POSTNIKOV PIECES AND BZ=pHOMOTOPY THEORY
NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Abstract. We present a constructive method to compute the cellularization*
* with
respect to Bm Z=p for any integer m 1 of a large class of Hspaces, nam*
*ely all those
which have a finite number of nontrivial Bm Z=phomotopy groups (the poi*
*nted
mapping space map*(Bm Z=p, X) is a Postnikov piece). We prove in particu*
*lar
that the Bm Z=pcellularization of an Hspace having a finite number of B*
*m Z=p
homotopy groups is a ptorsion Postnikov piece. Along the way we characte*
*rize the
BZ=prcellular classifying spaces of nilpotent groups.
Introduction
The notion of Ahomotopy theory was introduced by Dror Farjoun [9] for an arb*
*i
trary connected space A. Here A and its suspensions play the role of the sphere*
*s in
classical homotopy theory and so the Ahomotopy groups of a space X are defined
to be the homotopy classes of pointed maps [ iA, X]. The analogue to weakly con
tractible spaces are those spaces for which all Ahomotopy groups are trivial. *
* This
means that the pointed mapping space map *(A, X) is contractible, i.e. X is an*
* A
local space. On the other hand the classical notion of CW complex is replaced*
* by
the one of Acellular space. Such spaces that can be constructed from A by mean*
*s of
pointed homotopy colimits.
Thanks to work of Bousfield [2] and Dror Farjoun [9] there is a functorial wa*
*y to
study X through the eyes of A. The nullification PAX is the biggest quotient of*
* X
which is Alocal and CWAX is the best Acellular approximation of the space X.
Roughly speaking CWAX contains all the transcendent information of the mapping
space map *(A, X) since it is equivalent to map *(A, CWAX). Hence explicit com
putation of the cellularization would give access to information about map *(A,*
* X).
The importance of mapping spaces (in the case A = BZ=p) is well established sin*
*ce
Miller's solution to the Sullivan conjecture [17].
While there is a lot of literature devoted to computations of PAX, only very *
*few
computations of CWAX are available. For instance Chach'olski describes a strat
egy to compute the cellularization CWA(X) in [7]. This method has been success
fully applied in some cases (cellularization with respect to Moore spaces [21],*
* BZ=p
cellularization of classifying spaces of finite groups [10]), but it is in gene*
*ral difficult
to apply.
An alternative way to compute CWAX is the_following. The localization_map l :
X ! PAX provides an equivalence CWAX ' CWAP AX where as usual PA X denotes
___________
All three authors are partially supported by MEC grant MTM200406686, the thi*
*rd author is
supported by the program Ram'on y Cajal, MEC, Spain.
1
2 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
the homotopy_fiber of l. This equivalence gives a strategy_to_compute cellulari*
*zations
when CWAP AX is known. For instance if X is Alocal then P AX ' * and so
CWAX ' *. From the Ahomotopy point of view the next case in which the A
cellularization should be accessible is when X has only a finite number of Aho*
*motopy
groups, that is some iterated loop space nX is Alocal. Natural examples of sp*
*aces
satisfying this condition are obtained by considering the nconnected covers of*
* Alocal
spaces.
Let us specialize in Hspaces and A = Bm Z=p. P. Bousfield has determined the
fiber of the localization map X_!_PBm Z=pX (see [2]) when nX is Bm Z=plocal. *
*He
shows that for such an Hspace P Bm Z=pX is a ptorsion Postnikov piece F , who*
*se
homotopy groups are concentrated in degrees from m to m + n  1. As F is also an
Hspace (because l is an Hmap), we call it an HPostnikov piece. The cellulari*
*zation
of X (which is again an Hspace because CWA preserves Hstructures) therefore
coincides with that of a Postnikov piece. We do this in Section 3 and this enab*
*les us
to obtain our main result.
Theorem 4.1 Let X be a connected Hspace such that nX is Bm Z=plocal. Then
CWBm Z=pX ' F x K(W, m)
where F is a ptorsion HPostnikov piece with homotopy concentrated in degrees *
*from
m + 1 to n and W is an elementary abelian p group.
Thus, when X is an Hspace with only a finite number of Bm Z=phomotopy
groups, the cellularization CWBm Z=pX is a ptorsion HPostnikov piece! This is
not true in general if we do not assume X to be an Hspace. For instance, the
BZ=pcellularization of B 3 is a space with infinitely many nontrivial homotopy
groups [11].
For m = 1 there is a large class of Hspaces which is known to have some loca*
*l loop
space by previous work of the authors [6]: those for which the mod p cohomology
is finitely generated as an algebra over the Steenrod algebra. Hence we obtain*
* the
following.
