CONNECTED FINITE LOOP SPACES WITH MAXIMAL TORI
by
J. M. Moller and D. Notbohm
Abstract. Finite loop spaces are a generalization of compact Lie groups. *
*However,
they do not enjoy all of the nice properties of compact Lie groups. For e*
*xample having
a maximal torus is a quite distinguished property. Actually, an old conje*
*cture, due
to Wilkerson, says said that every connected finite loop space with a max*
*imal torus
is equivalent to a compact connected Lie group. We give some more evidenc*
*e for this
conjecture by showing that the associated action of the Weyl group on the*
* maximal
torus always represents the Weyl group as a crystallographic group. We al*
*so develop
the notion of normalizers of maximal tori for connected finite loop space*
* and prove
for a large class of connected finite loop spaces that a connected finite*
* loop space
with maximal tori is equivalent to a compact connected Lie group if it ha*
*s the right
normalizer of the maximal torus. Actually, in the cases under considerat*
*ion the
information about the Weyl group is sufficient to give the answer. All th*
*is is done
by first studying the analoguous local problems.
1. Introduction.
A loop space L := (L; BL; e) is a triple consisting of two spaces L and BL, *
*which
is pointed, and an equivalence e : BL! L between the loop space of BL and
L. The space BL is called the classifying space of L. A loop space L is called
finite if L is Zfinite, i.e. the integral cohomology H*(L; Z) is a finitely ge*
*nerated
graded Zmodul. A finite loop space L is called connected if L is a connected
space. For a compact Lie group G the triple (G; BG; e) is a finite loop space, *
*where
e : BG! G is the obviuos equivalence.
Following an old idea of Rector [18] several notions of (Lie) group theory c*
*an
be given in homotopy theoretic terms by means of classifying spaces (e.g. see [*
*18],
[17], [4], [5], [12]). For example, a homomorphism f : L! M of finite loop sp*
*aces
is a pointed map Bf : BL! BM. The homotopy fiber of Bf is denoted by M=L.
A homomorphism f : L! M is an isomorphism if Bf is a homotopy equivalence.
Inparticular, we are interested in maximal tori and the associated Weyl groups *
*of
connected finite loop spaces. We will use the following definition:
1.1 Definition. Let L be a connected finite loop space. A maximal torus of L is
a homomorphism f : T! L from a torus T into L which satifies the following two
conditions:
(1) The homogenous space L=T is Zfinite.
______________
1980 Mathematics Subject Classification (1985 Revision). 55P35, 55R35.
Key words and phrases. Finite loop space, pcompact group, classifying space*
*, maximal torus,
normalizer, Weyl group, covering.
Typeset by AM ST*
*EX
1
2
(2) The rank rk(L) of L equals the rank rk(T ) of T . The rank is defined t*
*o be
the transcendence degree of the polynomial ring H*(BL; Q) over Q.
The Weyl monoid WL := WL (T ) of a maximal torus f : T! L is the monoid
WL (T ) := { [w : BT! BT ] Bf O w ' Bf}
of all homotopy classes of self maps of BT over BL.
The classical maximal torus TG! G of a compact connected Lie group G passes
to a fibration G=TG! BT! BG, and an element w 2 WG of the Weyl group WG
of G give rise to a homotopy commutative diagram
BTG F_______w_______BTG_//
FFF xx
FFF xxx
## xx
BG :
Hence, the above definitions are the extract of the classical situation in term*
*s of
classifying spaces. We also show that every two maximal tori of a finite loop s*
*pace
are conjugated and that the definition of the Weyl group does not really depend*
* on
the chosen maximal torus (Proposition 3.5).
Usually, when talking about finite loop spaces, the Zfiniteness condition i*
*s re
placed by the condition of being homotopy equivalent to a finite CW complex
which is a slightly stronger condition. For our purpose Zfiniteness is suffici*
*ent.
Having a maximal torus is a quite distinguished property for finite loop spa*
*ces. It
is well known that there exist finite loop spaces which don't have a maximal to*
*rus.
Examples may be found in [18] or [17]. Actually, an old conjecture, which we co*
*uld
trace back to [19], says that a connected finite loop space a with maximal toru*
*s is
isomorphic to a compact connected Lie group (as loop space). We want to give so*
*me
more evidence for this conjecture. An integral representation W! Gl(n; Z) o*
*f a
finite group W is called crystallographic, if the associated rational represent*
*ation
W! Gl(n; Q) represents W as a reflection group. We define LTL := H2(BTL ; Z)
and L*TL := H2(BTL ; Z).
1.2 Theorem. Let L be a finite loop space with maximal torus T! L. Then the
following hold:
(1) The Weyl monoid WL is a group.
(2) The induced action of WL on L*TL represents WL as a crystallographic
group.
(3) We have H*(BL; Q) ~=H*(BT ; Q)WL .
The representation WL! Gl(L*TL ) is called the associated representation.
Examples of crystallographic representations W! Gl(n; Z) are given by acti*
*ons
of Weyl groups of compact connected Lie groups on the 2dimensional homology or
cohomology of the classifying space of the maximal torus. We say that a connect*
*ed
finite loop space L with maximal torus TL! L has the same Weyl group type as
a compact connected Lie group G if rk(L) = rk(G) =: n and if the two associated
representations WL ; WG ! Gl(n; Z) are conjugate. The following question is a
3
weaker and slightly different form of the above mentioned conjecture. Let L be a
connected finite loop space with the same Weyl group type as the compact con
nected Lie group G. Does this imply that L and G are isomorphic as loop spaces?
This conjecture can't be true. The compact connected Lie groups SO(2n + 1)
and Sp(n) are not isomorphic as loop spaces, because the existence of a homotopy
equivalence BSO(2n + 1) ' BSp(n) would imply that SO(2n + 1) and Sp(n) were
isomorphic as Lie groups [15]. But on the other hand they have the same Weyl
group type.
The right invariant to distinguish the isomorphism types of connected finite*
* loop
spaces with maximal tori should be given by the normalizer of the maximal torus.
Some evidence for this conjecture comes from [15] where this is proved for the *
*loop
spaces associated with compact connected Lie groups.
1.3 Definition. Let L be a connected finite loop space with maximal torus TL!
L, The normalizer of TL is a monomorphism N(TL )! L of finite loop spaces such
that N(TL ) fits up to homotopy into a fibration
BTL! BN(TL )! BWL
and such that
BTL ________BL//::
 uuuu
 uu
fflffluu
BN(TL )
commutes up to homotopy.
1.4 Proposition. Let L be a connected finite loop space with maximal torus
TL! L. Then there exists a normalizer N(TL ) of the maximal torus, and N(TL )
is isomorphic to a uniquely determined compact Lie group as loop space.
In order to proceed with the maximal torus conjecture we first want to fix s*
*ome
notation about compact connected Lie groups. For every compact connected Lie
group G there exists a finite covering K! Gs x T! G of compact Lie groups
where Gs is simply connected, where T is a torus and where K Gs x T is a finite
central subgroup.
