THE CENTRALIZER DECOMPOSITION OF BG
W.G. Dwyer
University of Notre Dame
x1. Introduction
Let G be a compact Lie group and p a fixed prime number. Recall that an
elementary abelian p-group is an abelian group isomorphic to (Z=p)r for some r.
Jackowski and McClure showed in [10] how to decompose the classifying space B G
at the prime p as a homotopy colimit of spaces of the form B CG (V ), where V *
*is
a nontrivial elementary abelian p-subgroup of G and CG (V ) is the centralizer *
*of
V in G (see x2). If the center of G is trivial then each of the centralizers *
*CG (V )
is a proper subgroup of G, and so in this case the decomposition theorem gives
an explicit way of gluing together B G, at least at p, from the classifying spa*
*ces
of smaller groups. In this paper we will use this decomposition to give parall*
*el
inductive proofs of three theorems about B G; the first two theorems are already
known but the third is probably new.
The prime p will be fixed in everything that follows. If X is a space, let L*
*Z=pX
denote the H Z=p-localization of X constructed by Bousfield [2]. A space is sai*
*d to
be H Z=p-local if the natural map X -! LZ=pX is an equivalence, or alternatively
if any map f : A -! B which induces an isomorphism on mod p homology also
induces an equivalence
f# : Map (B; X) -'!Map (A; X) :
If W and X are spaces, say that X is W -null if every map from W to X is canoni*
*cally
homotopic to a constant, in the sense that the map
(1.1) : X -! Map (W; X)
given by inclusion of constant maps is an equivalence (see [3] and [7]).
1.2 Theorem. (cf. [9], [16]) Let G be a compact Lie group and X a space which
is H Z=p-local and B Z=p-null. Then X is B G-null.
1.3 Miller's theorem. Suppose that Y is a finite complex. Miller [13] shows t*
*hat
Y is B G-null for any finite group G (this is the "Sullivan Conjecture"). Theo*
*rem
1.2 implies (see below) that the Bousfield-Kan p-completion Y ^pis B G-null for*
* any
compact Lie group G, and in this sense gives a generalization of Miller's theor*
*em
______________
The author was supported in part by the National Science Foundation.
Typeset by AM S-T*
*EX
1
2 W. DWYER
to compact Lie groups. Some kind of completion is definitely necessary here; one
way to see this is to use 1.2 and the arithmetic square [4] to compute that, if*
* G is
a connected compact Lie group, the 3-sphere S3 is usually not B G-null.
To derive the fact that Y ^pis B G-null from 1.2, observe that Y ^pis H Z=p-*
*local
by [2] and B Z=p-null by Miller's arguments. Note that Y ^pis equivalent to LZ=*
*pY
if Y is simply connected or more generally "Z=p-good" [1]. If Y is a finite com*
*plex
which is not Z=p-good, it seems to be unknown whether or not LZ=pY is BZ=p-null.
If G is a (topological) group, Z is a space, and f : B G -! Z is a map, say *
*that
f is null on finite p-groups if f . (B ae) is null homotopic for every finite p*
*-group P
and homomorphism ae : P -! G.
1.4 Theorem. (cf. [9, 3.3], [11, 3.11]) Let G be a compact Lie group and Z a
pointed connected space such that Z is H Z=p-local and Z is B Z=p-null. Then a
map f : BG -! Z is null homotopic if and only if it is null on finite p-groups.
Remark. If H be a compact Lie group such that ss0H is a p-group, the hypotheses
of 1.4 apply to the space Z = LZ=p(B H). To see this use 1.3 and note that B H *
*is
Z=p-good [1, VII, x5], so that by the fibre lemma [1, Ch. II] there is an equiv*
*alence
LZ=p(B H) = ((B H)^p) ' H^p:
Theorem 1.4 thus gives a criterion for maps between classifying spaces to be nu*
*ll
homotopic at p.
1.5 The functor PW . The statement of the final theorem requires some more ter-
minology from Bousfield [3] and Farjoun [7]. Suppose that W is some fixed space.
A map f : A -! B is said to be a PW -equivalence if f induces an equivalence
#
Map (B; X) -f-!Map (A; X)
for every W -null space X. Bousfield and Farjoun show that for any space X ther*
*e is
an associated W -null space PW (X) together with a natural PW -equivalence X -!
PW (X). It is easy to check from the definitions that if X0 is any other W -nul*
*l space
with a PW -equivalence X -! X0, then up to homotopy there is a unique equivalen*
*ce
X0 -! PW (X) which makes the appropriate diagram involving X commute. This
implies that a map f is a PW -equivalence if and only if PW (f) is an equivalen*
*ce.
1.6 A natural map. Suppose now that W = B Z=p and that q is a prime differ-
ent from p. Let LZ[1=p]X denote Bousfield's H Z[1=p]-localization of the space
X [2]. Direct checking with the definition shows that the Eilenberg-Mac Lane
spaces K(Z=q; n) and K(Q; n) (n 0) are W -null. It follows that if f : X -! Y
is a PW -equivalence then H *(f; Z=q) and H *(f; Q) are isomorphisms, hence that
H *(f; Z[1=p]) is an isomorphism, and hence that LZ[1=p]f : LZ[1=p]X -! LZ[1=p*
*]Y
is an equivalence. In particular, LZ[1=p]X -! LZ[1=p]PW (X) is an equivalence, *
*and
so the natural map PW (X) -! LZ[1=p]PW (X) gives up to homotopy a natural map
PW (X) ! LZ[1=p]X.