Proposition 4.2 Let X be a connected Hspace such that H*(X; Fp) is finitely ge*
*n
erated as algebra over the Steenrod algebra. Then
CWBZ=pX ' F x K(W, 1)
where F is a 1connected ptorsion HPostnikov piece and W is an elementary abe*
*lian
pgroup. Moreover, there exists an integer k such that CWBm Z=pX ' * for m k.
Our results allow explicit computations which we exemplify by computing the
BZ=pcellularization of the nconnected cover of any finite Hspaces (Propositi*
*on 4.3),
as well as the Bm Z=pcellularizations of the classifying spaces for real and c*
*omplex vec
tor bundles BU, BO, and their connected covers BSU, BSO, BSpin, and BString,
see Proposition 5.6.
POSTNIKOV PIECES AND BZ=pHOMOTOPY THEORY 3
1. A double filtration of the category of spaces
As mentioned in the introduction the condition that nX be Bm Z=plocal will
enable us to compute the Bm Z=pcellurarization of Hspaces. This section is de*
*voted
to give a picture of how such spaces are related for different choices of m and*
* n.
First of all we present a technical lemma which collects various facts needed*
* in the
rest of the paper.
Lemma 1.1. Let X be a connected space and m > 0. Then,
(1) If X is Bm Z=plocal then nX is Bm Z=plocal for all n 1.
(2) If X is Bm Z=plocal then it is Bm+s Z=plocal for all s 0.
(3) If X is Bm Z=plocal, then X is Bm+s Z=plocal for all s 1.
Proof.To prove (1) simply apply map *(BZ=p, ) to the path fibration X ! * ! X.
Statement (2) is given by Dwyer's version of Zabrodsky's lemma [8, Prop. 3.4]*
* to
the universal fibration BnZ=p ! * ! Bn+1Z=p.
Finally (3) is a direct consequence of Zabrodsky's lemma (now in its connected
version [8, Prop. 3.5]) applied to the universal fibration and using the fact t*
*hat X
BnZ=plocal implies map (BnZ=p, X)c ' X. 
Of course the converses of the previous results are not true. For the first s*
*tatement
take the classifying space of a discrete group at m = 1. For the second and th*
*ird
consider X = BU. It is a B2Z=plocal (see Example 1.4) space but neither BU not
BU are BZ=plocal. Observe that in fact nBU is never BZ=plocal. The next
result shows that this is the general situation for Hspaces. That is, if an H*
*space is
Bm+1 Z=plocal then either X is Bm Z=plocal or nX is never Bm Z=plocal 8n *
* 1.
Theorem 1.2. Let X be a Bn+1Z=plocal space such that kX is BnZ=plocal for
some k > 0. Then X is BnZ=plocal.
Proof.It is enough to prove the result for k = 2. Consider the fibration
K(Q, n + 1) ____P 2BnZ=pX ' X ____P BnZ=pX
where the fiber is a ptorsion EilenbergMac Lane space by Bousfield's descript*
*ion of
the fiber of the BnZ=pnullification [2, Theorem 7.2]. The total space is Bn+1*
*Z=p
local and so is the base by the previous lemma. Thus map *(Bn+1Z=p, K(Q, n + 1))
must be contractible as well, i.e. Q = 0. *
* 
The previous analysis leads to a double filtration of the category of spaces.*
* Let
n 0 and m 1. We introduce the notation
Snm= {X; nX isBm Z=plocal} .
Lemma 1.1 yields then a diagram of inclusions:
4 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
S01 ____S11____S21____. . .___Sn1 . . .
\ \ \ \
   
? ? ? ?
S02 ____S12____S22____. . .___Sn2 . . .
\ \ \ \
   
.? ? ? ?
.. ... ... ...
\ \ \ \
   
? ? ? ?
S0m ____S1m____S1m ____. ._.__Snm . . .
.. . . .
. .. .. ..
Example 1.3. Examples of spaces in every stage of the filtration are known.
(1) S01are the spaces that are BZ=plocal. This contains in particular any f*
*inite
space (by Miller's theorem [17, Thm. A]), and for a nilpotent space X (of
finite type with finite fundamental group) to be BZ=plocal is equivalen*
*t to
its cohomology H*(X; Fp) being locally finite by [22, Corollary 8.6.2].
(2) If X denotes the nconnected cover of a space X, then the homotopy fi*
*ber of
n1X ! n1X is a discrete space. Hence if X 2 S0mthen X 2 Sn1m.
(3) Observe that Snm Snkm+kfor all 0 k n.
(4) The previous examples provide spaces in every stage of the double filtra*
*tion.
Consider a finite space. It is automatically BZ=plocal. Its nconnected*
* cover
X lies in Sn11. Hence X 2 Snk1k+1for all 0 k n.