1.5 Definition. Let G be a compact connected Lie group.
(1) G satifies condition (Tp), if H*(G; Z) or, equivalently H*(BG; Z) is p
torsion free.
(2) G satifies condition (Ip), if H*(BG; Fp) ~=H*(BTG ; Fp)WG .
(3) G satisfies condition (Cp) if G satisfies (Tp) and Gs the condition (Ip*
*).
In [14] a compact connected Lie group G is called pconvenient if it satisfi*
*es (Cp)
. This condition was used there to prove homotopy uniqueness results for classi*
*fying
spaces of compact connected Lie groups. We will apply here similiar methods to
classify the isomorphism type of compact connected Lie groups as loop spaces us*
*ing
the normalizer. Each of the condition is a weaker than the following one. But f*
*or
an odd prime all three conditions are equivalent [14]. We also notice that U(2)
4
satisfies (I2) but not (C2) because SU(2) does not satisfy (T2). A complete lis*
*t of
simple simply connected compact Lie groups satifying (Cp) may be found in [14,
Chapter 1].
1.6 Theorem. Let G be a compact connected Lie group satisfying condition C2
and Tp for all primes. Let L be a connected finite loop space with maximal torus
TL! L. Then the following statements are equivalent:
(1) X has the same Weyl group type as G.
(2) N(TG ) and N(TL ) are isomorphic as Lie groups.
(3) BN(TG ) and BN(TL ) are homotopy equivalent.
(4) BG and BL are homotopy equivalent.
1.7 Remark. There is an interesting observation related to Proposition 1.4. Fo*
*r a
given extension N(T ) of an operation of a Weyl group W on a maximal torus T it*
* is
not known if there exist a connected finite loop space with these data. For exa*
*mple,
there are two extensions of the permutation representation of the symmetric gro*
*up
n ~=WU(n) on (S1)n = TU(n), namely the semi direct product and a nonsplitting
extension. But Theorem 1.6 says that only the semi direct product can occur as
the normalizer of a maximal torus of a connected finite loop space.
To study finite loop spaces, Dwyer and Wilkerson introduced the notion of p
compact groups in their influential paper [4]. A loop space X is a pcompact gr*
*oup
if the classifying space BX is pcomplete and if X is Fpfinite, i.e. H*(X; Fp*
*) is
finite [4]. The completion of a connected finite loop space always gives a pco*
*mpact
group. For nonconnected finite loop spaces one has to assume that the group of
the components is a finite pgroup.
The pcompact groups behave very much like compact Lie groups. For example,
a pcompact group always has a maximal torus in the sense of [4]. They also
constructed a normalizer N(TX ) of the maximal torus TX ! X which satisfies t*
*he
analogous properties as in Lemma 1.3 [4] (see also Section 2). The classifying *
*space
BN(TX ) fits into a fibration BTX ! BN(TX )! BWX . In particular, BN(TX )
is a fiberwise complete space with respect to this fibration. For details see S*
*ection
2.
For a fibration F! E! B we denote by EOpthe fiberwise padic completion
[3].
1.8 Proposition. Let L be a connected finite loop space.
(1) For any prime p the triple (L^p; BL^p; e^p) is a pcompact group.
(2) If T ! L is a maximal torus of L, then the completion gives rise to a
maximal torus Tp^! L^pand induces a natural isomorphism WL ~= WL^p
and a homotopy equivalence BN(TL )Op' BN(TL ^p).
The Weyl group WX of a pcompact group X acts on L*TX ^p:= H*(BTX ; Z^p)
and gives rise to a representation WX ! Gl(L*TX ^p). By the above lemma
the completion of a connected finite loop space with maximal torus gives a con
nected pcompact group with maximal torus. We say that a connected pcompact
group X has the padic Weyl group type of a compact connected Lie group G if
5
rk(X) = rk(G) =: n and if the associated representations WX ; WG ! Gl(n; Z^p)
are conjugate.
The next statement is the completed version of Theorem 1.6.
1.9 Theorem. Let G be a compact connected Lie group satifying condition (Tp)
if p is odd or (C2) if p = 2. Let X be a connected pcompact group. Then the
following conditions are equivalent:
(1) X has the same padic Weyl group type as G.
(2) BN(TG )Opand BN(TX ) are homotopy equivalent.
(3) BG^p and BX are homotopy equivalent, i.e. G^p and X are isomorphic
pcompact groups.
Using the padic result we are able to prove Theorem 1.6.
Proof of Theorem 1.6. Let us assume that (1) is satified, i.e. L and G have the
same Weyl group type. The Weyl group type of a connected finite loop space dete*
*r
mines the padic Weyl group type of the associated connected pcompact group. By
Theorem 1.9 this implies that BL^p' BG^pfor every prime. Moreover, we also have
a rational equivalence, because both spaces are rationally a product of Eilenbe*
*rg
MacLane spaces, and because H*(BL; Q) ~=H*(BTG ; Q)WG ~=H*(BG; Q) (Theo
rem 1.2). That is to say that BX and BG have the same adic genus. By assump
tion L has a maximal torus which implies that there exists a compact connected *
*Lie
group H such that BL ' BH [13]. Because the two associated representaions of the
Weyl groups are isomorphic, we also have K(BG) ~=K(BTG )WG ~=K(BTH )WH ~=
K(BH) as rings. The compact connected Lie group G satisfies the condition C2.
Because every qoutient of a Spingroup has 2torsion in the integral cohomology,
the finite cover Gsx T of G cannot contain a factor isomorphic to Spin(n) for s*
*ome
n. In this situation we can apply [14, 1.6] to conclude that BX ' BG. This is
condition (4).
If BL ' BG, then BN(TG )Op' BN(TL )Opfor any prime (Theorem 1.9). This
is really an equivalence of the associated fibrations, and implies therefore th*
*at
BN(TG ) ' BN(TL ) [15, proof of 3.6] which is condition (3). By [15, Theorem A],
a homoptopy equivalence BN(TG ) ' BN(TL ) establishes an isomorphism N(TG ) ~=
N(TL ) as compact Lie groups , which is condition(2).
If N(TG ) ~=N(TL ) as compact Lie groups, then it is obviuos that G and L ha*
*ve
the same Weyl group type. This completes a circle of implications.
The paper is organized as follows: Section 2 is devoted to pcompact groups.
From [4] we will recall how notions of (Lie) group theory translate to pcompact
groups. We also recall necessary results from [5] and [12]. In Section 3 we dis*
*cuss
the (integral) Weyl group of finite loop spaces and prove Theorem 1.2, Proposit*
*ion
1.4 and Proposition 1.8. Section 4 provides some more results about pcompact
groups necessary for the proof of Theorem 1.9. Inparticular, we study homotopy
classes of liftings of maps into a connected pcompact group to the maximal tor*
*us.
The last three sections are devoted to the proof of Theorem 1.9, which is split*
*ted
into several cases. First we discuss uniqueness results for products of unitary*
* groups
(Section 5), then for simply connected compact Lie groups (Section 6), and fina*
*lly
in the general case (Section 7). Section 7 also contains a proof of Theorem 1.9.
