CENTERS AND FINITE COVERINGS
OF FINITE LOOP SPACES
by
J. M. Moller and D. Notbohm
Abstract. The homotopy theoretic analogue of a compact Lie group is a pc*
*ompact
group, i.e a space X with finite modp cohomology and an loop structure g*
*iven by
an equivalence of the form X ' BX. The `classifying space' BX has to be a*
* p
complete space. We are concerned with the notions of centers and finite c*
*overings
of connected pcompact groups. In particular , we prove in this category *
*two well
known results for compact Lie groups; namely that the center of a connect*
*ed p
compact group is finite iff the fundamental group is finite and that ever*
*y connected
pcompact group has a finite covering which is a product of a simply conn*
*ected p
compact group and a torus. The latter statement also translates to connec*
*ted finite
loop spaces.
1. Introduction.
A finite loop space X is a triple (X; BX; e), in which e : X ! BX is an
equivalence from the space X into the loop space BX of the pointed space BX.
The loop space X is called finite if X is homotopy equivalent to a finite CW 
complex or if the integral homology H*(X; Z) is finitely generated as a graded
abelian group. The latter condition is a little weaker, but sufficient for pro*
*ving
most of the nice results about finite loop spaces.
Finite loop spaces are considered to be the homotopy theoretic generalisatio*
*n of
compact Lie groups. For a compact Lie group G the tripel (G; BG; e), consisting
of the compact Lie group, the classifying space BG and the natural equivalence
e : G! BG, is a finite loop space. Following an old idea of Rector [R1], name*
*ly
passing from a group to the associated classifying space, one would like to dev*
*elop
Lie group theory in terms of classifying spaces. This would give the chance to
extend all the beautiful results about Lie groups to the bigger class of finite*
* loop
spaces.
The maximal torus is one of the fundamental notions one has to define for fi*
*nite
loop spaces. A maximal torus of a finite loop space X is a map f : BTX ! BX
from the classifying space BTX of a torus TX into BX such that the homotopy
fiber X=TX of f is equivalent to a finite CW complex (or H*(X=T ; Z) is a fin*
*itely
generated graded abelian group) and such that TX and X have the same rank.
______________
1980 Mathematics Subject Classification (1985 Revision). 55P35, 55R35.
Key words and phrases. Finite loop space, pcompact group, classifying space*
*, maximal torus,
monomorphism, normalizer, centralizer, center, finite covering.
JMM thanks SFB 170 in G"ottingen for support and hospitality while this work*
* was done.
Typeset by AM ST*
*EX
1
2
The rank of X is defined to be the number of generators of the exterior algebra
H*(X; Q).
Let TG! G be a maximal torus of a compact Lie group G. Then, this definiti*
*on
is made up by extracting the basic properties of the associated fibration G=TG!
BTG! BG. Later we will change our point of view and reformulate the definition
of a maximal torus (see Section 2).
By work of Rector [R2], with help from McGibbon [McG] at the prime 2, it
turned out that there exist finite loop spaces which do not have a maximal toru*
*s.
There even exists a conjecture that every finite loop space with maximal torus
comes from a compact Lie group [W].
As usual completion makes life a lot easier. This also turns out to be true *
*in the
study of finite loop spaces. In a recent paper, Dwyer and Wilkerson [DW] defin*
*ed
a pcompact group to be a loop space (X; BX; e) such that BX is pcomplete
and such that X is Fpfinite, i.e. that H*(X; Fp) is a finite dimensional grad*
*ed
Fpvector space. They studied pcompact groups in great detail.
Here is a warning: In general you don't get a pcompact group just by the
completion of the classifying space of a finite loop space. For a pcompact gro*
*up X,
the fundamental group ss1(BX) is a finite pgroup, i.e. the group of the compon*
*ents
of X is also a finite pgroup. This is a restriction which comes into play. On *
*the
other hand the completion of the classifying space Bn of the symmetric group n
at an odd prime gives a highly connected space whose loop space is not Fpfinit*
*e.
Nevertheless pcompact groups are the right object to study finite loop spaces.
Let L be a finite loop space. The completion of the classifying space BLp of t*
*he
componets Lp lying over the pSylow subgroup of ss0(L) gives a pcompact group.
Moreover BLp! BL is a finite covering. The existence of a transfer allows to *
*carry
over a lot of the cohomological properties of BLp to BL (e.g. see [DW, Section*
* 2].
The results of Dwyer and Wilkerson [DW] as well as our experience show that
pcompact groups enjoy much of the rich internal structure of compact Lie group*
*s.
In particular , they showed that every pcompact group has a maximal torus and
a Weyl group and that, for a connected pcompact group, the rational cohomolgy
of the classifying space is given by the invariants of the Weyl group acting on*
* the
cohomology of the classifying space of the maximal torus. We will give an expli*
*cit
formulation of their result in Section 2.
Following the spirit of that influential paper we are here concerned with the
center of pcompact group and finite coverings of pcompact groups. To formulate
our results we first have to translate some of the basic notions of group theor*
*y in
terms of pcompact groups. In Section 2 we will recall the dictionary of [DW] *
*and
add some more translations.
1.1 Definition. In this definition X denotes a pcompact group.
(1) A pcompact torus of rank n is a pcompact group (T; BT; e) such that BT
is homotopy equivalent to an EilenbergMacLane space K(Z^pn; 2).
(2) A pcompact group X is called finite if BX is equivalent to an Eilenber*
*g
MacLane space K(ss; 1) of a finite pgroup of degree 1.
(3) A homomorphism g : Y ! X of pcompact groups is a pointed map
Bg : BY! BX. Two homorphisms g1; g2 : Y! X are conjugated if the
associated maps Bg1; Bg2 : BY! BX are freely homotopic.
3
(4) A homomorphism g : Y ! X is a monomorphism or equivalently Y is a
subgroup of X if the homotopy fiber X=Y of Bg is Fpfinite.
(5) For i = 1; 2, let gi : Yi! X be subgroups of the pcompact group X. T*
*hen,
Y1 is subconjugated to Y2 if there exists a homomorphism h : Y1! Y2 s*
*uch
that g2h and g1 are conjugated.
(6) A subgroup g : Z! X is central if the evaluation ev : map(BZ; BX)Bg!
BX is an equivalence.
(7) A central subgroup Z(X)! X is called the center of X if every central
subgroup Z! X is subconjugated to Z(X).
Before we state our first result we explain the motivation of some of these *
*def
initions. The third says that every homomorphism of groups is a loop map and
that two conjugated homomorphism induce homotopic maps between the associ
ated classifying spaces. Every inclusion H! G of compact Lie groups establish*
*es
a fibration G=H! BH! BG.
Analogously to the above definition of a maximal torus, the fourth part refl*
*ects
the fundamental properties of this fibration.
The sixth definition goes back to a theorem of Dwyer and Zabrodsky [DZ] on *
*the
one hand and the second author [N1] on the other hand. For any homomorphism
ae : P! G of a ptoral group P , i.e. a finite extension of a torus by a fini*
*te pgroup,
into a compact Lie group G , there exists a map BCG (ae)! map(BP; BG)Baewhich
becomes an equivalence after completion. Here, CG (ae) denotes the centralizer *
*of ae
in G.
The definition of a center might not be something the reader expects. In the
classical case conjugation acts trivially on the center. This also turns out to*
* be true
for pcompact groups (see Proposition 4.5 and 4.7).
1.2 Proposition. Every pcompact group has a center.
1.3 Theorem. Let X be connected pcompact group. Then the center Z(X)! X
is a finite subgroup of X if and only if the fundamental group ss1(X) is finite.
This theorem is the generalization of the analogous well known result about *
*semi
simple Lie groups. We can use this statement for the following definition. A co*
*n
nected pcompact group X is called semi simple if ss1(X) is finite or, equivale*
*ntly,
if the center Z(X) is a finite pcompact group.
In the classification of compact connected Lie groups, one first passes to a*
* finite
covering eG! G of a compact connected Lie group G, such that eGis a product of
a simply connecte Lie group and a torus. Then one splits the simply connected L*
*ie
group into a product of simple simply connected Lie groups. Our next statement
says that for pcompact groups at least the first step can always be carried ou*
*t.
1.4 Theorem. Let X be a connected pcompact group. Then there exist a simply
connected pcompact group Xs, a pcompact torus T and a homomorphism Xs x
T! X which establishes a fibration
BK! BXs x BT! BX :
4
Moreover, K is a finite pcompact group and K! Xs x T ! Xs is a central
monomorphism.
As mentioned earlier pcompact groups together with arithmetic square argu
ments will give you integral or global information. Our last statement is Theor*
*em
1.4 in the global case.
1.5 Theorem. Let L be a finite loop space. Then there exists a simply connected
finite loop space Ls, an integral torus T and a finite covering Ls x T! L wh*
*ich
establishes a fibration
BK! BLs x BT! BL :
Moreover, K is a finite abelian group.
These are the main results we can offer. On the way of proving these stateme*
*nts
we have to formulate and to prove several well known results about compact Lie
groups in the category of pcompact groups. Some of these are a triviality for *
*Lie
groups, but definitely not for pcompact groups (e.g see Section 2,3 and 4).
The paper is organized as follows: As already mentioned we recall the necess*
*ary
basic notions and the dictionary of [DW] in Section 2. A collection of well kn*
*own
results about compact Lie groups translated in terms of pcompact groups is the
content of Section 3. Section 4 is devoted to the notion of the center and the *
*proof
of Proposition 1.2. In Section 5 we prove Theorems 1.3 and 1.4, and the last se*
*ction
contains a proof of Theorem 1.5.
Completion is always meant in the sense of Bousfield and Kan [BK] and denot*
*ed
by U^p for a space U.
We denote by H*Q^p( ) := H*( ; Z^p) Q the cohomology with padic coefficien*
*ts
tensored over the integers with the rationals.
2. The dictionary.
In this section we recall the dictionary and some of the basic notion of [D*
*W].
The dictionary tells us how we have to translate notions of group theory and Lie
group theory in terms of finite loop spaces or pcompact groups. This provides *
*us
also with an appropriate language to formulate our results and proofs. Most of *
*the
notions are motivated by passing from groups to classifying spaces and extracti*
*ng
the basic properties in similiar way as in the definitions of Section 1.
2.1 Homotopy fixedpoints and proxy actions: Let G be a group acting
on a space X. The homotopy fixedpoint set XhG := mapG (EG; X) is defined to
be the mapping space of Gequivariant maps from a contractible CW complex EG
with a free Gaction into X. The homotopy fixed point set can also be interpret*
*ed
as the space of sections in the fiber bundle X! XhG ! BG, where XhG :=
EGxG X is the homotopy orbit given by the Borel construction. The Gequivariant
projection EG! * induces a map XG ~=mapG (*; X)! XhG .
