# Homotopy idempotent functors on classifying spaces

Natàlia Castellana and Ramón Flores

###### Abstract

Fix a prime $p$. Since their definition in the context of Localization Theory,
the homotopy functors $P_{B{\mathbb{Z}}/p}$ and $CW_{B{\mathbb{Z}}/p}$ have
shown to be powerful tools to understand and describe the mod $p$ structure of
a space. In this paper, we study the effect of these functors on a wide class
of spaces which includes classifying spaces of compact Lie groups and their
homotopical analogues. Moreover, we investigate their relationship in this
context with other relevant functors in the analysis of the mod $p$ homotopy,
such as Bousfield-Kan completion and Bousfield homological localization.

††N. Castellana: Departamento de Matemáticas, Universidad Autónoma de
Barcelona, 08193 Bellaterra, Spain; e-mail<EMAIL_ADDRESS>
R. Flores: Departamento de Estadística, Universidad Carlos III, 28029
Colmenarejo, Spain; e-mail<EMAIL_ADDRESS>(corresponding author)††
_Mathematics Subject Classification (2010):_ Primary 55P20; Secondary 55P80

##  1 Introduction

Let $A$ and $X$ be two connected topological spaces. The study of the
homotopical properties of $X$ that are visible through the mapping space ${\rm
map\,}(A,X)$ is called the $A$-homotopy theory of $X$ and was proposed by E.
Dror-Farjoun in [Far96]. In this context, it is particularly important to
describe the behaviour of the nullification $P_{\Sigma^{i}A}$ and the
cellularization ${CW}_{\Sigma^{i}A}$ (see definitions in Section 2), which are
functors that play in $A$-homotopy theory the same role as the connected
covers and Postnikov pieces play in classical ($S^{0}$-) homotopy theory.

Let $p$ be a prime. If $X$ is a space and we are interested in describing the
$p$-primary part of $X$ through its $A$-homotopy theory for some space $A$,
there are some choices of $A$ that become apparent. Probably the easiest one
are the Moore spaces $M({\mathbb{Z}}/p^{n},1)$ and their suspensions; this
task was undertaken in the nineties by Rodriguez-Scherer in the case of
cellularization [RS01] and Bousfield [Bou97], who did not only described
${P}_{M({\mathbb{Z}}/p^{m},n)}X$ for a wide number of spaces, but remarked the
close relationship between these functors and the $v_{n}$-periodic homotopy
theory.

In this paper we deal with the case $A=B{\mathbb{Z}}/p$. After Miller’s
solution of Sullivan conjecture [Mil84], and subsequent work of Lannes, Dwyer-
Zabrodsky and others, a number of new powerful tools were available to
researchers attempting to study the mapping space ${\rm
map\,}(B{\mathbb{Z}}/p,X)$, and overwhelming success was reached, particularly
for nilpotent spaces $X$. In the framework we are interested in, we should
emphasize the work of Neisendorfer [Nei95], where the author proves that the
functor ${P}_{B{\mathbb{Z}}/p}$ can often recover the $p$-primary homotopy of
$X$ from that of its $n$-connected cover, or [CCS07-2], about the
$B{\mathbb{Z}}/p$-homotopy of $H$-spaces.

The first motivation for our work came from two different sources: the study
undertaken by W. Dwyer in [Dwy96] concerning $B{\mathbb{Z}}/p$-nullification
of classifying spaces of compact Lie groups whose group of components is a
$p$-group, and its relationship with the homology decompositions of $BG$; and
Question 11 in Farjoun’s book ([Far96, page 175]), where he asked about the
cellularity of the $p$-completion of $BG$. This seemed a natural extension of
the problems considered by the second author concerning
$B{\mathbb{Z}}/p$-homotopy of finite groups (see [Flo07], [FS07], and [FF]),
so it was natural to investigate this structure with similar methods.

We show that under certain hypothesis, we are able to characterize the effect
of the nullification functor $P_{B{\mathbb{Z}}/p}$ by means of a fibration.