6
Completion and localisation are always meant in the sense of Bousfield and K*
*an
[3].
We also heard some rumour that Dwyer and Wilkerson have a result generalizing
Theorem 1.9.
2. pcompact groups.
In this section we recall the basic notions and the fundamental results about
pcompact groups from [4]. 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 [4], [5] and [12], we*
* omit
motivations.
2.1 Homomorphisms, monomorphisms, isomorphisms, subgroups and
exact sequences : A homomorphism f : Y ! X of pcompact groups is a
pointed map BY! BX. The map f is called an isomorphism if Bf : BY! BX
f g
is an equivalence. A sequence X! Y! Z of pcompact groups is short exact if
Bf Bg
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 a pcompact group X is a
monomorphism of pcompact groups.
2.2 pcompact toral groups : A pcompact toral group P is a pcompact
group P fitting into a short exact sequence T ! P ! ss, where T is a pad*
*ic
torus and where ss is a finite pgroup (as a pcompact group), i.e. Bss ' K(ss*
*; 1)
and ss is a honest finite group. For every pcompact toral group P , there exi*
*sts
a locally finite group P1 and a map BP1 ! BP which becomes an equivalence
after completion. P1 ! P is called the pdiscrete approximation of P [4, 6.4].
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 freely 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 : For a homorphism f : Y! X between pcompact groups,
the centralizer CX (f(Y )) is defined 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 [4, 5.1, 5.2 and 6.1].
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 ) is
7
a pcompact toral group whose component of the unit is given by TX [4]. Because
TX is self centralizing, there is a homomorphism TX ! CX (TX )
2.6 Theorem [4, 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.
(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 Proposition. Let X be a pcompact group, and let T! X be a monomor
phism of a pcompact torusinto X. Then the following conditions are equivalent:
(1) rk(T ) = rk(X).
(2) The homomorphism T! X is a maximal torus.
Proof. Let T! X be a subtorus satifying condition (1). By Theorem 2.6, there
exists a maximal torus TX ! X, and T! X loifts up to conjugation to a monom*
*or
phism T! TX . This is an isomorphism because of the rank condition, and T! X
is a maximal torus.
If TX ! X is a maximal torus, then rk(X) = rk(T ) (Theorem 2.9 (2)) and
hence, T! X satisfies condition (1).
2.8 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 [4, 9.5].
2.9 Theorem [4, 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.
8
2.10 Proposition. Let X be a connected pcompact group. Then H2(BX; Z^p)!
H2(BTX ; Z^p)WX is an isomorphism.
Proof. The statement is true for simply connected X as
H2(BTX ; Z^p)WX H2Q^p(BTX )WX ~=H2Q^p(BX) = 0 in this case.
For the general case, we recall [12] that there exists a diagram of exact se*
*quences
1______K//______TXs/x/TOOOq___TX//______1//
OOOO  
OOOOO  
fflfflq fflffl
1______K//______Xs/x/T _______X//_____/1/
of pcompact groups, where Xs is simply connected, where T is a pcompact torus
and where K! Xs x T is a central finite subgroup. The projection induces an
isomorphism WXs ~= WX =: W between the Weyl groups which we identify via this
map. Then, the upper horizontal line is W equivariant (up to homotopy). Passing
to 2dimensional cohomology and taking invariants establishes a diagram
1 oo____H2(BK;OZ^p) oo____H2(BTXs x BT ; Z^p)W oo____H2(BTX ; Z^p)W oo____1
OOOO OO OO
OOOOO  
 
1 oo____H2(BK; Z^p) oo_____H2(BXs x BT ; Z^p) oo_______H2(BX; Z^p) oo_____1
The middle terms in both rows are isomorphic to H2(BT ; Z^p). For the top row,
this follows from Theorem 2.9. Hence, the left and middle vertical arrows are
isomorphisms and so is the right one. This finishes the proof.
2.11 Proposition. Let p be an odd prime and X a pcompact group. If the
associated representation WX ! Gl(H2(BTX ; Z^p) is a pseudo reflection group
then X is connected.
Proof. Because WX is a padic pseudo reflection group, it is generated by elem*
*nets
of order coprime to p. By [12, 3.8] there exists an epimorphism WX ! ss0(X).
Because ss0(X) is a finite pgroup, this homomorphism is also the trivial map.
Thus, X is connected.
2.12 Remark. Let X be a pcompact group. The proof of Proposition 2.11 shows
that X is connected if any homorphism WX ! ss, from the Weyl group into a fin*
*ite
pgroup always has a kernel which is not a pseudo reflection group. For example,
this is true if p = 2 and if X has the 2adic Weyl group type of G, where G is a
quotient of a product of SU(n)'s, n 3 and of a torus. In particular this is tr*
*ue if
G satisfies condition (C2).
2.13 Normalizers, pnormalizers of maximal tori and ptoral Sylow
subgroups : 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
9
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
classifying spaces establishes a map BWX ! BHE(BTX ) which can be thought
of as being a classifying map of a fibration BTX ! BN(TX )! BWX . The total
space gives the the classifying space of the normalizer N(TX ) of TX . This is *
*always
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.
2.14 Proposition. Foe a pcompact group X, the map
H*(BX; Fp)! H*(BNp(TX ); Fp) is a monomorphism.
Proof. This follows from [4, proof of 2.3 and 9.13].
2.15 Centers : A subgroup Z! X of a pcompact group X is called central
[4] if the homomorphism CX (Z)! X is an isomorphism. The center Z(X) of X
is the maximal central subgroup of X [5, 1.2] [12, 4.3, 4.4]. To give an expli*
*cit
definition we use a result of Dwyer and Wilkerson [5, 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.
For connected pcompact groups, there is another desription of the center in
terms of the fixed point set of the Weyl group action on the maximal torus. Let
TX ! X be the maximal torus of a connected pcompact group X. The Weyl group
WX acts on the pdiscrete approximation T TX of TX . Here, we can consider T
as a honest locally finite group with a honest WX action. Then we define TXWX *
* to
be the pcompact group given by the equivalence BTXWX ' (BT WX )^pbetween the
classifying spaces.
2.16 Proposition. Let X be a connected pcompact group. If one of the following
three conditions is satisfied, then TXWX! X is the center of X.
(1) p is odd.
(2) TXWX is connected; i.e. TXWX is a pcompact torus.
(3) H*(BX; Fp) ~=H*(BTX ; Fp)WX .
Proof. If p is odd, this is proved in [5, 7.7]. If TXWX is a pcompact torus,*
* then
C := CX (TXWX is connected [12, 3.11] with Weyl group WC = WX . Therefore,
C! X induces an isomorphism in rational cohomology between the classifying
spaces (Theorem 2.9) and between the spaces itself, and is even an isomorphism
between connected pcompact groups 12,3.7.
10
If H*(BX; Fp) ~=H*(BTX ; Fp)WX , then0we have
H*(BCX (TXWX ); Fp) ~= H*(BTX ; Fp)W [14, 10.1]. Here, the subgroup W 0 WX
consists of all elements acting trivially on TXWX . That is to say that W 0= W*
*X .
