THE CENTER OF A p-COMPACT GROUP
W.G. Dwyer and C.W. Wilkerson
University of Notre Dame
Purdue University
x1. Introduction
Compact Lie groups appear frequently in algebraic topology, but they are rel*
*atively rigid
analytic objects, and partially for that reason are a challenge to understand h*
*omotopically.
Since H. Hopf (and H-spaces) topologists have aspired to escape the analytic st*
*raitjacket
by finding some homotopy theoretic concept which would capture enough of the id*
*ea of
"compact Lie group" to lead to rich and interesting structure theorems. In [12]*
* we came
up with our own candidate, the notion of p-compact group, and studied these obj*
*ects
to the extent of constructing maximal tori, Weyl groups, etc. In this paper we *
*continue
the study by looking at the idea of the "center" of a p-compact group and showi*
*ng that
two very different ways of defining the center are equivalent. This leads for *
*instance to
a reproof and generalization of a theorem from [15] about the identity componen*
*t of the
space of self homotopy equivalences of BG (G compact Lie). Along the way we fin*
*d various
familiar-looking elements of internal structure in a p-compact group X , enumer*
*ate the X 's
which are abelian in the appropriate sense, and construct what might be called *
*the "adjoint
form" of X .
Before describing in more detail the main results we are aiming at, we have *
*to introduce
some ideas from [12]. A loop space X is by definition a triple (X ; BX ; e) in *
*which X is a
space, BX is a connected pointed space (called the classifying space of X ), an*
*d e : X !
B X is a homotopy equivalence from X to the space B X of based loops in B X . A
p-compact group is a loop space X such that
(1) X is Fp-finite(in the sense that H*(X ; Fp) is finite dimensional),
(2) ss0X is a finite p-group, and
(3) ssiX (i 1) is a finitely generated module over the ring Zp of p-adic i*
*ntegers.
A homomorphism f : X ! Y between loop spaces is a pointed map Bf : BX ! BY; if X
and Y are p-compact groups the homomorphism f is said to be a monomorphism if t*
*he ho-
motopy fibre Y=f(X ) of Bf is Fp-finite(in this case, if the homomorphism f is *
*understood,
we will refer to X as a subgroup of Y). If f : X ! Y is a homomorphism of loop *
*spaces,
the centralizer of f(X ) in Y, denoted CY (f(X )), is the loop space Map (B X *
*; BY)Bf.
Here Map (B X ; BY)Bf is the component containing Bf of the space of (unpointed*
*) maps
_____________
The authors were supported in part by the National Science Foundation.
Typeset by AM S-*
*TEX
1
2 W. DWYER AND C. WILKERSON
from BX to BY. There is a natural loop space homomorphism CY (f(X ))!- Y, indu*
*ced
by evaluation of maps at the basepoint of BX .
A p-compact group X is said to be abelian if CX (id(X ))!- X is an equivale*
*nce, where
id : X!- X is the identity homomorphism. One of the first things we prove in t*
*he paper
is
1.1 Theorem. Any abelian p-compact group is equivalent to the product of a fini*
*te
abelian p-group and a p-compact torus.
Note that a p-compact torus is just the Fp-completion of an ordinary torus.
A homomorphism f : X ! Y of p-compact groups is said to be central if the ho*
*momor-
phism CY (f(X ))!- Y is an equivalence.
1.2 Theorem. Let X be a p-compact group. Then any central subgroup of X is abel*
*ian.
Moreover, X has (in an appropriate sense) a unique maximal central subgroup, de*
*noted
Cc(X ). If X is connected, then Cc(X ) is a subgroup of the maximal torus of X *
*and can
be described explicitly in terms of the normalizer of the torus.
The above maximal central subgroup Cc(X ) is called the p-compact center of *
*X . It
is possible to form a quotient p-compact group X =Cc(X ), and it turns out that*
* if X is
connected this quotient has a trivial p-compact center. The quotient X =Cc(X ) *
*is what we
call the adjoint form of X .
For a p-compact group X , Cc(X ) is one candidate for the "center" of X . T*
*here is
another candidate, already mentioned above. This is the loop space CX (id(X )),*
* which is
denoted Ch(X ) of X and called the homotopy center of X . We will prove the fo*
*llowing
theorem (see [11, x5] for what is in retrospect a special case).
1.3 Theorem. Let X be a p-compact group. Then the evident (x11) loop space homo-
morphism Cc(X )!- Ch(X ) is an equivalence.
This gives a generalization to non-simple groups of a fact which was proved *
*for G =
SO (3) in [9] and for general simple groups G in [15, Theorem 3].
1.4 Theorem. Let G be a compact connected Lie group. Then the natural map
B Center(G)!- Map (B G; BG)id
induces an isomorphism on homology with any finite coefficients.
1.5 Organization of the paper. Section 2 gives some general properties of p-com*
*pact groups,
including a recognition principle (2.15) for maximal tori, and x3 describes the*
* sense in which
certain p-compact groups have discrete approximations. Section 4 defines the n*
*otion of
maximal rank subgroup and gives an "infinitesimal" criterion (4.7) for the incl*
*usion of
such a subgroup to be an isomorphism. Section 5 contains the proof of 1.1, whi*
*le x6
contains a construction of the p-compact center and a study of the adjoint form*
*. In x7
there is an explicit calculation of the center of a connected p-compact group i*
*n terms of
the normalizer of a maximal torus; most of the interest here is at the prime 2,*
* where the
same complications arise for p-compact groups as for compact Lie groups. Sectio*
*n 7 also
gives the Weyl group structure of the centralizer of a subgroup of the torus (7*
*.6). Section 8
CENTER OF A p-COMPACT GROUP 3
shows that the homology decomposition theorem of Jackowski and McClure [14] can*
* be
extended to p-compact groups, and x9 illustrates how to use this decomposition *
*theorem to
construct inductive proofs; the actual statement proved in x9 is a generalizati*
*on of Miller's
Theorem [17]. There are a few technical results in x10, including a key splitti*
*ng theorem
(10.7). Finally, x11 has the proof of 1.3 and x12 the proof of 1.4.
1.6 Notation and terminology. In this paper p denotes a fixed prime number, Fp *
*the field
with p elements, Zp the ring of p-adic integers, and Qp the field Q Zp.
For a space X the symbol H *X denotes the mod p cohomology ring H *(X ; Fp) *
*and
H Qp(X ) the ring Q H*(X ; Zp). The space X is Fp-finiteif H *X is finite dim*
*ensional,
and f : X!- Y is an Fp-equivalence if it induces an isomorphismPH *Y ~= H*X .*
* If X
is Fp-finitethen the EulerPcharacteristic of X is the sum i(-1)irkFp HiX ; it*
* turns out
that this is the same as i(-1)irkQp HiQp(X ). The ring H*Qp(X ) serves as a r*
*eplacement
for rational cohomology in cases in which the actual rational cohomology ring m*
*ay be too
large to be useful [12, x4].
We assume that any space in this paper has the homotopy type of a CW-complex*
*; if not,
the space can be replaced by the geometric realization of its singular complex.*
* The word
"equivalence" stands for "homotopy equivalence". We will use Comp Fp(-) to deno*
*te the
homotopical Fp-completion functor constructed in [3] (see [12, x11] for a list *
*of some of the
properties of this functor); a space X is Fp-complete if the natural map X!- C*
*omp Fp(X )
is an equivalence.
1.7 Remark. Some of the results in this paper, including Theorem 1.1 and the fi*
*rst part
of Theorem 1.2, have been obtained independently by Moller and Notbohm in [18].
x2. Loop spaces and p-compact groups
In this section we will summarize a few basic facts about loop spaces and p-*
*compact
groups. Note that any topological group G is a loop space in a natural way: the*
* space BG
is the usual classifying space of G. For example, any discrete group is a loop *
*space. One
way to motivate some of the definitions below is to look at what they specializ*
*e to in the
case of discrete groups [12, x3].
2.1 Loop spaces. If X and Y are loop spaces, a homomorphism f : X ! Y is by
definition a pointed map B f : B X!- B Y; the space of homomorphisms X ! Y is *
*the
space Map *(B X ; BY) of pointed maps BX ! BY. An outer homomorphism X ! Y is a
map BX ! BY which does not necessarily preserve the basepoints. Two homomorphis*
*ms
f and g are homotopic if Bf and Bg are homotopic as pointed maps; the homomorph*
*isms
are conjugate if they are homotopic as outer homomorphisms, i.e., if Bf and Bg *
*are freely
homotopic. A homomorphism f is trivial if Bf is the constant map, and an equiva*
*lence if
B f is an equivalence. A short exact sequence
f g
(2.2) {1}!- X!- Y!- Z!- {1}
Bf Bg
of loop spaces is by definition a fibration sequence BX --! B Y -! BZ.
4 W. DWYER AND C. WILKERSON
If f : X ! Y is a homomorphism of loop spaces, the homogeneous space Y=f(X )*
* is
defined to be the homotopy fibre of B f over the basepoint of B Y; this space i*
*s denoted
Y=X if f is understood. The centralizer of f(X ) in Y, denoted CY (f(X )) (or C*
*Y (X ) if f
is understood) is the loop space of the mapping space component Map (B X ; BY)B*
*f. (The
map Bf is used as a basepoint in forming this loop space.) Evaluation at the ba*
*sepoint of
B Y gives a loop space homomorphism CY (X )!- Y, and the homomorphism f : X ! Y
is said to be central if the map CY (X )!- Y is an equivalence.
2.3 p-compact groups. A p-compact group is a loop space X such that X is Fp-fin*
*ite
and BX is Fp-complete, or equivalently [12, x2] a loop space X which satisfies *
*the three
conditions listed in x1. If X is a p-compact group and H ss0X a subgroup, the *
*inverse
image XH in X of H is a p-compact group with B (XH ) an appropriate covering s*
*pace
of B X ; for instance, the identity component X1 = X{1} of X is a p-compact gro*
*up. If
f : X ! Y is a homomorphism of p-compact groups, then f is said to be a monomor*
*phism
if Y=X is Fp-finiteand an epimorphism if Y=X = BZ for a p-compact group Z. In t*
*he
first case X is said to be a subgroup of Y.
2.4 Proposition. [12, 2.3, 9.11] For any p-compact group X the cohomology ring *
*H*B X
is noetherian. If f : X !- Y is a homomorphism of p-compact groups, then f i*
*s a
monomorphism if and only if under (B f)* the ring H*B X is finitely generated a*
*s a module
over H*B Y.
Note that a composite of monomorphisms is a monomorphism. In a short exact s*
*equence
as in 2.2 of p-compact groups, the homomorphism f is a monomorphism and g is an
epimorphism.
If G is a compact Lie group such that ss0G is a p-group, we will let G^ deno*
*te the p-
compact group with B(G^) = Comp Fp (B G)[12, x11]; as a space ^Gis just Comp Fp*
* G. A
p-compact torus is a p-compact group of the form ^T, where T = SO (2)r is an or*
*dinary
torus; more generally, a p-compact toral group G is a p-compact group such that*
* the
identity component G1 is a p-compact torus. We will work extensively with p-co*
*mpact
toral groups because they have good technical properties, some of which are lis*
*ted in the
next two propositions.
2.5 Proposition. [12, x6] Suppose that f : G ! X is a homomorphism of p-compact
groups, where G is a p-compact toral group. Then
(1) CX (G) is a p-compact group, and
(2) the map CX (G)!- X is a monomorphism.
2.6 Proposition. [12, 8.6] Suppose that f : G ! X is a homomorphism of p-compact
groups, where G is a p-compact toral group. Then f is central if and only if th*
*ere exists a
homomorphism G x X!- X which is homotopic to f on G x {1} and to idX on {1} x *
*X .
Remark. In saying that a homomorphism h : G x X!- X is "homotopic to f on G x *
*{1}",
for instance, we mean that the restriction of Bh to BG ~=BG x * BG x BX is hom*
*otopic
to Bf. Proposition 2.6 is true without the assumption that G is a p-compact tor*
*al group
(10.2), but we do not know how to give a similar generalization of 2.5.
CENTER OF A p-COMPACT GROUP 5
A loop space X is said to be abelian if the identity map of X is central. An*
*y p-compact
torus is abelian, as is more generally the product of a p-compact torus and a f*
*inite abelian
p-group. (In 5.2 we will show that any abelian p-compact group is of this secon*
*d form.)
2.7 Proposition. [12, 8.2] Let f : G ! X be a homomorphism of p-compact groups,
where G is an abelian p-compact toral group. Then f lifts naturally up to homot*
*opy to a
central homomorphism G!- CX (G).
2.8 Proposition. [12, 8.3] Suppose that f : G!- X is a central monomorphism of*
* p-
compact groups, where G is an abelian p-compact toral group. Then there is a p-*
*compact
group structure on the homogeneous space X =G and a natural short exact sequence
{1}!- G!- X!- X =G!- {1} :
Maximal tori and Weyl groups. If X is a p-compact group, a homomorphism f : T!-
X with T a p-compact torus is said to be a maximal torus for X if the map T!- *
*CX (T ) of
2.7 gives an equivalence between T and the identity component CX (T )1 of its c*
*entralizer.
Sometimes for brevity we will simply say that T is a maximal torus for X . For *
*a maximal
torus f : T!- X the Weyl space WX;f of T is defined to be the space of self-eq*
*uivalences
of BT over BX (constructed after replacing Bf : BT!- BX by an equivalent fibra*
*tion).