Theorem 4.1. Let $X$ be a connected space with finite fundamental group and
such that $(P_{B{\mathbb{Z}}/p}(X\langle 1\rangle))^{\wedge}_{p}\simeq*$. Then
there is a fibration

$L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})\rightarrow
P_{B{\mathbb{Z}}/p}(X)\rightarrow B(\pi_{1}(X)/T_{p}(\pi_{1}(X)))$

where $X_{p}$ is the covering of $X$ whose fundamental group is
$T_{p}(\pi_{1}(X))$, and $L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})$ denotes
Bousfield homological localization of $X_{p}$ with respect to
$H^{*}(-;{\mathbb{Z}}[\frac{1}{p}])$.

In particular, one can compute the homotopy groups of $P_{B{\mathbb{Z}}/p}(X)$
in terms of those of $X$ if $X$ is good enough. This result is quite general,
and in fact describes in a single statement a phenomenon which was previously
known for finite groups, $p$-compact groups and some compact Lie groups, but
not for $p$-local compact or Kač-Moody groups (Corollary 4.13); so, it can be
read then as a common property of a big family of homotopy meaningful spaces.
Moreover, finite loop spaces also satisfy this property (Corollary 4.12).

Another source of examples is the theory of infinite loop spaces. McGibbon
[McG] shows that infinite loop spaces satisfy the hypothesis of Theorem 4.1
(see Corollary 4.14).

We also obtain some interesting consequences of these results, including a
detailed analysis of the relationship between the
$B{\mathbb{Z}}/p$-nullification and ${\mathbb{Z}}[1/p]$-localization of these
spaces -which is very much in the spirit of [Dwy96, Section 6]\- and the
commutation of nullity functors on them, a situation that was discussed in
[RS00] in a general framework.

The second part of the paper deals with the effect of the cellularization
functor $CW_{B{\mathbb{Z}}/p}$ on classifying spaces of compact Lie groups. We
show a Serre-type dichotomy theorem.

Theorem 6.9. Let $G$ be a compact connected Lie group. If there exists a non
$p$-cohomologically central element of order $p$, then the
$B{\mathbb{Z}}/p$-cellullarization of $BG$ has infinitely many nonzero
homotopy groups. Otherwise, it has the homotopy type of a $K(V,1)$, where $V$
is a finite elementary abelian $p$-group.

This statement is in fact a consequence of a more general statement which
extends Proposition 2.3 in [FS07].

Theorem 6.1. Let $X$ be a connected nilpotent $\Sigma^{n}B{\mathbb{Z}}/p$-null
space for some $n\geq 0$. Then the $B{\mathbb{Z}}/p$-cellullarization of $X$
has the homotopy type of a Postnikov piece with homotopy groups are
concentrated in degrees $1$ to $n$ , or else it has infinitely many nonzero
homotopy groups. Moreover, if $X$ is $1$-connected of finite type, then the
fundamental group $\pi_{1}(CW_{B{\mathbb{Z}}/p}(X))$ is a finite elementary
abelian $p$-group.

This result opens the way to describe with precision (up to $p$-completion)
the $B{\mathbb{Z}}/p$-cellularization of $BG$ for an ample class of Lie groups
which includes p-toral groups and their discrete approximations, the 3-sphere,
extensions of elementary abelian groups by groups of order prime to $p$ -which
generalize [FS07, Corollary 3.3]-, or $BSO(3)$. In particular, we find
examples of both cases of the dichotomy statement. It is interesting to remark
here that we use intensively the fact that $CW_{A}$ preserves nilpotent spaces
(Lemma 2.5), a fact that was conjectured in [Far96], but which to our
knowledge has not been so far exploited in the literature.

Notation: Let $R$ be a commutative ring, $R_{\infty}(X)$ denotes Bousfield-Kan
$p$-completion of a space $X$ ([BK72]). When $R={\mathbb{Z}}/p$ for a prime
$p$, $R_{\infty}(X)$ will be used instead of $X^{\wedge}_{p}$. Moreover,
$L_{R}(X)$ denotes the $HR$-localization of Bousfield ([Bou75]). All spaces
are assumed to have the homotopy type of a $CW$-complex.