Next example provides a number of spaces living in S0mbut not obtained from t*
*he
first row of the filtration by taking nconnected covers. Of course their conn*
*ected
covers will be new examples of spaces living in Snm.
Example 1.4. Let E* be a homology theory. If "Ei(K(Z=pZ, n)) = 0 for i j then
the spaces Ei for i j representing the corresponding homology theory are BnZ=*
*p
local. If "Ej(K(Z=pZ, n  1)) 6= 0 then Ej is not Bn1Z=plocal. In particular *
*if E* is
periodic, it follows that the spaces {Ei} for i j are BnZ=plocal but none of*
* their
iterated loops are Bn1Z=plocal.
A first example of such a behavior is obtained from complex Ktheory, BU is B*
*2Z=p
local but BU and U are not BZ=plocal (see [18]). Note that real and quaternion*
*ic
Ktheory enjoy the same properties.
For every n, examples of homology theories following this pattern are given b*
*y p
torsion homology theories of type IIIn as described in [1]. The nth Morava Kt*
*heory
K(n)* for p odd is an example of such behavior with respect to EilenbergMac La*
*ne
spaces. The spaces representing K(n)* are Bn+1Z=plocal but none of their itera*
*ted
loops are BnZ=plocal.
POSTNIKOV PIECES AND BZ=pHOMOTOPY THEORY 5
Our aim is to provide tools to compute the Bm Z=pcellularization of any Hsp*
*ace
lying in the mth row of the above diagram. The key point is the following resu*
*lt of
Bousfield [2] determining the fiber of the localization map.
Proposition 1.5. Let n 0 and X be a connected Hspace X such that nX is
Bm Z=plocal. Then there is an Hfibration
F ____X ____PBm Z=pX
where F is a ptorsion HPostnikov piece whose homotopy groups are concentrated*
* in
degrees from m to m + n. 
Therefore, since F ! X is a Bm Z=pcellular equivalence, we only need to comp*
*ute
the cellularization of a Postnikov piece (which will end up being a Postnikov p*
*iece
again, see Theorem 3.6). Actually even more is true.
Proposition 1.6. Let X be a connected space such that CWBm Z=pX is a Postnikov
piece. Then there exists an integer n such that nX is Bm Z=plocal.
Proof.Let us loop once Chach'olski fibration CWBm Z=pX ! X ! P Bm Z=pC, see [7,
Theorem 20.5]. As P Bm Z=pC is equivalent to PBm Z=p C by [9, Theorem 3.A.1], *
*we
get a fibration over a Bm Z=plocal base space
CWBm Z=pX ____ X ____PBm Z=p C.
Now there exists an integer n such that nCWBm Z=pX is discrete, thus Bm Z=plo*
*cal.
Therefore so is nX. 
2.Cellularization of fibrations over BG
In general the cellularization of the total space of a fibration is very diff*
*icult to
compute. We explain in this section how to deal with this problem when the base
space is the classifying space of a discrete group. The first step applies to a*
*ny group,
in the second, see Proposition 2.4 below, we specialize to nilpotent groups.
ss
Proposition 2.1. Let r 1 and F _____E _____BG be a fibration where G is a
discrete group. Let S be the (normal) subgroup generated by all elements g 2 G*
* of
order pi for some i r such that the inclusion B _____BG lifts to E up to *
*an
unpointed homotopy. Then the pullback of the fibration along BS ____BG
f p
E0 _____E ____B(G=S)
 
  
 ss 
? ? p0 ?
BS ____BG ___B(G=S)
induces a BZ=prcellular equivalence f : E0! E on the total space level.
Proof.We have to show that f induces a homotopy equivalence on pointed mapping
spaces map *(BZ=pr, ). The top fibration in the diagram yields a fibration
f* r p* r
map *(BZ=pr, E0) ____ map *(BZ=p , E) ____ map *(BZ=p , B(G=S)).
6 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Since the base is homotopically discrete we only need to check that all compone*
*nts
of the total space are sent by p* to the component of the constant. Consider th*
*us a
map h : BZ=pr ! E. The composite p O h is homotopy equivalent to a map induced
by a group morphism ff: Z=pr ____G whose image ff(1) = g is in S by constructi*
*on.
Therefore p O h = p0O ss O h is nullhomotopic. *
* 
Remark 2.2. If the fibration in the above proposition is an Hfibration (in par*
*ticular
G is abelian), the set of elements g for which there is a lift to the total spa*
*ce forms
a subgroup of G. The central extension Z(D8) ,! D8 ! Z=2 x Z=2 of the dihedral
group D8 provides an example where the subgroup S is Z=2 x Z=2 but the element
in S represented by an element of order 4 in D8 does not admit a lift.