2.9 Theorem. [12, 9.5] Any p-compact group X has a maximal torus f : T!- X , u*
*nique
up to conjugacy. If T is such a maximal torus, then the space WX;f is homotopi*
*cally
discrete and ss0WX;f is a finite group (under composition). The order of ss0WX;*
*f is equal
to the Euler characteristic O(X =T ).
If f : T!- X is a maximal torus for X , the Weyl group WX;f is defined to b*
*e the group
ss0WX;f; this is sometimes denoted WX;T or WX and called the Weyl group of X .*
* The
normalizer N(T ) of T is the loop space whose classifying space is the Borel co*
*nstruction
of the action of WX;f on BT . It is clear that BN(T ) is up to homotopy a fibra*
*tion over
B WX with fibre B T , so that N(T ) is not a p-compact group in general unless*
* WX is a
p-group. However, the loop space Np(T ) obtained as the union of components of*
* N(T )
corresponding to a Sylow p-subgroup of WX is always a p-compact group. The p-co*
*mpact
group Np(T ) is called a "p-normalizer" of T . By construction, the homomorphis*
*m T!- X
factors as a composite
(2.9) T!- Np(T )!- N(T )!- X :
2.10 Proposition. [12, 9.9] If X is a p-compact group with maximal torus T , t*
*hen
the above homomorphism Np(T )!- X is a monomorphism, and the Euler characteris-
tic O(X =Np(T )) is relatively prime to p.
2.11 Remark. The loop space homomorphism N(T )!- X in 2.9 induces a group homo-
morphism f : WX = ss0N(T )!- ss0X . By definition the kernel of f is the Weyl *
*group of
the identity component X1 of X . The map f is onto; this comes down to showing *
*that if U
is any component of X =T then the homotopy fixed point set UhT is nonempty [12,*
* proof
6 W. DWYER AND C. WILKERSON
of 9.5], which is proved as in [12, proof of 8.11] by observing that O(U) 6= 0.*
* Since ss0X is
a finite p-group, it follows that the map ss0Np(T )!- ss0X is also onto.
Recall that an element of finite order in GL (r; Qp) or GL (r; Zp) is said t*
*o be a reflection
(or sometimes a pseudoreflection) if it pointwise fixes a codimension 1 subspac*
*e of (Qp )r.
The rank of a p-compact torus T is the number r such that T ~=G^ with G = SO (2*
*)r; this
equals the rank over Zp of the free module H2(B T; Zp).
2.12 Proposition. [12, 9.7] Let X be a connected p-compact group with maximal t*
*orus T ,
and let r be the rank of T . Then the natural action of the Weyl group WX on H2*
*(B T; Zp) is
faithful and and represents WX as a finite subgroup of GL (r; Zp) generated by *
*reflections.
2.13 Remark. The number r is 2.12 is called the rank of X ; it is also the numb*
*er of
generators in the polynomial algebra H *QpBX [12, 9.7]. Since H 2(B T; Zp) is t*
*he Zp-dual
of ss2B T , it follows that the natural action of WX on ss2B T is also faithful*
* and represents
WX as a group generated by reflections.
Euler characteristics. Here we extract two useful properties of Euler character*
*istics
from [12].
2.14 Proposition. Let f : X!- Y be a monomorphism of p-compact groups, G a p-
compact toral group, and g : G ! Y a homomorphism of loop spaces. Suppose that *
*one
of the following conditions holds.
(1) G is a p-compact toral group and O(Y=X ) is not divisible by p.
(2) G is a p-compact torus and O(Y=X ) is not zero.
Then g lifts up to conjugacy to a homomorphism "g: G!- X .
Sketch of proof. According to [12, 3.3] this is a question of proving that in t*
*he given cases
the homotopy fixed point set (Y=X )hG is nonempty. Case (2) then follows from c*
*ombining
[12, 4.7, 5.7, 6.7] (see also the proof of [12, 8.11]). Case (1) is proved in *
*essentially the
same way, but using [12, 4.6] and the inductive argument in the proof of [12, 4*
*.7] instead
of using [12, 4.7] itself.
2.15 Proposition. Suppose X is a p-compact group, T is a p-compact torus, and i*
* : T!-
X is a monomorphism. Then i is a maximal torus for X if and only if O(X =T ) 6=*
* 0.
Proof. If i is a maximal torus, then O(X =T ) 6= 0 by 2.9. Suppose then that O(*
*X =T ) 6= 0.
Let j : T 0-! X be a maximal torus for X ; note that j is also a monomorphism. *
*According
to 2.14, there are homomorphisms f : T!- T 0and g : T 0-! T such that j . f is*
* conjugate
to i and i . f0 is conjugate to j, and a simple argument using 2.4 shows that t*
*he composites
f . g and g . f are also monomorphisms. It is elementary to prove using 2.4 th*
*at any
monomorphism T!- T or T 0-! T 0is an equivalence; for instance, this applies t*
*o f . g and
g . f. It follows that f and g are equivalences, and hence that T is conjugate *
*in X to the
maximal torus T 0.
2.16 Remark. Let G be a compact Lie group with ss0G a p-group, and T!- G a Li*
*e-
theoretic maximal torus; it is a classical result that the Euler characteristic*
* O(G=T ) is
nonzero. An easy argument with the fibre lemma (cf. [12, proof of 5.7]) shows t*
*hat G=T
CENTER OF A p-COMPACT GROUP 7
is Fp-good and that G^=T^ is equivalent to the Fp-completion of G=T . It follo*
*ws that
O(G^=T^) 6= 0 and thus by 2.15 that ^Tis a maximal torus for ^G.
x3. Discrete approximations
Surprisingly, some questions about p-compact groups can be reduced to questi*
*ons about
discrete groups. This stems mostly from the fact that there is an effective way*
* of construct-
ing "discrete approximations" for p-compact toral groups.
Let Z=p1 denote the group Z[1=p]=Z = [nZ=pn. By definition, a p-discrete to*
*rus is a
discrete group A which is isomorphic to (Z=p1 )r for some r. A p-discrete toral*
* group is
a discrete group G with a normal subgroup A such that A is a p-discrete torus a*
*nd G=A
is a finite p-group. The p-discrete torus A is then the maximal divisible subgr*
*oup of G.
Any p-discrete toral group G can be expressed as an increasing union of finite *
*p-groups
[12, 6.19].
The closure G of a p-discrete toral group G is defined to be the loop space *
* Comp Fp(B G).
The completion map BG!- Comp Fp(B G) is an isomorphism on mod p cohomology [12,
6.9] and gives a homomorphism G!- G.
3.1 Proposition. If G is a p-discrete toral group, then G is a p-compact toral*
* group
and the homogeneous space G=G is an Eilenberg-MacLane space of type K(V; 1) for*
* some
rational vector space V .
Proof. All except the last statement is proven in [12, proof of 6.9]. By the a*
*rguments
there, and the fact that the completion functor Comp Fp(-) preserves products [*
*3, I, x7], it
is enough to show that if G = Z=p1 then the homotopy fibre of the map ffl : BG*
*!- BG is
of the indicated type. There is a short exact sequence
{1}!- Zp!- Qp !- Z=p1 !- {1}
of abelian groups, which gives rise to a fibration f : K(Z=p1 ; 1)!- K(Zp ; 2)*
* with fibre
K(Qp ; 1). The target of the map f is Fp-complete, and f induces an isomorphis*
*m on
mod p homology because the fibre of f has the mod p homology of a point. This i*
*mplies
that map f is in fact equivalent to the completion map ffl and gives the requir*
*ed calculation
of the homotopy fibre of ffl.
It is also possible to reverse the closure construction. A discrete approxi*
*mation for
a p-compact toral group G is defined to be a p-discrete toral group G together*
* with a
homomorphism ff : G!- G which induces an isomorphism H*B G!- H*B G.
3.2 Proposition. Any p-compact toral group G has a discrete approximation G!- *
*G.
If G and H are p-compact toral groups with discrete approximations G and H , t*
*hen
any homomorphism f : G!- H lifts uniquely to a homomorphism f : G !- H . The
homomorphism f is central if and only if its lift f is central, which is the ca*
*se if and only
if f(G ) is contained in the center of H .
Sketch of proof. The fact that discrete approximations exist is proven in [12, *
*6.8]. Suppose
that a : G !- G is a discrete approximation. The map B a : B G!- BG induces*
* an
8 W. DWYER AND C. WILKERSON
isomorphism on mod p homology and the target of Ba is Fp-complete; this implies*
* that
B a is equivalent to the Fp-completion map BG !- Comp Fp(B G) and thus (3.1) t*
*hat the
homotopy fibre of Ba is a space of type K(V; 1) for a rational vector space V .*
* The required
lifting result now comes from obstruction theory. The final statement comes ou*
*t of the
observation that f is central if and only if there is a suitable homomorphism G*
* x H!- H
(2.6), whereas the map fof K(ss; 1)'s is visibly central if and only if there i*
*s a corresponding
homomorphism G x H!- H ; one type of homomorphism can be obtained from the oth*
*er
by an appropriate closure/discrete approximation construction.
3.3 Remark. Note that by 3.2 any two discrete approximations of a p-compact tor*
*al group
G are canonically isomorphic (as groups). If G!- G is a discrete approximation*
*, then by
[12, 6.8] there is a short exact sequence
{1}!- (ss1G) Z=p1 !- G !- ss0G!- {1} :
The previous two propositions allow us to shift back and forth easily betwee*
*n p-compact
toral groups and their discrete approximations. Because of the following propos*
*ition, which
we think of as expressing the density in G of a discrete approximation, this sh*
*ifting process
is usually innocuous.
3.4 Proposition. (Density) [12, 6.7] Let G be a p-compact toral group, ff : G!-*
* G a
discrete approximation, and U an Fp-complete space (e.g., U = BX for a p-compac*
*t group
X ). Then B ff induces equivalences Map (B G; U)!- Map (B G; U) and Map *(B G*
*; U)!-
Map *(B G; U).
Remark. For example, if G is a p-discrete toral group and f : G!- X is a homom*
*or-
phism with X a p-compact group, then "passing to the closure" (i.e. applying th*
*e functor
Comp Fp(-) to Bf : BG!- BX ) gives a homomorphism f : G!- X .
Kernels and exact sequences. One of the most useful features of a p-discrete to*
*ral
group is that it has concrete elements, so that it is possible to describe kern*
*els (or later on
in x7 centers) as explicit subgroups.
Let G be a p-discrete toral group , x 2 G an element and f : G ! X a homomor*
*phism
of loop spaces. We will say that f(x) is trivial if the restriction of f to the*
* cyclic subgroup
of G generated by x is trivial (2.1). The kernel of f, denoted ker(f) is the s*
*ubset of G
given by {x | f(x) is trivial}. The kernel of f is said to be trivial if it con*
*tains only the
identity element of G.
3.5 Proposition. [12, x7]. Suppose that G is a p-discrete toral group, X a p-c*
*ompact
group, and f : G ! X a homomorphism. Then
(1) ker(f) is a normal subgroup of G,
(2) up to homotopy there is a unique homomorphism f0 : G= ker(f)!- X whi*
*ch
extends f,
(3) ker(f0) is trivial, and
(4) the induced homomorphism CX (G= ker(f))!- CX (G) is an equivalence.
The next proposition provides a simple way to recognize monomorphisms.
CENTER OF A p-COMPACT GROUP 9
3.6 Proposition. [12, 7.3] Suppose that G is a p-compact toral group, and f : G*
* ! X is
a homomorphism of p-compact groups. Let ff : G!- G be a discrete approximation*
*. Then
f is a monomorphism if and only if ker(f . ff) is trivial.
It is also useful to be able to recognize exact sequences.
3.7 Proposition. Consider a commutative diagram of loop spaces and homomorphisms
G ----! H ----! K
?? ? ?
y ?y ?y
G ----! H ----! K
in which the lower row is a sequence of p-compact toral groups and the upper ro*
*w is the
corresponding (3.2) sequence of their discrete approximations. Then the lower s*
*equence is
exact if and only if the upper one is.
Proof. Suppose that the upper sequence is exact; in this case the problem is to*
* show
that the induced fibre sequence of classifying spaces remains a fibre sequence *
*after Fp-
completion. By the fibre lemma [12, 11.7] this will be the case if the fundamen*
*tal group
of the base, namely K , acts nilpotently on the mod p homology of the fibre. S*
*ince the
mod p homology groups of the fibre are finite dimensional in each dimension [12*
*, x12], the
action of K on these homology groups must factor through the finite p-group obt*
*ained by
dividing K by its maximal divisible subgroup. This implies that the action is *
*nilpotent
[12, 11.6].
Suppose on the other hand that the lower sequence is exact. The map H!- K *
*is onto
by the description of discrete approximations given in 3.3: clearly ss0H!- ss0*
*K is onto,
and, since the cokernel of ss1H!- ss1K is a finite abelian p-group, (ss1H) Z*
*=p1 !-
(ss1K) Z=p1 is onto. Let G0 be the kernel of H!- K . Elementary algebra show*
*s that
G0 is a p-discrete toral group, and by the uniqueness provision of 3.2 the map *
*G!- H lifts
to a map i : G!- G0. Let G0 be the closure of G . The argument in the paragrap*
*h above
shows that G0-! H!- K is an exact sequence of p-compact toral groups, so by un*
*iqueness
of homotopy fibres the map G!- G0 induced by the closure of i is an equivalenc*
*e, and
hence (3.3) the map i is an isomorphism.