##  2 The cellularization and nullification functors

Let $A$ be a connected space. In this section we will define the functors
$CW_{A}$ and $P_{A}$, which are the main tools we use to describe the
$p$-primary structure of the spaces of interest in our work. Only some
particular features of these functors, that will be crucial in our further
developments, will be described while their relationship with Bousfield-Kan
completion will be studied in the next section. A thorough account to these
constructions can be found in [Far96].

######  Definition 2.1.

Let $A$ and $X$ be spaces. Then $X$ is called $A$-_null_ if the inclusion of
constant maps $X\hookrightarrow{\rm map\,}(A,X)$ is a weak equivalence.

This is equivalent to the condition that ${\rm map\,}_{*}(A,X)$ is weakly
contractible when $X$ is connected. Dror-Farjoun defines a coaugmented and
idempotent functor ${P}_{A}:\mathbf{Spaces}\rightarrow\mathbf{Spaces}$ where
${P}_{A}X$ is $A$-null for every $X$, and such that the coaugmentation
$X\rightarrow{P}_{A}X$ induces a weak equivalence ${\rm
map\,}({P}_{A}X,Y)\rightarrow{\rm map\,}(X,Y)$ for every $A$-null space $Y$.
The corresponding definitions in the pointed context are completely analogous.
Note that in the language of homotopy localization, $P_{A}$ is the
localization with regard to the constant map $A\rightarrow{*}$, and the
notation comes from Postnikov sections, which are in fact
$S^{n}$-nullifications. Moreover, a space $X$ such that $P_{A}X\simeq*$ is
called $A$-_acyclic_.

Now we consider the cellular construction, which is somewhat dual of the
previous construction, although not completely (see Theorem 2.3 below).

######  Definition 2.2.

Given pointed spaces $A$ and $X$, $X$ is said $A$-_cellular_ if it can be
built from $A$ by means of pointed homotopy colimits, possibly iterated.
Moreover, a map $X\rightarrow Y$ is said to be an $A$-_equivalence_ if it
induces a weak equivalence ${\rm map\,}_{*}(A,X)\rightarrow{\rm
map\,}_{*}(A,Y)$.

The $A$-cellularization (or $A$-cellular approximation) is a canonical way of
turning every space into an $A$-cellular space from the point of view of
$A$-equivalences, which generalizes the classic process of cellular
approximation. There exists an augmented endofunctor ${CW}_{A}$ of the
category of pointed spaces, such that for every space $X$ the augmentation
${CW}_{A}X\rightarrow X$ is an $A$-equivalence, and in initial among all maps
$Y\rightarrow X$ which induce $A$-equivalence. Unlike ${P}_{A}$, this functor
only makes sense in the pointed context ([Cha96, 7.4]), and can be
characterized in several ways [Far96, 2.E.8].

The remaining of the section is devoted to describe some properties of these
functors that we will frequently use later. We begin with a theorem of W.
Chachólski that can be considered the most powerful tool to compute
cellularization of spaces in an explicit way. The proof can be found in
[Cha96, 20.3].

######  Theorem 2.3.

Let $A$ and $X$ be pointed spaces, and let $C$ be the homotopy cofibre of the
evaluation $\bigvee_{[A,X]_{*}}A\rightarrow X$, where the wedge is taken over
all the homotopy classes of maps $A\rightarrow X$. Then $CW_{A}X$ has the
homotopy type of the fibre of the map $X\rightarrow P_{\Sigma A}C$.

Next we will describe two preservation properties, that will be used
extensively as we will frequently focus our interest in simply connected
spaces and, more generally, nilpotent spaces.

######  Lemma 2.4.

[Bou94, 2.9] If $X$ is $1$-connected then $P_{A}(X)$ is also $1$-connected.

In particular, note that, according to a famous result of Neisendorfer [Nei95,
Thm 0.1], there is no analogous result for higher degrees of connectivity.