CENTRALIZER DECOMPOSITION 3
1.7 Theorem. Let G be a compact Lie group such that ss0G is a p-group, and let
W = BZ=p. Then the natural map PW (B G) -! LZ[1=p](B G) is an equivalence.
Remark. Miller's theorem (1.3) implies that in the above situation the space BG*
* is
W -null and hence that the map B G -! PW (B G) is an equivalence. This shows
that there is a large difference between the spaces PW (B G) and PW (B G). T*
*he
functors PW and PW are related somewhat more closely if W itself is a suspen*
*sion
[8].
Remark. The three theorems above have corresponding forms that apply to p-
compact groups. For a version of 1.2 see [5, x9] and for a version of 1.4 see [*
*14, x5].
The analogue of 1.7 is proved by an argument very similar to the one in x6.
Remark. It is an interesting exercise to derive 1.2 from 1.7 (and 6.3), at leas*
*t in
the case in which the group G involved has ss0G a finite p-group.
Notation and terminology. We assume that all spaces have been replaced if nec-
essary by weakly equivalent CW-complexes (for instance, by the geometric real-
izations of their singular complexes). The word equivalence means "homotopy
equivalence". A space is said to be Z=p-acyclic if it has the mod p homology of*
* a
point; a map is a Z=p-equivalence if it induces an isomorphism on mod p homolog*
*y.
The author would like to thank E. Farjoun and the referee for their suggesti*
*ons.
x2. The Jackowski-McClure theorem
In this section we will briefly describe the main theorem of [10] and indica*
*te the
general way in which it can be used in inductive arguments.
Suppose that G is a compact Lie group. Let AG be the category in which the
objects are the nontrivial elementary abelian p-subgroups V of G; a morphism
V ! V 0in AG is a group homomorphism f : V -! V 0with the property that
there exists g 2 G such that f(x) = gxg-1 for all x 2 V . Note that specifyin*
*g a
morphism in AG does not involve choosing a particular such g. There is a functor
ff0Gfrom the opposite category AopGto G-spaces which assigns to V the coset spa*
*ce
G=CG (V ); if f : V -! V 0is realized by conjugation with g 2 G, then ff0G(f) a*
*ssigns
to the coset xCG (V 0) the coset xgCG (V ). Let E G be the total space of a uni*
*versal
principal G-bundle and ffG : AG ! Top the functor (E G x ff0G)=G. The following
two properties of this functor are easy to check.
(1) For each object V of AG , the space ffG (V ) is homeomorphic to EG=CG (*
*V )
and thus equivalent to B CG (V ).
(2) The unique G-maps ff0G(V ) ! * pass to compatible maps ffG (V ) -! B G.
These induce a map aG : hocolim ffG -! B G.
See [1, Ch. XII] for a discussion of homotopy colimits. Recall from [1, XII,*
* 3.2]
that if C is a category, the classifying space BC is defined to be hocolim(*C )*
*, where
*C : C -! Top assigns to each object of C the one-point space. The classify*
*ing
spaces B C and B Cop are homotopy equivalent [15, p. 86].
2.1 Theorem. [10] Suppose that G is a compact Lie group. Then
(1) the map aG : hocolim ffG -! B G is a Z=p-equivalence, and
(2) the classifying space B AG is Z=p-acyclic.
4 W. DWYER
Remark. Jackowski and McClure actually show that a certain cohomology spectral
sequence for H *(hocolim ffG ; Z=p) collapses; this is much sharper than 2.1(1)*
*. In
the course of this they show that the higher limits limiH j(ffG ) vanish for i *
*> 0.
For j = 0 these are the higher limits of the constant functor on AG with value *
*Z=p,
which by [1, XI, x5] are just the mod p cohomology groups of B AG . This gives
2.1(2).
Theorem 2.1 more or less states that at the prime p the homotopy type of BG *
*can
be constructed from the homotopy types of classifying spaces of smaller compact
Lie groups, in such a way that the shape of the gluing diagram, in other words *
*the
classifying space of AG , is trivial at p. This suggests using the theorem to *
*prove
statements about B G by induction on the size (e.g., the dimension) of G. There*
* is
one minor problem with this: the values of the functor ffG , which up to homoto*
*py
are the spaces BCG (V ) for elementary abelian p-subgroups of G, are not necess*
*arily
the classifying spaces of Lie groups smaller than G. In fact, if G contains a c*
*entral
subgroup of order p then the space BG appears among the values of the functor f*
*fG
and Theorem 2.1 amounts to a complicated but essentially circular construction *
*of
B G in terms of itself. This indicates that any inductive argument using 2.1 mu*
*st
treat centers in some special way. In practice this also involves treating disc*
*onnected
groups in a special way, since in general it is only if G is connected that div*
*iding
out by the center of G gives a quotient group with trivial center. One inducti*
*ve
scheme that fits this situation is described in the following proposition.
2.2 Proposition. Suppose that C is class of (topological) groups which has the
following three closure properties with respect to compact Lie groups G, H:
(1) If G is connected, the center of G is trivial, and H 2 C for all H of s*
*maller
dimension than G, then G 2 C.
(2) If G is connected with center C and G=C 2 C, then G 2 C.
(3) If G0 is the identity component of G and G0 2 C then G 2 C.
Then C contains every compact Lie group.