3.8 Remark. One consequence of 3.7 is the following: if G!- T!- K is a shor*
*t exact
sequence of p-compact toral groups in which T is a p-compact torus, then G is a*
*belian
and K is a p-compact torus. The proof of this amounts to inspecting the corresp*
*onding
exact sequence of discrete approximations and noticing that any subgroup of (Z=*
*p1 )r is
abelian, and any quotient group of (Z=p1 )r is (Z=p1 )s for some s.
3.9 One parameter subgroups. The p-discrete torus Z=p1 and its closure functio*
*n like
"one-parameter subgroups" of p-compact groups.
If f : X!- Y is a homomorphism of loop spaces, we will write f(X ) Y1 (and*
* say that
f(X ) lies in the identity component of Y) if f lifts to a homomorphism X!- Y1.
10 W. DWYER AND C. WILKERSON
3.10 Proposition. Suppose that f : Z=pn!- X is a homomorphism, where X is a
p-compact group . Then f(Z=pn) X1 if and only if f extends to a homomorphism
f0 : Z=p1 !- X .
Proof. First suppose that the extension f0 exists, and let A be the closure of *
*Z=p1 . Then
A is connected (proof of 3.1) and the fact that f(Z=pn) X1 follows from the fa*
*ct that
passing to closure gives an extension of f to a homomorphism A!- X . If f(Z=pn*
*) X1
then the required extension f0 exists by an inductive argument using [12, 5.5].
3.11 Proposition. Suppose that X is a p-compact group. Then X is connected if a*
*nd
only if every homomorphism f : Z=pn!- X (n 1) extends to a homomorphism f0 :
Z=p1 !- X .
Proof. If X is connected, the desired extensions exist by 3.10. Suppose conver*
*sely that
X is not connected, and let Np (T ) be a discrete approximation to a p-normaliz*
*er (2.10)
of a maximal torus T for X . By 2.11 the map Np (T ) = ss0Np (T )!- ss0X is*
* surjec-
tive. Since Np (T ) is a union of finite p-groups, this implies that there exis*
*ts a composite
homomorphism f : Z=pn!- Np (T )!- X such that f(Z=pn) * X1.
3.12 A slight generalization. At some points in this paper it will be convenien*
*t for us
to have a notion of discrete approximation somewhat more general than the one a*
*bove.
Definition. A loop space G is an extended p-compact torus if ss0G is finite and*
* the identity
component G1 is a p-compact torus. A discrete group G is an extended p-discrete*
* torus if
G has a normal subgroup T of finite index such that T is a p-discrete torus.
Note that if G is an extended p-discrete torus there is a unique normal p-di*
*screte torus
T in G such that G=T is finite; in fact T is the maximal divisible subgroup of *
*G. We will
denote this unique p-discrete torus by G(1). If f : G!- X is a homomorphism o*
*f loop
spaces and ss0X is a finite group, then f induces a homomorphism G=G(1)-! ss0X *
*and the
restriction of f to G(1)lifts uniquely to a homomorphism G(1)-! X1.
Definition. A discrete approximation for an extended p-compact torus G is a hom*
*omor-
phism f : G!- G, where G is an extended p-discrete torus and f induces
(1) an isomorphism G =G(1)!- ss0G, and
(2) an isomorphism H*B G1!- H*BG(1) .
It is not hard to see that the above notion specializes to the previous noti*
*on of discrete
approximation (3.2) if G is a p-compact toral group (i.e., if ss0G is a p-group*
*). The following
proposition is proved in exactly the same way as 3.2 (cf. [12, 6.8]).
3.13 Proposition. Any extended p-compact torus G has a discrete approximation G*
*!-
G. If G and H are extended p-compact tori with discrete approximations G and H *
*, then
any homomorphism f : G!- H lifts uniquely to a homomorphism f : G!- H .
In particular, any two discrete approximations for an extended p-compact tor*
*us are
canonically isomorphic (as ordinary discrete groups).
CENTER OF A p-COMPACT GROUP 11
x4. Monomorphisms of maximal rank
In this section we will work out some points connected with maximal rank sub*
*groups
of a p-compact group. The conclusion is a theorem which can be interpreted as *
*giving
an "infinitesimal" criterion for a homomorphism of connected p-compact groups t*
*o be an
equivalence (4.7).
4.1 Definition. A homomorphism f : X ! Y of p-compact groups is said to be of m*
*aximal
rank if f is a monomorphism and there is a maximal torus i : T ! X for X such t*
*hat f . i
is a maximal torus for Y.
4.2 Remark. It follows from the uniqueness of maximal tori (2.9) that if f : X *
*! Y is
of maximal rank then any maximal torus for X gives a maximal torus for Y. Simi*
*larly,
if f is a monomorphism then f is of maximal rank if and only if some maximal to*
*rus
T!- Y for Y lifts to a torus T!- X ; such a lift is necessarily a maximal tor*
*us for X (to
see this, combine 2.15 with the multiplicativity of the Euler characteristic in*
* the fibration
X =T!- Y=T!- X =Y, e.g. 10.6).
There is one simple way to construct homomorphisms of maximal rank.
4.3 Propositions. Let X be a p-compact group with maximal torus T!- X , suppose
that A is a p-compact toral group, and let j : A ! T be a monomorphism. Then t*
*he
natural monomorphism CX (A)!- X is of maximal rank.
Proof. There is a commutative diagram
CT (T )----! CT (A) ----! CX (A)
? ? ?
~=?y ?y ?y
=
T ----! T ----! X
in which the left vertical arrow is an equivalence because T is abelian. This p*
*rovides a lift
of T up to conjugacy to CX (A).
4.4 Lemma. Suppose that f : X ! Y is of maximal rank, i : T!- X is a maximal t*
*orus
for X , and f . i : T ! Y the corresponding maximal torus for Y. Then there is *
*a natural
monomorphism of Weyl groups WX;i-! WY;fi.
Remark. The above monomorphism WX;i!- WY;fidepends only up to inner automor-
phism on the choice of a torus T!- X for X . Sometimes for brevity we will say*
* that f
induces a monomorphism WX!- WY .
Proof of 4.4. By definition, the Weyl group WX;i is ss0WX;i, where WX;i is the *
*space
of self-maps over B X of a fibration u : U ! B X equivalent to i : B T !- BX .*
* Sim-
ilarly, WY;fi = ss0WY;fiwhere WY;fiis the space of self-maps over B Y of a fibr*
*ation
v : V ! B Y equivalent to fi : B T!- BY. We can assume without loss of gene*
*rality
that f : BX!- B Y is a fibration and choose v = f . u; in this case there is e*
*ssentially an
inclusion WX;i-! WY;fiof topological monoids and it is not hard to check that t*
*he in-
duced component homomorphism WX;i-! WY;fidoes not depend on the fibration choic*
*es
12 W. DWYER AND C. WILKERSON
involved. This gives the required map WX;i-! WY;fi; it remains to show that the*
* map is
a monomorphism.
As in [12, 9.5] the group WX;iis isomorphic to the set of components of the *
*homotopi-
cally discrete space (X =T )hT, and WY;fito the set of components of (Y=T )hT(s*
*ee [12, x10]
for a discussion of these homotopy fixed point sets). The map between these gro*
*ups can
be produced on the level of sets by applying the functor (-)hT to the map X =T!*
*- Y=T
induced by f. Let C be the component of (Y=X )hT corresponding to i : T ! X [1*
*2,
3.3]. Applying (-)hT to the fibration X =T!- Y=T!- Y=X [12, 10.6] shows th*
*at the
map (X =T )hT-! (Y=T )hTis an injection on ss0 if and only if the natural actio*
*n of ss1C
on ss0(X =T )hTis trivial. To prove this action is trivial, we will prove that*
* C is sim-
ply connected. Applying (-)hT to the fibration Y=X !- BX !- BY (with T "acti*
*ng"
[12, 10.8] trivially on base and total space) shows that C is the basepoint com*
*ponent of
CY (T )=CX (T ). This last homogeneous space is in fact homotopically discrete*
* because
CX (T )!- CY (T ) is a homomorphism of p-compact groups which is an equivalenc*
*e on
identity components, both identity components being equivalent to T (2.9).
If X is a space let cdFp X denote the largest integer n such that H nX 6= 0.*
* Similarly,
let cdQp X denote the largest integer n such that HnQpX 6= 0.
4.5 Lemma. If X is a p-compact group then cdFpX = cdQp X .
Proof. This follows from the Bockstein spectral sequence arguments in [4]. The *
*arguments
depend only on the fact that X is an Fp-finiteH-space.
4.6 Lemma. Suppose that f : X!- Y is a monomorphism of p-compact groups. Then
(1) cdFpY = cdFpX + cdFpY=X ,
(2) cdFpY = cdFpX if and only if the map f gives an equivalence between X a*
*nd a
union of components of Y, and
(3) If X is connected then cdQp Y = cdQp X + cdQpY=X .
Proof. The statements involving cdFp Y are from [12, 6.14]. The equality in (3*
*) follows
along the lines of [12, proof of 6.14] from an examination of the Serre spectra*
*l sequence of
the fibration X!- Y!- Y=X . Note that in this fibration the fundamental group*
* of the
base acts trivially on the (co)-homology of the fibre, since the fibration is p*
*ulled back from
the path fibration over the simply connected space BX .
4.7 Theorem. A monomorphism f : X ! Y between connected p-compact groups is an
equivalence if and only if f is of maximal rank and induces (4.4) a Weyl group *
*isomorphism
WX!- WY .
Proof. It is only necessary to show that if f is of maximal rank and induces a *
*Weyl group
isomorphism then f is an equivalence. Let T be a maximal torus for X , let W d*
*enote
WX;T ~=WY;T, and let R denote the ring of invariants (H *QpBT )W . Write S for *
*the ring
Qp R H *QpBT . According to [12, 9.7], H *Qp(X =T ) and H *Qp(Y=T ) are both *
*isomorphic
to S. By 4.6(3), cdQp X = cdQp Y and hence cdFp X = cdFp Y by 4.5. The result*
* now
follows from 4.6(2).
CENTER OF A p-COMPACT GROUP 13
x5. Abelian p-compact groups and central monomorphisms
The main goal of this section is to prove the following theorem.
5.1 Theorem. If f : X ! Y is a central monomorphism of p-compact groups, the X *
*is
equivalent to the product of a p-compact torus and a finite abelian p-group.
This is analogous to the theorem that if f : G ! H is a monomorphism of comp*
*act Lie
groups with f(G) contained in the center of H, then G is isomorphic to the prod*
*uct of a
torus with a finite abelian group. Applying 5.1 to the identity map of a p-comp*
*act group
X gives the following corollary (the "if" part of which is essentially obvious).
5.2 Corollary. A p-compact group X is abelian if and only if X is equivalent to*
* the
product of a p-compact torus and a finite abelian p-group.
Remark. A beautiful result of Bousfield [2, 6.9] implies that if X is a loop sp*
*ace which
is abelian in the sense of x2 then B X is equivalent to a product of Eilenberg-*
*MacLane
spaces. Given this result, it is possible to prove 5.2 by just enumerating the *
*products of
Eilenberg-MacLane spaces which can be classifying spaces of p-compact groups. W*
*e will
use a different approach.
The proof of 5.1 depends on a few lemmas.
5.3 Lemma. Suppose that f : X ! Z and g : Y!- Z are homomorphisms of loop
spaces and that g is central. Then there is up to conjugacy a unique homomorph*
*ism
(f; g) : X x Y!- Z which is conjugate to f on X x {1} and to g on on {1} x Y. *
*The
homomorphism (f; g) is central if f is.
Proof. This follows from the equivalence
Map (B X x BY; BZ) ~=Map (B X ; Map (B Y; BZ)) :
The assumption that g is central gives Map (B Y; BZ)Bg ~=B Z.
5.4 Lemma. Let G be a p-discrete toral group, X a p-compact group, and f : G!- *
* X
a homomorphism. Suppose that A is a cyclic p-group with generator a and i; j : *
*A!- G
two homomorphisms such i(a) commutes with j(a), f . i is conjugate to f . j, an*
*d f . i is
central. Then i(a)-1j(a) 2 ker(f).
Proof. Let s : A x A!- G be given by s(x) = i(x)j(x). By the uniqueness provis*
*ion of
5.3, the composite f . s is equal to the composite of f . i with the sum map A *
*x A!- A.
This implies directly that s(a-1; a) 2 ker(f).
5.5 Lemma. Let g : G!- H be a homomorphism of p-discrete toral groups and h : *
*H!-
X a homomorphism with ker(h) = {1}, where X is a p-compact group. Suppose that *
*h . g
is central. Then g is central.
Proof. Since h.g is central there is by 5.3 a homomorphism GxX!- X which is co*
*njugate
to h . g on G x {1} and to idX on {1} x X . This can be restricted to give a ho*
*momorphism
f : G x H!- X which is conjugate to h . g on G x {1} and to h on {1} x H. By *
*5.4,
for each x 2 G the element (x; g(x)-1) in G x H lies in ker(f). Since ker(f) is*
* a normal
subgroup of G x H which intersects {1} x H only in the identity element, it fol*
*lows from
elementary algebra that g(G) must lie in the center of H.