Therefore, BCX (TXWX )! BX is a homotopty equivalence and CX (TXWX )! X an
isomorphism.
3. The Weyl group of a connected finite loop space.
In this section L always denotes a connected finite loop space with maximal
torus f : TL! L. We are going to discuss the Weyl group WL of L. In particular
we will prove Theorem 1.1.
Passing to completion at a prime p we get a pcompact group L^pwith max
imal torus TL ^p! L^p and Weyl group WL^p. These pieces satify the identity
H*Q^p(BL^p) ~=H*Q^p(BTL ^p)WL^p by Theorem 2.9. That is to say that the monomor
phism H*Q(BL^p)! H*Q^p(BTL ^p) is a Galois extension of integral domains which
are integrally closed with Galois group WL^p. An integral domain is called inte*
*grally
closed if it is integrally closed in its field of fractions.
To get global information we have to study the rational situation. First we *
*have
to clarify and to fix some notions and notations about integral ring extensions.
We denote by F F (A) the field of fractions of an integral domain A. An integr*
*al
extension A! B of integral domains is called normal, separable or Galois if t*
*he
associated extension F F (A)! F F (B) of the field of fractions is normal, se*
*parable
or Galois.
3.1 Lemma. Let A! B be a Galois extension of integral domains A and B, and
let W be the Galois group of F F (A)! F F (B). If A and B are integrally clos*
*ed,
then A = BW .
Proof. First we claim that W acts on B. Because A! B is an integral extension,
for every b 2 B there exists a monic polynomial p(t) 2 A[t], i.e. the leading
coefficient is 1, with b as a root. Hence, for every w 2 W , the element w(b) *
*is
also a root of p(t); in particular w(b) is integral over A and over B. Because *
*B is
integrally closed, we have w(b) 2 B.
Now let b 2 BW . Then b is integral over A and b 2 F F (A). This implies that
b 2 A, which proves the idenity A = BW .
3.2 Proposition. Let L be a connected finite loop space with maximal torus
f : TL ! L. The map H*(BL; Q)! H*(BTL ) is a monomorphism and an
integral Galois extension of integrally closed domains.
Proof. Let A := H*(BL; Q) and B := H*(BTL ; Q). Then, A and B are polynomial
rings on n = rk(L) = rk(TL ) generators. In particular, they are integrally cl*
*osed
domains and noetherian. Because the homogenous space L=TL is Zfinite, a Serre
spectral sequence argument shows that B is a finitely generated Amodule. Becau*
*se
A and B have the same transcendence degree over Q, the map A! B is an
inclusion. This shows that A! B is an integral ring extension [9, IX, x1].
11
The characteristic of Q is zero, and therefore all extensions are separable.*
* So it
is only left to show that A! B is normal. Tensoring with Z^pyields
A Z^p ~=H*Q^p(BL^p) and B Z^p ~=H*Q^p(BTL ^p). Therefore, by Theorem 2.9,
A Z^p~= (B Z^p)WL^p! B Z^pis an integral Galois extension of integrally
closed domains. Now we can proceed as in [17, 3.5] to show that A! B is normal
and hence a Galois extension.
Proof of Proposition 1.8. Part (1) is obvious because Zfiniteness implies Fp
finiteness and because for a simply connected space Y we have (Yp^) ' (Y )^p[3,
VI, 6.5]. For (2), the same argument shows that TL ^p! L^pis a maximal torus.
The isomorphism between the Weyl groups is shown on the way of proving Theorem
1.2 below.
Proof of Theorem 1.2. Let WQ denote the Galois group of the extension
H*(BL; Q)! H*(BTL ; Q). The Weyl group Wp^ of the pcompact group L^pis
the Galois group of H*(BL; Q) Z^p! H*(BTL ; Q) Z^p(Theorem 2.9). So, the
functor Z^pinduces a group homomorphism WQ! Wp^. Because the completion
Tp^! L^pof the maximal torus of L is a maximal torus of L^p(Proposition 1.8),*
* both
groups WQ and Wp^ are faithfully represented by the action on H*(BT ; Q) Z^p~=
H*(BTp^; Z^p) Q. Because
^
(H*(BT ; Q) Z^p)WQ ~=H*(BL; Q) Z^p~= (H*(BTp^; Z^p) Q)Wp
the homomorphism WQ! Wp^ is an isomorphism.
In the diagram
1______/W/ ________[BT;/BT/]_________[BT;/BL]/
  
  
fflffl fflffl fflffl
1______Wp^//_____[BTp^;/BTp^]/_____[BTp^;/BL^p]/
both rows are exact by the definition of the Weyl group. i.e. the third term in*
* the
rows contains the qoutient of the first two terms as a subset. Here, W denotes *
*the
Weyl monoid of L. Therefore, pcompletion induces for every prime p a monoid
homomorphism W ! Wp^ which is injective. This map is also surjective. For
let wp 2 Wp^ for some prime p. We choose elements w0 2 WQ and wq 2 Wq^
corresponding under the isomorphisms Wq^ ~=WQ ~= Wp^. We realize w0 as a self
map of BTQ over BLQ and let also wq denote the associated self map of BTq^ over
BL^q. Since H2(BTp^; Z^p) Q ~= H2(BTQ ; Q) Z^pfor all primes, an arithmetic
square argument shows that there exists a self map w : BT! BT over BL such
that w^q' wq 2 [BTq^; BTq^]. Inparticular, we have W ~= Wp^, and W is a group,
which is part (1).
Because W is a subgroup of [BT; BT ], the integral cohomology group H2(BT ; *
*Z)
gives an invariant sublattice of H2(BT ; Q). Moreover, we have H*(BT ; Q)W ~=
H*(BT ; Q)WQ ~=H*(BL; Q). Because H*(BL; Q) is a polynomial ring, the action
12
of W on H2(BT ; Q) represents W as a reflection group. This proves the last two
parts of the statement.
The proof of Theorem 1.2 shows that the integral Weyl group WL and the padic
Weyl groups WL^p of a connected finite loop space L with maximal torus TL! L
are isomorphic for every prime p. We identitify all these Weyl groups via this
isomorphism. Using this fact one also can construct an integral normalizer of t*
*he
maximal torus TL! L, which is the claim of Proposition 1.4.
Fibrations of the form BTL! Y! BWL are classified by cohomology classes
in H3(BWL ; LTL ), where WL acts on LTL = H2(BTL ; Z) via the action on the
maximal torus. Fibrations over BW with fiber BTL ^pare classified by cohomol
ogy classes in H3(BWL ; LTL ^p). In both cases this follows from obstruction t*
*he
ory.Q Because WL is a finite group we have an isomorphism H3(BWL ; LTL ) ~=
p H3(BWL ; LTL ^p). This isomorphism is given by the product over all padic
fiberwise completion. Hence there exists a fibration BTL ! BN! BW , such
that fiberwise completion induces an equivalence BNOp' BN(TL^p). Moreover, by
[15, 3.2], the space BN is the classifying space of a uniquely determined compa*
*ct
Lie group N fitting into a short exact sequence 1! TL! N! W! 1. Inpar
ticular, the group N gives a finite loop space, which we call the normalizer of*
* the
maximal torus TL! L of L.
