bj2.
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 W1 on T1 and W2 on T2
are nontrivial. Inparticular, Ti! Yi is not a central subgroup. The composi*
*tion
BiY Bji as well as BiX Bji factor over BP1 x BY2. This follows from Remark 5.4.
Because Y2 is connected (Claim 1), the second coordinate, as a cyclic subgroup,
of both maps is subconjugate to T2 (Theorem 2.6). Hence, we can assume that
the second corodinate of ji : E! P1 x P2 takes image in T2. Analogously, we
can assume that the first cordinate also takes image in T1. That is to say that
ji : E! P1 x P2 represents E as a subgroup of the torus. By [13, 4.2], the t*
*wo
maps BiY Bj1 and BiY Bj2 are homotopic if and only if Bj1 and Bj2 differ by a
Weyl group element if and only if BiX Bj1 ' BiX Bj2. This proves Claim 2 for
E = Z=p.
Now let us assume that the rank of E is bigger than 1. Let E1 E denote the
first coordinate of E and E2 E a complement of E1, i.e. E ~=E1 E2. By what
we already proved we can assume that, for i = 1; 2, the restriction
j1E1 = j2E1 : E1! P1 x P2 is a subgroup of the maximal torus. Let j0i: E2!
CP1xP2 (E1) be the adjoint of ji. The centralizers X0 := CX (E1) and CY1xY2 (E1)
are subgroups of maximal rank, smaller than the centerfree group X, and the
normalizers of the maximal tori are given by CNX (E1) = N01x N02which again
splits as well as CP1xP2 (E1) = P10x P20. The latter centralizer is a ptoral *
*Sylow
subgroup of X0 and Y10x Y20. Moreover, Yi0:= CYi(iYiji(E1))! Yi is the subgro*
*up
associated to N0i. Therefore, by induction hypothesis, there exist an equivale*
*nce
21
'
Y10x Y20! X0 making the diagram
BY10x BY20
nn77 
nnnn 
nnn 
nn 
BP10x BP20 ______BN01x/BN02/P 
PPPPP 
PPPP 
P(( fflffl
BX0
commutative up to homotopy.
The two maps BiX Bj1; BiX Bj2 : BE1 x BE2! BX are homotopic if the
adjoints BE2! BX0 are homotopic. The same is true for Y1 x Y2. 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 Y1 x Y2 are isomorphic, i.e. there
exist a functor Ap(Y1 x Y2)! Ap(X), which is an isomorphism of categories.
Proof. By Proposition 2.10, every elementary abelian subgroup is subconjugate to
P1 x P2. Hence the statement follows from Claim 2.
Claim 4: Let j : E! P := P1 x P2 be a monomorphism. Then, the map j is a
special subconjugation of iY j if and only of it is a special subconjugation fo*
*r iX j.
j j0
Proof. Let E ! P be a special subconjugation of iY j and E ! P a special
subconjugation of iX j. By Claim 2, the composition iY j0 is conjugate to iY j.
Hence, the centralizer PX = CP (j0(E)), which is the ptoral Sylow subgroup of
CX (iX j(E)), is subconjugated to the ptoral Sylow subgroup PY = CP (j(E)) of
CY (iY j(E)) and vice versa. Therefore PY and PX are isomorphic. This proves t*
*he
statement.
j
Claim 5: Let E! P be a monomorphism, which is a special subconjugation of
'
iX j and iY j. Then, there exists an equivalence fE : BCY (E) ! BCX (E) making
the diagram
BCP (E)L
rr LLL
rrr LLL
xxrrrr LL&&
BCY (E) __________'__________BCX/(E)/
commutative up to homotopy.
Proof. Let E0 ~=Z=p E denote the first coordinate of E. By Claim 2 and Lemma
2.13 we can assume that E0! P is a toral subgroup and a special subconjugation
of E0! P! X and E0! P! Y . As shown in the proof of Claim 2 (using the
induction hypothesis), there exists a homotopy commutative diagram
BCP (E0)M
qq MM M
qqq MMM
xxqqqq MM&&
BCY (E0) __________'__________BCX/(E0)/ :
22
Passing to adjoints and taking centralizers finishes the proof of Claim 5.