A homotopy equivalence f : Y! X of Gspaces, which is also Gequivariant,
induces an homotopy equivalence Y hG! XhG between the homotopy fixedpoint
sets. This follows from the description as section spaces. This motivates the d*
*efini
tion of proxy actions. A proxy action of G on a space X is a Gspace Y such that
5
Y and X are homotopy equivalent. By XhG we denote the homotopy fixed point
set Y hG.
Proxy actions very often come up in homotopy theory. For example, let G be
a finite group and F! E! BG a fibration. Then G acts up to homotopy on
F and there exists a homotopy equivalent space F 0which realizes this "homotopy
action".
For a fibration F! E! X and every map BG! X the pull back diagram
FO______FhG//O_____BG//
OOO  
OOOOO  
fflffl fflffl
F _______E//_______X_//
establishes a proxy action on F . We think of FhG as the "Borel construction"*
* of
this proxy action. The homotopy fixedpoint set F hG is then given by the secti*
*on
space of the fibration F! FhG! BG.
2.2 pcompact groups : There is an equivalent definition of a pcompact
group [DW , Lemma 2.1, Remark]: A finite loop space (X; BX; e) is a pcompact
group, if X is Fpfinite and pcomplete and if ss0(X) is a finite pgroup. The *
*ratio
nal rank of a pcompact group X is defined to be the number of exterior generat*
*ors
of H*Q^p(X).
2.3 Isomorphisms and exact sequences: A homomorphism Y ! X of p
compact groups is an isomorphism if Bf : BY! BX is an equivalence. A sequence
f g
X! Y ! Z of finite loop spaces or pcompact groups is exact if the associa*
*ted
Bf Bg
sequence BX ! BY ! BZ is a fibration up to homotopy. In this case g is
f
called an epimorphism and Y! X a normal subgroup.
2.4 pcompact toral groups: We already defined what we understand by a
pcompact torus and by a finite pcompact group. A pcompact toral group P is
a pcompact group which fits into an exact sequence T! P! ss of pcompact
groups, where T is a pcompact torus and ss a finite pcompact group.
2.5 Elements of pcompact groups: An element of order pn of a pcompact
group X is a monomorphism Z=pn! X. A pth root of an element f : Z=pn! X
is an element f0 : Z=pn+1! X such that for the canonical homorphism j : Z=pn!
Z=pn+1 the composition f0j is conjugate to f. By [DW] any nontrivial pcompact
group contains an element of order p.
A non torsion element is a monomorphism S1^p! X.
2.6 Conjugation and subconjugation : Let f : Y! X be a monomorphism
of pcompact groups and i : P! X a pcompact toral subgroup. In Section 1 we
said that P is subconjugated to Y if there exists a homomorphism j : P! Y such
that fj and i are conjugated. If we think of BY! BX as being a fibration, the
induced map Bi : BP! BX establishes a proxy action on X=Y . The homotopy
6
fixedpoint set X=Y hP describes the lifts BP! BY over Bi : BP! BX. Let L
denote the set of homotopy classes of subconjugation of P into Y ; i.e. we ask *
*for
homotopy clases of lifts BP! BY . There exists a fibration
X=Y hP! map(BP; BY )L! map(BP; BX)Bi ' BCX (P )
which establishes an exact sequences of sets
ss1(BCX (P ))! ss0(X=Y hP)! L :
The last map is onto, the first set is a group which acts on the middle set. Th*
*at is
to say that L is given by the orbit of the action of ss1(BCX (P )) on ss0(X=Y h*
*P).
2.7 Discrete approximations and closures: Let T ~= (S1)n be a classical
torus and let T := {t 2 T : tpk = 1 for some k}. Then T is a discrete group,
isomorphic to (Z=p1 )n, and the natural inclusion T! T induces an Fpequivale*
*nce
BT ! BT . This is the generic example of a discrete approximation we have in *
*our
mind. Therefore in [DW] a pdiscrete torus of rank n is defined to be a discr*
*ete
group isomorphic to (Z=p1 )n and a pdiscrete toral group to be an extension of*
* a
pdiscrete torus by a finite pgroup.
A homomorphism f : P ! P from a pdiscrete toral group into a pcompact
toral group is a discrete approximation if Bf : BP ! BP is an Fpequivalence.
The pcompact toral group P is called the closure of P . Every pcompact toral
group has a pdiscrete approximation [DW, 6.8], and every pdiscrete toral gro*
*up
has a functorial closure [DW, 6.9 and 6.10].
Suppose that P and Q are pcompact toral groups with pdiscrete approxima
tions P! P and Q! Q. For any homomorphism f : P! Q there exists [DW,
Remark 6.11] a group homomorphism f : P! Q such that the diagram
f
P _____Q_//
x g
fflffl fflffl
P __f__Q_//
commutes up to conjugacy; in this situation we call f a discrete approximation *
*to
f. Note that the free homotopy set [BP; BQ] of conjugacy classes of homomor
phisms from P to Q is in a natural bijective correspondence with the set Rep (P*
* ; Q)
of conjugacy classes of homomorphisms from P to Q .
2.8 Centralizers: Let ae : P ! G be a homomorphism from a classical p
toral group P into a compact Lie group G. By results of [DZ] and [N1] there
exists a an Fpequivalence BCG (ae)! map(BP; BG)Bae. If ss0(G) is a finite *
*p
group, using a result of [BN], this can be translated to an equivalence BCG (a*
*e)^p!
map(BP; BG^p)Bae^p(see also [JMO]). Therefore, for a homorphism f : Y ! X
between pcompact groups, we define the centralizer CX (f(Y )) to be the loop s*
*pace
given by the triple
CX (f(Y )) := (map(BY; BX)Bf ; map(BY; BX)Bf ; id) :
7
The evaluation ev : BY x map(BY; BX)Bf ! BX establishes a homomorphism
Y x CX (f(Y ))! X of loop spaces. If Y is a pcompact toral group the central*
*izer
CX (f(Y )) is again a pcompact group and the evaluation CX (f(Y ))! X is a
monomorphism [DW, 5.1, 5.2 and 6.1].
2.9 Abelian pcompact group: A pcompact group A is called abelian if the
evaluation induces an isomorphism CA (id)! A. In particular , the adjoint of *
*the
evaluation gives a multiplication : AxA! A which also is a homomorphism. Let
A! X be a homomorphism from an abelian pcompact group into a pcompact
group. Taking adjoints this multiplication establishes a natural homomorphism
A! CX (A) which shows that A! X is subconjugated to CX (A). An easy ar
gument shows that every abelian ptoral group gives rise to an abelian pcompact
toral group.
2.10 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. The motivation of this definition comes from the fact th*
*at,
for a compact connected Lie group G the maximal torus is self centralizing, and
that therefore the centralizer of the maximal torus of a nonconnected compact L*
*ie
group is always a ptoral group.
2.11 Theorem [DW, 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.
2.12 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].
The fibration X=TX ! BTX ! BX establishes a proxy action of TX on the
homogenous space X=TX via BTX ! BX. Every element of the Weyl space can
be interpreted as a homotopy fixedpoint of this proxy action. That is to say t*
*hat
WT (X) = (X=TX )hTX .
Because all maximal tori of X are conjugated, the Weyl space as well as the
Weyl group does not depend essentially on the chosen maximal torus. If TX is
understood we denote the Weyl space by WX and the Weyl group by WX .
2.13 Theorem [DW, 9.5 and 9.7]. Let TX ! X be the maximal torus of a
connected pcompact group X.
(1) The rank n of TX is equal to the rank of X.
(2) The order of the Weyl group WX is equal to the Euler characteristic
O(X=TX ) of the homotopy fiber of BTX ! BX.
8
(3) The action of WX on BTX induces a representation
WX ! Aut(H*Q^p(BTX )) ~=Gl(n; Q^p)
which is a monomorphism whose image is generated by pseudoreflections.
(4) The map H*Q^p(BX)! H*Q^p(BTX )WX is an isomorphism.
This is the natural generalization of the well known results about compact c*
*on
nected Lie groups. By [CE] one cannot expect that the Weyl group is generated
by honest reflections.
2.14 Normalizers and pnormalizers of maximal tori: 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 establis*
*hes
a monoid homomorphism WX ! aut(BTX ) where aut(BTX ) denotes the monoid
of all self equivalences of BTX . Passing to classifying spaces establishes a *
*map
BWX ! Baut(BTX ) which we can be thought of as being a classifying map of
fibration BTX ! BN(TX )! BWX . The total space gives the the claasifying
space of the normalizer N(TX ) of TX . This construction is nothing but the Bor*
*el
construction.
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 restriction Np(TX )! X is a
monomorphism [DW 9.9].
There is a slightly different way to construct the normalizer for a connected
pcompact group X. The Weyl group WX acts only up to homotopy on BTX .
But because BTX is an EilenbergMacLane space we can replace this "homo
topy action" by a "real" action of WX on BTX . Moreover, we can assume that
BTX has a fixedpoint which we choose as basepoint. Then the evaluation ev :
map(BTX ; BX)Bi! BX is a fibration and an WX equivariant map where WX
acts on the mapping space via the action on the source and on BX trivially.
The equivalence BTX ' map(BTX ; BX)Bi is another realisation of the homo
topy action of WX as a real action. Obviously the evaluation extends to a map
BN(TX ) := EWX xWX map(BTX ; BX)Bi! BX. Analogously, we can define the
pnormalizer using the action of Wp on BTX . For a nonconnected pcompact group
one has to consider The action of WX on the component of the unit of CX (TX ) *
*or
on the universal cover of BCX (TX ) and then to carry out this construction.
Warning: The Borel construction EWX xWX BTX does not give the normalizer.
This always establishes a splitting fibration BTX ! EWX xWX BTX ! BWX
which is not true for the normalizer in general. The point is that one first ha*
*s to
turn the map BTX ! BX into a fibration.
The pnormalizer fits into an exact sequence TX ! Np(TX )! Wp and is the*
*re
fore a pcompact toral group.