Next we want to construct a map BN ! BL. By the construction of N,
for every prime p, there exists a map BN! BL^pwhich is an extension of the
map BTL ! BL^p and which induces an isomorphism H*Q^p(BN) ~= H*Q^p(BL).
The isomorphism between the cohomology groups follows from the Serre spectral
sequence for calculating the cohomology of BN and Theorem 2.9. The same argu
ment using Theorem 1.2 instead of Theorem 2.9 shows that there exists an isomor
phism H*(BN; Q) ~=H*(BL; Q). Because BLQ is a product of EilenbergMacLane
spaces, this last isomorphism can be realized by a map BN! BLQ . All coher
ence conditions for using an arithmetic square are controlled by cohomology and
are therefore satisfied by construction. This esatablishes a map BN! BL which,
again by construction, is an extension of BTL! BL. We collect the properties
of the group N, now called N(TL ), in the following statement, which also inclu*
*des
Proposition 1.4.
3.4 Proposition. Let L be a connected finite loop space with maximal torus
TL! L and Weyl group WL . Then the following holds:
(1) The triple (N(TL ); BN(TL ); e) is a finite loop space. In particular, *
*N(TL )
is isomorphic to a uniquely determined compact Lie group given as a fin*
*ite
extension of TL by WL .
(2) There exists a homomorphism N(TL )! L of finite loop spaces which is
an extension of TL! L.
Finally we show, as promised in the introduction, that every two maximal tori
of a finite loop space are conjugate.
g
3.5 Proposition. Let T! L be a homomorphism of a torus into a connected
f
finite loop space L with maximal torus TL ! L. Then T is subconjugate to
13
f
TL! L.
Proof. Let A := H*(BL; Q), B := H*(BTL ; Q) and C := H*(BT ; Q), and let
A^p, B^p and C^p denote the associated cohomology groups given by the theory
H*Q^p( ). After completion at a prime p, we already know that T is subconjugate*
* to
h^p
TL ^p! L^pvia a homomorphisms Tp^! TL ^p. Therefore, the diagram
*
BO__Bh__C^p//O
 OO
(*) Bf*  Bg* 
 
A ______A^p//
commutes. Let p(t) 2 A[t] be a monic polynomial which splits into linear facto*
*rs
over B. By the diagram (*), the polynomial Bg*(p(t)) 2 C[t] also splits into li*
*near
factors after completion at every prime. Now we can argue as in the proof of [1*
*7, 3.5]
to show that Bg*(p(t)) already splits into linear factors over C. Because A! *
*B is
an integral Galois extension, there exists an extension OE : B! C of Bg* : A!*
* C.
Because all involved spaces are rationally products of EilenbergMacLane spaces,
the map OE has a geometric realization Bh0 : BT0! (BTL )0 which is a lift of
Bg
BT0 ! BL0.
Over the adeles, the maps Bh0 and Bh^pdiffer only by an Weyl group element
[1]. Hence we can assume that they are equal over the adeles. In this case, t*
*he
coherence conditions for glueing all the maps together by an arithmetic square *
*are
satisfied. We get a map Bh : BT! BTL , such that the diagram
BT _______Bh_______BTL//E
EEEBgE xxx
EE xxBf
"" __xx
BL
commutes up to homotopy.
The following corollary is obvious.
3.6 Corollary. Let T1; T2! L be two maximal tori of a connected finite loop
space. Then T1 and T2 are conjugate.
In particular, this corollary says that the definition of the Weyl group doe*
*s not
depend in an essential way on the chosen maximal torus.
4. The map [ ; BTX ]! [ ; BX].
In this section, X is a pcompact group with a fixed maximal torus f : TX !*
* X.
For any pcompact toral group P , the Weyl group WX acts on the set [BP; BTX ]
of homotopy classes of maps and establishes a map
F : [BP; BTX ]=WX ! [BP; BX] :
14
4.1 Proposition. Let P be a pcompact toral group. The map
F : [BP; BTX ]=WX ! [BP; BX]
is an injection.
Remark. In [11; 2.3], a similiar slightly weaker result is proved with differen*
*t meth
ods.
For the proof of this result we will use the following construction. Let A *
*be
a pcompact abelian group and ff : A! TX a homomorphism. Both classifying
spaces BA and BTX are loop spaces and carry a multiplication . Because ff is a
homomorphism this establish a commutative diagram
BA x BA BffxidBTX_x/BA/ _idxBfBTXfx_BTX//
 (idxBff) 
fflfflBff fflffl fflffl
BA ____________BTX//_____________BTX_________ :
Taking adjoints in the vertical line yields a diagram
BA ________Bff_______BTX// ________Bf__________BX_//OO
(*)   e
fflffl fflffl 
map(BA; BA)id ______map(BA;/BTX/)Bff ______map(BA;/BX)BfOBff/ :
The right vertical arrow is given by the evaluation. Now let w 2 WX be an elem*
*ent
of the Weyl group. The diagram
BTX x BTX ______BTX//
wxw w
fflffl fflffl
BTX x BTX ______BTX//
commutes up to homotopy and, taking adjoints again and combining it with (*),
establishes another homotopy commutative diagram, namely
BAOO_Bff__BTX//_______map(BA;/BTX/)Bff ________map(BA;/BX)BfOBff/
OOO OOOO
OOOO w map(id;w) OOOO
OOO fflffl fflffl OO
BA wOBff_BTX//______map(BA;/BTX/)wOBff ______map(BA;/BX)BfOwOBff/ :
The component of map(BA; BTX ) in the bottom row is determined by the homo
topy commutative square
BTX _______map(BA;/BX)BfOBff/
w 
fflffl fflffl
BTX ______map(BA;/BX)BfOwOBff/ :
Now we are prepared for the proof of proposition 4.1
15
Proof of proposition 4.1. Let ff; fi : P! TX be two homomorphism such that
f O ff and f O fi are conjugate. First of all this implies that ff and fi have*
* the
same kernel which we can divide out. This follows from [16, x2]. There is only
discussed the case of maps with source given by the classifying space of a hone*
*st
ptoral group, but the same arguments are applicable in our situation. Hence, we
can assume that ff and fi are monomorphisms and that P is abelian as a subgroup
of TX [12, 3.1 and 3.5]. We get the following diagram
e
BTX ! map(BP; BX)Bff ! map(BP; BX)fOBff ! BX
?? fl fl
yw flfl flfl
e
BTX ! map(BP; BTX )Bfi ! map(BP; BX)fOBfi ! BX :
The top and the bottom row give two maximal tori of map(BP; BX)fOBff. Hence
the left vertical arrow exists and is given by a self equivalence. Because the *
*outer
square commutes this self equivalence is given by an element w 2 WX . The above
considerations imply that w O Bff ' Bfi which finishes the proof.