The second preservation property concerns to cellularization and it answers
question $7$ stated by Dror-Farjoun in his book [Far96, p.175]. It is
remarkable that the analogous problem in the category of groups was solved in
[FGS07].

######  Lemma 2.5.

If $X$ is a nilpotent space then $CW_{A}(X)$ is also nilpotent.

###### Proof.

Apply [BK72, V.5.2] to the fibration $CW_{A}(X)\rightarrow X\rightarrow
P_{\Sigma A}C$ in Theorem 2.3. ∎

From the definitions, one can check that if $X$ is $A$-null then
$CW_{A}(X)\simeq*$ since $*\hookrightarrow X$ is an $A$-equivalence. In
general, the $A$-cellularization functor also preserves $\Sigma^{n}A$-nullity
for $n\geq 1$.

######  Lemma 2.6.

Let $X$ be a space which is $\Sigma^{n}A$-null for some $n\geq 1$ then
$CW_{A}(X)$ is also $\Sigma^{n}A$-null.

###### Proof.

Again from Theorem 2.3 we have a fibre sequence $CW_{A}(X)\rightarrow
X\rightarrow P_{\Sigma A}(X)$. Since the base space is $\Sigma A$-null, it is
also $\Sigma^{n}A$-null for any $n\geq 1$. The result follows since the
nullification functor $P_{\Sigma A}$ preserves then the fibration [Far96,
3.D.3]. ∎

If we specialize now to $A=B{\mathbb{Z}}/p$, which is the case of interest in
this paper, and we turn our attention to Eilenberg-MacLane spaces, it is
interesting to observe that given an arbitrary group $G$, the
$B{\mathbb{Z}}/p$-nullity properties of $K(G,n)$ for small values of $n$ imply
the $B{\mathbb{Z}}/p$-nullity for $every$ value of $n$, as well as some group-
theoretic features of $G$.

######  Lemma 2.7.

Let $G$ be an abelian discrete group. $K(G,2)$ is $B{\mathbb{Z}}/p$-null if
and only if $p$ is invertible in $G$ and $K(G,n)$ is $B{\mathbb{Z}}/p$-null
for all $n\geq 1$.

###### Proof.

We only need to show that if $K(G,2)$ is $B{\mathbb{Z}}/p$-null then $p$ is
invertible in $G$ and $K(G,n)$ is $B{\mathbb{Z}}/p$-null for all $n\geq 1$.
Since $K(G,1)\simeq\Omega K(G,2)$ is $B{\mathbb{Z}}/p$-null,
$Hom({\mathbb{Z}}/p,G)=[B{\mathbb{Z}}/p,BG]_{*}=0$. Therefore $G$ has no
elements of order $p$. Then, multiplication by $p$ gives rise to a short exact
sequence $0\rightarrow G\stackrel{{\scriptstyle p}}{{\rightarrow}}G\rightarrow
G/pG\rightarrow 0$. Now consider the induced fibration $K(G,1)\rightarrow
K(G/pG,1)\rightarrow K(G,2)$. Since both $K(G,1)$ and $K(G,2)$ are
$B{\mathbb{Z}}/p$-null, by [Far96, 3.D.3], we see that $B(G/pG)$ is also
$B{\mathbb{Z}}/p$-null. Therefore $G/pG$ has no elements of order $p$, so it
must be trivial. That is $G\stackrel{{\scriptstyle p}}{{\rightarrow}}G$ is an
isomorphism and $p$ is invertible in $G$.

A standard transfer argument (see e.g. [Bro82, Prop III.10.1]) shows that
$\tilde{H}^{*}(B{\mathbb{Z}}/p;G)$ is trivial. In particular, ${\rm
map\,}_{*}(B{\mathbb{Z}}/p,K(G,n))$ is weakly contractible for all $n\geq 1$.
∎

We finish this preliminary section by describing a context in which we can
obtain information about the homology and homotopy groups of the
cellularization.

