BOUSFIELD LOCALIZATIONS OF CLASSIFYING SPACES OF
NILPOTENT GROUPS
WILLIAM G. DWYER, , EMMANUEL DROR FARJOUN, , AND DOUGLAS C.
RAVENEL,
1.Introduction
Let G be a finitely generated nilpotent group. The object of this paper is to
identify the Bousfield localization LhBG of the classifying space BG with respe*
*ct
to a multiplicative complex oriented homology theory h*. We show that LhBG is
the same as the localization of BG with respect to the ordinary homology theory
determined by the ring h0. This is similar to what happens when one localizes
a space X with respect to a connected ring theory E: it follows from results of
Bousfield [Bou79 , Theorem 3.1] that LE X is the localization of X with respect*
* to
ordinary homology with coefficients in the ring E0. The point in this paper is *
*that
we do not require the spectrum h to be connected.
Our main result is
Theorem 1. Let G be a finitely generated nilpotent group, and let h* be a mul
tiplicative complex oriented homology theory. Then LhBG = LR BG, where R is
the ring h0 and LR is localization with respect to the ordinary homology theory
determined by R.
The hypothesis that h* be multiplicative is not essential to any of our argum*
*ents.
We include it mainly to avoid cumbersome statements, and because most complex
oriented theories of interest, such as Morava Ktheory, are multiplicative. By *
*mod
ifying somewhat the results it seems that one could remove the assumption that *
*h*
is multiplicative. However, do not consider this case. Complex orientability is*
* used
in an essential way, in the proof of Theorem 3.
Our method of proof is to begin with finite pgroups and proceed by induction*
* on
the order of the group. We show that if G has a normal subgroup H such that BH
is h*local, G=H = Z=p, and B(G=H) is Rlocal for R = h0, then BG is h*local.
We do this by studying the fibration
BH ! BG ! B(G=H) = BZ=p
To pass to arbitrary finitely generated nilpotent groups, we use the arithmetic
square decomposition of LhX due to Mislin and Bousfield.
Our main task is to study the fibration displayed above. In general it is not*
* true
that if the base and fibre of a fibration are local with respect to some homolo*
*gy
theory, then the total space is also local. For example in the fibration
S2 ! RP 2! BZ=(2)
____________
All three authors were partially supported by the USIsrael Binational Scien*
*ce Foundation,
and the first and third authors by the National Science Foundation.
0
BOUSFIELD LOCALIZATIONS OF CLASSIFYING SPACES OF NILPOTENT GROUPS 1
both fiber and base are local with respect to ordinary integral homology, but t*
*he
total space is not [BK72 ], [DDK77 ].
Our technique for dealing with this problem is to use the following lemma. If*
* B
is a space and C is a class of fibrations over B, say that C has h*accessible *
*fibres if
any h*equivalence (over B) between fibrations in C induces an h*equivalence on
fibres.
Lemma 2. Let h* be an arbitrary homology theory, and consider the diagram
F _______Ew_______wBss
 