Next we will discuss under which conditions the map is also a surjection. We
have the following generalisation of well known results of Borel.
4.2 Theorem. Let X be a connected pcompact group with maximal torus f :
TX ! X. Then the following holds:
Bf*
(1) The map H*(BX; Z^p) ! H*(BTX ; Z^p)WX is an isomorphism if and
only if H*(BX; Z^p) is ptorsion free.
(2) If H*(BX; Z^p) is ptorsion free, every elementary abelian subgroup V!
X is subconjugated to TX .
(3) Let A be an abelian pcompact group. If H*(BX; Fp) ~=H*(BTX ; Fp)WX ,
then the map F : [BA; BTX ]=WX ! [BA; BX] is a bijection.
Proof. If H*(BX; Z^p) is ptorsion free, then H*(X; Z^p) is also ptorsion free,
H*(X; Fp) ~= E(__y1; :::; __yr) is an exterior algebra, H*(BX; Fp) ~= Fp[__x1; *
*:::; __xr] and
H*(BX; Z^p) ~=Z^p[x1; :::; xr] are polynomial algebras, r is equal to the rank *
*rkQ (X)
Bf*
of X and H*(BX; Z^p) ! H*(BTX ; Z^p)WX is a monomorphism. All this follows
from results of Hopf and Borel (e.g. see [8]).
Let y 2 H*(BX; Z^p). If Bf* (y) 0 mod p then y 0 mod p. Otherwise we
would get an algebraic relation among the generators of the image of H*(BX; Fp)*
*!
H*(BTX ; Fp) which contradicts the fact that H*(BTX ; Fp) is a finitely generat*
*ed
module over H*(BX; Fp). On the other hand, for every x 2 H*(BTX ; Z^p)WX ,
there exist n 2 N and z 2 H*(BX; Z^p) such that Bf* (z) = pnx. This follows from
Theorem 2.9. If n > 0, then z is divisible by p by the above considerations, a*
*nd
Bf* (p1 z) = p1 x. This proves (1).
By (1) the map H*(BX; Fp)! H*(BTX ; Fp) is an injection. For every el
ementary abelian pgroup V , the algebra H*(BV ; Fp) is an injective object in
the category of unstable algebras over the Steenrod algebra. Thus, every map
H*(BX; Fp)! H*(BV ; Fp) lifts to a map H*(BTX ; Fp)! H*(BV ; Fp). Because
16
in this situation the modp cohomology classifies the maps up to homotopy and
because every algebraic map has a realisation [10] this proves part (2).
The third part is proved in [14, 10.1].
5. pcompact groups with the Weyl group type of unitary groups.
If X is a connected pcompact group with the same padic Weyl group type
as a compact connected Lie group G, then both have the same rank and we can
identitify the two maximal tori as well as the Weyl groups. That is to say that*
* there
exists a homomorphism TG! X which is a maximal torus with Weyl group WG .
Here, we have to complete TG in order to get a pcompact torus. As already said
in Section 4, the Weyl group WG acts on the set [BY; BTG ] of homotopy classes *
*of
maps for any pcompact group Y .
5.1 Theorem. Let G be a product of unitary groups, and let X be a connected
pcompact group with the same padic Weyl group type as G. Then for any abelian
pcompact group A the following holds:
(1) [BA; BTG ^p]! [BA; BX] is surjective.
(2) [BA; BTG ^p]=WG! [BA; BX] is a bijection.
(3) For any homomorphism g : A! TG , the centralizer CX (g) is connected
and has the padic Weyl group type of CG (g) which is a product of unit*
*ary
groups.
Proof. By Proposition 4.1, the first statement implies the second one.
Two prove (1) and (3) we first assume that A is a finite abelian pcompact
group, i.e A is really a finite abelian pgroup. Hence, A is isomorphic to a pr*
*oduct
A ~=A1 x . .x.Ar such that Ai ~=Z=pki. Moreover, every map BA! BTG ^plifts
to a map BA! BTG and any map BA! BG^pto BG. By [6] in both cases the
lifts are induced by a homomorphism A! TG or A! G of groups. For the proof
of (1) and (3) we will use an induction over the number of factors of A.
Let A ~= Z=pk. Then the map g : A! X factors through the maximal torus,
because every finite cyclic group is subconjugated to the maximal torus (Theorem
2.6). This proves (1) in the case of a cyclic group.
Because A is a subgroup of TG , the centralizer CX (A) contains TG which pl*
*ays
also the role of the maximal torus of CX (A). Let WC denote the Weyl group of
CX (A) which is isomorphic to the isotropy group Iso(g) := {w 2 WG w OBg ' Bg}
By the assumptions this is also the Weyl group of CG (A) which is again a produ*
*ct
of unitary groups. Therefore, TGWC is a pcompact torus and contains A. Moreove*
*r,
TGWC ! CX (A) is the center (Proposition 2.16 (2)), CX (A) ~= CCX (A)(TGWC ) *
*~=
CX (TGWC ) is a connected pcompact group [12, 3.11] and has the padic Weyl
group type of CG (A) which is a product of unitary groups. This is condition (3*
*).
Now let A be an abelian finite pgroup. Then A ~=A0 x A1 split into a product
such that A1 ~= Z=pk and such that A0 has less factors than A. Let g : A! X
be a homomorphism. By induction hypothesis the restriction gA0 : A0! X
is subconjugate to the maximal torus, and CX (g(A0)) is connected and again of
the padic Weyl group type of a product of unitary groups. Passing to adjoints
yields a homomorphism __g: A1! CX (g(A0)). We can apply again the induction
17
hypothesis to show that A1 is subconjugate to TG in CX (g(A0)) which proves the
first statement and that CX (g(A)) = CCX (g(A0))(__g(A1)) is connected and of t*
*he
padic Weyl group type of a product of unitary groups which is part (3).
Finally let A be an abelian pcompact toral group.S Let A(k)! A be the
subgroup of elements of order pk. Then A(1) := k A(k)! A is a pdiscrete
approximation. The map map(BA; BX)! limkmap(BA(k); BX) is an equiv
alence. Moreover, for every fixed homomorphism f : A ! X, the sequence
map(BA(k); BX)BfBA(k) stabilizes [4, 6.1, 6.7]. This proves the statement in t*
*he
general case.
For odd primes there is a more general result of this form.
5.2 Theorem. Let p be an odd prime, G a compact connected Lie group satisfying
condition (Tp), and X a connected pcompact group with the same padic Weyl
group type as G. Then the following holds:
(1) [BA; BTG ^p]! [BA; BX] is an epimorphism.
(2) [BA; BTG ^p]=WG! [BA; BX] is an isomorphism.
(3) For any homomorphism g : A! TG , the centralizer CX (g) is connected
and has the padic Weyl group type of CG (g). Moreover CG (g) satisfies
condition (Tp).