The next lemma is a variation of Dwyer's version of Zabrodsky's Lemma in [8].
f
Lemma 2.3. Let F ! E  ! B be a fibration over a connected base, and A a
connected space such that A is F local. Then any map g : E ! A which is homot*
*opic
to the constant when restricted to F factors through a map h : B ! A up to unpo*
*inted
homotopy and moreover g is pointed nullhomotopic if and only if h is so.
Proof.Since A is F local, we see that the component of the constant map map **
*(F, A)c
is contractible and therefore the evaluation at the base point map (F, A)c ! A *
*is an
equivalence. By Proposition 3.5 in [8], f induces a homotopy equivalence
map (B, A) ' map (E, A)[F].
where map (E, A)[F]denotes the space of those maps E ! A which are homotopic to
the constant when restricted to F .
We restrict now to the component of the constant map c : E ! A. There is only
one component in the pointed mapping space sitting over c since any map homotop*
*ic
to the constant map is also homotopic by a pointed homotopy. The result follows*
*. 
i ss
Proposition 2.4. Let r 1 and F _____E _____BG be a fibration where G is a
nilpotent group generated by elements of order pi with i r. Assume that for e*
*ach
of these generators x 2 G, the inclusion B _____BG lifts to E up to unpoint*
*ed
homotopy. If F is BZ=prcellular then so is E.
Proof.Chach'olski's description [7] of the cellularization CWBZ=pr(E) as the ho*
*motopy
fiber of the compositeWf : E ____C ____P BZ=pr(C) where C is the homotopy cof*
*iber
of the evaluation map [BZ=pr,E]BZ=pr ____E tells us that E is cellular if th*
*e map
f is nullhomotopic. Observe that if f is nullhomotopic then the fiber inclusi*
*on
CWBZ=pr(E) ____E has a section and therefore E is cellular since it is a retra*
*ct of a
cellular space ([9, 2.D.1.5]).
As the existence of an unpointed homotopy to the constant map implies the exi*
*s
tence of a pointed one, we work now in the category of unpointed spaces. Remark
that for any map g : Z ____E from a BZ=prcellular space Z, the composite f O g
is nullhomotopic since g factors through the cellularization of E. In particul*
*ar the
composite f Oi is nullhomotopic. By Lemma 2.3 there exists ~f: BG ____P BZ=pr*
*(C)
such that ~fO ss ' f and, moreover, f is nullhomotopic if and only if ~fis so.
POSTNIKOV PIECES AND BZ=pHOMOTOPY THEORY 7
We first assume that G is a finite group and show by induction on the order o*
*f G
that ~fis nullhomotopic. If G = p, then the existence of a section s : BG __*
*__E
implies that f O s = ~fis nullhomotopic since BG = BZ=p is cellular.
Let {x1, . .,.xk} be a minimal set of generators which admit a lift. Let H E*
* G
be the normal subgroup generated by x1, . .,.xk1 and their conjugates by powers
of xk. There is a short exact sequence H ____G ____Z=pa where the quotient gr*
*oup
is generated by the image of xk. Consider the fibration F ! E0 ! BH obtained
by pulling back along BH ! BG and denote by h : E0 ! E the induced map
between the total spaces. The inclusions in G of two conjugate subgroups are (f*
*reely)
homotopic and so H satisfies the assumptions of the proposition. Thus the induc*
*tion
hypothesis tells us that E0is cellular and therefore f Oh is nullhomotopic. Thi*
*s implies
that the restriction of ~fto BH is nullhomotopic. Consider the following diagram
B( \ H) ____BH
  @@*
? ? f@R~
B() ______BG ____P BZ=pr(C)

  `
? ? f0
BZ=pa = BZ=pa
By Lemma 2.3, it is enough to show that f0is nullhomotopic. But again by Lemma *
*2.3
applied to the fibration on the left, we see that f0 is nullhomotopic since ~fr*
*estricted
to is so. Therefore ~fis nullhomotopic.
Assume now that G is not finite. Any subgroup of G generated by a finite numb*
*er
of elements of order a power of p has a finite abelianization, and must therefo*
*re be
finite itself by [20, Theorem 2.26]. Thus G is locally finite, i.e. G is a filt*
*ered colimit
of finite nilpotent groups generated by elements of order pi for i r. Likewis*
*e, BG
is a filtered homotopy colimit of BS where S are finite groups (generated by fi*
*nite
subsets of the set of generators) which verify the hypothesis of the propositio*
*n. The
total space E can be obtained as a pointed filtered colimit of the total spaces*
* obtained
by pulling back the fibration. By the finite case situation they are all cellu*
*lar and
therefore so is E. 