14 W. DWYER AND C. WILKERSON
5.6 Lemma. Suppose that P is a p-compact toral group, X is a p-compact group, a*
*nd
f : P ! X is a central monomorphism. Then P is equivalent to the product of a p*
*-compact
torus and a finite abelian p-group.
Proof. Let P be a discrete approximation (3.2) for P , and f the composite of f*
* with the
map P ! P . Since ker(f) is trivial (3.6), it follows from 5.5 that P is abelia*
*n and hence
for algebraic reasons isomorphic to the product of a finite abelian p-group and*
* a group of
the form (Z=p1 )r. This implies that P , which is equivalent to the closure (3.*
*1) of P , has
the desired form.
Proof of 5.1. Let i : T ! X be a maximal torus for X with discrete approxima*
*tion
T ! T . Let ibe the composite of i with T ! T . The composites f . i and f . ia*
*re central
by 2.6. Suppose that the action of the Weyl group W of the identity component *
*X1 of
X on H 2(B T ; Zp) is nontrivial. In this case there must be a self homotopy e*
*quivalence
B w : BT ! BT , not homotopic to the identity, such that Bi . Bw is homotopic t*
*o Bi. Such
a w lifts uniquely (3.2) to an automorphism w : T ! T, not equal to the identit*
*y, such
that i. w is conjugate to i. Since w is not the identity, there must be some el*
*ement x 2 T
such that w(x) 6= x. By 5.4, then, (x-1w (x)) 6= 1 belongs to ker(f . i), which*
* is impossible
(3.6) because f . i is a monomorphism. This shows that W acts trivially on H 2(*
*B T ; Zp)
and hence, since this action is faithful [12, 9.7], that W itself is trivial. B*
*y 4.7 (applied to
T!- X1), the p-compact group X1 is a torus. Since X is thus a p-compact toral *
*group,
the desired result is given by 5.6.
x6. p-compact centers and adjoint forms.
Let f : C ! X be a central monomorphism of p-compact groups. The map f is sa*
*id
to be a p-compact center for X (or more informally C is said to be a p-compact *
*center
for X ) if for any central monomorphism g : A ! X of p-compact groups there exi*
*sts
up to homotopy a unique homomorphism h : A ! C such that f . h is homotopic to
g. This signifies that up to homotopy f is a terminal object in the category o*
*f central
monomorphisms A ! X .
6.1 Theorem. Any p-compact group X has a p-compact center f : C ! X . Suppose
that f is such a p-compact center and that g : A ! X is a central homomorphism *
*in which
A is a p-compact toral group; let L(g; f) be the space of all homomorphisms h :*
* A!- C
such that f . h = g. Then L(g; f) is contractible.
Remark. Note above that if A!- X is a central monomorphism, then A is necessar*
*ily a
p-compact toral group (5.1).
6.2 Remark. Let (B f)fib: (B C)fib-! BX be a pointed fibration equivalent to Bf*
*. By the
above phrase the space of all homomorphisms h : A!- C such that f . h = g we m*
*ean the
space of all basepoint preserving maps Bh : BA!- (B C)fibsuch that (B f)fib. B*
*h = Bg.
Theorem 6.1 implies that p-compact centers are homotopically unique in a ver*
*y strong
sense. For this reason we will sometimes speak of the p-compact center of X and*
* denote
it by Cc(X ). According to 2.8 there is a natural p-compact group structure on *
*X =Cc(X ).
CENTER OF A p-COMPACT GROUP 15
6.3 Theorem. If X is a connected p-compact group then the p-compact center of X*
* =Cc(X )
is trivial.
Remark. If X is a connected p-compact group, the quotient X =Cc(X ) is called t*
*he adjoint
form Xad of X . Note that if X is not connected then X =Cc(X ) may have a nont*
*rivial
p-compact center; consider, for instance, a nonabelian finite p-group or the Li*
*e group
O (2).
Let X be a p-compact group, C a p-discrete toral group, and f : C!- X a cen*
*tral
homomorphism such that ker(f) = {1}. We will say that f is a p-discrete center *
*for X if
the induced map f : C!- X is a p-compact center for X . Such a p-discrete cent*
*er is just a
discrete approximation to Cc(X ) and has very similar uniqueness properties. In*
* the course
of proving 6.1 we will give the following description of the p-discrete center *
*of a p-compact
group X ; for connected X a more explicit description appears in x7.
6.4 Theorem. Let X be a p-compact group, T a maximal torus for X , Np(T ) the i*
*nverse
image in the normalizer of T of a p-Sylow subgroup of the Weyl group of X , and*
* Np (T )
a discrete approximation for Np(T ). Let C Np (T ) be the set of elements x 2*
* Np (T )
such that !- X is central. Then C is a subgroup of Np (T ) and C!- X is a p*
*-discrete
center for X .
6.5 Lemma. Let G be a p-compact toral group and g : G!- X a monomorphism of
p-compact groups. Suppose that that H is a p-compact toral group, and that h : *
*H!- X
is a central homomorphism. Then if h lifts to any homomorphism "h: H!- G it li*
*fts to
exactly one, in the sense that the space (cf. 6.2) of all such lifts "his contr*
*actible.
Proof. Suppose that h lifts to a homomorphism "h: H!- G. Let i : H!- H be a d*
*iscrete
approximation, and let K H be the kernel (3.5) of h . i. It is clear from 3.*
*6 that K is
also the kernel of "h. i, since otherwise, after (3.2) lifting "h. i to a map H*
* !- G for some
discrete approximation G of G, we would find a nontrivial element in the kerne*
*l of the
composite G!- G!- X . Let H0 be the closure (3.1) of H =K . By naturality of *
*the closure
construction there is a homomorphism H!- H0. The map h extends (3.5, 3.4) to a*
* central
monomorphism h0: H0!- X . Similarly, "hextends to a monomorphism "h0: H0!- G wh*
*ich
is a lift of h0. An argument using 5.5 and discrete approximations shows that "*
*h0is central.
Consider the diagram (2.8)
BG ----! B(G="h0(H0))
? ?
B g?y ?y
B h
BH ----! BX ----! B(X =h0(H0))
in which the right hand square is a homotopy fibre square and the composite of *
*the two
lower arrows is null homotopic. Let E be the homotopy pullback of BG!- BX over*
* Bh.
It follows that there is a homotopy equivalence over B H between E and B H x (X*
* =G),
and so the space of lifts up to conjugacy of h into G (i.e. the homotopy fibre*
* over B h
of Map (B H; BG)!- Map (B H; BX )) is equivalent to Map (B H; X =G). This las*
*t space is
16 W. DWYER AND C. WILKERSON
equivalent by evaluation at the basepoint of BH to X =G itself [12, 6.1 and 5.3*
*]. There is
now a 3 x 3 square
=
F ----! X =G ----! X =G
?? ? ?
y ?y ?y
Map *(B H; BG){Bh}----! Map (B H; BG){Bh}----! BG
?? ? ?
y ?y ?y
Map *(B H; BX )Bh----! Map (B H; BX )Bh ----! BX
in which the rows and columns are fibration sequences; a subscript "{B h}" deno*
*tes restrict-
ing to the components of the indicated mapping space that cover Bh up to homoto*
*py. The
space F is the desired space of lifts "h, and is clearly contractible.
6.6 Lemma. Let X be a p-compact group, P a p-discrete toral group, and f : P ! *
*X a
monomorphism. Let C P be the set of all elements x 2 P such that f() is cen*
*tral.
Then C is a subgroup of P and f(C) is central.
Proof. By 5.5, C is contained in the center of P . It is enough to show that if*
* {Coe}oe2Sis
a countable collection of subgroups of the center of P such that f(Coe) is cent*
*ral for each
oe 2 S, then the subgroup of the center of P generated by the Coeis central. F*
*or this,
observe first that if C1 and C2 are two subgroups of the center of P such that *
*f(C1) and
f(C2) are central, then C1 + C2 is central; this is a consequence of 5.3 and 3.*
*5. Secondly,
apply a homotopy limit argument [12, 6.20] to show that if C0 C1 C2 . .i.s an
increasing chain of subgroups of the center of P such that f(Ci) is central for*
* each i, then
f([iCi) is central.
Proof of 6.1 and 6.4. We will use the notation of 6.4. By 6.6 the group C is a *
*subgroup of
the center of Np (T ) and f(C ) is central. Let C!- X be the closure of C ; by*
* density the
map C!- X is central and by 3.6 it is a monomorphism (cf. 2.10). Suppose that *
*f : A!- X
is a central homomorphism in which A is a p-compact toral group; for instance, *
*f might
be a central monomorphism (5.1). Since O(X =Np(T )) is relatively prime to p (2*
*.10), the
map f lifts by 2.14 to a central (5.5) homomorphism g : A!- Np(T ). Let A!- A*
* be a
discrete approximation. By 3.2 the composite A!- A!- Np(T ) lifts to a homomo*
*rphism
g : A!- Np (T ) such that, by 2.6 and the definition of C , f(A ) C . Passing*
* to closures
gives a lift A!- C of the homomorphism f. This proves in particular that C is *
*a p-compact
center for X . The statement about L(g; f) follows from 6.5.
6.7 Remark. The subgroup C of Np (T ) considered in the above proof is contain*
*ed in
the center of Np (T ) (5.5). Let T be the maximal divisible subgroup of Np (T *
*), so that
T is a discrete approximation to the maximal torus T . If X is connected the *
*quotient
group Np (T )=T WX acts faithfully on ss2T (2.13) and hence faithfully by con*
*jugation
on T = ss2T Z=p1 (3.3). It follows that if X is connected the group C is co*
*ntained
in T and so, after passing to closures, that the map Cc(X )!- X lifts to a uni*
*que (6.5)
homomorphism Cc(X )!- T .
CENTER OF A p-COMPACT GROUP 17
6.8 Lemma. Suppose that
i
{1}!- A!- X!- Y!- {1}
is a short exact sequence of p-compact groups in which A is an abelian p-compac*
*t toral
group and Y is connected. Then the monomorphism i is central.
Proof. Since A is abelian the identity component Map (B A; BA)idof the space of*
* self ho-
motopy equivalences of B A is equivalent to B A. This implies that any fibrati*
*on over a
simply connected base with fibre BA is principal. In particular the fibration o*
*f classifying
spaces corresponding to the above short exact sequence is principal, and the co*
*rresponding
action map BAxB X!- BX has the properties required to show that i is central (*
*2.6).
Proof of 6.3. Let T be a maximal torus for X . As remarked above (6.7), the hom*
*omorphism
Cc(X )!- X lifts to a homomorphism Cc(X )!- T ; this lift is a central (5.5) *
*monomor-
phism (3.6). Let X 0denote X =Cc(X ) and let T 0denote T=Cc(X ) (see 2.8). By 3*
*.8 the
p-compact group T 0is a p-compact torus. Consider the map of short exact sequen*
*ces
{1} ----! Cc(X ) ----! T ----! T 0----! {1}
? ? ?
=?y ?y ?y :
{1} ----! Cc(X ) ----! X ----! X 0----! {1}
Taking vertical fibres in the induced diagram of classifying spaces shows that *
*X =T ~=X 0=T 0,
so that T 0-! X 0is a maximal torus for X 0(2.15). Suppose that C0!- X 0is a p-*
*compact
center and let C!- X be the "inverse image" of C0 in X , i.e., let B C be give*
*n by the
homotopy fibre square
BC ----! BX
? ?
(6.9) ?y ?y
B C0 ----! B X 0
It is clear from a homotopy group calculation that C is a p-compact toral group*
*. Since X 0
is connected, the map C0!- X 0lifts to a monomorphism C0!- T 0as above, and s*
*o the
map C!- X lifts to a monomorphism C!- T (the fact C!- T is a monomorphism fo*
*llows
from the fact that T=C is equivalent to T 0=C0). By 3.8, then, C is abelian. Th*
*ere is an
evident short exact sequence C!- X!- X 0=C0, and applying 6.8 to it shows tha*
*t the map
C!- X is central. As a consequence, the homomorphism C!- X lifts to a homomor*
*phism
C!- Cc(X ) and so the composite C!- X!- X 0is trivial. An argument with dis*
*crete
approximations, for instance, now shows that C0!- X 0is trivial and thus that C*
* is trivial
(3.7, 3.6).
x7. Computing p-compact centers
In this section we will give an explicit way to compute the p-compact center*
* of a p-
compact group X in terms of the normalizer of a maximal torus for X . Suppose t*
*hat X is
18 W. DWYER AND C. WILKERSON
a p-compact group of rank r (2.13), T is a maximal torus for X , and W = WX is *
*the Weyl
group of X . Let N(T ) be the normalizer of T , N (T )!- N(T ) a discrete appr*
*oximation in
the sense of x3.12, and T the maximal divisible subgroup N (T )(1)of N (T ), so*
* that T is a
discrete approximation in the usual sense for N(T )1 = T .
The Weyl group W acts on T by conjugation in the group extension
(7.1) {1}!- T!- N(T )!- W!- {1}
and on BT by monodromy in the fibration
(7.2) BT!- BN(T )!- BW :
The fibration 7.2 is obtained from the exact sequence 7.1 by passing to classif*
*ying spaces
and taking fibrewise Fp-completion; under this construction the two actions cor*
*respond.