2.15 Kernels and monomorphisms: Let f : Y ! X be a homomorphism
of pcompact groups, let P := Np(TY ) be the pnormalizer of some maximal torus
9
TY! Y and P! P the pdiscrete approximation of P which is a "real" discrete
group. Every element a 2 P generates a cyclic subgroup P of finite order a*
*nd
induces a sequence of homomorphism Z=pk! P ! P! Y ! X of pcompact
groups (don't mind that P is not a pcompact group). Then we define the prekern*
*el
by preker(f) := {a 2 P : BfB ' *}. This definition goes back to [I] and is
denoted in [DW] as the kernel of f. The set preker(f) is a normal subgroup of P
[N2] or [DW]. We define the kernel ker(f) of f to be the closure of preker(f) *
*which
is then a "normal"_pcompact toral subgroup of P , i.e. there exists a pcompa*
*ct
toral group P and an exact sequence ker(f)! P! Q of pcompact toral groups.
This is proved in [DW ,7.2]. But there is only treated the case of the pdisc*
*rete
approximations. Passing to closures establishes the described result. For det*
*ails
see also [N2], where only the case of Y being a compact Lie group is handled, *
*but
all the arguments also apply in our situation. The definition of ker(f) does n*
*ot
depend essentially on the chosen maximal torus and pnormalizer, because all p
normalizers are conjugated. In [DW; 8.11] this is proved for maximal tori, but
similiar arguments also apply to pnormalizers.
We say the ker(f) is trivial if Bker(f) is contractible. In Section 1 we def*
*ined
f to be a monomorphism, if the homotopy fiber X=Y of Bf is Fpfinite.
In classical group theory every homomorphism can be made into an monomor
phism by dividing out the kernel. A similiar statement is true in the category*
* of
pcompact groups.
2.16 Proposition. Let P be a pcompact toral group or a pdiscrete toral group
and f : P! X be a homomorphism into a pcompact group. Let K := ker(f) be
the_kernel and Q := P=K be the quotient. Then f factors over a homomorphism
f : Q! X with trivial kernel. Moreover, map(BQ; BX)B__f! map(BP; BX)Bf is
a homotopy equivalence.
Proof. For the case of a pdiscrete toral group see [DW , Lemma 7.5], and for
the case of a pcompact toral group this follows by [N2] or by using pdiscrete
approximations and taking closures.
2.17 Theorem. For a homomorphism f : Y ! X of pcompact groups the fol
lowing three conditions are equivalent:
(1) f is a monomorphism.
(2) H*(BY ; Fp) is a finitely generated H*(BX; Fp)module.
(3) The kernel ker(f) is trivial.
Proof:. The equivalence of (1) and (2) is proved in [DW , 9.11]. Let P! Y *
* be
a pnormalizer of a maximal torus of Y . By [DW , 7.3] the restriction fP i*
*s a
monomorphism if and only if ker(f) is trivial. This shows that (1) implies (3).
Now let ker(f) be trivial. Then, by what is already said, H*(BP ; Fp) is a fini*
*tely
generated H*(BX; Fp) module. The algebra H*(BX; Fp) is noetherian [DW ,2.3].
Therefore, the submodule H*(BY; Fp) H*(BP ; Fp) [DW , proof of Theorem 2.3]
is also finitely generated over H*(BX; Fp) which is condition (2). This complet*
*es
a circle of implications.
For a different prove of the equivalence of (2) and (3), which is not that m*
*uch
in the spirit of [DW] see [N2, Theorem 1.2]. There is only treated the case o*
*f Y
10
being a compact Lie group. The major tool is a theorem of Quillen which says th*
*at,
for a compact Lie group G, the cohomology H*(BG; Fp) is detected by elementary
abelian subgroups. Because H*(BY ; Fp) is noetherian there is a similiar result*
* in
our case [R3]. All the other arguments of [N2] can be carried over to the case*
* of
pcompact groups.
In particular , Theorem 2.17 implies that the composition of two monomorphis*
*ms
is always a monomorphism and that the first is a monomorphism if the composition
is one.
For later purpose we will mention a slightly mor general situation, where the
kernel of a map BX! U of a map into a space U can be defined. A space U is
called BZ=plocal if the evaluation ev : map (BX; U)! V is an equivalence, a*
*nd
almost BZ=plocal if the evaluation induces an equivalence map(BZ=p; U)const' U
between the component of the constant map and U. Then U is almost BZ=plocal
if and only if the loop space U is BZ=plocal. In [N2] the definition of a ker*
*nel
is given for maps BG! U where G is a compact Lie group and U a pcomplete
almost BZ=plocal space. But all the arguments and all constructions also work
for maps BX! U where X is a pcompact group and U is a pcomplete almost
BZ=plocal space. In particular the kernel is a normal subgroup of Np(TX ).
2.18 Cohomological dimension : For an Fpfinite space X, Dwyer and Wilk
erson define the modp cohomological dimension cdFp(X) as the largest integer i
such that eHi(X; Fp) does not vanish. If the total reduced cohomology of X is z*
*ero,
then cdFp(X) = 1.
Analogously we define the rational cohomological dimension cdQ^p using the t*
*he
cohomology theory H*Q^p( ). For a pcompact group X we get cdQ^p(X) = cdFp(X)
(see Lemma 3.2).
3. Lie theory for pcompact groups.
This section contains a collection of basic results to be used later. All of*
* these
results have Lie group analogues that are wellknown if not blatantly obvious. *
*We
begin by investigating abelian pcompact groups and covering spaces of pcompact
groups, then turn to monomorphisms into pcompact toral groups, mod p dimen
sion, Weyl groups of nonconnected pcompact groups and finish by showing that
the centralizer of a pcompact torus in a connected pcompact group is connecte*
*d.
3.1 Proposition. Any abelian pcompact group is isomorphic to a product of a
pcompact torus and a finite abelian group.
Proof. Let A be an abelian pcompact group and i : T! A a maximal torus. It
suffices to show that A is a pcompact toral group for, by [DW, Remark 8.5], A*
* will
then have the desired form. The centralizer CA (T ) is a pcompact toral group *
*by
the defintion of maximal torus; in fact the canonical lift (see 2.9) j : T! *
*CA (T )
of i takes T isomorphically to the identity component of CA (T ). Denoting pr*
*e
11
___
composition with Bi by Bi , we have a diagram
___
BCA (A) _Bi__BCA_(T/)/OO
qq 
Be1 ' Be2qqqq Bj
fflfflxxqqq 
BA oo__Bi_____BT
where e1 and e2 are evaluation_homomorphisms._ Both triangles in the diagram
are commutative, i.e. Be2 O Bi = Be1 and Be2 O Bj = Bi. This implies that
ss*(Be2) maps ss*(BCA (T )) onto ss*(BA) and that ss2(Bi) : ss2(BT )! ss2(BA)*
* is
an epimorphism with a right inverse. Hence ss2(BA) is a free Z^pmodule and A a
pcompact toral group.
The completed odd sphere (S2n1 )^p; np  1, is homotopy commtutative as an
Hspace but nonabelian as a pcompact group (when n > 1).
For later reference we record a lemma that can be extracted from Kane [K, x3*
*x4]
(who credits Browder [B1] with the original idea).
3.2 Lemma. Let X be a connected Hspace such that ssi(X); i 1; is a finitely
generated Z^pmodule and H*(X; Fp) is finite. Then:
(1) Any connected covering space of X has the same properties.
(2) H*Qp(X) is finite dimensional, HdQp(X) ~=Qp and H>dQp(X) = 0 where d =
cdFp(X).
3.3 Corollary. Suppose that X is a connected pcompact group, that Y is a
connected space, and that Y ! X is a covering map. Then Y is a pcompact
group.
Proof. The given data amounts to a fibration
Y! X! BQ
where Q is a quotient of the finitely generated Z^pmodule ss1(X). The projecti*
*on
map in this fibration is a loop map, for [BX; B2Q] = [X; BQ], and therefore Y i*
*s a
loop space. Lemma 3.2 shows that Y is in fact a pcompact group (see 2.2).
In Section 2 we explained what we mean by the pdiscrete approximation f :
X ! G of a homomorphism f : X! G of pcompact toral groups.
3.4 Proposition. Suppose that f : X! G is a homomorphism of pcompact
toral groups and that f is a discrete approximation to f. Then
(1) f is a monomorphism , f is a monomorphism
(2) f is an isomorphism , f is an isomorphism
(3) f is central, f is central
Proof. (1) is a consequence of Theorem 2.15. The key observation is that preker*
*(f)
is the usual algebraic kernel of f.
12
Statement (2) follows easily from the commutative diagram
_Bf___//
BX BG
Bx  Bg
fflffl fflffl
BX _Bf___BG//
where the vertical maps become homotopy equivalences after completion at p.
(3) Let CG (X ) denote the algebraic centralizer in G of f(X ). The homoto*
*py
fibre of BG ! BG being K(ss1(G) Q; 1) implies that the homotopy fibre of
BCG (X) = BCG (X ) = map (BX ; BG)B(fi)! map (BX ; BG )Bf = BCG (X )
is K(H0(BX ; ss1(G) Q); 1); in particular, CG (X ) is a discrete approximation
to CG (X). Hence BCG (X) ! BG is a homotopy equivalence if and only if
BCG (X )! BG is a homotopy equivalence if and only if CG (X ) = G .
3.5 Proposition. Let X be a pcompact group, G a pcompact toral group, and
f : X! G a monomorphism. Then:
(1) X is a pcompact toral group.
(2) If f is central, X is an abelian pcompact group.
(3) If G is a pcompact torus and X is connected, X is a pcompact torus.
(4) If G is a pcompact torus, so is G=X
Proof. We first prove (3). Under the assumptions in (3), the homogeneous space
G=X is connected and the fundamental group is a finitely generated Z^pmodule.
Let BY denote the universal covering space of G=X. The loop space Y = BY is
equivalent to a component of (G=X) which is a covering space of the connected
pcompact group X; hence Y is also a connected pcompact group by Corollary 3.3.
Moreover, because BY is a covering of an Fpfinite space, the Sullivan conject*
*ure
[M] shows that all homomorphisms Z=p! Y are trivial. Thus Y is itself triv*
*ial
(see 2.5). Consequently G=X = K(ss1(G=X); 1) is, by Fpfiniteness, a pcompact
torus and so is X by the exact homotopy sequence.
(1) Let f0 : X0! G0 be the restriction of f to the identity components. *
*It
suffices, by (3), to show that also f0 is a monomorphism. But that follows from*
* the
fact that ker(f0)! ker(f) is a monomorphism.
Now that we know X is a pcompact toral group, (2) follows from Proposition *
*3.4,
because, with notation from that proposition, f(X ) ~=X is abelian if f is a c*
*entral
monomorphism. In this case, G =X is easily seen to be a discrete approximation
to the pcompact group G=X. As any quotient of a pdiscrete torus is again a
pdiscrete torus [F, Theorem 23.1], this proves (4).
The combination of Proposition 3.1 and Proposition 3.5 shows that if f : X!*
* A
is a monomorphism and A is an abelian pcompact group, so is X.