Proof. We argue as in the above proof. Only one argument has to be replaced. We
use the same notation as above. In this case the TGWC may not be a torus. But by
[14, 10.1] we have
H*(BCG (g(A)); Fp) ~=H*(BTG ; Fp)Iso(g)
where Iso(g) := {w 2 WG w O Bg ' Bg} is the isotropy group of g. In particular,
CG (g(A)) is connected with Weyl group WC ~=Iso(g), the modp cohomology is
concentrated in even degrees, and CX (g(A)) is a pcompact group with the same
Weyl group type as CG (g(A)). Because WC is a reflection group and because p is
odd, CX (g(A)) is connected (Lemma 2.11). Using this argument the induction of
the proof of Theorem 5.1 works and gives a proof of the statement.
One part of Theorem 1.9 for products of unitary groups is contained in the
following statement:
Theorem 5.3. Let X be a connected pcompact group with maximal torus, and
let G be a product of unitary groups. If BN(TX ) and BN(TG )Opare homotopy
equivalent then the following holds:
(1) BX is ptorsion free, i.e. the padic cohomology is ptorsion free.
(2) H*(BX; Z^p) ~=H*(BTX ; Z^p)WX .
(3) BX and BG^pare homotopy equivalent.
Proof. Because the two normalizers BN(TX )Opand BN(TG )Opare equivalent the
space X has the same padic Weyl group type as G. The modp cohomology of
BN(TG ) is detected by elementary abelian psubgroups [7]. The map
H*(BX; Fp)! H*(BN(TX ); Fp) is a monomorphism (Proposition 2.14). Thus,
18
the cohomology H*(BX; Fp) is also detected by elementary abelian psubgroups.
By Theorem 5.1, we know that every elementary abelian psubgroup is subcon
jugate to the maximal torus. This implies that H*(BX; Fp) is concentrated in
even degrees and that H*(BX; Z^p) is ptorsion free. By Theorem 4.2, it fol
lows that H*(BX; Z^p) ~= H*(BTG ; Z^p)WG . Because H*(BTG ; Z^p)WG Fp ~=
H*(BTG ; Fp)WG we get isomorphisms
H*(BX; Fp) ~= H*(BTG ; Fp)WG ~= H*(BG; Fp). Because G is a product of uni
tary groups we can apply [14, 1.3] to conclude that BX and BG^pare homotopy
equivalent. That is that we proved all three statements.
For later purpose we need for odd primes a slightly more general result.
5.4 Theorem. Let p be an odd prime and G a product of unitary groups and
SU(p)'s. Let X be a connected pcompact group with the same padic Weyl group
type as G. If BN(TX ) and BN(TG )Opare homotopy equivalent then the following
holds:
(1) BX is ptorsion free, i.e. the padic cohomology is ptorsion free.
(2) H*(BX; Z^p) ~=H*(BTX ; Z^p)WX .
(3) BX and BG^pare homotopy equivalent.
Proof. The proof is the same as above. We only have to ensure that the modp
cohomology of BN(TG ) is detected by elementary abelian subgroups. But this is
also true for SU(p) [14, 12.6].
5.5 Proposition. Let G be a product of unitary groups, or, if p is odd, let G
be compact connected Lie group satifying condition (Tp). If X is a connected p
compact group having the same padic Weyl group type as G, then BN(TX ) and
B(NTG )Opare homotpy equivalent.
Proof. For the proof we have to discuss extensions, i.e. fibrations, of the form
(*) BTG ^p! BNOp! BWG :
Fibrations of this form are classified by cohomology classes in H3(BWG ; ss2(BT*
*G ^p))
[15, 3.1]. If p is odd and if G satifies the condition (Tp), the obstruction g*
*roup
vanishes [14, 5.3] and hence, there is only one fibration up to fiber homotopy *
*equiv
alence. Hence, we have BN(TX ) ' B(NTG )Op.
Now let G be a product of unitary groups. Difficulties only arrise for p = 2*
* be
cause in this case there is more than one extension in general. By Proposition *
*2.10,
the map H2(BN(TX ); Fp)! H2(BTX ; Fp)WG is an epimorphism. We can apply
[14, 11.5], which implies that BN(TX ) is equivalent to the fiberwise completio*
*n of
the classifying space of the semi direct product TG o WG ~=N(TG ). This proves
the statement for products of unitary groups.
Finally we are prepared to prove Theorem 1.9 for products of unitary groups:
Proof of Theorem 1.9 for products of unitary groups. Condition (1) implies
(2) by Proposition 5.5, (2) implies (3) by Theorem 5.2, and (3) implies (1) bec*
*ause
X and G are isomorphic pcompact groups.
19
6. Connected pcompact group with the Weyl group type of pseudo
simply connected or simply connected compact Lie groups..
A pseudo simply connected compact Lie group is a product of simple simply
connected compact Lie groups not isomorphic to a SU(n) and unitary groups.
That is that we replace the special unitary factors in a simply connected compa*
*ct
Lie group by the associated unitary groups.
6.1 Proposition. Let G be a pseudo simply connected compact Lie group satifying
the condition (Cp). Then there exists an elementary abelian subgroup V! TG ,*
* a
compact connected Lie group H which is a product of unitary groups and SU(p)'s
and an exact sequence
1! K! H! CG (V )! 1 ;
where K is a finite central subgroup of H of order coprime to p. The centraliz*
*er
CG (V ) is of maximal rank and the index of WG : WCG (V ) is also coprime to
p. The two groups H and CG (V ) have the same Weyl group type, BN(TH )Op'
BN(TCG (V ))Op, and the modp cohomology of BN(TG ) is detected by elementary
abelian pgroups.
Proof. This statement is Lemma 5.2 of [14]. Only the last three assertions need*
* a
remark. They follow from what is already said and from the fact that the modp *
*co
homolgy of BN(TH ) is detected by elementary abelian subgroups [7] [14, 12.6].
6.2 Theorem. Let G be a pseudo simply connected compact Lie group satisfying
condition (Cp). Let X be a connected pcompact group with the same Weyl group
type as G. Then the following holds:
(1) H*(BX; Z^p) ~=H*(BTG ; Z^p)WG .
(2) BN(TG )Opand BN(TX ) are homotopy equivalent.
(3) BX and BG^pare homotopy equivalent.
Proof. For p = 2, a pseudo simply connected compact Lie group is a product
of unitary groups. This case we already discussed in the last section. Theref*
*ore
we assume that p is odd. Part (2) follows from Proposition 5.5. The modp
cohomology of BN(TX ) ' BN(TG )Opis detected by elementary abelian subgroups.
This follows from Proposition 6.1. Now, using Theorem 5.2 instead of Theorem
5.1, we can argue analogously as in Theorem 5.3 or Theorem 5.4 to show that
H*(BX; Z^p) ~= H*(BTG ; Z^p)WG , and, using again [14, 1.3], that BX ' BG^p.
This is Part (3) and also implies Part (1).
Finally we discuss in this section the case of simply connected compact Lie
groups.
6.3 Theorem. Let G be a simply connected compact Lie group satisfying condi
tion (Cp). Let X be a connected pcompact group with the same Weyl group type
as G. Then the following holds:
(1) H*(BX; Z^p) ~=H*(BTG ; Z^p)WG .