Sometimes the existence of the "local" sections defined for every generator p*
*ermits
to construct a global section of the fibration. By a result of Chach'olski [7,*
* Theo
rem 4.7] the total space of such a split fibration is cellular since F and BG a*
*re so.
This is the case for an Hfibration and E is then weakly equivalent to a product
F x BG.
A straightforward consequence of the above proposition (in the case when the
fibration is the identity of BG) is the following characterization of the BZ=pr*
*cellular
classifying spaces. For r = 1 we obtain R. Flores' result [10].
Corollary 2.5. Let r 1 and G be a nilpotent group generated by elements of or*
*der
pi with i r. Then BG is BZ=prcellular. 
8 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Example 2.6. The quaternion group Q8 of order 8 is generated by elements of ord*
*er 4.
Therefore BQ8 is BZ=4cellular. We do not know an explicit way to construct BQ8
as a pointed homotopy colimit of a diagram whose values are copies of BZ=4.
The previous technical propositions allow us to state the main result of this*
* section.
It provides a constructive description of the cellularization of the total spac*
*e of certain
fibrations over classifying spaces of nilpotent groups.
Theorem 2.7. Let G be a nilpotent group and F ____E ____BG be a fibration with
BZ=prcellular fiber F . Then the cellularization of E is the total space of a *
*fibration
F ____CWBZ=pr(E) ____BS where S / G is the (normal) subgroup generated by the
ptorsion elements g of order piwith i r such that the inclusion B ____BG*
* lifts
to E up to unpointed homotopy.
Proof.By Proposition 2.1 pulling back along BS ! BG yields a cellular equivalen*
*ce
f in the following square:
f
ES _____E
 
 
 
? ?
BS ____BG
By Proposition 2.4 the total space ES is cellular and therefore ES ' CWBZ=pr(E)*
*. 
Corollary 2.8. Let G be a nilpotent group and S / G be the (normal) subgroup
generated by the ptorsion elements g of order piwith i r. Then CWBZ=prBG ' B*
*S.
Moreover when G is finitely generated, S is a finite pgroup.
Proof.We only need to show that S is a finite pgroup. Notice that the abeliani*
*zation
of S is ptorsion, then S is also a torsion group (see [23, Cor. 3.13]). Moreov*
*er, since
G is finitely generated, by [23, 3.10], S is finite. *
* 
In fact in that case the previous result also holds when the base space is an
EilenbergMac Lane space K(G, n).
i ss
Proposition 2.9. Let F _____E _____K(G, n) be a fibration where G is a finite*
*ly
generated group by elements of order pi where i r and n > 1. Assume that for
each generator x 2 G, the inclusion K(, n) _____K(G, n) lifts to E. If F *
*is
BZ=prcellular then E is so. 
3. Cellularization of nilpotent Postnikov pieces
In this section we compute the cellularization with respect to BZ=pr of nilpo*
*tent
Postnikov pieces. The main difficulty lies in the fundamental group, so it wil*
*l be
no surprise that these results hold as well for cellularization with respect to*
* Bm Z=pr
with m 2. We will often use the following closure property [9, Theorem 2.D.11*
*].
Proposition 3.1. Let F ! E ! B be a fibration where F and E are Acellular.
Then so is B. 
POSTNIKOV PIECES AND BZ=pHOMOTOPY THEORY 9
Example 3.2. [9, Corollary 3.C.10] The Eilenberg Mac Lane space K(Z=pk, n) is
BZ=prcellular for any integer k and any n 2.
The construction of the cellularization is performed by looking first at the *
*universal
cover of the Postnikov piece. We start with the basic building blocs, the Eilen*
*berg
Mac Lane spaces. For the results on the structure on infinite abelian groups, w*
*e refer
the reader to Fuchs' book [12].
Lemma 3.3. An EilenbergMac Lane space K(A, m) with m 2 is BZ=prcellular
if and only if A is a ptorsion abelian group.
Proof.That A must be ptorsion is clear. Assume thus that A is a ptorsion grou*
*p.
If A is bounded it is isomorphic to a direct sum of cyclic groups. Since cellul*
*arization
commutes with finite products K(A, m) is BZ=prcellular when A is a finite dire*
*ct sum
of cyclic groups. Taking a (possibly transfinite) telescope of BZ=prcellular s*
*paces we
obtain that K(A, m) is so for any bounded group.
In general A splits as a direct sum of a divisible group D and a reduced one *
*T .