Specializing to the case of pcompact torus groups we obtain
13
3.6 Proposition. Let S and T be pcompact torus groups and f : S! T a
homomorphism.
(1) f is a monomorphism , T=S is a pcompact torus ,
ss1(f) is a split injective homomorphism
(2) If cdFp(S) = cdFp(T ), then f is a monomorphism if and only if f is an
isomorphism.
(3) If ss1(f) is injective, then there exists a finite abelian pgroup K an*
*d a
factorization f0 : S=K! T of f which is a monomorphism.
Proof. (1) The proof of Proposition 3.5 shows that if f : S! T is a monomorph*
*ism
then T=S is a pcompact torus; the converse is clear. The other biimplication i*
*s a
direct consequence of the exact homotopy sequence.
(2) follows immediately from Proposition 3.4.
(3) Denoting by K := ker(f) the kernel of f we get (2.15) a commutative diag*
*ram
S __f____T//==
 
  0
fflfflf
S=K
of homomorphisms between pcompact torus groups where f0 is a monomorphism.
As ss1(f) is assumed to be injective, the fundamental group functor shows that
ss1(K) = 0, i.e. that K is a finite abelian pgroup.
3.7 Proposition. Let f : X! Y be a monomorphism between two connected
pcompact groups such that H*Qp(f) : H*Qp(Y )! H*Qp(X) is an isomorphism. Then
f is a homotopy equivalence.
Proof. Lemma 3.2 shows that cdFp(X) = cdFp(Y ) and therefore any monomorphism
from X to Y is a homotopy equivalence [DW, Proposition 6.14, Remark 6.15].
We are next aiming at the homotopy theoretic equivalent of the statement that
connected abelian subgroups of compact connected Lie groups have connected cen
tralizers.
In 3.83.11 below, X denotes a pcompact group.
3.8 Proposition. Suppose that i : T! X0 is a maximal torus for the connected
component X0 of X. Then T! X0! X is a maximal torus for X and there
exists a short exact sequence
1! WT (X0)! WT (X)! ss0(X)! 1
relating the Weyl groups.
Proof. Mapping BT into the universal covering map BX0! BX produces another
covering map BCX0 (T )! BCX (T ) showing that CX (T ) is a pcompact toral gr*
*oup
with T ~= CX0 (T ) as its identity component.
14
Let w : BT! BT be an element in the Weyl space of T! X. As both Bi and
Bi O w are lifts in the diagram
BX0;;
x x 
x 
x fflffl
BT ______BX//
there exists by covering space theory a unique covering translation (w) 2 ss0(X)
such that
BT ________w________BT//
Bi  Bi
fflffl (w) fflffl
BX0 G_______________/BX0/
GGG ww
GGG www
## ww
BX
commutes. Clearly, : WT (X)! ss0(X) is a homomorphism with WT (X0) as
kernel. Surjectivity of follows from the fact that any morphism from any p
compact torus into X0 factors through the maximal torus T (Theorem 2.11).
So, for a pcompact toral group, the Weyl group agrees with the group of com
ponents.
In the following corollary, Np(T )! X denotes the pnormalizer of a maximal
torus T! X.
3.9 Corollary. The homomorphism ss0(Np(T ))! ss0(X) is surjective.
Proof. The pnormalizer is a pcompact toral group with T as its identity compo*
*nent
and its group of components is a pSylow subgroup Wp of the Weyl group WT (X).
When viewing ss0(Np(T )) = Wp as the group of covering translations of BX0 over
BX, the homomorphism ss0(Np(T ))! ss0(X) becomes the restriction of to Wp.
Since ss0(X) is a finite pgroup, the restriction of the epimorphism to Wp rem*
*ains
an epimorphism.
The next lemma can be viewed as a converse to [DW, Proposition 5.5].
3.10 Lemma. Suppose that, for any integer n 1, any homomorphism Z=pn! X
can be extended to Z=pn+1 . Then X is connected.
Proof. Assume X is not connected. Any discrete approximation N to the p
normalizer Np(T ) is an extension of a discrete approximation T to the maximal
torus of X by ss0(Np(T )). Since ss0(Np(T )) maps onto ss0(X) by Corollary 3.9,*
* N
contains some cyclic subgroup Z=pn such that the homomorphism
Z=pn ,! N ! ss0(Np(T ))! ss0(X)
is nontrivial. The corresponding homomorphism of pcompact groups
Z=pn! Np(T )! X
15
is then nontrivial on ss0. This homomorphism can not be extended to the pdiscr*
*ete
torus Z=p1 for then it would factor through T (Theorem 2.11).
The proof of the final of the auxilliary results is very much in the spirit *
*of the
proofs of [DW, Proposition 5.4, Proposition 5.5].
3.11 Proposition. Let S be a pcompact torus and S! X a homomorphism. If
X is connected, so is the centralizer CX (S) of S in X.
Proof. Let n be an arbitrary natural number and Z=pn! CX (S) a homomorphism.
It suffices to show (Lemma 3.10) that this homomorphism extends to Z=pn+1 , or,
equivalently, that the adjoint f : Z=pn x S! X extends to Z=pn+1 x S. Consider
the commutative diagram
map (B(Z=pn+1 ) x BS; BX) ______map/(B(Z=pn)/x BS; BX)
 
 
fflffl fflffl
map (B(Z=pn+1 ); BX) ____________map/(B(Z=pn);/BX)
of restriction fibrations. The homotopy fibre over BfB(Z=pn) of the bottom map
can [DW, Lemma 10.6, Lemma 10.7] be identified to the homotopy fixed point
n+1
set (Xp1 )h(Z=p ) for some proxy action of Z=pn+1 on Xp1 . This homotopy
fibre is Fpfinite [DW, Theorem 4.5, Proposition 5.7] with Euler characteristic
[DW, Lemma 5.11] equal to pr (here we use that X is connected) where r is
the rational rank of X. Similarly, the homotopy fibre over Bf of the top map is
h(Z=pn+1)
homotopy equivalent to the homotopy fixed point set ( Xp1 )hS where
S is a discrete approximation to S. We have just seen that Xp1 h(Z=pn+1)is
Fpfinite with nonzero Euler characteristic so by [DW, Theorem 4.7], and [DW,
Proposition 5.7] in order to handle the Fpcompleteness problem, the homotopy
fibre of the top map is nonempty.
In [DW] the Euler characteristic of a homogenous spaces X=Y turns out to be
a quite useful invariant in the study of pcompact groups.. In classical Lie g*
*roup
theory this invariant is not that much used. The next statement has a straight
forward proof in classical Lie group theory using the the associated Lie algebr*
*as.
3.12 Proposition. Let f : Y! X be a monomorphism of pcompact groups such
that f induces an isomorphism ss0(Y )! ss0(X) between the components and such
that the Euler characteristic O(X=Y ) = 1. Then f is an isomorphism.
The proof is based on two lemmas.
3.13 Lemma. Let Y! X be a monomorphism of pcompact groups. If the Euler
characteristic O(X=Y ) 6 0 mod p then every pcompact toral subgroup P! X of
X is subconjugate to Y .
Proof. The homomorphism P ! X establishes a proxy action on X=Y . If the
homotopy fixedpoint set X=Y hP is non empty,Si.e. if for example O(X=Y hP) 6= *
*0,
then P is subconjugated to Y . Let P = kPk! P be a pdiscrete approximation
written as the union of finite pgroups. Then O(X=Y hPk) O(X=Y ) 6 0 for every
16
k [DW, Theorem 4.6 and Proposition 5.7]. This implies that Pk is subconjugated
to Y , and so is P . Passing to the closure, proves that P also is subconjugate*
*d to
Y .
3.14 Lemma. Let f : Y! X be a monomorphism of pcompact groups. Then f
induces an isomorphism N(TY ) ~=N(TX ) if and only if O(X=Y ) = 1.
Proof. By Lemma 3.13 the condition O(X=Y ) = 1 implies that Np(TY ) ~=Np(TX ).
In particular , T := TY ~=TX , and WY ! WX is a monomorphism because the
Weyl groups acts effictively on the maximal tori. The maps T! Y! X defines a
proxy action of T on the fibration Y=T! X=T! X=Y . In the associated fibrat*
*ion
Y=T hT! X=T hT! X=Y hT
the first two terms are homotopically discrete. We have Y=T hT ' WY and X=T hT '
WX . Therefore X=Y hT is also homotopically discrete and 1 = O(X=Y hT) =
WX =WY  This implies that WY ~= WX and that N(TY ) ~=N(TX ).
An isomorphism N := N(TY ) ~=N(TX ) of loop spaces establishes the diagram
Y=N ______X=N// ______X=Y//
 
 
fflffl= fflffl
BN _______BN//
 
 
fflffl fflffl
X=Y ______/BY/ ______BX/:/
The fibration Y=N! BN! BY is oriented because ss1(BN)! ss1(BY ) is an
epimorphism. Thus, the top horizontal fibration is also oriented. Now the mul*
*ti
plicativity of the Euler characteristic shows that 1 = O(X=N) = O(Y=N)O(X=Y ) =
O(X=Y ), which proves the second half of the statement.
Proof of 3.12. The monomorphism f : Y! X establishes a diagram of fibrations
X0=Y0 __'___X=Y// _________*//
  
  
fflffl fflffl fflffl
BY0 ________BY//_______Bss0(Y/)/
Bf0 Bf '
fflffl fflffl fflffl
BX0 _______/BX/ ______Bss0(X)// :
Here, Y0 and X0 denote the components of the unit. The right lower vertical arr*
*ow
is an equivalence by assumption. Thus, the upper left arrow is also an equiva
lence. Hence Y ! X is an isomorphism if and only if Y0! X0 is an isomor
phism. For the latter map the Euler characteristic condition is also satisfied*
*. By
Lemma 3.14, the monomorphism f0 induces an isomorphism WY0 ~=WX0 between
the Weyl groups. Moreover, Y0 and X0 are connected. Hence, by Theorem 2.11,
17
H*Q^p(BX0) ~=H*Q^p(BY0). By Proposition 3.7 this implies that f0 : Y0! X0 is *
*an
isomorphism and so is f : Y! X.
This finishes the collection of assorted basic facts about pcompact groups *
*ex
tending the uncanny similarity with compact Lie groups.
4. The center of a pcompact group.
In this section we define the center of a pcompact group and show that any
central monomorphism factors through this center.
Throughout this section, X and Z denote pcompact groups. Let i : T! X be
a maximal torus for X; its centralizer CX (T ) is a pcompact toral group with *
*T as
its identity component.