######  Lemma 2.8.

If $R$ is a ring of coefficients and $\tilde{H}^{*}(A;R)=0$, then
$\tilde{H}^{*}(CW_{A}(X);R)=0$. If $X$ is nilpotent and $R\subset{\mathbb{Q}}$
then $\pi_{i}(CW_{A}(X))\otimes R=0$ for $i>0$.

###### Proof.

Under the hypothesis of the theorem, $K(R,n)$ is $A$-local for $n>0$, then the
space ${\rm map\,}_{*}(CW_{A}(X),K(R,n))$ is weakly contractible. By Lemma
2.5, we can apply [BK72, V.3.1]. ∎

##  3 $B{\mathbb{Z}}/p$-homotopy and $p$-completion

We devote this section to the description of the behaviour of the functors
$CW_{A}$ and $P_{A}$ with respect to the $p$-completion functor of Bousfield
and Kan. In particular, if $\eta\colon X\rightarrow X^{\wedge}_{p}$ is the
$p$-completion, we want to characterize when the maps $CW_{A}(\eta)$ and
$P_{A}(\eta)$ are mod $p$ equivalences. This will be fundamental in our
approach to the $B{\mathbb{Z}}/p$-nullification and
$B{\mathbb{Z}}/p$-cellularization of classifying spaces, which will be
undertaken in the last two sections and is the main goal of our note.

A first approximation to these kind of questions appears in the work of Miller
in the solution of the Sullivan Conjecture, which implies immediately a
statement about $B{\mathbb{Z}}/p$-nullity.

######  Theorem 3.1.

[Mil84, Thm 1.5] Let $W$ be a connected space with
$\tilde{H}^{*}(W;{\mathbb{Z}}[\frac{1}{p}])=0$ and let $X$ be a nilpotent
space. Then $\eta\colon X\rightarrow X^{\wedge}_{p}$ is a $W$-equivalence.

######  Corollary 3.2.

If $X$ is a nilpotent space, the $p$-completion $\eta\colon X\rightarrow
X^{\wedge}_{p}$ is a $\emph{B}{\mathbb{Z}}/p$-equivalence.

Observe that if $X$ is $1$-connected, we can $p$-complete our target space, if
necessary, before computing $CW_{B{\mathbb{Z}}/p}X$. This statement, and the
fact that the $B{\mathbb{Z}}/p$-cellularization is constructed using copies of
$B{\mathbb{Z}}/p$ as pieces, may lead to think that $CW_{B{\mathbb{Z}}/p}X$ is
always a $p$-complete space. Next lemma shows that this is true in certain
cases but, as we will see in Example 3.4, not always.

######  Lemma 3.3.

If $X$ is a nilpotent space, then $CW_{B{\mathbb{Z}}/p}(X)$ is $p$-complete if
and only if
$\tilde{H}_{*}(CW_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p};{\mathbb{Q}})=0$.

###### Proof.

Since $B{\mathbb{Z}}/p$ is both $\mathbb{Q}$-acyclic and
$\mathbb{F}_{q}$-acyclic for $q\neq p$, $CW_{B{\mathbb{Z}}/p}(X)$ is so
([Far96, D.2.5] or Lemma 2.8), and then the rationalization and
$q$-completions of $CW_{B{\mathbb{Z}}/p}(X)$ are trivial. By Lemma 2.5,
$CW_{B{\mathbb{Z}}/p}(X)$ is also a nilpotent space, so it admits a Sullivan
arithmetic square decomposition. The result follows. ∎

######  Example 3.4.

Consider the space $X=K({\mathbb{Z}}/p^{\infty},2)$. $X$ is
$B{\mathbb{Z}}/p$-cellular by [CCS07-2, Lemma 3.3], but it is not $p$-complete
since $X^{\wedge}_{p}\simeq K({\mathbb{Z}}^{\wedge}_{p},3)$ and
$\tilde{H}^{*}(X^{\wedge}_{p};{\mathbb{Q}})\neq 0$. In fact, the
$p$-completion $\eta\colon K({\mathbb{Z}}/p^{\infty},2)\rightarrow
K({\mathbb{Z}}^{\wedge}_{p},3)$ induces a $B{\mathbb{Z}}/p$-cellular
equivalence, then $CW_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})\simeq X$. On the other
hand, taking for example $p=2$ and $X=B\Sigma_{3}$, the classifying space of
the symmetric group in three letters, it is not nilpotent, and the
cellularization is not complete. See [FS07, Example 2.6] for details.