   
f  g  
  
u u Lhss 
F 0_____wLhE ______Bw
in which each row is a fiber sequence. Suppose that B and F are both h*local, *
*and
that there exists some class C of fibrations over B which has h*accessible fib*
*res and
contains both ss and Lhss. Then E is h*local.
Proof.Since C has h*accessible fibres and h*(g) is an equivalence, h*(f) is al*
*so
an equivalence. The space F is h*local by assumption, and F 0is h*local since*
* it
is the homotopy fiber of a map between h*local spaces. Therefore f, being and
h*equivalence between h*local spaces, is an equivalence. It follows that_g_is*
* also
an equivalence and E is h*local. __
In order to use this lemma, we show in the next section that if h* is a multi
plicative complex oriented homology theory with the property that h0 is a vector
space over Z=p, then the class of all fibrations over BZ=p has h*accessible fi*
*bres.
2.Fibrations over BZ=p
In this section we prove the following theorem. The results and arguments are
inspired by the work of Kriz [Kri].
Theorem 3. Let h* be a multiplicative complex oriented homology theory such
that the ring h0 is a mod p vector space. Suppose that E and E0 are fibrations *
*over
BZ=p with fibres F and F 0, respectively, and that f : E ! E0 is a map over BZ=p
which induces an isomorphism h*E ~=h*E0. Then f also induces an isomorphism
h*F ~=h*F 0.
Recall that the homotopy coequalizer of a pair of maps f; g : X ! Y is obtain*
*ed
by taking the cylinder X x [0; 1] and gluing one end to Y by f and the other end
to Y by g. This construction is sometimes also called the double mapping cylind*
*er
of f and g. Given maps of pairs f; g : (X; A) ! (Y; B), the homotopy coequalizer
of f and g is the pair (Z; C), where Z is the homotopy coequalizer of the two m*
*aps
X ! Y and C is the homotopy coequalizer of the two maps B ! C. A diagram of
pairs equivalent to
(X; A)_____w(Y;fB);_g__w(Z; C)
is said to be a homotopy coequalizer diagram. The following lemma is elementary.
Lemma 4. Let
(X; A)_____w(Y;fB);_g__w(Z; C)
2WILLIAM G. DWYER, UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN 46556 USA, EMMANUEL*
* DROR FARJOUN, HEBREW UNIVERSITY, JERUSALEM, ISRAEL, AND DOUGL@
be a homotopy coequalizer diagram. Suppose that h* is a homology theory. Then
there is a natural long exact sequence
. ..! hi(X; A) f*g*!hi(Y; B) ! hi(Z; C) ! hi1(X; A) ! . . .
Lemma 5. Let G be a group of order p with generator g, and V a mod p vector
space with an action of G. Then the endomorphism (1  g) of V is nilpotent (in
the sense that for some integer k, (1  g)k = 0). In particular, the kernel of
(1  g) : V ! V is nontrivial.
Proof.It is possible to choose k = p, since, in view of the fact that we are_wo*
*rking
mod p, (1  g)p = 1  gp = 0. __
Lemma 6. Suppose that h* is a multiplicative complex oriented homology theory.
Consider a homotopy fibre square
E" _____w"E0g

q q0
 
u f u
E _____wE0
in which q and q0 are principal S1bundles. If f induces on isomorphism on h*,
then so does g.
Proof.Let 0be the complex line bundle over E0associated to q0and the complex
line bundle over E associated to q. Denote the Thom spaces of these bundles by
M() and M(0) respectively. There is a map of cofibration sequences
"E_______wEq ______M()w
  
  
g f  M(f)
  
u q0 u u
E"0_______wE0 _____wM(0)
The map h*(f) is an isomorphism by hypothesis. By the Thom isomorphism for
h*, the map h*(M(f)) can be identified with h*(f) and so it too is an isomorphi*
*sm.
The fact that h*(g) is an isomorphism follows from looking at long exact homolo*
*gy_
sequences and using the five lemma. __
Proof of Theorem 3.Let G denote the group Z=p, and g 2 G some chosen gener
ator. We can assume that F and F 0are Gspaces, and that f is obtained up to
homotopy by taking the Borel construction fi(f) on a Gmap F ! F 0. (One way
to obtain a suitable Gspace equivalent to F , for instance, is to take the pul*
*lback
over E ! BG of the universal cover of BG.)
Suppose that X is a Gspace (in our case either F or F 0). Note that the homo*
*topy
coequalizer of the Gmaps 1; g : G ! G is the circle S1 with the usual rotation
action of G. More generally, the homotopy coequalizer of 1; g : XxG ! XxG is the
product Gspace X x S1 Taking Borel constructions gives a homotopy coequalizer
diagram
fi(X x G)______fi(XwxuG);v____fi(Xwx S1)
BOUSFIELD LOCALIZATIONS OF CLASSIFYING SPACES OF NILPOTENT GROUPS 3
where u and v are the appropriate induced Borel construction maps. It is clear
that fi(X x S1) is the total space of a principal S1bundle over fi(X), in fact*
*, the
total space of the pullback along the map fi(X) ! BG of the usual principal S1
bundle fi(S1) ! BG. The Borel construction fi(X x G), on the other hand, can be
identified up to homotopy with X itself in such a way that u and v can be ident*
*ified
with the original maps 1 and g.
Let "Edenote the Borel construction B(G; F x S1) and "E0the Borel construc
tion B(G; F 0x S1). According to the above considerations we have a homotopy
coequalizer diagram
1; g
(F 0; F_)___w(F 0; F_)__w(E"0; "E)
Since h*(E0; E) vanishes by assumption, h*(E"0; "E) vanishes by Lemma 6. Lemma 4
then implies that the endomorphism (1  g*) of h*(F 0; F ) is an automorphism, *
*but
by Lemma 5 this can happen only if h*(F 0; F ) = 0, in other words, only_if_the*
* map
F ! F 0induces an h*isomorphism. __
3. Localization of classifying spaces
We begin by recalling a result of Bousfield [Bou82 ] about localizations of B*
*Z=p.
(In that paper he actually determines LhK(A; n) for any homology theory h* and
any abelian group A.) We say that a space is h*acyclic if the reduced homology
"h*(X) vanishes, or equivalently if Lh(X) is contractible.
Lemma 7. If h* is a multiplicative homology theory, then the space BZ=p is h*
acyclic if p is invertible in h0 and h*local otherwise. Equivalently, LhBZ=p =
LR BZ=p, where R = h0 and LR denotes localization with respect to H*(; R).
With Theorem 3 in hand we can prove the following.
Theorem 8. Suppose that h* is a multiplicative complex oriented homology theor*
*y,
and that G is a finite pgroup. The space BG is h* acyclic if p is invertible i*
*n h0
and h*local otherwise.
Proof.If p is invertible is h0 it is obvious from the AtiyahHirzebruch spectral
sequence that BG is h*acyclic, so we assume that p is not invertible in h0 and
prove that BG is h*local. Suppose first that h0 is a Z=pvector space. We argue
by induction on the order of G. Let H G be a normal subgroup of index p.
The space BZ=p is h*local by Lemma 7, and so there is a diagram of fibration
sequences:
BH _______BG_w _______BZ=pw
   