Let s 2 W be a reflection (2.12). The order ord(s) of s must divide the ord*
*er of the
multiplicative group of roots of unity in Qp *, so that ord(s) = 2 if p = 2 and*
* ord(s)
divides (p - 1) if p is odd. Since H 2(B T; Zp) is the dual over Zp of ss2B T *
*= (Zp )r, a
nonidentity element s 2 W is a reflection if and only if the fixed point set of*
* the action
of s on ss2B T = ss1T contains a subgroup isomorphic to (Zp )r-1, or equivalent*
*ly if and
only if the action of s on T = ss1T Z=p1 (see 3.3) contains a subgroup isomor*
*phic to
(Z=p1 )r-1.
7.3 Definition. If s 2 W is a reflection,
(1) the fixed point set F (s) of s is the fixed point set of the action of *
*x on T by
conjugation,
(2) the singular hyperplane H(s) of s is the maximal divisible subgroup of *
*F (s) (so
that H(s) ~=(Z=p1 )r-1),
(3) the singular coset K(s) of s is the subset of T given by elements of th*
*e form xord(s),
as x runs through elements of N (T ) which project to s in W , and
(4) the singular set oe(s) of s is the union oe(s) = H(s) [ K(s).
7.4 Remark. The singular coset K(s) is genuinely a coset of H(s) in T. On the o*
*ne hand,
suppose that x and x0 are two elements of N (T ) which project to s. Then x = *
*x0a for
some a 2 T, and it is easy to calculate directly that xord(s)= (x0)ord(s)(a), w*
*here is the
endomorphism of T which sends a to the product of the elements sia, 1 i ord(s*
*). The
image of is a divisible subgroup of T which is pointwise fixed by s and thus c*
*ontained
in H(s). It follows that xord(s)and (x0)ord(s)are in the same coset of H(s). Co*
*nversely, if
y = xord(s)2 K(s) and a 2 H(s), it is possible (because H(s) is divisible) to f*
*ind b 2 H(s)
with bord(s)= a. Then ay = (bx)ord(s).
Our main theorem in this section is the following one.
7.5 Theorem. Let X be a connected p-compact group, T a maximal torus for X , and
T!- T a discrete approximation for T . Define C T by
C = \soe(s)
CENTER OF A p-COMPACT GROUP 19
where the intersection is indexed by reflections s 2 W . Then C!- X is a p-dis*
*crete center
(6.4) for X .
This is actually a corollary of a more general calculation. Suppose that A *
* T is a
subgroup. Let WX (A) denote the Weyl group of CX (A), and WX (A)1 the Weyl grou*
*p of
the identity component CX (A)1 of CX (A). There are inclusions WX (A)1 WX (A) *
* W ,
where the last follows from x4.
7.6 Theorem. Let X be a connected p-compact group with maximal torus T and Weyl
group W . Suppose that A T is a subgroup. Then
(1) WX (A) is the subgroup of W consisting of the elements which, under the*
* conju-
gation action of W on T, pointwise fix the subgroup A, and
(2) WX (A)1 is the subgroup of WX (A) generated by those elements s 2 WX (A*
*) such
that s 2 W is a reflection and A oe(s).
7.7 Remark. The above theorems raise some natural questions about the structure*
* of oe(s).
By definition there are inclusions
H(s) oe(s) = H(s) [ K(s) F (s) ;
and K(s) is a coset of H(s). If p is odd, then s acts on T =H(s) ~=Z=p1 with o*
*nly the
identity as a fixed point (this follows from the fact that at an odd prime a p-*
*adic root of
unity has a nontrivial reduction mod p), and so the above chain of inclusions c*
*ollapses to
H(s) = oe(s) = F (s). If p = 2 the situation is more complex. In this case the *
*reflection s is
of order 2 and acts on T=H(s) ~=Z=21 by inversion; the fixed subgroup here is *
*of order 2,
and this potentially allows H(s) to be of index 2 in F (s). There are three po*
*ssibilities
(we list with each one an example of a compact Lie group G such that the possib*
*ility is
realized in ^G):
(1) H(s) ( oe(s) = F (s): SU(2).
(2) H(s) = oe(s) ( F (s): SO(3).
(3) H(s) = oe(s) = F (s): U(2)
Note that in all cases oe(s) is a subgroup of T.
Remark. The above description of oe(s) shows that if p is odd, then
(1) the p-discrete center C of X ( 7.5) is the center of N (T ) (i.e. the s*
*ubgroup of T
which is fixed by the conjugation action of W ), and
(2) the group WX (A)1 of 7.6 is the subgroup of WX (A) generated by the ref*
*lections
contained in WX (A).
If p = 2 then the p-discrete center of X is contained in the center of N (T ) b*
*ut is not
necessarily equal to it, unless of course the center of N (T ) is trivial. This*
* last possibility
actually comes up if X is the 2-compact group DI(4) from [11]; in this case X i*
*s of rank 3 and
it is possible to verify that the action of the Weyl group W on the elements of*
* exponent 2
in T gives an epimorphism W!- GL (3; F2) (see [11, x4] for instance). This imp*
*lies that
the center of N (T ) is trivial, hence that the p-discrete center C of X is tri*
*vial and that
the p-compact center Cc(X ) is trivial.
20 W. DWYER AND C. WILKERSON
Proof of 7.5 (given 7.6). Let A be a subgroup of T . By 7.6 and 4.7 the map A!*
*- X
is central if and only if A oe(s) for each reflection s in the Weyl group W of*
* X (recall
that W is generated by such reflections). The result follows from the descript*
*ion of the
p-discrete center of X provided by 6.4, and the fact (6.7) that since X is conn*
*ected the
p-discrete center is contained in T.
7.8 Lemma. Suppose that X is a connected p-compact group, A is a p-discrete tor*
*us and
i : A ! X is a homomorphism. Then CX (i(A)) is connected.
Proof. It is enough (3.11) to show that any homomorphism Z=pn!- CX (A) (n 0)
extends to a homomorphism Z=p1 !- CX (A) or equivalently that any homomorphism
Z=pn xA!- X extends to a homomorphism Z=p1 xA!- X . This is easily seen to be*
* true
if X is a p-compact torus. For the general case, then, it is enough to choose a*
* maximal
torus T!- X and show that any homomorphism f : Z=pn x A!- X lifts up to conju*
*gacy
to a homomorphism "f: Z=pn x A!- T . To simplify notation, let C denote Z=pn. *
*Such a
lift "fexists if and only if the homotopy fixed point set
(X =T )h(CxA)~=((X =T )hC)hA
is nonempty (see [12, 3.3, 10.5]). The "left translation action" of C via f on *
*X =T (see [12,
3.3]) extends to an action of the connected loop space X and so is homotopicall*
*y trivial,
i.e., each element of C acts by a self-map of X =T which is homotopic to the id*
*entity.
This implies that the Lefschetz number of the action of C on X =T is equal to t*
*he Euler
characteristic of X =T and so in particular is nonzero (2.9). This Lefschetz nu*
*mber is equal
to the Euler characteristic (X =T )hC[12, 4.5, 5.7], and so O((X =T )hC) is non*
*zero. By [12,
4.7, 5.7], the space ((X =T )hC)hAis not empty.
7.9 Proof of 7.6(1). Let W 0 W be the group of elements which, under the conjug*
*ation
action of W on T, pointwise fix the subgroup A, and let C denote the centralize*
*r of A in
N (T ). By calculation there is an exact sequence (of discrete groups)
{1}!- T!- C!- W 0-! {1} :
The homomorphism N (T )!- X induces a homomorphism C!- CX (A) which immediate*
*ly
gives the inclusion W 0 WX (A). Conversely, if WX (A) is not contained in W 0, *
*then there
are elements a 2 A and w 2 WX (A) such that w . a 6= a. Let C A be the cyclic
subgroup generated by a, let i : C!- T be the inclusion homomorphism, and j : *
*C!- T
the composite of i with the conjugation action of w. Clearly i and j become co*
*njugate
when composed with the homomorphism T !- CX (A), and so it follows from 5.4 th*
*at
a(w . a)-1 belongs to the kernel of T!- CX (A). This is impossible, since the *
*composite
T!- CX (A)!- X has a trivial kernel.
7.10 Lemma. Let X be a p-compact group, A an abelian p-discrete toral group, and
f : A!- X a homomorphism. Let U and V be two subgroups of A and U . V A the
subgroup generated by both. Then there is a natural map
(7.11) CX (U . V )!- CCX (V )(U)
CENTER OF A p-COMPACT GROUP 21
which is an equivalence.
Proof. Multiplication gives a surjective homomorphism U x V!- U . V , an induc*
*ed map
B U x BV!- B (U . V ), and hence a map
Map (B (U . V ); BX )!- Map (B U x BV; BX ) = Map (B U; Map (B V; BX )*
*) :
Restricting to appropriate components provides the required natural map 7.11. S*
*ince all
of the elements in the kernel of the map U x V !- U . V are also in the kerne*
*l of the
composite U x V!- U . V!- X , the fact that the map 7.11 is an equivalence fo*
*llows from
3.5.
7.12 Lemma. In the situation of 7.6, let s 2 W be a reflection. Then s 2 WX (A)*
*1 if and
only if s 2 WCX (H(s))(A)1.
Proof. The maximal rank map CCX (H(s))(A)!- CX (A) clearly carries WCX (H(s))(*
*A)1
into WX (A)1; this gives one of the implications. Suppose then that s 2 WX (A)1*
*. Let Z be
the centralizer of H(s) in CX (A)1. Then s 2 WZ1 because s 2 WZ by 7.9 and WZ1 *
*= WZ
because Z is connected (7.8). However, by 7.10 the p-compact group Z1 equals th*
*e identity
component of CCX (H(s))(A), and so it follows that s 2 WCX (H(s))(A)1.
Proof of 7.6(2). By [12, 9.7] the subgroup WX (A)1 of W is generated by the ref*
*lections it
contains, so proving (2) comes down to showing that a reflection s 2 W belongs *
*to WX (A)1
if and only if A oe(s).
Suppose first that A oe(s). To show that s 2 WX (A)1 is enough to show that
s 2 WX (oe(s))1, and it is this second inclusion that we will prove. By 7.12, w*
*e can assume
that H(s)!- X is central. If H(s) = oe(s), then WX (oe(s))1 = WX (H(s))1 = WX1*
* = WX ,
so s 2 WX (oe(s))1 for trivial reasons and we are done. If H(s) 6= oe(s), then*
* p = 2 and
oe(s) = H(s) x Z=2 (see 7.7). Choose t 2 K(s) oe(s) to generate the second fa*
*ctor of
oe(s) = H(s)xZ=2, and let x 2 N (T ) be an element, with image s 2 WX , such th*
*at x2 = t.
There is a commutative diagram of loop spaces and homomorphisms
Z=2 ----! Z=4 ----! Z=21
? ? ?
t?y x?y f?y
T ----! N (T )----! X
in which the map f is obtained from 3.10 by using the connectivity of X . Takin*
*g centralizers
of everywhere in this diagram and using 7.10 for the equivalence CX () ~*
*=CX (oe(s))
gives another commutative diagram
Z=2 ----! Z=4 ----! Z=21
? ? ?
t?y x?y f?y
T ----! CN (T)()----! CX (oe(s))
This shows that x, which represents s 2 WX (oe(s)), lies in the identity compon*
*ent of
CX (oe(s)) (3.10).
22 W. DWYER AND C. WILKERSON
Conversely, suppose that A is not contained in oe(s). If A is not contained *
*in F (s) then
s =2WX (A) by 7.6(1), and the result we want, that s =2WX (A)1, certainly follo*
*ws. If
A F (s) but A * oe(s), then p = 2, H(s) = oe(s), and F (s) ~=H(s) x Z=2 (see 7*
*.7). By
7.12, in showing s =2WX (A)1, we can replace X by CX (H(s)) or, equivalently, a*
*ssume
that H(s)!- X is central. By 7.10, then, CX (A) is equivalent to CX (F (s)). *
* Choose
x 2 N (T ) so that the image of x in W is s and such that x2 = 1; this is possi*
*ble because
K(s) = H(s) in the current situation and so 1 2 K(s). If s 2 WX (A)1 then x li*
*es in
the identity component of CX (F (s)) and so the homomorphism f : Z=2!- CX (F (*
*s))
representing x lifts to a homomorphism Z=2!- CX (F (s))1. Since the Euler char*
*acteristic
O(CX (F (s))1=T ) is non-zero (2.9), f lifts further (2.14) to a homomorphism Z*
*=2!- T
and even (3.2) to an actual group homomorphism Z=2!- T. This implies that f(Z=*
*2) is
central in CX (F (s)), since every element of order 2 in T is contained in F (s*
*) and so is
central in CX (F (s)). However it is impossible for f to be central in this way*
*, since x acts
nontrivially by conjugation on T and so x is not in the center of the evident s*
*ubgroup of
N (T ) which is a discrete approximation to the 2-normalizer of the torus in CX*
* (F (s)) (see
5.5). This contradiction establishes that A is contained in oe(s).
x8. A homology decomposition theorem
Suppose that X is a p-compact group. Let AX denote the category whose object*
*s are
the pairs (V; f), where V is a nontrivial elementary abelian p-group and f : V!*
*- X is a
conjugacy class of monomorphisms. A morphism (V; f)!- (V 0; f0) in this categ*
*ory is a
injection i : V!- V 0such that f0 . i is conjugate to f. There is a functor *
*ffX : AopX-!