We proceed now to a systematic study of the induced map
$CW_{B{\mathbb{Z}}/p}(\eta)\colon CW_{B{\mathbb{Z}}/p}(X)\rightarrow
CW_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})$. We want to show under which conditions
it becomes a mod $p$ equivalence. The first step is a reduction concerning the
fundamental group, for which we need the following definition.

######  Definition 3.5.

We say that an element $x\in\pi_{1}(X)$ lifts to $X$ if there exists a
homotopy lift

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\emph{B}(\langle
x\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\langle
x\rangle}}$$\textstyle{\emph{B}(\pi_{1}(X)).}$

######  Proposition 3.6.

Let X be a connected space. There is a fibration

$CW_{B{\mathbb{Z}}/p}(X)\rightarrow X\rightarrow Z$

with $\pi_{1}(Z)\cong\pi_{1}(X)/S$, where $S$ is the normal subgroup generated
by the elements of order $p$ which lift to $X$.

###### Proof.

The fibration in the proposition is the one constructed by Chachólski (see
Theorem 2.3) where $Z=P_{\Sigma B{\mathbb{Z}}/p}(C_{X})$. The subgroup $S$ is
constructed in [CCS07-2, Prop. 2.1] in a way that $E\rightarrow X$ is a
$B{\mathbb{Z}}/p$-cellular equivalence, where $E$ is the homotopy pullback

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Bi}$$\textstyle{B(\pi_{1}(X)).}$

By construction $\pi_{1}(E)\cong S$ is generated by elements of order $p$
which lift to $E$. Then the Chachólski’s cofibre $C_{E}$ (see Theorem 2.3) is
$1$-connected and $P_{\Sigma B{\mathbb{Z}}/p}(C_{E})$ is too by Lemma 2.4.
Since $E\rightarrow X$ is a $B{\mathbb{Z}}/p$-equivalence, from the following
diagram of fibrations

$\textstyle{CW_{B{\mathbb{Z}}/p}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{CW_{B{\mathbb{Z}}/p}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}(C_{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}(C).}$

where $C$ is Chachólski’s cofibre for $X$, we see that the fundamental group
of $P_{\Sigma B{\mathbb{Z}}/p}(C)$ is $\pi_{1}(X)/S$. ∎

######  Corollary 3.7.

Let $X$ be a connected space such that $\pi_{1}(X)$ is generated by elements
of order $p$ which lift to $X$. There is a bijection
$[B{\mathbb{Z}}/p,CW_{B{\mathbb{Z}}/p}(X)]\cong[B{\mathbb{Z}}/p,X]$ between
unpointed homotopy classes of maps.

###### Proof.

Since $CW_{B{\mathbb{Z}}/p}(X)\rightarrow X$ is a $B{\mathbb{Z}}/p$-homotopy
equivalence, there is a bijection
$[B{\mathbb{Z}}/p,CW_{B{\mathbb{Z}}/p}(X)]_{*}\cong[B{\mathbb{Z}}/p,X]_{*}$
between pointed homotopy classes of maps. The following diagram

$\textstyle{[B{\mathbb{Z}}/p,CW_{B{\mathbb{Z}}/p}(X)]_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[B{\mathbb{Z}}/p,X]_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[B{\mathbb{Z}}/p,CW_{B{\mathbb{Z}}/p}(X)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[B{\mathbb{Z}}/p,X]}$

shows that the quotient map is also a bijection since the induced morphism on
fundamental groups $\pi_{1}(CW_{B{\mathbb{Z}}/p}(X))\rightarrow\pi_{1}(X)$ is
an epimorphism by Proposition 3.6. ∎

We can get information about the fundamental group of the cellularization
since being $B{\mathbb{Z}}/p$-cellular imposes some restrictions on the
fundamental group of the space.