Proof. We prove by induction on the dimension of the compact Lie group G that
G 2 C. Condition (1) implies that the trivial group is in C. Suppose that that G
is of dimension d and assume inductively that H 2 C for each compact Lie group
H of dimension less than d (this is certainly true if d=0). Let G0 be the ident*
*ity
component of G and C the center of G0. Then G0=C 2 C by property (1) and
induction, G0 2 C by (2), and hence G 2 C by (3).
2.3 Finite groups. In the examples that come up in this paper, C will typicall*
*y be
the class of all topological groups G such that B G has some appropriate proper*
*ty.
Statement 2.2(1) is then proved using 2.1. Statements 2.2(2)-(3) are proved by
more direct arguments involving facts about finite groups; for instance, obtain*
*ing
2.2(3) usually involves knowing something about the finite group G=G0. We will
prove the necessary statements about finite groups by a subsidiary initial indu*
*ction
that depends on the following elementary proposition.
2.4 Proposition. Suppose that C is class of (topological) groups which has the
following two closure properties with respect to finite groups G, H:
(1) If G has no central subgroup of order p, and H 2 C for all H of smaller
CENTRALIZER DECOMPOSITION 5
order than G, then G 2 C.
(2) If G has a central subgroup C of order p and and G=C 2 C, then G 2 C.
Then C contains every finite group.
x3. The fibration principle
In this section we will discuss some basic homotopy theoretic observations t*
*hat
turn out to be useful in proving Theorems 1.2, 1.4 and 1.7. We first discuss ma*
*pping
spaces for which the domain is the total space of a fibration (3.1), and then m*
*apping
spaces for which the domain is a homotopy colimit (3.6). We end by explaining
how to tie these two discussions together (3.11).
3.1 Maps from the total space of a fibration.
3.2 Proposition. (Fibration Principle) Suppose that f : E -! B is a fibration
over a connected base B with fibre F , and that X is some space. Then there is
another naturally associated fibration fX : EX -! B with fibre Map (F; X) suc*
*h that
the space of sections of fX is equivalent to Map (E; X).
The way to understand this proposition is to picture E as a fibre bundle ove*
*r B
and notice that giving a map E -! X amounts to giving, for each point b 2 B, a
map from a copy of F to X. We will sketch a direct proof (see also 3.11). For*
* a
topological treatment of fibrewise function spaces, see [12, Ch. 9].
Proof of 3.2 (Sketch). Let G = Aut (F ) denote the monoid of self homotopy equi*
*v-
alences of F . The monoid G acts from the left on F , and associated to this a*
*ction
is a universal fibration u : E(F ) -! B G with fibre F . Since u is universal t*
*here is
a map c : B -! B G, unique up to homotopy, such that the pullback of u over c is
equivalent to the fibration f. We denote such a pullback B xBG E(F ); the map c
is understood in this notation.
The monoid G also acts from the right (by composition) on M = Map (F; X). As-
sociated to this action is a fibration v : E(M) -! B G with fibre M. The evalua*
*tion
map F x M -! X induces a map
(3.3) e : E(F ) xBG E(M) -! X :
Consider now the category C of spaces over B G. An object in this category is
a space Y together with a map g : Y -! BG; a morphism h : Y -! Y 0is a map
of spaces such that g0h = g. For any object Y of C let (Y ) denote the mapping
space
Map (Y xBG E(F ); X)
and (Y ) the space of sections of the fibration Y xBG E(M) -! Y . The map e
above (3.3) induces a map
t(Y ) : (Y ) -! (Y )
which gives a natural transformation between the two indicated functors Cop !
Top . To prove the proposition we will show that t(B) is an equivalence.
6 W. DWYER
The map t(Y ) is an equivalence if Y is a point or more generally if Y is a
contractible space; in this case the domain and range of t(Y ) are each equival*
*ent
to M = Map (F; X). Suppose that
Y1 ----! Y2
?? ?
y ?y
Y3 ----! Y4
is a homotopy pushout diagram of spaces over B G. It is not hard to see that the
induced diagram
Y1 xBG E(F ) ----! Y2 xBG E(F )
?? ?
y ?y
Y3 xBG E(F ) ----! Y4 xBG E(F )
is also a homotopy pushout diagram. The natural transformation t then gives a
map of squares
(Y4) ----! (Y2) (Y4) ----! (Y2)
?? ? t ? ?
y ?y -! ?y ?y
(Y3) ----! (Y1) (Y3) ----! (Y1)
in which each square is a homotopy pullback square (because mapping constructio*
*ns
like and convert homotopy pushouts to homotopy pullbacks [1, XII 4.1]). It
follows that if t(Yi) is an equivalence for i 3 then t(Y4) is also an equivale*
*nce.
Both and convert disjoint unions to products,`so if {Yff} is a collection of
spaces over B G with disjoint union Y = ffYff, then t(Y ) is an equivalence i*
*f each
t(Yff) is. Let C be the smallest homotopy invariant class of spaces over B G wh*
*ich
contains all contractible spaces, is closed under homotopy pushouts, and is clo*
*sed
under disjoint unions. By the discussion above, t(Y ) is an equivalence for ea*
*ch
space Y in C. It is clear by induction on dimension that C contains every spac*
*e Y
over B G such that the underlying space of Y is a finite dimensional CW-comple*
*x.
Suppose that Y is an infinite complex over B G, and let Yn be the n-skeleton of*
* Y .