A ptorsion divisible group is a direct sum of copies of Z=p1 , which is a unio*
*n of
bounded groups, thus K(D, m) is cellular. Now T has a basic subgroup P < T which
is a direct sum of cyclic groups and the quotient T=P is divisible. So K(T, m) *
*is the
total space of a fibration
K(P, m) ! K(T, m) ! K(D, m)
When m 3 we are done because of the above mentioned closure property Proposi
tion 3.1. If m = 2 we have to refine the analysis of the fibration because K(D,*
* m  1)
is not cellular. However, as D is a union of bounded groups D[pk], the space K(*
*T, 2)
is the telescope of total spaces Xk of fibrations with cellular fiber K(P, 2) a*
*nd base
K(D[pk], 2). We claim that these total spaces are cellular (and thus so is K(T,*
* 2)) and
proceed by induction on the bound. Consider the subgroup D[pk] < D[pk+1] whose
quotient is a direct sum of cyclic groups Z=p. Therefore Xk+1 sits in a fibrati*
*on
K( Z=p, 1) ! Xk ! Xk+1
where fiber and total space are cellular. We are done. *
* 
We are now ready to prove that any ptorsion simply connected Postnikov piece*
* is
a BZ=prcellular space.
Proposition 3.4. A simply connected Postnikov piece is BZ=prcellular if and on*
*ly
if it is ptorsion.
Proof.Let X be a simply connected ptorsion Postnikov piece. For some integer m,
the mconnected cover X is an EilenbergMac Lane space, which is cellular by
Lemma 3.3. Consider the principal fibration
K(ssm (X), m  1) ____X ____X
If m 3 both X and K(ssm (X), m  1) are cellular. It follows that X
is cellular by the closure property Proposition 3.1. The same argument shows t*
*hat
X<2> is cellular.
10 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Let us thus look at the fibration X<2> ! X ! K(ss2X, 2). The same discussion *
*on
the ptorsion group ss2X as in Lemma 3.3 will apply. If this is a bounded group*
*, say by
pk, an induction on the bound shows that X is actually the base space of a fibr*
*ation
where the total space is cellular because its second homotopy group is pk1bou*
*nded,
and the fiber is cellular because it is of the form K(V, 1) with V a ptorsion *
*abelian
groups whose torsion is bounded by pr. The closure property Proposition 3.1 ens*
*ures
that X is then cellular.
If ss2X is divisible, X is a telescope of cellular spaces, hence cellular. If*
* it is reduced,
taking a basic subgroup B < ss2X yields a diagram of fibrations
X<2> _____Y _______K(B, 2)
  
  
? ? ?
X<2> _____X _____K(ss2X, 2)
 
  
  
? ? ?
*____ K(D, 2)=== K(D, 2)
which exhibits X as the total space of a fibration over K(D, 2) with D divisibl*
*e and
BZ=prcellular fiber. Therefore writing D as a union of bounded groups as in t*
*he
proof of Lemma 3.3, X is a telescope of cellular spaces, therefore it is BZ=pr*
*cellular
as well. 
Remark 3.5. The proof of the proposition holds in the more general setting when*
* X
is a ptorsion space such that X is BZ=pr cellular for some m 2. The propo*
*sition
corresponds to the case when some mconnected cover X is contractible.
Recall from [13, Corollary 2.12] that a connected space is nilpotent if and o*
*nly if its
Postnikov system admits a principal refinement . .!.Xs ! Xs1 ! . .!.X1 ! X0.
This means that each map Xs+1 ! Xs in the tower is a principal fibration with f*
*iber
K(As, is 1) for some increasing sequence of integers is 2. We are only inter*
*ested
in finite Postnikov pieces, i.e. nilpotent spaces that can be constructed in a*
* finite
number of steps by taking homotopy fibers of kinvariants Xs ! K(As, is).
The key step in the study of the cellularization of a nilpotent finite Postni*
*kov piece
is the analysis of a principal fibration (given in our case by the kinvariants*
*).
Theorem 3.6. Let X be a ptorsion nilpotent Postnikov piece. Then there exists a
fibration
X<1> ____CWBZ=prX ____BS
where S is the (normal) subgroup of ss1(X) generated by the elements g of order*
* pi
with i r such that the inclusion B ! Bss1X admits a lift to X up to unpoin*
*ted
homotopy.
Proof.By Proposition 3.4 the universal cover X<1> is cellular and there is a fi*
*bration
X<1> ____X ____BG where G = ss1(X) is nilpotent. The result follows then from
Theorem 2.7. 
POSTNIKOV PIECES AND BZ=pHOMOTOPY THEORY 11
4. Cellularization of Hspaces
In this section we will apply the computations of the cellularization of pto*
*rsion
nilpotent Postnikov systems to determine CWBZ=pX when X is an Hspace. We
prove:
Theorem 4.1. Let X be a connected Hspace such that nX is BZ=plocal. Then
CWBZ=pX ' Y x K(W, 1)
where Y is a simply connected ptorsion HPostnikov piece with homotopy concen
trated in degrees n and W is an elementary abelian pgroup.