4.1 Lemma. Let f : Z! X be a central monomorphism.
(1) There exists a central monomorphism g : Z! CX (T ) such that
CX<(T<)
gxxx 
xx 
xx fflffl
Z ___f____X//
commutes up to conjugacy; in particular, Z is abelian.
f
(2) The composition U ! Z! X is a central monomorphism for any p
compact group U and any monomorphism U! Z.
Proof. (1) Choose, as in [DW, Lemma 8.6], a homomorphism h : Z x X! X with
f = h(Z x *) and h(* x X) equal to the identity map on X. Let g : Z! CX (T )
h
and j : T! CX (Z) be the adjoints of Z x T! Z x X! X. Note that g is a
lift of f and j is a lift of i; in particular, both g and j are monomorphisms s*
*o Z is
a pcompact toral group by Proposition 3.5. Centrality of g is now a consequence
[DW, Lemma 8.6] of the commutative diagram
BZ VVVVV
 VVVVVVBgV
 VVVVV
fflffl VV**
BZ x map (BT; BCXO(Z))BjO ______map/(BT;/BX)Bii4______BCX_(T_)__4
 iiiiiii
 iiii 'i
 iii
map (BT; BCX (Z))Bj
where the horizontal arrow takes (z; v), z 2 BZ and v : BT ! BCX (T ) =
map (BZ; BX)Bf , to the map BT 3 t! v(t)(z). The upward slanting arrow
is induced by the homotopy equivalence BCX (Z)! BX. The central subgroup Z
is abelian by Propositon 3.5.
18
(2) Note first that also U is abelian, in particular a pcompact toral group*
*, by
Proposition 3.1 and Proposition 3.5. The commutative diagram
BCX (Z)J ________________/BCX/(U)
JJJ ttt
' JJJJ ttt
$$ yytt
BX
of restriction homomorphisms, shows that the right evaluation fibration admits a
section. By [DW, Lema 8.6], this implies that U! X is central.
Let now Z! Z and C! CX (T ) be discrete approximations. For any subgroup
A Z(C ), let A! X denote the homomorphism of loop spaces defined as the
composite A ,! Z(C ) ! C ! CX (T ) ! X. As usual, if also B < Z(C ), AB
denotes the subgroup generated by A and B.
4.2 Lemma. Suppose that A and B are subgroups of Z(C ) such that the homo
morphisms A! X and B! X are central. Then also AB! X is central.
Proof. The abelian group structure on the pdiscrete toral group Z(C ) can be u*
*sed
to define an epimorphism A x B! AB. We have CX (A x B) ~=CX (AB) by [DW,
Lemma 7.5]. Furthermore, CX (A x B) ~=CCX (A)(B) ~=CX (B) ~=X by adjointness
and centrality.
4.3 Definition. The pdiscrete center of X is the set
Z (X) := {t 2 Z(C )  ! X is central}:
where stands for the finite cyclic subgroup of Z(C ) generated by t. By Lem*
*ma
4.2, Z(X) is actually a subgroup of Z(C ); in particular Z(X) is an abelian pd*
*iscrete
toral group. The center of X, denoted Z(X), is defined as the closure of Z (X).
The center Z(X) of X enjoys a pleasant universal property.
4.4 Theorem. Let X be a pcompact group.
(1) The center Z(X) is an abelian pcompact group and Z(X)! X is a
central monomorphism.
(2) For any central monomorphism f : Z! X there exists a monomorphism
g : Z! Z(X) such that
Z(X)<<
gzzz 
z 
zz fflffl
Z ___f___X_//
commutes up to conjugacy.
Proof. The center Z(X) is abelian by its very definition and the homomorphism
Z(X)! CX (T )! X is, as the composition of two monomorphisms, a monomor
phism. To see that this homomorphism is central, choose [DW, Proposition 6.7,
19
Proposition 6.21] a finite subgroup A < Z (X) < Z(C ) such that the restriction
homomorphism CX (Z(X))! CX (Z (X))! CX (A) is an isomorphism. As A is
finite and for each element t 2 A, ! X is central, a finite induction usi*
*ng
Lemma 4.2, shows that A! X is central. Hence CX (Z(X)) ~= CX (A) ~= X, i.e.
Z(X)! X is central.
Any discrete approximation g : Z! C to g : Z! CX (T ) factors through t*
*he
center Z(C ) of C by Proposition 3.4. By Lemma 4.1, the homomorphism
g
! Z! C ! X
is central for any z 2 Z , i.e. g(z) 2 Z (X) for all z 2 Z . This means that g *
*factors
through Z (X) so g factors through Cl(Z (X)) = Z(X).
Let, for example G be a pcompact toral group. Since the evaluation homomor
phism CG (G)! G is a central monomorphism [DW, Proposition 5.1, Proposition
5.2, Theorem 6.1], it factors through the center Z(G). On the other hand, as Z(*
*G)
is a pcompact toral group (even abelian), the central monomorphism Z(G)! G
admits a factorization through CG (G) (see 2.9). It follows (use e.g. Proposi*
*tion
3.6) that the abelian pcompact groups Z(G) and CG (G) are isomorphic. (It is a
tempting conjecture that such an isomorphism exists for any pcompact group. In
the case of compact connected Lie groups this is proved by [JMO].)
Let G be a connected compact Lie group with Lie group theoretic center Z(G).
The central monomorphism Z(G^p)! G^pfactors [DW, Lemma 8.6] through the
centralizer BCG^p(G^p) = map (BG^p; BG^p)B1 which is homotopy equivalent [JM
O] to BZ(G)^p. On the other hand Z(G)^p! G^pis obviously a central subgroup
[DW, Lemma 8.6]. By the universal property of the center (Theorem 4.4) it foll*
*ows
that Z(G^p) ~=Z(G)^p.
Let j : Z! X be a pcompact toral subgroup. The pull back diagram
(X=Z)hZ Bj____BZ//
(*) Bj 
fflfflBj fflffl
BZ ________/BX/
establishes a proxy action of Z on X=Z.
4.5 Proposition. Let j : Z! X be an abelian pcompact toral subgroup. Then Z
is central if and only if the fibration X=Z! (X=Z)hZ! BZ is fiber homotopic*
*ally
trivial. Moreover, if this is the case, we have (X=Z)hZ ' X=Z.
Proof. The identity id : Z! Z subconjugates Z into Z. This implies that there
exists a natural section s : BZ! (X=Z)hZ of the fibration (X=Z)hZ! BZ. We
can apply the functor map(BZ; ) to the pull back diagram (*) which yields anoth*
*er
pullback diagram
M _____________map(BZ;/BZ)id/
(**)  
fflffl fflffl
map(BZ; BZ)id ______map(BZ;/BX)Bj/ :
20
The space M map(BZ; (X=Z)hZ consists of some components and contains at
least the component of s. The mapping spaces map(BZ; BZ)id are homotopy
equivalent to BZ via the evaluation.
If j : Z! X is central we have map(BZ; BX)Bi ' BX which implies that M is
connected and that map(BZ; (X=Z)hZ )s ' (X=Z)hZ . Using this equivalence and
taking the adjoint we can construct the middle arrow in the diagram of fibratio*
*ns
X=ZO ______X=Z/x/BZ ______BZ//OO
OOOO  OOOO
OOOOO  OOOO
fflffl OO
X=Z _______(X=Z)hZ//_______BZ// :
By construction the diagram commutes and gives the desired trivialization.
If the fibration X=Z! (X=Z)hZ ! BZ is fiber homotopically trivial, there
exists a unique section sc : BZ! (X=Z)hZ` for every element c 2 ss0(X=Z) ~=
ss0((X=Z)hZ ), and the mapping space c map (BZ; (X=Z)hZ )sc is equivalent to
(X=Z)hZ . Hence, in the pull back diagram (**) we have M ' (X=Z)hZ This
implies that map(BZ; BX)Bj ' BX and that Z is a central subgroup.
The last statement of the proposition follows from the Sullivan conjecture [*
*M].
This finishes the proof.
The homotopy fixed point set (X=Z)hZ measures the different ways you can
subconjugate Z into Z. Because Z ! X is central, the fundamental group
ss1(BCX (Z)) ~= ss1(BX) acts transitively on ss0(X=ZhZ ) ~= ss0(X=Z). The last
statement and the remarks of 2.6 say that, up to homotopy there is only one way
to do it. That is that "conjugation by elements" of X acts trivially on the cen*
*ter.
Asume from now on that X is connected. Then the maximal torus i : T! X is
selfcentralizing, i.e. T = CX (T ) (Theorem 2.11). Thus any central monomorphi*
*sm
f : Z! X will (Lemma 4.1) factor through a monomorphism g : Z! T . These
monomorphisms extend [DW, Proposition 8.3] to a commutative diagram
ZOOO_g___T//_____T=Z//
OOOO  
OOOOO i i=Z
f fflffl fflffl
Z _____X_//_____X=Z//
where the rows are exact sequences of pcompact groups.
4.6 Proposition. Let f : Z! X be a central monomorphism into a connected
pcompact group X. Then:
(1) i=Z : T=Z! X=Z is a maximal torus for X=Z.
(2) X=T and X=Z_T=Zare homotopy equivalent homogeneous spaces.
(3) WT (X) and WT=Z (X=Z) are isomorphic groups.
(4) The center of X=Z is Z(X)=Z.
Going to extremes, we take the central monomorphism Z(X)! X and form the
pcompact group P X = X=Z(X) with the maximal torus P i : P T = T=Z(X)!
X=Z(X) = P X.
21
4.7 Corollary. The center of P X is trivial.
For the proof of Proposition 4.6 we need some lemmas.
4.8 Lemma. Let Z! X be a central monomorphism into a connected pcompact
group X. Then
(1) BZhT = BZ x BT and BZhT = map (BT; BZ).
(2) (X=T )hZ = X=T x BZ and (X=T )hZ = X=T .
Proof. (1) This follows immediately from the commutative diagram
BZ x BT ________BT//__________BX//
pr2  
fflffl fflffl fflffl
BT _________B(T=Z)//______B(X=Z)//
where the left square is induced from the commutative square
Z x T _______/T/
 
 
fflffl fflffl
T ________T/=Z/
with the top homomorphism given by group multiplication. The expression for the
homotopy fixed point set now follows from the Sullivan conjecture [M].