The fact that Y 2 C then follows from the fact that there is a homotopy pushout
diagram
` ` ` id+ id `
( n Yn) ( n Yn) - ---! n Yn
? ?
id+s?y ?y
`
nYn - ---! Y
in which the map s is a shift map derived from the inclusions Yn -! Yn+1 .
One case is particularly interesting. The following proposition is closely r*
*elated
to work of Zabrodsky as reformulated by Miller [13, 9.5].
CENTRALIZER DECOMPOSITION 7
3.4 Proposition. Suppose that f : E -! B is a fibration over a connected base
B with fibre F , and that X is a space. If the map : X -! Map (F; X) is an
equivalence (1.1), then the restriction map f# : Map (B; X) -! Map (E; X) is*
* an
equivalence.
Proof. Let g : B -! B be the identity fibration. Under the stated hypotheses the
map f itself induces an equivalence between the fibration fX of 3.2 and the fib*
*ration
gX (which is the projection X x B -! B). The induced map between spaces of
sections, which is essentially f# , is then also an equivalence.
Restricting attention to individual mapping space components gives a more sp*
*e-
cialized variant of 3.4. If F and X are spaces, let Map (F; X)[F]denote the spa*
*ce of
maps F -! X which are homotopic to constant maps. More generally, if f : E -! B
is a fibration over a connected base B with fibre F , let Map (E; X)[F] denote *
*the
space of those maps E -! X which are homotopic to constant maps when restricted
to F .
3.5 Proposition. Suppose that f : E -! B is a fibration over a connected base
B with fibre F , and that X is a space. If the map : X -! Map (F; X)[F] is
an equivalence, then the restriction map f# : Map (B; X) -! Map (E; X)[F] is *
*an
equivalence.
3.6 Maps from a homotopy colimit. There are mapping space results roughly
parallel to the above ones with the notion "total space of a fibration" replace*
*d by
the notion "homotopy colimit". The analogue of 3.2 is the following proposition*
* of
Bousfield and Kan.
3.7 Proposition. [1, XII, x4] Suppose that C is a small category, X a space, a*
*nd
fl : C -! Top a functor. Let Map (fl; X) : Cop -! Top be the functor which as*
*signs
to each object c the space Map (fl(c); X). Then there is an equivalence
Map (hocolim fl; X) -~!holim Map (fl; X) :
In a situation like that of 3.4 this gives the following.
3.8 Proposition. Let C be a small category, X a space, and fl : C -! Top a
functor. Assume that for each object c of C, the map : X -! Map (fl(c); X) is
an equivalence (1.1). Then the unique natural transformation fl ! *C induces an
equivalence
Map (B C; X) = Map (hocolim (*C ); X) -~!Map (hocolim fl; X) :
Proof. By assumption the functor Map (fl; X) (see 3.7) is equivalent to Map (*C*
* ; X).
The homotopy limit of this constant functor is Map (hocolim (*C ); X).
3.9. As above, restricting attention to individual mapping space components giv*
*es
a specialized variant. If C is a small category and fl : C -! Top is a funct*
*or,
then by the definition of homotopy colimit [1, XII, x2] there is a natural map
fl(c) -! hocolim fl for each object c of C. If X is a space, let Map (hocolim *
*fl; X)[fl]
denote the space of maps f : hocolim fl -! X such that for each object c of C t*
*he
restriction of f to fl(c) is homotopic to a constant map.
8 W. DWYER
3.10 Proposition. Let C be a small category, X a space, and fl : C -! Top
a functor such that for each object c of C, the map : X -! Map (fl(c); X)[fl(*
*c)]
is an equivalence. Then the unique natural transformation fl ! *C induces an
equivalence
Map (B C; X) = Map (hocolim (*C ); X) -~!Map (hocolim fl; X)[fl]:
3.11 Total spaces vs. homotopy colimits. Propositions 3.2 and 3.7 are tied
together by the fact that the total space of a fibration over B with fibre F is
equivalent to the homotopy colimit of a functor whose values are all equivalent*
* to
F and whose domain category has classifying space equivalent to B. Suppose for
simplicity that B is the geometric realization of a simplicial complex K. Let *
*CB
be the category in which an object is a (closed) simplex oe of B and there is a*
* single
morphism oe -! oe0 if oe is contained in oe0 (there are no other morphisms). *
*The
classifying space BCB is the geometric realization of the barycentric subdivisi*
*on of
K and so is homeomorphic to B. If f : E -! B is a fibration with fibre F , then
there is a functor flE : CB -! Top which sends oe to f-1 (oe), and it is poss*
*ible to
check that hocolim flE is equivalent to E. This is obvious if B itself is a s*
*implex,
and in general one can make an induction, based on homotopy pushouts, over the
skeletal filtration of B (cf. proof of 3.2). By 3.7 there is an equivalence
Map (E; X) -! holim Map (flE ; X) :
Now consider the following proposition.
3.12 Proposition. Suppose that C is a small category and that fl : C -! Top i*
*s a
functor which sends each object of C to a space equivalent to Z and each morphi*
*sm
of C to an equivalence. Then
(1) the natural map hocolim fl -! B C is up to homotopy a fibration with fi*
*bre
Z, and
(2) the space of sections of this fibration is equivalent to holim fl.
Applying this proposition to the functor Map (flE ; X) shows that hocolim fl*
*E is
up to homotopy the total space of a fibration over B with fibre Map (F; X) and
space of sections equivalent to Map (E; X). (Note [15, p. 91] that the classi*
*fying
space BCopBis equivalent to BCB , and hence to B). This gives a proof of 3.2 wh*
*ich
uses 3.7.