Proof.The fibration in Bousfield's result Proposition 1.5 yields a cellular equ*
*ivalence
between a connected ptorsion HPostnikov piece F and X. Theorem 3.6 thus appli*
*es.
Moreover, as F is an Hspace as well, the subgroup S is abelian generated by el*
*ements
of order p. Therefore we have a fibration F <1> ! CWBZ=pF ! K(W, 1) which admits
a section (summing up the local section). The cellularization therefore splits.*
* 
This result applies for Hspaces satisfying certain finiteness conditions.
Proposition 4.2. Let X be a connected Hspace such that H*(X; Fp) is finitely g*
*en
erated as algebra over the Steenrod algebra. Then
CWBZ=pX ' F x K(W, 1)
where F is a 1connected ptorsion HPostnikov piece and W is an elementary abe*
*lian
pgroup. Moreover there exists an integer k such that CWBm Z=pX ' * for m k.
Proof.In [6] the authors prove that if H*(X; Fp) is finitely generated as algeb*
*ra over
the Steenrod algebra then nX is BZ=plocal for some n 0. Hence Theorem 4.1
applies and we obtain the desired result. In addition Lemma 1.1 shows that X is
Bn+s+1Z=plocal for any s 0, which implies the second part of the result. *
* 
The technique we propose in this paper is not only a nice theoretical tool wh*
*ich
provides a general statement about how the BZ=pcellularization of Hspaces look
like. Our next result shows that one can actually identify precisely this new *
*space
when dealing with connected covers of finite Hspaces. Recall that by Miller's *
*theorem
[17, Thm. A], any finite Hspace X is BZ=plocal and therefore CWBZ=p(X) ' *. T*
*he
universal cover of X is still finite and thus CWBZ=p(X<1>) is contractible as w*
*ell. We
can therefore assume that X is 1connected. The computation of the cellularizat*
*ion
of the 3connected cover is already implicit in [4].
Proposition 4.3. Let X be a simply connected finite Hspace and let k denote the
rank of the free abelian group ss3X. Then CWBZ=pX<3> ' K( kZ=p, 1). For n 4,
up to pcompletion, the universal cover of CWBZ=p(X) is weakly equivalent to*
* the
2connected cover of (X[n]).
Proof.By Browder's famous result [5, Theorem 6.11] X is even 2connected and its
third homotopy group ss3X is free abelian (of rank k) by J. Hubbuck and R. Kane*
*'s
theorem [14]. This means we have a fibration
K( kZp1, 1) ____X<3> ____PBZ=pX<3>
12 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
which shows that CWBZ=pX<3> ' K( kZ=p, 1). We deal now with the higher con
nected covers. Consider the following commutative diagram of fibrations
F _____________F _________*
  
  
  
? ? ?
X[n] _________X _______Xw
w
  ww
  w
? ?
PBZ=p( X[n]) ___PBZ=p(X)____X
where F is a ptorsion Postnikov piece by [2, Thm 7.2] and the fiber inclusions*
* are
all BZ=pcellular equivalences because the base spaces are BZ=plocal. Therefore
CWBZ=p(X) ' CWBZ=p(F ) ' F <1> x K(W, 1)
We wish to identify F <1>. Since the fibrations in the diagram are nilpotent, *
*by [3,
II.4.8] they remain fibrations after pcompletion. By Neisendorfer's theorem [1*
*9] the
map PBZ=p(X) ! X is an equivalence up to pcompletion, which means that
PBZ=p( (X[n]))^p' *. Thus Fp^' ( (X[n]))^p. Notice that (X[n]) is 1connected
and its second homotopy group is free by the above mentioned theorem of Hubbuck
and Kane (which corresponds up to pcompletion to the direct sum of k copies of
the Pr"ufer group Z=p1 in ss1F ). Hence F <1> coincides with (X[n])<2> up to*
* p
completion. 
To illustrate this result we compute the BZ=2cellularization of the successi*
*ve con
nected covers of S3. The only delicate point is the identification of the funda*
*mental
group.
Example 4.4. Recall that S3 is BZ=2local since it is a finite space. Thus the
cellularization CWBZ=2S3 is contractible. Next the fibration
K(Z21, 1) ! S3<3> ! PBZ=2S3<3>
shows that CWBZ=2S3<3> ' K(Z=2, 1). Finally since S3[4] does not split as a pro*
*duct
(the kinvariant is not trivial), we see that CWBZ=2S3<4> ' K(Z=2, 3). Likewise*
*, for
any integer n 4, we have that CWBZ=2S3 is equivalent to the 2completion of
the 2connected cover of (S3[n]). The same phenomenon occurs at odd primes.