(2) Composition of maps yields a commutative diagram of mapping spaces
map (BZ; BT )Bg x map (BZ; BZ)B1 __O___map/(BZ;/BT )Bg
 
 
fflffl fflffl
map (BZ; BX)Bf x map (BZ; BZ)B1 __O___map/(BZ;/BX)Bf
equivalent to a commutative diagram
BT x BZ ______/BT/
 
 
fflffl fflffl
BX x BZ ______BX//
of classifying spaces. Restrict to the fibre of the fibration to the left and o*
*btain a
trivialization
X=T x BZ ______BT//
pr2 
fflffl fflffl
BZ ____Bf___/BX/
of (X=T )hZ . The expression for the homotopy fixed point set now follows from *
*the
Sullivan conjecture [M].
Recall from Proposition 1.6, that ss1(g) : ss1(Z)! ss1(T ) is a (split) mo*
*nomor
phism. Let ss1(T )W denote the set of invariant elements for the action of the*
* Weyl
group W := WT (X) on ss2(BT ) = ss1(T ).
22
4.9 Lemma. ss1(Z) ss1(T )W .
Proof. For any element w of the Weyl space, i.e. for any map w : BT! BT over
BX, the two maps Bg; w OBg : BZ! BT are both lifts of Bf : BZ! BX. Since
the space of lifts, (X=T )hZ = X=T , is connected, Bg and w O Bg are homotopic
over BX. In particular, ss2(Bg) = ss2(w O Bg) = ss2(w) O ss2(Bg).
Proof of 4.6.(1)(3). First note that T=Z is a pcompact torus by Proposition 1*
*.5
and that the centralizer CX=Z (T=Z) ~=CX=Z (T ) is connected by Proposition 1.1*
*1.
Next map BT into the fibration BX! B(X=Z) to obtain [DW, Lemma 10.6] the
fibration
BZhT! map (BT; BX)! map (BT; B(X=Z))
containing the subfibration (the base space here is 1connected)
BZ! BCX (T )! BCX=Z (T )
with connected total space; here we used Lemma 4.8 to identify the fibre. A com
parison of fibrations now shows that CX=Z (T ) ~=CX (T )=Z ~=T=Z and thus T=Z is
a maximal torus for X=Z.
The commutative diagram immediately above Proposition 4.6 induces a homo
topy equivalence
X=T __'___X=Z_T=Z//
of homogeneous spaces and shows that
BT ______B(T=Z)//
Bi B(i=Z)
fflffl fflffl
BX ______B(X=Z)//
is a pull back. Naturality of pull backs now determines a homomorph*
*ism
WT=Z (X=Z)! WT (X) of Weyl groups. This map is injective for WT (X) acts
(Lemma 4.9) on ss1(T )=ss1(Z) = ss1(T=Z)iwherejWT=Z (X=Z) is faithfully present*
*ed
(Theorem 2.13). But as WT=Z (X=Z) = O X=Z_T=Z= O (X=T ) = WT (X) by (The
orem 2.13) it is in fact a group isomorphism.
In the above proof of 4.6 and elsewhere in this paper, we need to restrict f*
*ibration
maps to connected components of the total spaces. To that end, we make a general
remark: Let a
Fc! E! B
c2ss0(F)
be a fibration of based spaces; the fibre F is written as the disjoint union of*
* its
connected components Fc. Let E0 and B0 be the base point components of E and B
and let @ : ss1(B)! ss0(F ) be the boundary map in the exact homotopy sequenc*
*e.
Then a
Fc! E0! B0
c2@ss1(B)
is again a fibration.
Yet two more lemmas, not without independent interest, however, are needed
before the proof of the final assertion of Proposition 4.6.
23
4.10 Lemma. Let Z ! X be a central monomorphism into a connected p
compact group X. Then the induced map of (based or free) homotopy sets
[BG; BZ]! [BG; BX]
is injective for any pcompact toral group G.
Proof. It suffices to consider the case of based maps. The fibres of [BG; BZ]!
[BG; BX] are the orbits of a group action
[BG; B(X=Z)] x [BG; BZ]! [BG; BZ]
associated to the fibration BZ! BX! B(X=Z). But the group [BG; B(X=Z)]
= [BG; X=Z] is trivial by the Sullivan conjecture [M].
4.11 Lemma. Let Z! X a central monomorphism into a connected pcompact
group X. Then the homomorphism
Z! X! X=A
is also central for any central monomorphism A! X.
Proof. Both monomorphisms A! X and Z! X factor through the center Z(X)
by Theorem 4.4. Exactly as in the proof of Lemma 4.8, consider the trivializati*
*on
of BAhZ
BA x BZ __B_____BZ(X)// ___Bz_____BX//
pr1  
fflffl fflffl fflffl
BZ _________B(Z(X)=A)// ______B(X=A)//
where z : Z(X)! X is the canonical monomorphism and is the restriction
A x Z! Z(X) x Z(X)! Z(X) to A x Z of the abelian structure on Z(X).
For any ' 2 [BZ; BA], define 1 + ' 2 [BZ; BZ(X)] to be the composite map
1x' B
BZ ! BZ x BZ ! BZ x BA ! BZ(X)
where is the diagonal. Identifying the homotopy sets involved with the corre
sponding sets of homomorphisms of discrete approximations, one sees that '!
1 + ' is injective.
Using the above trivialization of BAhZ to describe the fibre BAhZ *
* as
map (BZ; BA) we obtain the fibration
map (BZ; BA)! map (BZ; BX)! map (BZ; B(X=A))
by mapping BZ into the fibration defining B(X=A). The homotopy sequence ends
with the exact sequence
@
ss1(map (BZ; B(X=A)))! [BZ; BA]! [BZ; BX]
24
of sets. The last map, given by '! Bz O (1 + '), is an injection by the abo*
*ve
remarks and Lemma 4.10. Thus the boundary map @ is constant so the above
fibration contains the subfibration
BA! BCX (Z)! BCX=A (Z)
of connected spaces. Since BCX (Z) ' BX by centrality, comparison shows that
BCX=A (Z) ' B(X=A), i.e. that Z is central X=A.
(The connectedness condition in Lemma 4.10 and Lemma 4.11 can be relaxed a
little. In 4.10 (4.11) it suffices to require that ss0(Z)! ss0(X) (ss0(Z(X))!*
* ss0(X))
be surjective.)
We are now ready for the proof of the final statement of Proposition 4.6.
Proof of 4.6.(4). The pdiscrete center of X=Z is a subgroup of the discrete ap*
*prox
imation T =Z to T=Z. Suppose t 2 T is such that tZ 2 Z (X=Z) meaning that the
homomorphism ! T=Z! X=Z is central. So is then ! X! X=Z by
[DW, Lemma 7.5]. Mapping B() into the fibration BX! B(X=Z) yields the
fibration
BZh! map (B(); BX)! map (B(); B(X=Z)):
This time the base space component BCX=Z () is simply connected being homo
topy equivalent to B(X=Z) by centrality. Thus this fibration contains the subfi*
*bra
tion
BZ! BCX ()! B(X=Z)
showing, by comparison, that BCX () ' BX or, equivalently, that t is in Z (X*
*).
Hence Z (X=Z) Z (X)=Z .
Conversely, if t 2 Z (X), then ! T ! X is central. So is then !
X! X=Z by Lemma 4.11 and ! T=Z! X=Z by [DW, Lemma 7.5]. Thus
Z (X)=Z Z (X=Z).
Next we will generalize another basic property of compact Lie groups, namely
that any finite normal subgroup of a compact connected Lie group is central.
4.12 Proposition. Let K! X! Y be an exact sequence of pcompact groups,
where K is finite and X is connected, i.e. : K! X is a normal subgroup. Then,
: K! X is a central.
Proof. We apply the functor map(BK; ) to the fibration BK! BX! BY . This
yields a diagram of fibrations
FO______________M//OO____ev______BK//
OOOO  
OOOOO  
fflffl ev fflffl
F _______map(BK;/BX)B/ _______BX//

  
  
fflffl fflffl ev fflffl
* ______map(BK;/BY/)const __'___BY// :
25
The space M consists of some, but a finite number, components of map(BK; BK)
and contains at least the component of the identity. Because K is a finite grou*
*p,
the fiber F is homotopically discrete given by a disjoint union of a finite num*
*ber of
qoutients of K. The homogenous space X=CX ((K)) ' F is connected because X
is connected. Therefore X=CX ((K)) ' * which implies that K! X is central.
Finally we will look at the center from two more points of view, which also *
*could
be used for the definition.
The first views the center as a maximal central subgroup. We say the center
Z(X) of X is a central subgroup which satisfies the condition of Theorem 4.4 (2*
*).
For the construction of the center one considers the set of all conjugacy class*
*es of
central subgroups. This is partially ordered by the relation given by subconjug*
*ation.
All central subgroups are subconjugated to CX (T ) which is a pcompact toral g*
*roup.
We can choose a maximal element Z(X) in the poset of all central subgroups. If
there exists a central subgroup Z of X which is not subconjugated to Z(X), we c*
*an
construct a proper extension of Z(X) which is also central by using the argumen*
*ts
of the proof of Lemma 4.2. This contradicts the maximality of Z(X). The univers*
*al
property of the center ensures that both definitions give the same.
The second views the center as the kernel of an adjoint representation. For
a compact Lie group G the center Z(G) is also given as the kernel of the adjoint
representation G! Aut(G) of G into the automorphisms of G given by conjugatio*
*n.
Similiarly one can proceed for pcompact groups.
The free loop space fibration
X ' BX! BX := map (S1; BY )! BX
of a pcompact group X has a classifying map : BY! Baut(Y ) which is called
the classifying map of the adjoint representaion of Y . For a compact Lie group*
* this
construction gives the induced map BG! BAut(G)! Baut(G).
To speak about kernels it is necessary that the target Baut(X) of the map i*
*s p
complete and is almost BZ=plocal. The latter condition means that the evaluati*
*on
induces an equivalence map (BZ=p; Baut(X))const' Baut(X) (see Section 2).
4.13 Proposition. Let Y be a pcomplete Fpfinite space. Then the following
holds:
(1) Baut(Y ) is almost BZ=plocal.
(2) If Y is a loop space in addition, then Baut(Y ) is also pcomplete.
Proof. For (1) it is sufficient to show that aut(Y ) ' Baut(Y ) is BZ=plocal.
Taking adjoints we get
map (BZ=p; aut(Y )) ' map (BZ=p x Y; Y )F
' map (Y; map (BZ=p; Y ))__F
' aut(Y; Y ) :
Here F denotes the set of homotopy classes_of maps f : BZ=p x Y! Y such that
fY is a homotopy equivalence, and F is the set of homotopy classes of the adjo*
*ints
of F . The last equivalence is a consequence of the Sullivan conjecture [M].
26
Condition (2) follows from a combination of [BK; VI 5.4, 7.1, 7.2].