######  Lemma 3.8.

If $X$ is a $B{\mathbb{Z}}/p$-cellular space, its fundamental group is
generated by elements of order $p$ which lift to $X$. Moreover, if $X$ is a
finite type $1$-connected space,then $\pi_{1}(CW_{B{\mathbb{Z}}/p}(X))$ is a
finitely generated abelian generated by elements of order $p$ which lift to
$CW_{B{\mathbb{Z}}/p}(X)$.

###### Proof.

Let $S$ be the normal subgroup of $\pi_{1}(X)$ generated by elements of order
$p$ which lift to $X$. Consider the pullback diagram

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Bi}$$\textstyle{B\pi_{1}(X).}$

By [CCS07-2, Prop 2.1], the map $E\rightarrow X$ is a
$B{\mathbb{Z}}/p$-cellular equivalence. Since $X$ is
$B{\mathbb{Z}}/p$-cellular, there exists a map $f\colon E\rightarrow
CW_{B{\mathbb{Z}}/p}(E)$ such that $i\circ f\simeq id$ where $i\colon
CW_{B{\mathbb{Z}}/p}(E)\rightarrow E$. In fact, this implies that $p\colon
E\rightarrow P_{\Sigma B{\mathbb{Z}}/p}(C_{E})$ is nullhomotopic, $p\simeq
p\circ i\circ f\simeq*\circ f\simeq*$, therefore
$CW_{B{\mathbb{Z}}/p}(E)\simeq E\times P_{B{\mathbb{Z}}/p}(\Omega C_{E})$. But
this implies that $E$ is $B{\mathbb{Z}}/p$-cellular since
$CW_{B{\mathbb{Z}}/p}(E)$ is $B{\mathbb{Z}}/p$-acyclic, and then $E\simeq X$.
In particular, $\pi_{1}(X)=S$.

To prove the second statement, it remains to prove that
$\pi_{1}(CW_{B{\mathbb{Z}}/p}(X))$ is a finitely generated abelian group.
Since $X$ is $1$-connected, then the Chachólski’s cofibre $C_{X}$ is
$1$-connected and $P_{\Sigma B{\mathbb{Z}}/p}(C_{X})$ is too by Lemma 2.4.
Then, we see that $\pi_{2}(P_{\Sigma B{\mathbb{Z}}/p}(C_{X}))\cong
H_{2}(P_{\Sigma B{\mathbb{Z}}/p}(C_{X});{\mathbb{Z}})$ is a quotient of
$H_{2}(C_{X};{\mathbb{Z}})$, which in turn is a quotient of the finitely
generated group $H_{2}(X;{\mathbb{Z}})$. ∎

Next we need a technical lemma which describes the somewhat intrincate
relationship between completion and nullification and that is a key result to
understand under which conditions $P_{A}(\eta)\colon P_{A}(X)\rightarrow
P_{A}(X^{\wedge}_{p})$ is a mod $p$ equivalence (Corollary 3.11).

######  Lemma 3.9.

Let $A$ be a connected space, and let $X$ such that $P_{A}(X^{\wedge}_{p})$
and $P_{A}(X)$ are $p$-good spaces. Assume that $P_{A}(X)^{\wedge}_{p}$ and
$P_{A}(X^{\wedge}_{p})^{\wedge}_{p}$ are $A$-null spaces. Then the
$p$-completion map $\eta_{X}\colon X\rightarrow X^{\wedge}_{p}$ induces a mod
$p$ equivalence $P_{A}(\eta)\colon P_{A}(X)\rightarrow P_{A}(X^{\wedge}_{p})$.