(Spaces) given by
ffX (V; f) = Map (B V; BX )Bf :
By definition, ffX (V; f) is equivalent to BCX (g(V )) for any g : V!- X conta*
*ined in the
conjugacy class f.
Evaluation at the basepoint of BV gives a map
ffX (V; f)!- BX
which is natural in (V; f) [10, x6]; together these induce a map
aX : hocolimffX!- B X :
In this section we will prove the following theorem.
8.1 Theorem. For any p-compact group X the map aX : hocolimffX!- B X induces an
isomorphism on mod p cohomology.
Remark. This is called a homology decomposition theorem because it shows how to*
* express
B X up to mod p homology in terms of classifying spaces of other p-compact grou*
*ps; if X
has trivial p-compact center, these other p-compact groups will be strictly sma*
*ller than
X (cf. proof of 9.10). A decomposition theorem like this was first proved for c*
*lassifying
spaces of compact Lie groups in [14]. We will follow the more algebraic treatm*
*ent from
[10].
CENTER OF A p-COMPACT GROUP 23
Proof of 8.1. Let R denote the object H *BX of the category K of unstable algeb*
*ras over
the mod p Steenrod algebra, and let AR be the category whose objects are the pa*
*irs (V; g),
where V is a nontrivial elementary abelian p-group and g : R!- H*B V is a map *
*in K which
makes H *BV into a finitely generated R-module. A morphism (V; g)!- (V 0; g0*
*) in this
category is an injection i : V!- V 0such that (B i)*.g0= g. Let T V : K!- K b*
*e the functor
which is left adjoint to tensor product with H*B V ; this functor is studied by*
* Lannes in [16] .
There is a functor ffR : AR!- K which assigns to (V; g) the object T (V; R)g, *
*also denoted
TgV(R), which is the summand or "component" of T V(R) corresponding to the map g
[10, x3]. Let T!- X be a maximal torus, and let S = H *BNp(T ). The monomorphi*
*sm
(2.10) Np(T )!- X induces a map i : R!- S. Since X =Np(T ) is Fp-finiteand ha*
*s Euler
characteristic prime to p, the Becker-Gottlieb transfer construction (as genera*
*lized in [5])
provides a left inverse t : S!- R for i, such that t is both a map of R-module*
*s and a map
of modules over the mod p Steenrod algebra. Moreover, the object S 2 K has a no*
*ntrivial
"center" in the sense of [10, x4] because any discrete approximation Np (T ) to*
* Np(T ) has a
nontrivial group theoretic center [10, 1.4]. By [10, 1.2], then, the natural ma*
*p R!- limffR
is an isomorphism, and the higher limits limiffR vanish for i > 0. (Note that t*
*hese higher
limits are calculated in the category of graded mod p vector spaces.)
For any elementary abelian p-group V there is a map from the set of conjuga*
*cy classes
of homomorphisms f : V!- X to the set of K-maps g : H*B V!- R; the map assig*
*ns
to f the cohomology map (B f)* induced by Bf. According to [16, 3.1.4] the map *
* is a
bijection. Since a homomorphism f : V!- X is a monomorphism if and only if (f)*
* makes
H *BV into a finitely generated module over R (2.4), it follows that gives an *
*isomorphism
of categories : AX!- AR .
If V is an elementary abelian p-group and f : V !- X is a homomorphism, *
*then
the mapping space component Map (B V; BX )Bf is the classifying space of a p-co*
*mpact
group and so in particular it is Fp-complete and has finite dimensional mod p c*
*ohomology
groups. By [16] then, the natural map f : T (V; R)(f) !- H *Map (B V; BX )Bf *
*is an
isomorphism; here f is adjoint to the map on cohomology induced by the evaluati*
*on map
B V x Map (B V; BX )Bf!- BX . These isomorphisms combine to give a natural equ*
*ivalence
from the functor ffR to the composite H*(ffX ) . . It follows then from the rem*
*arks above
about ffR that limiH*(ffX ) = 0 for i > 0 and that the natural map limH *(ffX )*
*!- H*(B X )
is an isomorphism. The theorem can now be derived by a straightforward applicat*
*ion of
the spectral sequence of [3, XII, 5.8]; recall that this is a first quadrant sp*
*ectral converging
to H*(hocolim ffX ) which at E2 contains the groups limiH*(ffX ).
The above proof of 8.1 reveals a few properties of AX which we will use late*
*r on.
8.2 Proposition. If X is a p-compact group, then the nerve (or underlying space*
* [3, XI,
x2]) Nerve(AX ) of the category AX has the mod p (co-)homology of a point.
Proof. An inspection of [3, XI, x6] shows that HiNerve (AX ) is isomorphic to l*
*imifi, where
fi is the constant functor on AopXwith value Fp. Since fi is naturally equival*
*ent to the
functor H0(ffX ), this proposition follows from the vanishing result referred t*
*o in the proof
of 8.1.
24 W. DWYER AND C. WILKERSON
8.3 Proposition. If X is a p-compact group, then AX is equivalent to a category*
* with a
finite number of objects and a finite number of morphisms between any two objec*
*ts.
Proof. By definition the set of morphisms between any two objects of AX is fini*
*te, so it is
enough to show that up to isomorphism AX has only a finite number of objects. A*
*s in the
proof of 8.1, such objects correspond to pairs (V; g), where V is a nontrivial*
* elementary
abelian p-group and g : H *BX!- H *BV makes H *BV a finitely generated modul*
*e over
H *BX . Since H*B X is a finitely generated, i.e. noetherian, algebra over Fp, *
*an elementary
Poincare series argument shows that any such pair (V; g) must have the Krull di*
*mension
of H *BV (which equals rkFpV ) less than or equal to the Krull dimension of H *
**BX ; in
particular, up to isomorphism there are only a finite number of possible V . Fo*
*r any given
V , a K-map g : H *BX!- H *BV is determined by the images under g of some fini*
*te set
of algebra generators for H *BX ; since H *BV is finite in each dimension, ther*
*e are only a
finite number of choices for these images.
x9. An inductive principle and the "Sullivan Conjecture"
In this section we will describe an inductive principle which can sometimes *
*be used in
conjunction with 8.1 to prove statements about p-compact groups. We then show t*
*hat the
principle leads to a proof for p-compact groups of an analogue of Miller's Theo*
*rem (the
"Sullivan Conjecture") [17].
9.1 Definition. A class Cl of p-compact groups is said to be saturated if it sa*
*tisfies the
following five conditions.
(1) Cl is closed under equivalences, in the sense that if f : X!- Y is a h*
*omomorphism
of p-compact groups which is an equivalence, then X 2 Cl if and only if*
* Y 2 Cl.
(2) The trivial p-compact group belongs to Cl.
(3) If the identity component X1 of X is in Cl, then X 2 Cl.
(4) If X is connected and X =Cc(X ) 2 Cl, then X 2 Cl.
(5) If X is connected, Cc(X ) is trivial, and Y 2 Cl for all p-compact grou*
*ps Y such
that cdFpY < cdFpX , then X 2 Cl.
9.2 Theorem. Any saturated class Cl contains all p-compact groups.
Proof. We will work by induction on cdFp X to show that X 2 Cl. If cdFp X = 0 *
*then
X1 is contractible and X 2 Cl by the first three conditions. Suppose that cdFpX*
* > 0 and
that Y 2 Cl for all Y with cdFpY < cdFpX . Let C be the p-compact center of X1.*
* Then
X1=C 2 Cl by (5), X1 2 Cl by (4), and hence X 2 Cl by (3).
9.3 Theorem. (Sullivan conjecture for p-compact groups) Let X be a p-compact gr*
*oup
and U a space which is Fp-complete and Fp-finite. Then inclusion of constant ma*
*ps gives
an equivalence U!- Map (B X ; U).
Before proving 9.3 we will list a few lemmas. If X and U are spaces, write "*
*X ? U" if
inclusion of constant maps gives an equivalence U!- Map (X ; U). The first thr*
*ee lemmas
are relatively elementary.
9.4 Lemma. [2, x2-3] Let F!- E!- B be a fibration. If F ? U and B ? U then E *
*? U.
CENTER OF A p-COMPACT GROUP 25
9.5 Lemma. Let C be a category and F : C!- (Spaces) a functor. If Nerve(C) ? U*
* and
F (c) ? U for each object c of C, then hocolimF ? U.
Proof. See [3, XII, x4] and observe that the homotopy limit of a constant funct*
*or on C
with value U is the space Map (Nerve(C); U).
9.6 Lemma. [12, 11.13] If U is an Fp-complete space and X!- Y is an Fp-equival*
*ence,
then X ? U if and only if Y ? U.
9.7 Lemma. If X is a p-compact toral group and U is a space which is Fp-complet*
*e and
Fp-finite, then BX ? U.
Proof. By density (3.4) we can assume instead that X is a p-discrete toral grou*
*p. In this
case X is an increasing union of finite p-groups [12, 6.19], and B X is equival*
*ent to the
corresponding sequential homotopy colimit of classifying spaces of finite p-gro*
*ups [3, XII,
3.6]. Since BG ? U for any finite p-group G by Miller's Theorem, the lemma foll*
*ows from
9.5.
Let Cl be the class of all p-compact groups X such that B X ? U for each spa*
*ce U
which is Fp-complete and Fp-finite. It is clear that Cl satisfies (1) and (2) o*
*f 9.1. In order
to complete the proof of 9:3, i.e., to show that Cl is the class of all p-compa*
*ct groups, it is
enough by 9.2 to prove the following three propositions. In each of them X is a*
* p-compact
group and U is a space which is Fp-complete and Fp-finite.
9.8 Proposition. If B(X1) ? U, then BX ? U.
Proof. There is a fibration sequence BX1!- B X!- B ss0X . Since Bss0X ? U by *
*[17], the
proposition follows from 9.4.
9.9 Proposition. Suppose that X is connected and that Z = X =Cc(X ). If B Z ? *
*Y,
then BX ? Y.
Proof. There is a fibration sequence BCc(X )!- BX!- B Y. Since BCc(X ) ? Y by*
* 9.7,
the proposition follows from 9.4.
9.10 Proposition. If X is connected, Cc(X ) is trivial, and BY ? U for all Y su*
*ch that
cdFp Y < cdFpX , then BX ? U.
Proof. Let f : V !- X be a monomorphism with V a nontrivial elementary abelia*
*n p-
group. Since Cc(X ) is trivial the homomorphism f cannot be central, and so (b*
*y 4.6)
cdFp CX (f(V )) < cdFpX . It follows that ffX (V; f) ? U for each object (V; f)*
* of AX , and
so hocolimffX ? U by 8.2 and 9.5. By 9.6, BX ? U.
x10. A few technical results
In this section we begin by proving a collection of results which are more o*
*r less imme-
diate corollaries of 9.3. Some of these are interesting because they are genera*
*lizations for
arbitrary p-compact groups of results that were previously known only for p-com*
*pact toral
groups. We finish by giving a splitting result (10.7) for extensions of a p-com*
*pact torus by
a connected p-compact group in adjoint form; this will be used in the proof of *
*11.7.
26 W. DWYER AND C. WILKERSON
10.1 Proposition. (cf. [12, 6.1 and 5.3]) If f : X ! Y is a trivial homomorphis*
*m between
p-compact groups, then f is central.
Proof. Let f : BX!- BY be the constant map. The proposition follows from the f*
*act that
the loop space Map (B X ; BY)f is equivalent to Map (B X ; Y), which, by 9.3, *
*is in turn
equivalent to Y.
10.2 Proposition. (cf. 2.6) A homomorphism f : X ! Y between p-compact groups is
central if and only if there is a homomorphism : X x Y ! Y which restricts to *
*f on
X x {1} and to the identity on {1} x Y.
Proof. This is the same as the proof of [12, 8.6]. If f is central then an eval*
*uation map
gives the required homomorphism . If exists, then it induces an equivalence be*
*tween
CY (f(X )) and CY (g(X )), where g : X!- Y is the trivial homomorphism. This *
*second
centralizer is equivalent to Y by 10.1.
10.3 Lemma. (cf. 3.5) Let
f
{1}!- X!- Y!- Z!- {1}
be a short exact sequence of p-compact groups and g : Y!- U a homomorphism suc*
*h that
g.f is a trivial homomorphism. Then up to homotopy there is a unique homomorphi*
*sm g0:
Z!- U which extends g, and the natural map CU (g0(Z))!- CU (g(Y)) is an equiv*
*alence.
Proof. This is the same as the proof of [12, 7.5]. Let h : X!- U be the trivia*
*l homomor-
phism. The idea is to express the space of maps B Y!- B U which up to homotop*
*y are
trivial on BX as the space of sections of a fibration over BZ with fibre Map (B*
* X ; BU)Bh,
and then to use 10.1 to identify this fibration as the product BZ x BU!- BZ.
10.4 Proposition. Suppose that
f
{1}!- X!- Y!- Z!- {1}
is a short exact sequence of p-compact groups, and that g : U!- Y is a central*
* homomor-
phism of p-compact groups. Then the composite f . g : U!- Z is also central.
Proof. By 10.2 it is enough to construct an appropriate homomorphism : U x Z!-*
* Z.
Consider the corresponding homomorphism U x Y!- Y which expresses the fact th*
*at
g is central. The adjoint of the associated map of classifying spaces is a map*
* B U!-
Map (B Y; BY)idwhich induces (by composing points in the target with Bf) a map *
*BU!-
Map (B Y; BZ)Bf. By 10.3, however, Map (B Y; BZ)Bf is equivalent via precompos*
*ition
with Bf to Map (B Z; BZ)id, and it is easy to check that the adjoint of the res*
*ulting map
B U!- Map (B Z; BZ)idgives the required homomorphism .