Because a pcompact group enjoys the properties of the last proposition we c*
*an
speak about the kernel K := ker()! Np(TX )! X of the map .
4.14 Proposition. The subgroup j : K! X is the center of X.
Proof. We have to show two things, namely that K is a central subgroup of X and
that every central subgroup of X is subconjugated to K. The universal property
of the center, stated in Theorem 4.4, then proves the statement.
Let Z! X be a central subgroup. The product map BZ x BX! BX estab
lishes a map BZ x BX! BX which fits into a pull back diagram of fibrations
BZ x BX ______BZ//
 
 
fflffl fflffl
BX _________BX// :
The upper row is the trivial fibration and shows that the composition BZ! BX!
Baut(X) is null homotopic. The central subgroup Z! X is subconjugated to
Np(TX ) and therefore also subconjugated to K.
As a subgroup of Np(TX ) the kernel K is a pcompact toral group. The proxy
action of K on X established by the pull back diagram
BK x X _____BK_//
 
 Bj
fflffl fflffl
BX _______BX//
is trivial. Hence, we have XhK ' X. Taking adjoints establishes the equivalenc*
*es
map (BK; BX)Bj ' map (BK x S1; BX)fBKx* 'Bj
' map (BK; map (S1; BX))fOev'Bj
' (BK x X! BK)
' XhK :
Here ( ) denotes the section space of the bundle. This shows that the evaluation
ev : map (BK; BX)Bj! BX is a homotopy equivalence and that therefore K! X
is central.
5. The finite covering.
Throughout this section, X denotes a connected pcompact group with maximal
torus i : T! X, Weyl group W := WT (X) and center Z(X).
Our first goal is to to obtain a description, rationally at least, of the su*
*bgroup
ss1(Z(X)) of ss1(T ). It was shown in Lemma 4.9 that the fundamental group of t*
*he
center is contained in the W invariant subgroup of the fundamental group of T .
27
5.1 Proposition. The index of ss1(Z(X)) in ss1(T )W is finite.
Proof. We show that dim Qp(ss1(Z(X)) Q) dim Qp(ss1(T )W Q).
Let S be a pcompact torus with mod pdimension equal to the rank of the free
finitely generated Z^pmodule ss1(T )W . There exists, since [BS; BT *
*] =
Hom (ss1(S); ss1(T )), a homomorphism e : S! T such that the image of the i*
*n
duced monomorphism ss1(e) is ss1(T )W .
Composition with i : T! X produces a homomorphism i_: CT (S)! CX (S)
of centralizers. An adjointness argument, bearing in mind that CX (T ) ~=T , sh*
*ows
that CT (S) ~= T (Theorem 2.11) is a maximal torus for the connected (Proposi
tion 3.10) pcompact group CX (S). Consider the homomorphism WT (CX (S))!
WT (X) of Weyl groups determined by the diagram T! CX (S)! X. Both Weyl
groups are faithfully presented in ss1(T ) (Theorem 2.13), so this homomorphism*
* is
injective. It is also surjective. To see this, note that because ss1(S) = ss1*
*(T )W is
invariant under W , w O e ' w for any fibre selfmap w of BT over BX; in other
words, the mapping space component BCT (S) = map (BS; BT )Be is mapped to
itself under postcomposition with w. Hence we obtain a commutative diagram
BTOO____________w____________BT//OO
'  '
 w 
BCT (S)L_____________________BCT/(S)/
LLL rr
LLL rrr
LL && xxrrrr
BCX (S)
showing that w is in the Weyl group WCT(S)(CX (S)). Theorem 2.13 now implies
~=
that the monomorphism (see 2.8) CX (S)! X induces an isomorphism H*Qp(X) !
H*Qp(CX (S)) and therefore (Proposition 3.7) CX (S)! X is an isomorphism. This
means that ie : S! X is central.
As shown in Proposition 3.6, e : S ! T factors through a monomorphism
e0 : S=K! T for some finite abelian pgroup K. The composition of e0 with
i : T ! X remains central [DW, Lemma 7.5] and is, as the composition of
monomorphisms, a monomorphism. Thus, by the universal property of the center
(Theorem 4.4), there exists a monomorphism S=K! Z(X) and hence (Proposi
tion 3.6) dim Qp(ss1(Z(X)) Q) dim Qp(ss1(S=K) Q) = dim Qp(ss1(S) Q) =
dim Qp(ss1(T )W Q) as required.
5.2 Corollary. The monomorphism Z(X) ! X induces an isomorphism
~=
ss1(Z(X)) Q ! ss1(X) Q of vector spaces over Qp.
Proof. We have
ss2(BZ(X)) Q ~=(ss2(BT ) Q) WT(X) ~=H2Qp(BT )WT(X) ~=H2Qp(BX)
~=ss2(BX) Q
by Proposition 5.1 and Theorem 2.13.
The following theorem is an immediate consequence of Corollary 5.2.
28
5.3 Theorem. The center of X is isomorphic to a finite abelian pgroup if and
only if the fundamental group ss1(X) is finite.
Recall from Lemma 3.3 that connected covering spaces of connected pcompact
groups are pcompact groups. Thus, in particular, the universal covering space
X<1> of X is a pcompact group.
Choose [DW, Lemma 8.6] a homomorphism X x Z(X)! X extending the
identity map on X and the central monomorphism Z(X)! X. The composite
homomorphism
ss : X<1> x Z(X)0! X x Z(X)0! X x Z(X)! X
is investigated more carefully in the following main result.
5.4 Theorem. For any connected pcompact group X, there exists a short exact
sequence of the form
ss
K! X<1> x Z(X)0! X
pr1
where K is a finite abelian pgroup and K! X<1> x Z(X)0 ! X<1> is a central
monomorphism.
Proof. The exact homotopy sequence for ss together with Corollary 5.2 immediate*
*ly
show the existence of the short exact sequence and also that K is a finite abel*
*ian
pgroup. Proposition 4.12 and Lemma 4.11 show that : K! X<1> x Z(X)0 and
pr1 O : K! X<1> are central homomorphisms.
The commutative diagram
X<1>=K __________B(Z(X)0)_//__________BX_//OO
OO
  OOOO
  OOOO
fflfflB fflffl Bss
BK ________BX<1>/x/B(Z(X)0) _____BX_//

pr1OB  pr1 
fflffl fflffl fflffl
BX<1> _____________BX<1>________________*//
of interlocking fibrations shows that X<1>=K is homotopy equivalent to X=Z(X)0,
in particular Fpfinite. Thus pr1 O : K! X<1> is a monomorphism.
5.5 Corollary. Let i1 : S! X<1> be a maximal torus for the universal covering
pcompact group X<1>. Then:
(1) (i1 x 1Z(X)0 )=K : (S x Z(X)0)=K! (X<1> x Z(X)0)=K is a maximal
torus for X = (X<1> x Z(X)0)=K.
(2) Z(X) = (Z(X<1>) x Z(X)0)=K.
Proof. (1) By [DW, Lemma 7.5] and since any homomorphism into the abelian
pcompact group Z(X)0 is central,
CX<1>xZ(X)0 (S x Z(X)0) = CX<1>(S x Z(X)0) x CZ(X)0 (S x Z(X)0)
= CX<1>(S) x Z(X)0
29
is a pcompact toral group with S x Z(X)0 as its identity component. Thus
S x Z(X)0 is a maximal torus for X<1>. (More generally, the maximal torus of
a product is the product of the maximal tori.) Now point (1) follows from Propo
sition 4.6.
(2) Let S x Z0! S x Z(X)0 be a discrete approximation. For any pair (s; z)*
* 2
S x Z0,
C<(s;z)>(X<1> x Z(X)0) = C(X<1>) x Z(X)0
by a computation similar to the one above. Thus Z (X<1>xZ(X)0) = Z (X<1>)xZ0
and point (2) follows from Proposition 4.6.
A fundamental theorem of Browder [B2] says, when translated into the present
context, that the first nonzero homotopy group of a connected pcompact group
occurs in an odd dimension. So for example, ss2(X) = ss2(X<1>) = 0 always.
5.6 Corollary. Let X be a connected pcompact group with maximal torus
i : T! X and Weyl group W . Then
(1) The homomorphism ss1(i) : ss1(T )! ss1(X) is surjective and the rank *
*of
the kernel equals the rational rank of the universal covering pcompact
group X<1> of X.
(2) X=T is simply connected and ss2(X=T ) is a free finitely generated Z^p
module.
(3) X and X<1> have isomorphic Weyl groups.
(4) H*Qp(BX) ~=H*Qp(BX<1>)W H*Qp(B(Z(X)0))
Proof. By Corollary 5.5, the maximal torus of X has the form T = (S x Z(X)0)=K
where S! X<1> is a maximal torus for X<1>. Proposition 4.6 tells that
X=T ' X<1>_x_Z(X)0__S'xXZ(X)<1>=S
0
which is simply connected and has second homotopy group isomorphic to ss1(S)
since ss2(X<1>) = 0 by Browder's theorem. This proves (1), and (2) is just a
reformulation of (1) using the exact homotopy sequence.
In order to prove (3), note that
h(SxZ(X)0) i j
X<1>_x_Z(X)0__ ' (X<1>=S) hS hZ(X)0 ' (X<1>=S) hS
S x Z(X)0
where the last homotopy equivalence comes from the fact that the action of the *
*di
visible abelian group Z(X)0 on the homotopically discrete Weyl space of X<1> mu*
*st
be essentially trivial; cfr. [DW, Proposition 8.10]. Taking groups of componen*
*ts,
we obtain the first of the isomorphisms
WS (X<1>) ~=WSxZ(X)0 (X<1> x Z(X)0) ~=WT (X)
while the second one follows from Proposition 4.6.
The final assertion follows by expressing H*Qp(BX) as a ring of invariants (*
*The
orem 2.13).
In classical Lie group theory, the order of the center of a simply connected*
* Lie
group divides the order of the Weyl group. In particular , at large primes, ev*
*ery
compact connected Lie group splits into a product of a simply connected one and
a torus. The same statement also is true for pcompact groups.
30
5.7 Theorem. Let X be a connected pcompact group. If (p; WX ) = 1, then
X ~= X<1> x Z(X)0 is isomorphic to the product of the universal cover X<1> and
the connected component Z(X)0 of the center Z(X) of X.