Claim 6: There exists a natural transformation FY! FX , which is an isomor
phism on objects.
j
Proof. Let E! P be a special subconjugation of jY := iY j and jX := iX j. Th*
*en
we define
OEE := H*(fE ; Fp) : H*(BCY (E); Fp)! H*(BCX (E); Fp) :
Let E1! E2 be a morphism of Ap(Y ) ~= Ap(X). By Lemma 2.13, for i = 1; 2,
there exist monomorphisms ji : Ei! P , which are special lifts of ji;Y and ji*
*;X
such that the diagram
BE1 F_______________BE2_//
FFF xx
FFF xxx
## xx
BP
commutes up to homotopy. Passing to centralizers gives another diagram, namely
BCY (E2) _____________________BCY_(E1)//
pp 88  qq88 
ppp  qqq 
pp  q q 
pp  qq 
BCP (E2)0N____________________BCP/(E1)/M 
NNNN fE2 MM MM fE1
NNN  MMM 
N && fflffl M&& fflffl
BCX (E2) _____________________BCX/(E1)/ :
The two triangle commute up to homotopy (Claim 5) as well as the upper and
the lower parallelogram. For any pcompact group U with ptoral Sylow subgroup
PU! U, the map H*(BU; Fp)! H*(BPU ; Fp) is an injection (Proposition 2.10).
The diagonal arrows induce injections in modp cohomology, because we always
chose special subconjugation. Therefore, the square commutes at least in modp
cohomology, and the maps E establish a natural transformation FY! FX , which
is an isomorphism on the objects.
Claim 7: There exists an isomorphism
~=
OE : H*(BX; Fp) ! H*(BY ; Fp)
OE
of algebras over the Steenrod algebra such that the composition H*(BX; Fp)!
H*(BY ; Fp)! H*(BYi; Fp equals the map H*(BX; Fp)! H*(BYi; Fp) induced
by the inclusion BY1! BX.
Proof. We have the following sequence of isomorphisms:
H*(BX; Fp) ~= lim FX ~= limFY ~= H*(BY ; Fp) :
Ap(X) Ap(Y )
23
The first and last isomorphism follow from Theorem 2.14 and the middle isomor
phism from Claim 6. The second part of the statement follows from the construct*
*ion
of the natural transformation FY! FX .
Claim 8: The composition Y2! X! X=Y1 is a homotopy equivalence.
Proof. In the pull back diagram
E _______BY1//
 
 
fflffl fflffl
BY2 ______BX//
we want to calculate the modp cohomology of E via the EilenbergMoore spectral
sequence. The E2 is given by
E2 = T orH*(BX;Fp) (H*(BY1; Fp); H*(BY2; Fp))
~=T orH*(BY1;Fp)(H*(BY1; Fp); Fp)
~=T orFp(Fp; Fp) :
The functor FpH*(BYi; Fp) is exact. This implies the second isomorphism and
also, together with Claim 7, the first isomorphism. Therefore, E is modp acycl*
*ic.
Moreover E is pcomplete, because all the other involved spaces are pcomplete,
Hence, E ' *. The map between the fibers is obviously given by the composition
Y2! X! X=Y1. This proves the statement.
'
Claim 9: There exists an equivalence BY1xBY2 ! BX, such that the diagram
BN1 x BN2 '_____BNX//
 
 
fflffl fflffl
BY1 x BY2 _______BX//
commutes up to homotopy.
Proof. Using Claim 8, the same arguments as in the proof of Step 2 are applicab*
*le
(see Remark 5.4). This establishes an equivalence BY1 x BY2! BX with the
desired properties. In this case the proof even simplifies a little, because BY*
*1 and
BY2 are centerfree connected groups, which is to say that, for i = 1; 2, the ma*
*pping
spaces map(BYi; BYi)id ' BZ(Yi) are contractible. .
Claim 9 finishes the induction for the proof of Step 5, which was the last s*
*tep to
complete the proof of Theorem 5.3.
6. Simply connected and centerfree connected pcompact groups.
In this section we want to prove Theorem 1.2 and Theorem 1.3. We call a
finite extension N of a pcompact torus T simple if the associated representati*
*on
N=T! Gl(H2(BT ; Z^p) Q is irreducible.