   
f  g 
   
  
u u  
F 0_______LhBGw _____BZ=p:w
It thus follows immediately from Lemma 2 and Theorem 3 that BG is h*local.
Now consider a general h of the specified type. For a prime q, let h=q denote*
* the
smash product of the spectrum h representing h* with a mod q Moore spectrum
denoted here by M. Clearly h=q is still complex orientable: This is true since
a complex orientation for a spectrum E is a class x 2 E2(CP 1) with certain
4WILLIAM G. DWYER, UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN 46556 USA, EMMANUEL*
* DROR FARJOUN, HEBREW UNIVERSITY, JERUSALEM, ISRAEL, AND DOUGL@
properties. One has a map E ! E ^ M induced by the unit in M, and one can use
the image of x under this map as a complex orientation for E ^ M.
Alternatively, E is complex orientable iff it is an MUmodule spectrum. If E *
*is
an MUmodule spectrum, so is E ^ M.
If X is a space, let XQ denote the localization of X with respect to rational
homology. Since (BG)Q = *, it follows from Proposition 7.2 of [Bou82 ] that we
have a fibration sequence
Q i Q j
LhBG ! qLh=qBG ! q Lh=qBG = pt:
Q
where q runs though the primes not invertible in h0. We know from above that
Lh=pBG = BG and that Lh=qBG = pt:for q 6= p. It follows that LhBG = BG_as_
claimed. __
Slightly more generally we have
Theorem 9. Suppose that G be a finite nilpotent group and that h is a multipli*
*ca
tive complex oriented homology theory. Then LhBG = LR BG, where LR is as in
Lemma 7. In particular, if no prime dividing the order of G is invertible in h*
*0,
then BG is h*local.
Proof.TheQgroup G is theQdirect product of its Sylow psubgroups Gp, so we have
BG ' pBGp and LhBG ' pLhBGp. The factors in this second product can
be identified with the help of Theorem 8. There is a similar product formula_for
LR BG. __
We now turn to the proof of the main theorem.
Proof of Theorem 1.It is shown by Bousfield in [Bou82 ] that for any space X,
LhX ' LhLR X where LR is localization with respect to H*(; R). It is easy to
check that a map of spaces is an isomorphism on H*(; R) if and only if it is an
isomorphism on pH*(; Z=p R) as well as an isomorphism on H*(; Q R).
Let P be the set of all primes which are not invertible in R. It follows that a*
* map
of spaces is an isomorphism on H*(; R) if and only if it is an isomorphism on
H*(; p2PZ=p), as well as, if Q R 6= 0, an isomorphism on H*(; Q). Since
BG is a nilpotent space, the results of [DDK77 ] imply that if Q R = 0 there *
*is
an equivalence Y
LR BG ' LZ=pBG
p2P
while if Q R 6= 0 there is a homotopy fibre square
Q
LR BG _______w p2PLZ=pBG