(2) BN(TG )Opand BN(TX ) are homotopy equivalent.
(3) BX and BG^pare homotopy equivalent.
20
Proof. We split G = G1 x G2 into a product where G1 is the product of all facto*
*rs
isomorphic to a SU(n) and where G2 collects the other factors. By the assumptio*
*ns,
SU(2) does not occur for p = 2. By Proposition 2.16, we have Z(X) ~=(TGWG )^p=
Z(G1)^px Z(G2)^p.
For each n there exists an exact sequence 1! Z(SU(n))! SU(n) x S1!
U(n)! 1. Let T be a torus with one S1 for each factor in G1, and let Z(G1)!
T be the inclusion established by the above exact sequence. This gives rise to
commutative diagrams of exact sequences
1 _____Z(G1)_//_____TG1/x/TOxOTG2 ______T__G1x/TG2/:= T__G_____1//
OOO 
OOOOO  
fflffl __ ffl__ffl
1 _____Z(G1)_//______G1/x/T x G2 ________G/1x/G2 := G _______/1/
and __ __
1 ______G//_____G//_____T//_____1/;/
__
where Z(G1)!_ G2 is the trivial homomorphism, and where T := T=Z(G1). The
quotient G is a pseudo simply connected compact Lie group_which also satifies
condition (Cp). Moreover, the epimorphism G x T! G as well as the inclusion
induces an isomorphism WG ~= W__Gand TG xT! T__Gand TG! T__Gare equivariant.
Working with X instead of G one gets similiar exact sequences, namely
1 ______Z(G1)^p//_____TG1/x/TOx TG2 ______T__G1x/TG2/:= T__G_____1//
OOOO 
OOOOO  
fflffl fflffl
1 ______Z(G1)^p//________X/x/Tp^ ________________/___X/__________1//
and ___ __
1______X//______X//_____T//_____1/:/
___ __ ___ __^
This shows_that_X_has_the_same_Weyl_group type as G and hence that BX ' BG p.
The maps BX ! BT and BG ! BT are totally determined by H2( ; Z^p). And
by construction, this establishes an commutative diagram
BG^p ______B__G^p//___B__T^p//

  
  fflffl
fflffl ___fflffl __^
BX ______BX_//______BT/p/
where the right and left vertical arrows are homotopy equivalences. And so is t*
*he
left one, which is part (3). The first and the second statement obviously foll*
*ow
from this fact.
21
7. Connected pcompact group with the Weyl group type of a general
compact connected Lie group.
As mentioned in the introduction, for every compact connected Lie group G,
q
there exists a finite covering 1! K! eG! G! 1, where Ge ~= Gs x T i*
*s a
product of a simply connected Lie group Gs and a torus T . The group K eG is
central subgroup. This covering we call universal finite. The group Ge is uniqu*
*ely
determined but not the map q. One can compose q with a finite self covering of *
*T .
Every universal finite covering establishes two associated commutative diagr*
*ams
of exact sequences of compact Lie groups:
__
Ks ! Gs ! Gs=Ks := G s
?? ? ?
y ?y ?y
K ! Gs x T ! G
?? ? ?
y ?y ?y
___ __
K = K=Ks ! T ! T=Ks := T
and
Ks ________ Ks ! *
?? ? ?
y ?y ?y
K ! Gs x T ! G
?? ? fl
y ?y flfl
___ __
K ! Gs x T ! G ;
__ __ __
where Ks := K \ (Gs x {0})._Because H1(G s; Z) = 0,_the sequence G s! G ! T
induces isomorphisms H1(T ; Z) ~=H1(G; Z) and H2(BT ; Z)_~=H2(BG; Z)._On the
other hand these isomorphisms_determine the_maps G ! T and BG ! BT , and
therefore, the fibration BG s! BG ! BT . __
If G satisfies_condition (Tp), then, by [14, 4.1], BGs x BT and BG s x T as *
*well
as BGs_and BG s are_ptorsionfree. The order Ks of Ks is coprime to p, because
ss2(BG s) ~=H2(BG ; Z) ~=Ks.
7.1 Theorem. Let G be a compact connected Lie group satisfying condition (Cp).
Let X be a connected pcompact group with the same Weyl group type as G. Then
the following holds:
(1) H*(BX; Z^p) ~=H*(BTG ; Z^p)WG .
(2) BN(TG )Opand BN(TX ) are homotopy equivalent.
(3) BX and BG^pare homotopy equivalent.
Proof. The finite covering
K  ! Gs x T  ! G ;
22
Gs simply connected and T a torus, establishes a fibration
BKp ! BGs^px BTp^ ! BG^p;
where Kp is the pSylow subgroup of K. This fibration is classified by a map
BG^p! BBKp. Since H2(BG; Z^p) ~= H2(BTG ; Z^p)WG ~= H2(BX; Z^p) (Proposi
tion 2.10) and since BG and BX are simply connected, we also get an isomorphism
in 2dimensional cohomology with coefficients Z=pr for all r. Therefore, there *
*is a
corresponding map BX ! BBKp classifying the fibration
BKp! BY! BX :
The total space BY is simply connected and pcomplete. Hence, Y := BY is a
connected pcompact group [12, 3.3], and Y! X is a finite covering of connect*
*ed
pcompact groups. The fibration fits into
BKpO ______BGs^px/BTp^/ _______BG^p//
OOOO OO OO
OOOO  
OOO  
BKpO ______BTGs/^px/BTp^ ______BTG/^p/
OOOO
OOOOO  
fflffl fflffl
BKp ___________BY//____________BX// :
The Weyl group WG = WGs acts as Weyl group on BT Gs^px BTp^! BY . Thus,
Y has the same padic Weyl group type as Gs^px Tp^. By Theorem 6.3 , the spaces
BGs x BT and BY are homotopy equivalent. Moreover, this equivalence fits into
a diagram
BKpO ______BGs^px/BTp^/ ______BG^p//
OOO
(*) OOOOOO '
fflffl
BKp ___________BY//___________BX// :
We have to fill in the arrow in the right collumn. This is done by the followi*
*ng
trick. The top row is a principal fibration [2, 7.2]. the composition BKp! BY*
*!
BX is null homotopic and, because X has finite modp cohomology, the map
BKp! map(BKp; BX)constis an equivalence [16]. By a lemma of Zabrodsky [20]
(for details see also [2, 9.8 and the proof of 9.7]), we can fill in the arrow *
*in the
right column making the diagram commutative up to homotopy. This arrow is an
equivalence, which is the third statement. The first two statements follow obvi*
*uosly
from the third.
Proof of Theorem 1.9. Part (1) implies (3) by Theorem 7.1. The implications
from (3) to (2) and from (2) to (1) are obvious.
23
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Matematisk Institut, Universitetsparken 5, DK2100 Kobenhavn O, Denmark,
email: moller@math.ku.dk.
24
Mathematisches Institut, Bunsenstr. 35, 37073 G"ottingen, Germany,
email: notbohm@cfgauss.unimath.gwdg.de.