24
6.1 Lemma. Let N be a finite extension of a pcompact torus T , such that N=T
acts as a pseudo reflection group on H2(BT ; Z^p) Q. Then, up to order, there
exists at most one splitting N ~=N1 x ::: x Nn into simple finite extensions of
pcompact tori.
Proof. The existence of a splitting into simple factors shows that we have spli*
*ttings
W := N=T ~= W1 x ::: x Wn for the quotient, LT := H2(BT ; Z^p) Q ~=L1 ::: Ln
for the 2dimensional homology of BT and T ~= T1 x ::: x Tn for the torus T its*
*elf.
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 pcompact
group. Then, the following holds:
(1) The normalizer NX ~=N1 x ::: x Nn splits, up to order, uniquely into a
product of simple factors. For each factor, the qoutient Wi := Ni=Ti ac*
*ts
on H2(BTi; Z^p) as a pseudo reflection group.
(2) The pcompact group X ~= X1 x ::: x Xn splits into a product of simple
centerfree connected pcompact groups.
(3) For a splitting X ~=X1 x ::: x Xn into simple centerfree connected p
compact groups, the normalizers of the maximal tori of the factors are
determined by the factors of NX , i.e , after reordering the factors, t*
*here
exist isomorphisms NXi ~=Ni such that the diagrams
BNXi ' BNi _____BNX_//
 
 
fflffl fflffl
BXi ___________BX//
commute up to homotopy.
Proof. The action of WX on L := H2(BTX ; Z^p) represents W as a pseudo reflect*
*ion
group, and the lattice L is centerfree. Therefore, by [16, 1.3], the lattice L*
* ~=
L1 ::: Ln splits into a direct sum of sublattices, such that Wi acts trivially
on Lj for i 6= j. Let Ti := TLi be the padic torus associated to Li. Moreove*
*r,
T ~= TL ~= T1 ::: Tn is the torus associated to L.
Q The fibrationQBTX ! BNX ! BWX isQclassified by a map cN : BWX '
iBWi! BHE(BTi). Let c0 : BWX ! iBHE(BTi) be the classifying
map of the product of the fibrations BTi! BN0! BWi given by the semi direct
product. Since BSHE(BTi) ' B2Ti, the difference between cN and c0 is measured
by a cohomology class in
Y Y Y
H3(BWX ; ss3(B2Ti)) ~=H3(BWX ; Li) ~= H3(BWi; Li) :
i 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
25
BTX ! BNX ! BWX . 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 NX into simple factors, whi*
*ch 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 pcompact
group. Then, the following holds:
(1) The normalizer NX ~=N1 x ::: x Nn splits, up to order, uniquely into a
product of simple factors. For each factor, the qoutient Wi := Ni=Ti ac*
*ts
on H2(BTi; Z^p) as a pseudo reflection group.
(2) The pcompact group X ~= X1 x ::: x Xn splits into a product of simple
simply connected pcompact groups.
(3) For a splitting X ~=X1 x ::: x Xn into simple simply connected pcompact
groups, the normalizers of the maximal tori of the factors are determin*
*ed by
the factors of NX , i.e , after reordering the factors, there exist iso*
*morphisms
NXi ~=Ni such that the diagrams
BNXi ' BNi _____BNX_//
 
 
fflffl fflffl
BXi ___________BX//
commute up to homotopy.
___
Proof. Let X := X=Z(X) be the associated centerfree connected pcompact group.
The center Z := Z(X) of X is a finite group [12, 5.3]. All these spaces fit in*
*to a
principal fibration ___
BZ! BX! BX :
___
with classifying_map BX ! B2Z [2, 7.2]. Hence, there exists an isomorphism
ss2(BX ) ~=ss1(BZ) ~=Z. Q ___ ___ Q ___
By Theorem 6.2, there exist splittingsQN__X~= iN i of N__Xand X ~= iX i o*
*f X
into simple centerfree factors. Let Z ~= i Zi be the_associated_splitting. The*
* above
classifying map also splits into a productQof maps BX i! B2Zi. The product of*
* the
fibers Xi gives a splitting of X ~= iXi into simpleQsimply connected pcompact
groups. This also establishes a splitting of NX ~= iNXi 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.
26
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email: notbohm@cfgauss.unimath.gwdg.de.