Proof. Because p is coprime to the order of WX , we have an isomorphism
H*(BX; Z^p) ~=H*(BTX ; Z^p)WX [DMW]. In particular , H*(BX; Z^p) is torsion
free. Moreover, the WX module H2(BTX ; Z^p) ~=M1 M2 splits into a direct sum
where M1 is a fixedpoint free WX module and M2 ~= H2(BTX ; Z^p)WX is given
by the fixedpoints. Classifying spaces of pcompact tori are EilenbergMacLane
spaces. Therefore we can realize the summands by maps Bji : BTi! BTX ,
i = 1; 2. Both tori, T1 and T2, inherit an WX action and the maps can be reali*
*zed
by equivariant maps.
Next we want to show that T2! TX ! X is central. The centralizer CX (T2)
is connected and of maximal rank (Proposition 3.11). By construction WCX (T2)=
WX . Hence, by Theorem 2.13, the map BCX (T2)! BX is rationally an equiva
lenc, and by Proposition 3.7 a homotopy equivalence. This shows that T2! X is
central. In particular , we have T2 ~=Z(X)0.
Let det : X! T be the generalized determinant. That is that T is a pcompa*
*ct
torus of the same rank as the free Z^pmodule H2(X; Z^p) and that Bdet is given*
* by
an isomorphism H2(X; Z^p) ~=H2(BT ; Z^p). By the above remarks the fiber of Bdet
Bdet
is given by the universal cover X<1>. The composition BZ(X)0! BX ! BT
is a homotopy equivalence, because H2(BX; Z^p) ~=H2(BTX ; Z^p)WX . We identify
Z(X)0 and T via this equivalence. Hence Bdet has a left inverse given by the ce*
*ntral
map BZ(X)0! BX. The adjoint of BX<1>! BX ' map (BZ(X)0; BX)Bj2
establishes an equivalence of fibrations
BX<1>O ______BX<1>/x/BZ(X)0 ______BZ(X)0//
OOOO
OOOOO ' '
fflffl Bdet fflffl
BX<1> _____________BX//_______________/BT/ :
This finishes the proof.
The proof of Theorem 5.7 obviously has the following corollary:
5.8 Corollary. Let X be a connected pcompact group. If p does not divide the
order of the Weyl group, then the center Z(X) is connected. In particular , if *
*X is
semi simple, then Z(X) is trivial.
6. Finite coverings of connected finite loop spaces.
In this section we will prove Theorem 1.5 which says that every connected fi*
*nite
loop space has a finite covering which splits into a product of a simply connec*
*ted
finite loop space and a torus. To do this we use the results of the last sectio*
*n which
give us a splitting at each prime. An arithmetic square argument will complete *
*the
proof.
Let L = (L; BL; e) be a connected finite loop space. Then completion at a
prime p gives a pcompact group L^p= (L^p; BL^p; e^p). The rational cohomology
31
H*(BL; Q) ~=Q[x1; :::; xn]Qis a polynomial algebra of generators xi of even deg*
*ree
2ri. We define d(X) := iri.
The 2dimensional cohomology H2(BL; Z) is torsionfree of rank s, because BL
is simply connected. Let T be a torus of the same rank and let Bdet : BL! BT
be the generalized determinant established by a chosen isomorphism H2(BL; Z) ~=
H2(BT ; Z).
6.1 Proposition. There exists an unstable Adams operation k : BT! BT and
a map BT! BL which gives after completion a central subgroup Tp^! L^psuch
that the diagram
yBL<<
yyy Bdet
yy k fflffl
BT ______BT//
commutes up to homotopy.
Proof. Let p be a prime. Passing to completion and by Theorem 5.4 we get a
commutative diagram
BZ(L^p)0O______BL^p<1>/x/BZ(L^p)0O _______BL//
 Bdet
 kp fflffl
BTp^ __________________________________BTp^// :
The composition Bg : BZ(L^p)0! BTp^ of the upper row and Bdet is rationally
an equivalence because BL^p<1> is 3connected. In particular , the fiber is giv*
*en by
the kernel of g which is a finite abelian pgroup Kp. Hence, there exists only *
*one
obstruction for a left inverse of Bg contained in H2(BTp^; ss1(Kp)) ~=H2(BTp^; *
*Kp).
Let kp := Kp. By an Adams map kp : BTp^! BTp^ this obstruction is mapped
to zero which proves the existence of the left vertical arrow.
By Theorem 2.13 and by a theorem of Chevalley [C] the order of the Weyl
group WL^p is equal to d(L) for every prime. If p is coprime to d(X), then BL^p*
*is
equivalent to the product of BL^p<1> x BZ(L^p)0, and the left vertical arrow ex*
*ists
with kp = 1. That is to sayQthat only for a finite number of primes kp is unequ*
*al
to 1. The product k := p kp is a finite number. Unstable Adams operations of
any degree can be realized as self maps of BT and commute up to homotopy. This
establishes a commutative diagram
BL^;;
vv
vvv Bdet
vv k fflffl
BT ^ ______BT/^/ :
Q
Here, BL^ := pBL^pdenotes the product of all padic completions.
Rationally BL is a product of rational EilenbergMacLane spaces. The map
Bdet* : H*(BT ; Q)! H*(BL; Q) is an isomorphism on the 2dimensional genera
tors of the polynomial ring H*(BL; Q). In particular, there exists a right inve*
*rse of
32
Bdet* which can be realized by a map BTQ! BLQ . Adding the Adams operation
k into the picture we also get a diagram commutative up to homotopy
BLQ;;
xx
xxx Bdet
xx k fflffl
BTQ ______BTQ// :
The coherence conditions for using the arithmetic square are satisfied by const*
*ruc
tion. This establishes the desired diagram of the statement. The centrality of *
*the
lift BT! BL is already proved.
The universal cover L<1>! L of a connected finite loop space L is also a f*
*inite
loop space, and passes to map BL<1>! BL between the classifying spaces. This
follows analogously as in Corollary 3.3. Actually, the proof of Proposition 3.2*
* and
Corollary 3.3 is the padic version of an integral argument.
Proof of Theorem 1.5. Let L<1>! L be the universal cover of the finite loop
space L. By Proposition 6.1 and the proof we can choose an Adams operation
k : BT! BT and a central lift Bg : BT ^! BL^ . The adjoint of BL<1>^!
BL^ ' map (BT ^; BL^ )Bg establishes a diagram commutative up to homotopy
BL<1>^ x BT ^ ______BL^//
 
 Bdet
fflffl k fflffl
BT ____________/BT/ :
The left vertical map is the projection on the second factor.
Rationally the map BL<1>Q! BLQ extends to a map BL<1>Q x BTQ! BLQ
where the restriction on the second factor is given by the left inverse of Bdet*
*. In this
case everything follows from the fact that all invoved spaces are rationally pr*
*oducts
of EilenbergMacLane spaces. Again, adding the Adams operation k : BT! BT
into the picture establishes the analogous diagram for the rationalisations of *
*the
spaces. The coherence conditions for glueing together are satisfied because ove*
*r the
adeles the homotopy classes of the maps are controlled by cohomology. This prov*
*es
the statement.
References
[BN] D. Blanc and D. Notbohm, Mapping spaces of compact Lie groups and
padic completion, Proc. Amer. Math. Soc. 117 (1993), 251258.
[BK] A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localiz*
*a
tions, Lecture Notes in Mathematics 304, SpringerVerlag, BerlinHeid*
*el
bergNew York, 1972.
[B1] W. Browder, The cohomology of covering spaces of Hspaces, Bull. Amer.
Math. Soc. 65 (1959), 140141.
[B2] W. Browder, Torsion in Hspaces, Ann. Math. 74 (1961), 2451.
33
[C] C. Chevalley, Invariants of finite groups generated by reflections, A*
*mer.
J. Math. 77 (1955), 778782.
[CE] A. Clark and J. Ewing, The realization of polynomial algebras as coho
mology rings, Pacific J. Math. 50 (1974), 425434.
[DMW] W.G. Dwyer, H.R. Miller and C.W. Wilkerson, Homotopical uniqueness
of classifying spaces, Topology 31 (1992), 2946.
[DW] W.G. Dwyer and C.W. Wilkerson, Homotopy fixed point methods for Lie
groups and finite loop spaces, Preprint.
[DZ] W. Dwyer and A. Zabrodsky, Maps between classifying spaces, Algebraic
Topology, Barcelona 1986 (J. Aguade and R. Kane, eds.), Lecture Notes*
* in
Mathematics 1298, SpringerVerlag, BerlinHeidelberg, 1987, pp. 1061*
*19.
[F] L. Fuchs, Infinite Abelian Groups. Vol. I, Pure and Applied Mathemati*
*cs
36, Academic Press, New YorkLondon, 1970.
[I] K. Ishiguro, Retracts of classifying spaces of compact connected Lie *
*groups,
Preprint.
[JMO] S. Jackowski, J.McClure and R. Oliver, Homotopy classification of sel*
*f
maps of BG via Gactions. Part I, Ann. of Math.(2) 135 (1992), 18322*
*6.
[K] R.M. Kane, The Homology of Hopf Spaces, NorthHolland Mathematical
Library 40, Elsevier Science Publishers B.V., AmsterdamNew YorkOx
fordTokyo, 1988.
[McG] C.A. McGibbon, Which group structures on S3 have a maximal torus?,
Geometric Applications of Homotopy Theory I, Proceedings, Evanston
1977 (M.G. Barratt and M.E. Mahowald, eds.), Lecture Notes in Mathe
matics 657, SpringerVerlag, BerlinHeidelbergNew York, 1978, pp. 353
 360.
[M] H.R. Miller, The Sullivan conjecture on maps from classifying spaces,
Ann. Math.(2) 120 (1984), 3987.
[N1] D. Notbohm, Maps between classifying spaces, Math. Z. 207 (1991), 153
 168.
[N2] D. Notbohm, Kernels of maps between classifying spaces, Math. Gott.
Heft 17 (1992).
[R1] D.L. Rector, Subgroups of finite dimensional topological groups, J. P*
*ure
Appl. Algebra 1 (1971), 253273.
[R2] D.L. Rector, Loop structures on the homotopy type of S3, Symposium
on Algebraic Topology, Seattle 1971 (P.J. Hilton, ed.), Lecture Notes
in Mathematics 249, SpringerVerlag, BerlinHeidelbergNew York, 1971,
pp. 99  105.
[R3] D.L. Rector, Noetherian cohomology rings and finite loop spaces with *
*tor
sion, J. Pure Appl. Algebra 32 (1984), 191227.
[W] C. Wilkerson, Rational maximal tori, J. Pure Appl. Algebra 4 (1974),
261272.
Matematisk Institut, Universitetsparken 5, DK2100 Kobenhavn O, Denmark,
email: moller@math.ku.dk.
Mathematisches Institut, Bunsenstr. 35, D3400 G"ottingen, Germany,
email: notbohm@cfgauss.unimath.gwdg.de.