Remark. The first statement of 3.12 is a form of Quillen's Theorem B [15, p. 97*
*].
Statement 3.12(2) can be proved by using the interpretation of homotopy limit in
[6, 2.12]. This identifies holim fl up to homotopy as the mapping space Map ("**
*C; fl),
where "*Cis a "free resolution" of the functor *C (i.e, a CW-functor [6, 1.16] *
*weakly
equivalent to *C ) and the maps are computed in the category of functors C ! To*
*p .
The space hocolim ("*C) is equivalent to B C. One proves by skeletal induction *
*([6,
1.16], cf. proof of 3.2) that if A : C ! Top is any CW-functor then the space
Map (A; fl) is equivalent in a natural way to the space of sections of the fib*
*ration
EA -! hocolim A, where EA is determined by the (homotopy) pullback diagram
EA ----! hocolim fl
?? ?
y ?y :
hocolim A ----! hocolim(*C ) = BC
CENTRALIZER DECOMPOSITION 9
x4. Maps into spaces which are B Z=p-null
In this section we will prove 1.2. Suppose that X is a space which is H Z=p-*
*local
and BZ=p-null. Let C be the class of all topological groups G with the property*
* that
: X -! Map (B G; X) is an equivalence; it is necessary to prove that C contai*
*ns
all compact Lie groups. Recall that, by the definition of what it means to be
"H Z=p-local", any Z=p-equivalence A -! B induces an equivalence Map (B; X) -~!
Map (A; X).
4.1 Lemma. If G is a discrete locally finite group (i.e. a union of finite gro*
*ups)
then G 2 C.
Remark. The following argument is a prototype of the argument below for compact
Lie groups.
Proof of 4.1. We prove using 2.4 and induction on the order of G that any finite
group G belongs to C. The group Z=p belongs to C by definition. If G has a cent*
*ral
subgroup C of order p then applying the 3.4 to the fibration
B C -! B G -! B(G=C)
and using the induction hypothesis shows that G 2 C. If G has no central subgro*
*up
C of order p, then for each object V of AG the space ffG (V ) has the homotopy *
*type
of B H, where H is of smaller order than G. By 2.1(1), the map hocolim ffG -! B*
* G
induces an equivalence
Map (B G; X) -~!Map (hocolim ffG ; X):
By induction and 3.8 the natural transformation ffG -! *AG induces an equivale*
*nce
Map (B AG ; X) -~!Map (hocolim ffG ; X) :
However 2.1(2) guarantees that B AG is Z=p-acyclic, so that the map
: X -! Map (B AG ; X)
is an equivalence. Tracing through the various identifications shows that
: X -! Map (B G; X)
is also an equivalence.
The passage to general locally finite groups is by a standard homotopy colim*
*it
argument [13, proof of 9.8].
4.2 Lemma. If G is an abelian compact Lie group, then G 2 C.
Proof. The group G is isomorphic to the product of a finite abelian group with
a torus. Let D G be the group of elements of order a power of p, considered
as a discrete group. It is not hard to see by explicit calculation that the map
B D -! B G induces an isomorphism on mod p homology and therefore an equiva-
lence Map (B G; X) -~!Map (B D; X). The desired result follows from 4.1.
10 W. DWYER
There are now three steps to carry out, which correspond to the three hypoth*
*eses
of 2.2.
Step I. Suppose that G is a connected compact Lie group of dimension d with a
trivial center, and that H 2 C for all compact Lie groups H of dimension less t*
*han
d. It is necessary to prove that G 2 C. By 2.1(1), the map hocolim ffG -! B G
induces an equivalence
Map (B G; X) -~!Map (hocolim ffG ; X):
Since G is connected and has trivial center, for each object V of AG the space
ffG (V ) has the homotopy type of B H, where H is of dimension less than d. By
induction and 3.8 the natural transformation ffG -! *AG induces an equivalence
Map (B AG ; X) -~!Map (hocolim ffG ; X) :
However 2.1(2) guarantees that B AG is Z=p-acyclic, so that the map
: X -! Map (B AG ; X)
is an equivalence. Tracing through the various identifications shows that
: X -! Map (B G; X)
is also an equivalence.
Step II. Suppose that G is connected with center C, and that G=C 2 C. It is
necessary to show that G 2 C. There is a fibration sequence
B C -! B G -! B(G=C)
in which by 4.2 the fibre B C belongs to C. By 3.4, then, the restriction map
Map (B(G=C); X) -! Map (B G; X)
is an equivalence. The result now follows from the fact that G=C 2 C.
Step III. Suppose that the identity component G0 of G belongs to C; it is neces*
*sary
to show that G 2 C. There is a fibration sequence
B G0 -! B G -! B ss0G :
By 3.4 and the assumption on G0 the restriction map
Map (B ss0G; X) -! Map (B G; X)
is an equivalence. The result now follows from the fact (4.1) that ss0G 2 C.
CENTRALIZER DECOMPOSITION 11
x5. Maps null on finite p-groups
In this section we will prove 1.4, or more accurately a slight generalizatio*
*n of it.
If G is a topological group and X is a space, let Map (B G; X)[p]denote the spa*
*ce
of all maps f : B G -! X which are null on finite p-groups. Let Z be a pointed
connected space such that Z is H Z=p-local and Z is B Z=p-null. Define C to be
the class of all topological groups G with the property that the map
: Z -! Map (B G; Z)[p]
is an equivalence (1.1). What we will prove is the following.