 
 
 
 
u Q u
(BG)Q _____w( p2PLZ=pBG)Q :
We will carry out the proof by showing that LR BG is h*local, so that LhBG '
LhLR BG ' LR BG. To do this we will show that all of the constituents in the
above formulas for LR BG are h*local, and then appeal to the fact that the cla*
*ss
of h*local spaces is closed under homotopy inverse limit constructions.
BOUSFIELD LOCALIZATIONS OF CLASSIFYING SPACES OF NILPOTENT GROUPS 5
Now according to [BK72 , VI 2.6, 2.2 and IV x2] the space LZ=pBG ' (Z=p)1 BG
can be identified as B(G^p), where G^p= limG=psG is the plowercentralseries
completion of G. In particular LZ=pBG is equivalent to the homotopy inverse lim*
*it
of the tower {B(G=psG)}s. If p 2 P then each space in this tower is h*local
(Theorem 8), andQso LZ=pBG is h*local by homotopy inverse limit closure. By the
same principle, p2PLZ=pBG is h*local.
We can complete the proof by showing that if Q R 6= 0 then any space W
local with respect to rational homology is also local with respect to h*. Given*
* the
definition of what it means for a space to be h*local, we have to show that any
h*equivalence f : X ! Y induces a bijection f# : [Y; W ] ! [X; W ] (where the
brackets indicate homotopy classes of maps). However, by [Bou82 , 3.3], such an
f is a rational equivalence, so the fact that f# is a bijection follows from th*
*e_fact
that W is local with respect to rational homology. __
4.possible extensions and related problems
It was shown above that the Bousfield localization with respect to certain ho
mology theories of the classifying space BG of a finitely generated nilpotent g*
*roup
G is the same as the localization with respect to a classical homology theory w*
*ith
appropriate coefficients. The question remains open for other (non finitely gen*
*er
ated) nilpotent groups and other localization functors. Using the fact that K(F*
*; 2),
where F is any free abelian group, is local with respect to complex K theory it
is not hard to see that so is K(G; 1) for any abelian group G and in fact one c*
*an
show that theorem 1 holds for any abelian group.
To go beyond EilenbergMacLane spaces the following is a natural possible ex
tension of the main results above.
Let N be a nilpotent space whose homotopy groups vanish above certain dimen
sion n. Is it true that any Bousfield homological localization of N is equivale*
*nt to
its localization with respect to a well chosen classical homology theory?
Similar question arise beyond the realm of Bousfield homological localization.
Namely, one may ask for analogues of the above questions for an arbitrary homo
topical localization Lf with respect to an arbitrary map f. In that case it is *
*not
true that the localization will be the same as the localization with respect to*
* a well
chosen classical homology. This is because the map BZ=p2 ! BZ=p, induced by
the quotient group map, is in fact a homotopy localization map, but it is not an
homological localization map. But one does expect that an arbitrary localization
LfN of a nilpotent space N as above will also be a nilpotent space with vanishi*
*ng
homotopy groups above a certain dimension that depends only on n.
References
[BK72] A. K. Bousfield and D. M. Kan. Homotopy Limits, Completions and Localiza*
*tions,
volume 304 of Lecture Notes in Mathematics. SpringerVerlag, 1972.
[Bou79]A. K. Bousfield. The localization of spectra with respect to homology. T*
*opology, 18:257
281, 1979.
[Bou82]A. K. Bousfield. On homology equivalences and homological localizations *
*of spaces.
American Journal of Mathematics, 104:10251042, 1982.
[DDK77]W. G. Dwyer, E. Dror, and D. M. Kan. An arithmetic square for virtually *
*nilpotent
spaces. Illinois Journal of Mathematics, 21:242254, 1977.
[Kri] I. Kriz. Morava Ktheory of classifying spaces: some calculations. To ap*
*pear in Topology.
[Kur56]A. G. Kurosh. The Theory of Groups, Volume Two. Chelsea Publishing Compa*
*ny, New
York, 1956.
6WILLIAM G. DWYER, UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN 46556 USA, EMMANUEL*
* DROR FARJOUN, HEBREW UNIVERSITY, JERUSALEM, ISRAEL, AND DOUGL@
UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN 46556 USA
HEBREW UNIVERSITY, JERUSALEM, ISRAEL
UNIVERSITY OF ROCHESTER, ROCHESTER, NY 14627 USA