10.5 Proposition. Let
f
{1}!- X!- Y!- Z!- {1}
be a short exact sequence of p-compact groups, U a p-compact group, and g : U!-*
* X a
homomorphism. Then
(1) if f . g is central then g is central, and
(2) if g is central and Z is connected then f . g is central
CENTER OF A p-COMPACT GROUP 27
Proof. Let h : U!- Z denote the trivial homomorphism. Evaluation at the basepo*
*int of
B U gives a map of fibration sequences
F ----! Map (B U; BY)B(f.g)----! Map (B U; BZ)Bh
? ? ?
a?y b?y c?y
B X ----! BY ----! B Z
where F denotes the space of maps from BU to BX which after composition with Bf*
* are
homotopic to B(f . g). Note that if F is connected then F = Map (B U; BX )Bg. B*
*y 10.1
the map c is an equivalence. It follows that if b is an equivalence if and onl*
*y if a is an
equivalence. If f . g is central then b is an equivalence, so that a is an equi*
*valence, F is
connected, and g is central. This gives (1). If Z is connected then F is connec*
*ted by the
long exact homotopy sequence of the upper fibration; thus if g is central, the *
*map a is an
equivalence, so b is an equivalence too and f . g is central. This gives (2).
10.6 Lemma. Suppose that X!- Y!- Z is a fibration sequence of Fp-finitespaces*
*, such
that Z is connected and the action of ss1Z on H*X is nilpotent. Then O(Y) = O(X*
* )O(Z).
Proof. It follows from [12, 4.13] that the desired multiplicativity holds in E2*
* of the fibration
Serre spectral sequence, and a standard inductive argument shows that this pers*
*ists to
E1 .
Remark. Lemma 10.6 does not necessarily hold for a general fibration sequence o*
*f Fp-finite
spaces. For example, let p be odd and consider the fibration
S2!- RP 2-! RP 1 :
The mod p cohomology Euler characteristics of base and total space are 1, while*
* that of
the fibre is 2.
10.7 Proposition. Suppose that
{1}!- X!- Y!- A!- {1}
is a short exact sequence of connected p-compact groups, where X has a trivial *
*p-compact
center and A is a p-compact torus. Then the sequence is equivalent to the produ*
*ct sequence
{1}!- X!- X x A!- A!- {1} :
Proof. Let T be a maximal torus for Y and T!- T a discrete approximation. Let *
*f be the
composite T!- Y!- A, and denote ker(f) by K and T=K by Q. Let K and Q denote*
* the
closures of K and Q respectively. Observe that Q , as a quotient of T, is a p-d*
*iscrete torus
and hence that Q is a p-compact torus. There is a commutative diagram of loop s*
*paces
and homomorphisms (3.5)
K ----! T ----! Q
?? ? ?
y ?y ?y
X ----! Y ----! A
28 W. DWYER AND C. WILKERSON
which on passing to closures gives a map of short exact sequences
{1} ----! K ----! T ----! Q ----! {1}
? ? ?
(10.8) ?y ?y ?y
{1} ----! X ----! Y ----! A ----! {1}
By construction the map Q!- A has trivial kernel, and so Q!- A is a monomorph*
*ism
(3.6). The composite K!- X!- Y has trivial kernel (since T!- Y is a monomorp*
*hism)
and so K !- X has trivial kernel and K!- X is a monomorphism. Taking vert*
*ical
homotopy fibres in the diagram of classifying spaces induced by 10.8 gives a fi*
*bre sequence
X =K!- Y=T!- A=Q
in which all three spaces are Fp-finiteand in which it is straightforward to ve*
*rify that the
fundamental group of the base, which is a quotient of ss1A, acts nilpotently on*
* the homology
of the fibre. By multiplicativity of Euler characteristic (10.6), O(Y=T ) = O(X*
* =K)O(A=Q).
Since O(Y=T ) 6= 0, it follows that O(X =K) 6= 0 6= O(A=Q) and hence (2.15) tha*
*t Q is
a maximal torus for A and (use in addition 10.6) that the identity component K1*
* is a
maximal torus for X . Since A is itself a p-compact torus, the uniqueness prop*
*erty of
maximal tori implies that the map Q!- A is an equivalence. As in 3.8, K is a p*
*roduct of
a p-compact torus and a finite abelian p-group; since X is connected and a maxi*
*mal torus
in a connected p-compact group is self-centralizing [12, 9.1], we conclude that*
* the finite
abelian p-group factor is trivial, K is connected, and K itself is a maximal to*
*rus for X .
Let C!- T be the p-discrete center of Y (6.7). By 7.5, C = \soe(s), where*
* s runs
through all reflections in the Weyl group W of Y. By 10.5 the intersection C \ *
*K is trivial,
since X has a trivial p-discrete center. Let T W be the fixed point set of the *
*conjugation
action of W on T, and Tnrm the image of the norm map T!- T which sends an elem*
*ent
of T to the sum of its W -conjugates. Then Tnrm T W, C T W, the quotient T W=*
*Tnrm
is finite because it is annihilated by |W |, and T W=C is finite by 7.7. The qu*
*otient T!- Q
is clearly equivariant with respect to the conjugation action of W on T and the*
* trivial
action of W on Q . Under this trivial action of W on Q , the norm map Q!- Q *
*is just
multiplication by |W | and so is surjective, because Q is divisible. As a conse*
*quence, the
composite Tnrm!- T!- Q is surjective. Since Q has no proper subgroups of finit*
*e index, it
follows that the maps T W!- Q and C!- Q are surjective too, this last map in *
*fact being
an isomorphism because as mentioned above the intersection C \ K is trivial. Pa*
*ssing to
closures shows that the homomorphism Y!- A induces an equivalence Cc(Y)!- A. *
*The
composite homomorphism Cc(Y) x X!- Cc(Y) x Y!- Y (where the second map comes
from 2.6) is the required equivalence of Y with Cc(Y) x X ~= A x X . It is easy*
* to check
that this equivalence is compatible with the original short exact sequence.
x11. Homotopy centers
In this section we will prove 1.3. Let X be a p-compact group. The multipl*
*ication
homomorphism (2.6):
c : Cc(X ) x X!- X
CENTER OF A p-COMPACT GROUP 29
gives a classifying space map Bc : BCc(X ) x BX!- B X . The adjoint of Bc is a*
* map
B Cc(X )!- Map (B X ; BX )id, or equivalently a loop space homomorphism jX : C*
*c(X )!-
Ch(X ). This is the homomorphism which will turn out to be an equivalence.
11.1 Remark. Let X be a p-compact group. The construction in [12, proof of 8.3]*
* shows
that the fibration sequence BCc(X )!- BX!- BXad extends one step further to t*
*he right
to give a fibration sequence
(11.2) B X!- BXad!- B 2Cc(X ) :
It follows directly from Theorem 1.3 that 11.2 is equivalent to the universal f*
*ibration
B X!- E!- B(Aut (B X )id)
with fibre BX over a simply connected space; in particular, the total space of *
*this universal
bundle is BXad.
The next proposition shows that in order to prove that jX is an equivalence *
*it is enough
to prove that Ch(X ) is a p-compact group.
11.3 Proposition. Suppose that X is a p-compact group and that Ch(X ) is a p-co*
*mpact
group. Then the map jX : Cc(X )!- Ch(X ) is an equivalence.
Proof. The composition operation on Map (B X ; BX )id gives a homomorphism Ch(X*
* ) x
Ch(X )!- Ch(X ) which shows (10.2) that Ch(X ) is abelian and hence (5.2) that*
* Ch(X )
is a p-compact toral group. Consider the evaluation map
Bh : BCh(X ) x BX = Map (B X ; BX )idx BX!- BX :
This induces a homomorphism h : Ch(X ) x X!- X whose restriction to Ch(X ) x {*
*1}
is a central homomorphism Ch(X )!- X (2.6). By 6.1 this central homomorphism *
*lifts
uniquely to a homomorphism kX : Ch(X )!- Cc(X ). The homomorphism diagrams
h c
Ch(X ) x X ----! X Cc(X ) x X ----! X
? ? ? ?
kX xid?y id?y jX xid?y id?y
c h
Cc(X ) x X ----! X Ch(X ) x X ----! X
commute up to homotopy, the first by the uniqueness provision of 5.3 and the se*
*cond
by construction. It follows directly that jX . kX : Ch(X )!- Ch(X ) is homotop*
*ic to the
identity, and from the uniqueness provision of 6.1 that kX . jX is also homoto*
*pic to the
identity.
The above proof also gives a slightly more specialized result which we will *
*use later on.
30 W. DWYER AND C. WILKERSON
11.4 Proposition. Suppose that X is a p-compact group, and that the identity co*
*m-
ponent Ch(X )1 is a p-compact torus. Then the restriction of jX : Cc(X )!- C*
*h(X ) to
identity components gives an equivalence Cc(X )1!- Ch(X )1.
Let Cl be the class consisting of all p-compact groups X such that Ch(X ) is*
* a p-compact
group. By 11.3, Theorem 1.3 is equivalent to the statement that Cl contains all*
* p-compact
groups. It is clear that Cl satisfies (1) and (2) of 9.1. By 9.2, then, in orde*
*r to prove 1.3 it
is enough to prove the following three propositions. In each one, X is a p-comp*
*act group.
11.5 Proposition. If Ch(X1) is a p-compact group, then so is Ch(X ).
11.6 Proposition. Suppose that X is connected, and let C = Cc(X ). If Ch(X =C) *
*is a
p-compact group, then so is Ch(X ).
11.7 Proposition. Suppose that X is connected, and that Cc(X ) is trivial. If C*
*h(Y) is
a p-compact group for each p-compact group Y with cdFp Y < cdFpX , then Ch(X ) *
*is a
p-compact group.
The proofs of 11.5 and 11.6 amount to fairly standard homotopy theoretic cal*
*culations;
in order to organize these calculations we set up some notation.
If U is a space, let Aut(U) denote the space of (unpointed) self homotopy eq*
*uivalences
of U; this is a loop space, and B Aut(U) is the classifying space for fibration*
*s with fibre
equivalent to U. If f : E!- B is a fibration, let Aut(f) denote the space of c*
*ommutative
diagrams
E ----! E
? ?
(11.8) f?y f?y
B ----! B
in which both of the horizontal arrows are equivalences. Let AutB (f) denote th*
*e subspace
of Aut(f) consisting of diagrams as above in which the map B!- B is the identi*
*ty. The
spaces Aut (f) and Aut B(f) are loop spaces such that B Aut(f) and B AutB (f) c*
*lassify
certain types of iterated fibrations [7].
If X and Y are spaces let [X ; Y] denote ss0 Map (X ; Y). Suppose that B is *
*a connected
space and that f : E!- B is a fibration with fibre F. We will denote by c(f) t*
*he element of
[B; BAut (F)] consisting of classifying maps for f, and by C(f) the subset of [*
*B; BAut (F)]
obtained by taking the orbit of c(f) under the precomposition action of ss0 Aut*
*(B) on
[B; BAut (F)].
11.9 Proposition. [7] Let f : E!- B be a fibration with fibre F over the conne*
*cted space
B. Then there is a fibration sequence
Map (B; BAut (F))C(f)-! B Aut (f)!- BAut (B) :
Remark. The above fibration sequence is associated to the composition action of*
* Aut(B)
on Map (B; BAut (F)C(f).
If X and U are spaces with U connected, let Map (X ; U)0 denote the subsp*
*ace of
Map (X ; U) consisting of maps which are homotopic to constant maps. Write X ?*
*0 U
if inclusion of constant maps gives an equivalence U!- Map (X ; U)0 (cf. x9).
CENTER OF A p-COMPACT GROUP 31
11.10 Proposition. Suppose that B is a connected space, that f : E!- B is a fi*
*bration
with fibre F, and that F ?0 B. Then the forgetful map u : Aut(f)id-! Aut (E)id(*
*cf. 11.8)
is an equivalence.
Proof. Let Aut B(f){idE}denote the subspace of Aut B(f) consisting of maps of E*
* over
B which as maps of E are homotopic to the identity, and let Aut (f){idE;idB}den*
*ote the
subspace of Aut(f) consisting of commutative diagrams 11.8 in which the upper a*
*rrow is
homotopic to the identity map of E and the lower one to the identity map of B. *
*Consider
the map of fibration sequences
u0
AutB(f){idE}----! Aut (f){idE;idB}----! Aut(B)id
? ? ?
= ?y u?y v?y
w
AutB(f){idE}----! Aut(E)id ----! Map (E; B)f
in which u0like u is a forgetful map and the maps v and w are given by composit*
*ion with f
(on the left and right, respectively). Since F ?0 B, the argument used in the p*
*roof of 10.3
shows that v is an equivalence. The fibration sequence then shows that u is an *
*equivalence
and also that the domain of u, being connected, is the identity component Aut (*
*f)id of
Aut (f).
The next two lemmas are proved by direct homotopy group calculation, and dep*
*end on
the following observation.
