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 BG
at the prime p as a homotopy colimitof 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 casethe 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 parallel
inductive proofs of three theorems about BG; the first two theorems arealready
known but the third is probably new.
The prime pwill be fixed in everything that follows. If X is a space,let LZ=*
*pX
denote the HZ=p-localization of X constructed by Bousfield [2].A space is said *
*to
be HZ=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 Wto X is canonical*
*ly
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 HZ=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 that
Y!is!B G-null for any finite group G (this is the "Sullivan Conjecture"). Theor*
*em
1.2!implies (see below) that the Bousfield-Kan p-completion Y ^pis B G-null for*
*any
compact!Lie!group G, and inthis sense gives a generalization of Miller's theorem
!
The authorwas supported in part by the National Science Foundation.
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2 W. DWYER
to compact Lie groups. Some kind of completion is definitelynecessary 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 S 3is usually not BG-null.
To derive the fact that Y ^pis B G-null from 1.2,observe that Y ^pis H Z=p-l*
*ocal
by [2] and BZ=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 (top ological) group, Z is a space, and f : BG ! Z is a map, say *
*that
f is null on finite p-groups if f (Bae) is null homotopic for every finite p-gr*
*oup 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 HZ=p-local and Z is BZ=p-null. Then a
map f :B G ! Z isnul l 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 equiva*
*lence
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 finaltheorem 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 there*
* 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 ! X 0, then up to homotopythere is a unique equivalence
X0 ! PW (X) which makes theappropriate 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 = BZ=p and that q is a prime differ-
ent from p. Let LZ[1=p]X denote Bousfield's H Z[1=p]-localization ofthe 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 thatLZ[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, a*
*nd
so the natural map PW (X) ! LZ[1=p]PW(X ) gives up to homotopy a natural map