5.1 Theorem. The class C contains all compact Lie groups G.
In particular, if G is a compact Lie group then the space Map (B G; Z)[p]is *
*con-
nected. This implies that every map B G -! Z which is null on finite p-groups *
*is
homotopic to a constant map, which is 1.4.
5.2 Lemma. If G is a locally finite group (i.e., a union of finite p-groups) t*
*hen
G 2 C.
Remark. As in x4, this is a prototype of the proof below for compact Lie groups.
Proof. We prove using 2.4 and induction on the order of G that any finite group*
* G
belongs to C. If G has a central subgroup C of order p then applying the 3.5 *
*to
the fibration
B C -! B G -! B(G=C)
and using the induction hypothesis shows that G 2 C. If G has no central subgro*
*up
of order p, for each object V of AG the space ffG (V ) has the homotopy type *
*of
B K for a group K of order smaller than G. By 2.1(1), the map hocolim ffG -! B G
induces an equivalence a#G : Map (B G; Z) -! Map (hocolim ffG ; Z). Moreover*
*, a
check with the definitions shows that the composite maps (cf. 3.9)
ffG (V ) -! hocolim ffG -aG-!BG
are obtained up to homotopy by applying the classifying space construction to
homomorphisms K -! G. It follows in the notation of 3.9 (with fl = ffG ) that a*
*#G
induces an equivalence from Map (B G; Z)[p]to a union of components of the space
Map (hocolim ffG ; Z)[fl]. By induction, 3.10, and 2.1(2), the map
: Z -! Map (hocolim ffG ; Z)[fl]
is an equivalence. Tracing through the various identifications gives the desir*
*ed
equivalence Z -! Map (B G; Z)[p].
The result follows for arbitrary locally finite groups by a homotopy colimit*
* cal-
culation [13, proof of 9.8].
12 W. DWYER
5.3 Lemma. If G is an abelian compact Lie group, then G 2 C.
Proof. This follows from 5.2: as in 4.2, there is a locally finite p-group D a*
*nd a
homomorphism D -! G which induces an equivalence
Map (B G; Z) -~!Map (B D; Z) :
The proof of 5.1 now has three steps, which correspond to the three steps of*
* 2.2.
Step I. Suppose that G is a connected compact Lie group of dimension d with a
trivial center, and that K 2 C for all compact Lie groups K of dimension less t*
*han d.
It is necessary to prove that G 2 C. By 2.1(1), the map hocolim ffG -! B G indu*
*ces
an equivalence a#G : Map (B G; Z) -! Map (hocolim ffG ; Z). Since G is conne*
*cted
and has trivial center, for each object V of AG the space ffG (V ) has the homo*
*topy
type of B K, where K 2 C. Moreover, a check with the definitions shows that the
composite maps (cf. 3.9)
ffG (V ) -! hocolim ffG -aG-!BG
are obtained up to homotopy by applying the classifying space construction to
homomorphisms K -! G. It follows in the notation of 3.9 (with fl = ffG ) that a*
*#G
induces an equivalence from Map (B G; Z)[p]to a union of components of the space
Map (hocolim ffG ; Z)[fl]. By induction, 3.10, and 2.1(2), the map
: Z -! Map (hocolim ffG ; Z)[fl]
is an equivalence. Tracing through the various identifications gives the desir*
*ed
equivalence Z -! Map (B G; Z)[p].
Step II. Suppose that G is connected with center C, and that G=C 2 C. It is
necessary to show that G 2 C. This is the same as the second step in the proof *
*of
1.2, but uses 5.3 and 3.5 instead of 4.2 and 3.4.
Step III. Suppose that the identity component G0 of G belongs to C; it is neces*
*sary
to show that G 2 C. This is the same as the third step in the proof of 1.2, but*
* uses
5.2 and 3.5 instead of 4.1 and 3.4.
x6. Calculating PW (B G)
In this section W will denote the fixed space B Z=p. The goal is to compute
PW (B G) when G is a compact Lie group such that ss0G is a p-group.
6.1 Remark. As described in [3], the space PW (B G) is built by constructing a *
*nested
collection of spaces X , one for each countable ordinal . The space X0 is B G; *
*if
is a limit ordinal then X = [0< X0 ; if = 0+ 1 is a successor ordinal then X
is obtained from X0 by adjoining cones on all maps of W and its suspensions in*
*to
X0 . The space PW (B G) is then the union or colimit [ X (note that the number
of spaces in this union is uncountable).
CENTRALIZER DECOMPOSITION 13
6.2 Lemma. For a 1-connected space X the following three conditions are equiv-
alent:
(1) X is Z=p-acyclic.
(2) For each i 2, ssiX is a module over Z[1=p].
(3) The natural map X -! LZ[1=p]X is an equivalence.
Moreover, these conditions imply
(4) X is W -null.
Proof. The equivalence of (1) and (2) follows from Serre "mod-C" theory, since X
is Z=p-acyclic if and only if the reduced integral homology groups of X are uni*
*quely
p-divisible, which if ss1X is trivial is the case if and only if the homotopy g*
*roups
of X are uniquely p-divisible. The equivalence between (2) and (3) is from [1, *
*V,
x3] (note that since X is simply connected the Bousfield localization LZ[1=p]X *
*is
equivalent to the Bousfield-Kan space Z[1=p]1 (X) [2]). The fact that (2) impli*
*es
(4) results for instance from a direct calculation with obstruction theory. Not*
*e that
(4) does not imply the others, e.g., the n-sphere (n 2) is 1-connected and sat*
*isfies
(4) (by 1.3), but does not satisfy (1).