11.11 Remark. A connected space U is the classifying space BG of a p-compact to*
*ral group
G if and only if
(1) ss1U is a finite p-group,
(2) ss2U is a finitely generated free module over Zp (since Zp is a princ*
*ipal ideal
domain, this is equivalent to the condition that ss2U be a finitely gen*
*erated torsion
free module over Zp) and
(3) ssi(U) = {0} for i > 2.
If U is both a loop space and the classifying space of a p-compact toral group *
*then U is
the classifying space of an abelian p-compact toral group (2.6).
11.12 Lemma. Suppose that Y is a connected space with H iY finite for i = 1; 2 *
*and
H 1(Y; Zp) = 0. Let U be connected space such that U is the classifying space *
*of a p-
compact toral group and let f : Y!- U be a map. Then each component of Map (Y*
*; U)f
is the classifying space of a p-compact toral group.
11.13 Remark. The above lemma applies if Y = BX for a p-compact group X .
11.14 Lemma. If G is a p-compact toral group, then Aut(B G)id= BCh(G) is the cl*
*as-
sifying space of a p-compact toral group.
Proof of 11.5. By 11.3, Ch(X1) is a p-compact toral group. The fibration
f
B(X1)!- BX!- Bss0X
32 W. DWYER AND C. WILKERSON
satisfies the conditions of 11.10, so the map
Aut(f)id-! Aut (B X )id= BCh(X )
is an equivalence. Looping down the fibration sequence from 11.9 then gives a *
*fibration
sequence for BCh(X ) in which the fibre is a disjoint union of classifying spac*
*es of p-compact
toral groups (11.12) and the base is Aut(B ss0X )id~=B C, where C is the center*
* of ss0X . It
follows from the long exact homotopy sequence of a fibration (cf. 11.11) that B*
*Ch(X ) is
the classifying space of a p-compact toral group.
Proof of 11.6. The fibration
f
BCc(X )!- BX!- B(X =Cc(X ))
satisfies the conditions of 11.10 (see 10.1), so the map
Aut(f)id-! Aut (B X )id= BCh(X )
is an equivalence. Looping down the fibration sequence from 11.9 then gives a *
*fibration
sequence for BCh(X ) in which the base is contractible (6.3, 11.3) and the fibr*
*e is the clas-
sifying space of a p-compact toral group (11.12, 11.13). This gives the desired*
* result.
To prove the final proposition we will need a concept from [6]. A map f : X*
*!- Y
of spaces is said to be centric if composition with f induces an equivalence Au*
*t(X )id-!
Map (X ; Y)f.
11.15 Lemma. Let X be a p-compact group, V an elementary abelian group, f : V!-*
* X
a monomorphism, and i : CX (V )!- X the induced monomorphism. Then the map
B i : BCX (V )!- BX is centric.
Proof. This is the same as the proof of [6, 4.2]. The rough idea is that, sinc*
*e V is cen-
tral in CX (V ), the centralizer of CX (V ) in X is the same as the centralizer*
* of CX (V )
in the centralizer of V in X . Encoding this in mapping space terms gives the*
* stated
equivalence.
The following is clear from the complex given in [3, XI, x6] for computing h*
*igher derived
functors of the inverse limit.
11.16 Lemma. Let A be a category, R a noetherian ring and ff : A!- (R-modules)*
* a
functor. Suppose that A is equivalent to a category with a finite number of obj*
*ects and a
finite number of morphisms between any two objects, and that ff(x) is a finitel*
*y generated
R-module for any object x of A. Then limiff is a finitely generated R-module f*
*or each
i 0.
Proof of 11.7. The Fp-equivalence aX of 8.1 gives an equivalence [12, 11.3]
~=
a#X: Map (B X ; BX ) -! Map (hocolim ffX ; BX ) :
CENTER OF A p-COMPACT GROUP 33
As explained in [3, XII, x4] the target of this equivalence is the same as holi*
*mfiX , where
fiX : AX!- (Spaces) sends (V; f) to Map (ffX (V; f); BX ). For each object (V;*
* f) of AX let
eV;f: ffX (V; f) = Map (B V; BX )Bf!- BX
be given by evaluation at the basepoint of BV . The definition of aX implies th*
*at a#Xcarries
the component Aut (B X )id of Map (B X ; BX ) to the basepoint component of hol*
*im fi0X,
where for each object (V; f) of AX , fi0X(V; F ) is the pointed space given by
fi0X(V; F ) = Map (ffX (V; f); BX )eV;f:
Since Cc(X ) is trivial, each space ffX (V; f) is the classifying space of s*
*ome p-compact
group Y with cdFp Y < cdFpX (cf. proof of 9.10). Consequently, by 11.3 and the *
*given
hypotheses, Aut (ffX (V; f))id is the classifying space of a p-compact toral gr*
*oup, in fact,
of an abelian p-compact toral group (11.11). By 11.15, each space fi0X(V; f) i*
*s also the
classifying space of an abelian p-compact toral group. There is a second quadra*
*nt homotopy
spectral sequence [3, XI, x7] [1]
E2i;j= lim-issjfi0X) ssj-iholim fi0X= ssj-iAut (B X )id= ssj-iBCh(X )
The E2-term of the spectral sequence has these properties:
(1) E2i;j= 0 for j 3, becauseQssjfi0X(V; f) = 0 for all (V; f) and all j *
*3.
(2) E20;2= limss2fi0X (V;f)ss2fi0X(V; f) is a torsion-free module over Z*
*p, because
ss2fi0X(V; f) is a torsion-free module over Zp for each (V; f).
(3) E2i;jis a finitely generated module over Zp for j = 1; 2 and all i 0, *
*by 8.3 and
11.16.
It follows from these observations and the structure of the spectral sequence t*
*hat ssiAut (B X )id=
0 for i 3, that ss2 Aut(B X )idis a finitely generated torsion-free (equivalen*
*tly, free) mod-
ule over Zp, and that ss1 Aut(B X )id is a finitely generated module over Zp. *
*(Note that
this fundamental group is necessarily abelian, because Aut(B X ) is a loop spac*
*e.) We will
be done if we can show that ss1 Aut(B X )id is finite, since then Aut (B X )id *
*= B Ch(X )
will be the classifying space of a p-compact toral group. It is clear in any c*
*ase that the
identity component Ch(X )1 is a p-compact torus, since the nonzero homotopy of *
*this com-
ponent consists at most of a single finitely generated free module over Zp in d*
*imension 1.
Since Cc(X ) is trivial by assumption, it follows from 11.4 that Ch(X )1 is con*
*tractible, and
hence that Aut(B X )idis of type K(M; 1) for some finitely generated module M o*
*ver Zp.
Therefore B(Aut (B X )id) is of type K(M; 2). We will prove directly that M = 0*
*, which is
certainly enough to show that M is finite. Pick a homomorphism Zp!- M and cons*
*ider
the fibration sequence
BX!- E!- K(Zp ; 2)
obtained from the induced map K(Zp ; 2)!- K(M; 2) by pulling back the universa*
*l fibration
over K(M; 2) = B (Aut (B X )id). This gives a short exact sequence of p-compac*
*t groups
which by 10.7 is a product sequence. This implies that the above fibration is a*
* product
fibration and, by the meaning of universality, that the homomorphism Zp!- M is*
* trivial.
Since this is true for any homomorphism Zp!- M, M itself is trivial.
34 W. DWYER AND C. WILKERSON
x12. Relationship to Lie groups
In this section we will prove 1.4; this gives one explicit connection betwee*
*n the theory
described in this paper and the homotopy theory of compact Lie groups.
Let G be a compact Lie group and Ca(G) the algebraic center of G, so that Ca*
*(G) is a
compact abelian Lie group. The multiplication homomorphism Ca(G) x G!- G induc*
*es
a map on classifying spaces whose adjoint is a map e1 : BCa(G)!- Map (B G; BG)*
*id(cf.
[15, x4]). This is the natural map referred to in the statement of 1.4.
Recall from x2 that if G is compact Lie group with ss0G a p-group, G^ denote*
*s the p-
compact group obtained from G by Fp-completion. There is a natural homomorphism
ffi : G!- ^G.
12.1 Lemma. Let G be a connected compact Lie group, A a p-discrete toral group,*
* and
f : A!- G a homomorphism. Then the maps
h
Map (B G; BG)id-! Map (B G; BG^)Bffi
k
Map (B A; BG)Bf!- Map (B A; BG^)B(ffi.f)
induced by composition with ffi are Fp-equivalences.
Proof. We will only treat the map h;Qthe case of the map k is similar but simpl*
*er. Let
Comp 0Fp(B G) denote the product q6=pComp Fq(B G). By [15, 3.1], the complet*
*ion map
ffl : BG!- Comp 0Fp(B G) combines with Bffi to give an Fp-equivalence
Map (B G; BG)id-! Map (B G; BG^)Bffix Map (B G; Comp 0Fp(B G))ffl
To finish the proof it is enough to show that the space
Y
(12.2) Map (B G; Comp 0Fp(B G))ffl~= Map (B G; Comp Fq(B G))fflq
q6=p
has the mod p homology of a point, where fflq : BG!- Comp Fq(B G) is the Fq-co*
*mpletion
map. However, as described in [12, 11.10] each completion map fflq induces an e*
*quivalence
Map (Comp Fq(B G); Comp Fq(B G))id-! Map (B G; Comp Fq(B G))fflq
whose domain, by 1.3, is the classifying space of an abelian q-compact group. I*
*t follows
that the space in 12.2 is a nilpotent (even simple) space with uniquely p-divis*
*ible homotopy
groups, and thus has the desired acyclicity property.
12.3 Lemma. Suppose that G is a connected compact Lie group and K a subgroup
of G such that ss0K is a p-group and the inclusion K!- G induces an isomorphi*
*sm
H *BG!- H*B K. Then K = G.
Proof. It follows from application of the Eilenberg-Moore spectral sequence or *
*the fibre
lemma [12, 11.7] to appropriate path fibrations that the map K!- G is an Fp-eq*
*uivalence.
Since K and G are both orientable manifolds, this implies that K and G have the*
* same
dimension; the fact that G is connected now easily gives K = G.
CENTER OF A p-COMPACT GROUP 35
12.4 Lemma. [15, A.4] Suppose that G is a connected compact Lie group, K a subg*
*roup
of G which is a finite p-group, and C the centralizer of K in G. Then ss0C is *
*a finite
p-group.
12.5 Theorem. Suppose that G is a connected compact Lie group and i : K G a
subgroup of G which is a finite p-group. Then K lies in the center of G if and *
*only if the
composite K!- G!- ^Gis central.
Proof. If K lies in the center of G, then applying the Fp-completion functor to*
* the map of
classifying spaces induced by the multiplication map K xG!- G gives the homomo*
*rphism
KxG^-! G^required for showing that K!- ^Gis central (2.6). If K!- ^Gis centra*
*l, consider
the square
Map (B K; BG)Bi ----! Map (B K; BG^)B(ffi.i)
?? ?
y ?y
BG ----! B ^G
where the vertical maps are induced by evaluation at the basepoint of B K. Her*
*e the
two horizontal maps are Fp-equivalences (the upper one by 12.3) and the right h*
*and map
is an equivalence by assumption. It follows that the map Map (B K; BG)Bi!- BG*
* is an
isomorphism on homology. Let H be the centralizer of K in G. The main result of*
* [13]
implies that the natural map BH!- Map (B K; BG)Bi is an Fp-equivalence, and it*
* follows
that the inclusion H!- G induces an Fp-equivalence BH!- BG. By 12.4 and 12.5,*
* then,
H = G.
12.6 Lemma. Let G be a connected compact Lie group and C the subgroup of Ca(G)
given by elements whose order is a power of p. (We are treating C as a discrete*
* group).
Then the composite homomorphism C!- G!- ^Gis a p-discrete center for ^G.
Proof. Let T!- G be a Lie-theoretic maximal torus for G; as in 2.16, the induc*
*ed homo-
morphism ^T-! ^Gis a maximal torus for ^G. Let T be the subgroup of T given by *
*elements
with order a power of p. The composite of the inclusion T!- T with the complet*
*ion map
ffi : T!- ^Tis a discrete approximation for T^. By ordinary Lie group theory*
*, Ca(G) is
contained in T and so C is the set of elements x 2 T such that x lies in the ce*
*nter of G. By
12.5 this is the same as the set of elements x 2 T such that the composite !*
*- T!- G^
is central; by 6.4 and 6.7, this set is the p-discrete center of ^G.
Proof of 1.4. It is enough to show that for each prime p the map e1 : B Ca(G)!-
Map (B G; BG)idis an Fp-equivalence. Fix p as usual, and let C Ca(G) be the gr*
*oup of
elements of order a power of p. There are maps
u a e1 h ~=
B C!- BC (G) -! Map (B G; BG)id-! Map (B G; BG^)Bffi- Map (B ^G; BG^)id
where u is induced by the inclusion C!- Ca(G), h is given by composition with *
*Bffi, and
the last equivalence is also induced by composition (on the other side) with Bf*
*fi. A straight-
forward argument using 1.3 and 12.6 shows that the composite BC!- Map (B G; BG*
*^)Bffiis
an Fp-equivalence. By inspection the map u is an Fp-equivalence (recall that Ca*
*(G) is the
product of a torus and a finite abelian group), and by 12.1 the map h is an Fp-*
*equivalence
too. It follows that e1 is an Fp-equivalence.
36 W. DWYER AND C. WILKERSON
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University of Notre Dame, Notre Dame, Indiana 46556
Purdue University, West Lafayette, Indiana 47907
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