5. Cellularization with respect to Bm Z=p
All the techniques developed for fibrations over BG apply to fibrations over *
*K(G, n)
when n > 1 and we get the following results.
Lemma 5.1. Let n 2 and X be a connected space. Then
CWBnZ=pr(X) = CWBnZ=pr(X) .
POSTNIKOV PIECES AND BZ=pHOMOTOPY THEORY 13
Proof.For n > i consider the following fibrations over BnZ=prlocal base spaces:
X ____X ____K(ssi(X), i)
We see that CWBnZ=pr(X) = CWBnZ=pr(X). 
Proposition 5.2. Let m 2 and X be a ptorsion nilpotent Postnikov piece. Then
there exists a fibration
X ____CWBm Z=pX ____K(W, m)
where W is a ptorsion subgroup of ssm (X) with torsion bounded by pr.
Theorem 5.3. Let X be a connected Hspace such that nX is Bm Z=plocal. Then
CWBm Z=pX ' F x K(W, m)
where F is a ptorsion HPostnikov piece with homotopy concentrated in degrees *
*from
m + 1 to n and W is an elementary abelian pgroup.
Example 5.4. Let X denote "Milgram's space", see [16], the fiber of Sq2 : K(Z=2*
*, 2) !
K(Z=2, 4). This is an infinite loop space. By Proposition 3.4 we know it is alr*
*eady
BZ=2cellular. Let us compute the cellularization with respect to Bm Z=2 for hi*
*gher
m's. Since the kinvariant is not trivial, we see that CWB2Z=2X ' CWB3Z=2X '
K(Z=2, 3).
We compute finally the cellularization of the (infinite loop) space BU and it*
*s 2
connected cover BSU with respect to EilenbergMacLane spaces Bm Z=p. By Bott
periodicity this actually tells us the answer for all connected covers of BU.
Example 5.5. First of all, recall from Example 1.4 that BU is B2Z=plocal since
eK*(B2Z=p) = 0 and its iterated loops are never BZ=plocal. Therefore CWBm Z=p(*
*BU)
is contractible if m 2. Since BU ' BSU x BS1 the same property holds for BSU.
We now compute the Bm Z=pcellularization of BO and its connected covers BSO,
BSpin, and BString.
Proposition 5.6. Let m 2. Then
(i)CWBm Z=p(BO) ' CWBm Z=p(BSO) ' CWBm Z=p(BSpin) ' *,
(ii)CWBm Z=p(BString) ' * if m > 2,
(iii)CWB2Z=p(BString) ' K(Z=p, 2) and map *(B2Z=p, BString) ' Z=p.
Proof.In [15] W. Meier proves that real and complex Ktheory have the same acyc*
*lic
spaces, hence BO is also B2Z=plocal. Therefore CWBm Z=p(BO) is contractible f*
*or
any m 2. The 2connected cover of BO is BSO and there is a splitting BO '
BSO x BZ=2, so that CWBm Z=p(BSO) ' *.
The 4conected cover of BO is BSpin. It follows from the fibration
BSpin ! BSO w2!K(Z=2, 2)
that that fiber of BSpin ! BSO is BZ=2. Since BSO and BZ=2 are B2Z=plocal,
so is BSpin. Therefore CWBm Z=p(BSpin) is contractible.
14 NAT`ALIA CASTELLANA, JUAN A. CRESPO, AND J'ER^OME SCHERER
Finally, the 8connected cover of BO is BString. It is the homotopy fiber of
p1=4
BSpin ! K(Z, 4), where p1 denotes the first Pontrjagin class. Consider the fib*
*ration
K(Z, 3) ! BString ! BSpin
where the base space is Bm Z=plocal for m 2. Together with the exact sequence
1 1
Z ! Z[__] ! Z=p , this implies that
p
CWBm Z=p(BString) ' CWBm Z=p(K(Z, 3)) ' CWBm Z=p(K(Z=p1 , 2))
which is contractible unless m = 2, when we obtain K(Z=p, 2). This yields the e*
*xplicit
description of the pointed mapping space map *(B2Z=p, BString). 
Observe that the iterated loops of the mconnected covers of BO and BU are ne*
*ver
BZ=plocal. Hence we know that their cellularization with respect to BZ=p must *
*have
infinitely many nonvanishing homotopy groups by Proposition 1.6.
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Nat`alia Castellana and J'er^ome Scherer Juan A. Crespo
Departament de Matem`atiques, Departament de Economia i de Hist`oria
Universitat Aut`onoma de Barcelona, Econ`omica,
E08193 Bellaterra, Spain Universitat Aut`onoma de Barcelona,
Email: natalia@mat.uab.es, E08193 Bellaterra, Spain
jscherer@mat.uab.es Email: JuanAlfonso.Crespo@uab.es,