6.3 Remark. If X is a connected space with the property that H 1(X; Z[1=p]) = 0,
e.g., X = B G with ss0G a p-group, then LZ[1=p]X is 1-connected [1, VII, 3.2] [*
*2]
and satisfies the conditions given in 6.2 [1, V, x3]. In particular it follows*
* from
Theorem 1.7 that if G is a compact Lie group with ss0G a p-group then all of the
mod p homology of BG can be killed by the iterated cone adjunction process of 6*
*.1.
6.4 Remark. Lemma 6.2 leads to the following recognition principle. Let X be
a connected space such that H 1(X; Z[1=p]) is trivial, and suppose that we can
construct a PW -equivalence f : X -! Y such that Y is 1-connected and Z=p-
acyclic. Then the natural map PW (X) -! LZ[1=p]X is an equivalence. To see this,
note first that H* (f; Z[1=p]) is an isomorphism (1.6). The desired result now *
*follows
from the commutative diagrams
X ---f-! Y X - -f--! Y
?? ? ? ?
y ~?y ?y ~?y
LZ[1=p]X ---~-! LZ[1=p]Y PW (X) - -~--! PW (Y )
in which by 6.2 the indicated vertical arrows are equivalences (for instance, 6*
*.2(4)
implies that Y ~ PW (Y )).
6.5 Lemma. [3, 2.5] The class of PW -equivalences is closed under homotopy col-
imits, in the sense that if C is a small category, fl; fl0 : C -! Top are fu*
*nc-
tors, and o : fl -! fl0 is a natural transformation which gives a PW -equival*
*ence
oc : fl(c) -! fl0(c) for each object c of C, then hocolim o : hocolim fl -! hoc*
*olim fl0 is
a PW -equivalence.
6.6 Remark. Note that 6.5 is proved with 3.7. This lemma implies in particular
that PW -equivalences are stable under "cobase change", i.e., if X -! X0 is a P*
*W -
equivalence and X -! Y is a map, then the natural inclusion of Y in the homotopy
pushout of the diagram X0- X -! Y is also a PW -equivalence.
14 W. DWYER
6.7 Remark. The class of Z=p-equivalences is also closed under arbitrary homoto*
*py
colimits [1, XII, 5.7].
6.8 Lemma. Let C be a small category such that B C is connected, fl; fl0 : C -!
Top functors, and o : fl -! fl0 a natural transformation. Suppose that for*
* each
object c of C the map oc : fl(c) -! fl0(c) is a map between connected spaces wh*
*ich
induces a surjection of fundamental groups. Then the map hocolim o : hocolim fl*
* -!
hocolim fl0 is also a map between connected spaces which induces a surjection of
fundamental groups.
Proof. This follows from the van Kampen theorem and the explicit construction of
the homotopy colimit in [1, XII, x5].
6.9 Lemma. [3, 2.9] Suppose that X is a connected space. Then PW (X) is con-
nected, and the map X -! PW (X) induces a surjection of fundamental groups.
Remark. This is clear from the description of PW (X) in 6.1.
Let C be the class of (topological) groups with the property that the map
PW (B G) -! LZ[1=p](B G) is an equivalence. As usual, there are three steps (*
*cf.
2.2) involved in proving that C contains every compact Lie group G such that ss*
*0G
is a p-group. We have to modify Step I slightly to handle a technical issue co*
*n-
nected with the center (see Step II). The following theorem guarantees that the
main inductive step does not leave the class of compact Lie groups G such that
ss0G is a p-group.
6.10 Theorem. [11, A.4] Suppose that G is a compact Lie group such that ss0G
is a p-group, and that K is a finite p-subgroup of G (for example, K might be an
elementary abelian p-subgroup of G). Then ss0CG (K) is also a p-group.
Step I. Suppose that G is a connected compact Lie group of dimension d with no
element of order p in its center. Assume that H 2 C for all H of dimension less
than d such that ss0H is a p-group. We have to show that G 2 C. Let Y be the
space which fits into the homotopy pushout diagram
hocolim ffG - aG---!B G
?? ?
y ?y
hocolim PW (ffG )- ---! Y
in which the left hand vertical arrow is induced by the natural map ffG -! PW (*
*ffG ).
By 6.9, 6.8, and the van Kampen theorem, Y is 1-connected. By 6.5 the left hand
vertical map is a PW -equivalence, and so (6.6) the map BG -! Y also is. Part (*
*1) of
2.1 gives that the upper horizontal arrow in the above diagram is a Z=p-equival*
*ence.
Since G has no central element of order p, each space ffG (V ) (V an object of *
*AG )
is of form BH for some H which by induction (6.10) belongs to C, and so (6.3) e*
*ach
space PW (ffG (V )) is Z=p-acyclic. It follows from 6.7, applied to the unique *
*natural
transformation PW (ffG ) -! *AG , that the induced map hocolim PW (ffG ) -! B AG
is a Z=p-equivalence, and thus by 2.1(2) that hocolim PW (ffG ) is Z=p-acyclic.*
* A
Meyer-Vietoris sequence calculation now gives that Y is Z=p-acyclic, and so the
desired result is a consequence of 6.4.
CENTRALIZER DECOMPOSITION 15
6.11 PW and fibrations. Recall from [3, x4] that given a fibration sequence F *
*-!
E -! B over a connected base B, it is possible to apply PW fibrewise to obtain
another fibration sequence PW (F ) -! E -! B. There is a PW -equivalence E -! E
of spaces over B which on fibres gives the natural map F -! PW (F ). If E -! B
is a principal fibration, then so is E -! B (cf. [3, x3]); in particular, taki*
*ng E = *
and F = G shows that if G is a topological group or more generally a loop space
then PW (G) is also a loop space.
6.12 Lemma. Let G -! E -! B be a principal fibration sequence over a connected
base B. Assume that the classifying space B PW (G) is W -null. Then the sequence
PW (G) -! PW (E) -! PW (B) is also up to homotopy a fibration sequence.
Proof. This is a restatement of [3, 4.3].
6.13 Lemma. If G is a locally finite p-group then PW (B G) is contractible. In
particular, G 2 C.
Proof. Note that G 2 C if and only if PW (B G) is contractible, since "H*(B G; *
*Z[1=p])
vanishes (G is a union of finite p-groups) and so LZ[1=p](B G) is contractible.*
* We
first prove by induction on the order of G that if G is a finite p-group then P*
*W (B G)
is contractible. This is clear if G is trivial or if G = Z=p. Otherwise, there *
*exists a
cyclic group oe of order p in the center of G and a corresponding fibration seq*
*uence
Boe -! B G -! B(G=oe) = BK
with PW (B K) contractible by induction. Applying PW fibrewise (6.11) thus giv*
*es
a PW -equivalence B G -! B K, which shows that PW (B G) is contractible too.
The statement for a general locally finite group G follows 6.5 and the fact *
*that
B G can be expressed as the filtered homotopy colimit of the classifying spaces*
* of
the finite subgroups of G (cf. [1, XII, 3.5] or [13, 9.8]).
6.14 Lemma. If G is an abelian compact Lie group such that ss0G is a p-group,
then G 2 C.
Proof. As in the proof of 4.2, let D G be the group of elements of order a pow*
*er
of p, considered as a discrete group. Let f : B D -! BG be the map induced by
the inclusion D -! G. Then F is a Z=p-equivalence and (by the assumption on
ss0G) ss1(f) is a surjection. Let Y be the space with fits into the homotopy pu*
*shout
diagram
B D ----! B G
?? ?
y ?y
PW (B D) ~ * ----! Y
in which PW (B D) is contractible by 6.13. By the van Kampen theorem Y is
1-connected. Clearly Y is Z=p-acyclic, and by 6.6 the map B G -! Y is a PW -
equivalence. The desired result follows from 6.4.
Step II. Suppose that G is a connected compact Lie group. Let C0 be the center *
*of
G, and C C0 the inverse image in C0 of the p-torsion subgroup of ss0C0. Assume
that G=C 2 C. (Observe that the center of G=C, which is isomorphic to C0=C, has
16 W. DWYER
no nontrivial element of order p, so that in the modified inductive scheme we a*
*re
following the group G=C will be handled in Step I.) It is necessary to prove th*
*at
G 2 C. Since C is abelian, the fibration sequence
B C -! B G -! B(G=C)
is principal. By 6.14 and 6.3, PW (B C) is a 1-connected Z=p-acyclic space. B*
*y a
Serre spectral sequence argument, the classifying space BPW (B C) (see 6.11) is*
* also
Z=p-acyclic, and so (6.2) is W -null. Lemma 6.12 thus shows that there is a fib*
*ration
sequence
PW (B C) -! PW (B G) -! PW (B(G=C))
From 6.3 it is clear that the base and fibre here are Z=p-acyclic. This implies*
* the
space PW (B G) is Z=p-acyclic. The space PW (B G) is 1-connected (6.9) and so t*
*he
desired result follows from 6.4.
Step III. Suppose that G is a compact Lie group such that ss0G is a p-group, and
assume that the identity component G0 2 C. It is necessary to prove that G 2 C.
Consider the fibration
(6.15) PW (B G0) -! X -! Bss0G
obtained by applying PW fibrewise (6.11) to the fibration B G0 -! B G -! B ss0*
*G.
There is a PW -equivalence B G -! X . By induction and 6.3, the space PW (B G0)
satisfies the conditions given in 6.2, and in particular the higher homotopy gr*
*oups
of this space are uniquely p-divisible. This implies that the (twisted) cohomol*
*ogy
groups H i(B ss0G; ssjPW (B G0)) vanish for i > 0 and j 2 and hence by obstruc*
*tion
theory that the fibration 6.15 has a section s : B ss0G -! X . Since PW (B G0*
*) is
Z=p-acyclic (6.2), H *(s; Z=p) is an isomorphism; since PW (B G0) is 1-connecte*
*d,
ss1(s) is an isomorphism. Let Y be the space which fits into the homotopy pusho*
*ut
diagram
Bss0G ---s-! X
?? ?
y ?y
PW (B ss0G) ~ * ----! Y
in which PW (B ss0G) is contractible by 6.13. Clearly Y is simply connected and*
* Z=p-
acyclic; by 6.6 the map X -! Y is a PW -equivalence. The composite BG -! X -!*
* Y
is then also a PW -equivalence, and the desired result follows from 6.4.
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University of Notre Dame, Notre Dame, Indiana 46556
Processed September 22, 1994.