###### Proof.

Let $\epsilon\colon P_{A}(X^{\wedge}_{p})\rightarrow(P_{A}(X))^{\wedge}_{p}$
be the unique map up to homotopy such that the right square of the following
diagram commutes:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{X}}$$\scriptstyle{\iota_{X}}$$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\scriptstyle{\iota_{X^{\wedge}_{p}}}$$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\iota_{X})^{\wedge}_{p}}$$\textstyle{P_{A}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{A}(\eta_{X})}$$\textstyle{P_{A}(X^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{P_{A}(X)^{\wedge}_{p}.}$

Note that $\epsilon$ exists because $P_{A}(X)^{\wedge}_{p}$ is $A$-null by
hypothesis. The left square commutes by naturality, so
$(\iota_{X})^{\wedge}_{p}\circ\eta_{X}\simeq\epsilon\circ
P_{A}(\eta_{X})\circ\iota_{X}$. But also,
$(\iota_{X})^{\wedge}_{p}\circ\eta_{X}\simeq\eta_{P_{A}(X)}\circ\iota_{X}$ by
naturality of the completion. Because of the universal property of the
nullification functor, $\epsilon\circ P_{A}(\eta_{X})\simeq\eta_{P_{A}(X)}$.
Since $P_{A}(X)$ is $p$-good, $\eta_{P_{A}(X)}^{*}$ is an isomorphism in mod
$p$ cohomology. In particular, $\epsilon^{*}$ is a monomorphism and
$P_{A}({\eta}_{X})^{*}$ is an epimorphism.

Now consider the following commutative diagram:

$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\scriptstyle{\iota_{X^{\wedge}_{p}}}$$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\eta_{X})^{\wedge}_{p}}$$\scriptstyle{(\iota_{X})^{\wedge}_{p}}$$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\iota_{X^{\wedge}_{p}})^{\wedge}_{p}}$$\textstyle{P_{A}(X^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{P_{A}(X)^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{A}(\eta_{X})^{\wedge}_{p}}$$\textstyle{P_{A}(X^{\wedge}_{p})^{\wedge}_{p}.}$

That is
$(\iota_{X^{\wedge}_{p}})^{\wedge}_{p}\circ(\eta_{X})^{\wedge}_{p}\simeq
P_{A}(\eta)^{\wedge}_{p}\circ\epsilon\circ\iota_{X^{\wedge}_{p}}$. But we also
have
$(\iota_{X^{\wedge}_{p}})^{\wedge}_{p}\circ(\eta_{X})^{\wedge}_{p}\simeq(\eta_{P_{A}(X^{\wedge}_{p})})\circ\iota_{X^{\wedge}_{p}}$.
By hypothesis $P_{A}(X^{\wedge}_{p})^{\wedge}_{p}$ is $A$-null, then the
universal property of the nullification functor implies that
$P_{A}({\eta}_{X})^{\wedge}_{p}\circ\epsilon\simeq\eta_{P_{A}(X^{\wedge}_{p})}$.
Since $P_{A}(X^{\wedge}_{p})$ is $p$-good,
$(\eta_{P_{A}(X^{\wedge}_{p})})^{*}$ is an isomorphism and hence
$(P_{A}(\eta_{X})^{\wedge}_{p})^{*}$ is a monomorphism. Therefore
$P_{A}(\eta_{X})^{*}$ is so, and we are done. ∎

######  Remark 3.10.

If $X$ has finite fundamental group, then both $P_{A}(X^{\wedge}_{p})$ and
$P_{A}(X)$ are $p$-good spaces since they also have finite fundamental groups.

######  Corollary 3.11.

If $X$ is a $1$-connected space and $A$ is such that
$\tilde{H}_{*}(A;{\mathbb{Z}}[\frac{1}{p}])=0$ then $P_{A}(\eta)\colon
P_{A}(X)\rightarrow P_{A}(X^{\wedge}_{p})$ is a mod $p$ equivalence.

###### Proof.

If $X$ is $1$-connected then $X^{\wedge}_{p}$ is also $1$-connected and both
spaces are $p$-good. Moreover the $B{\mathbb{Z}}/p$-nullification of a
$1$-connected space is also $1$-connected. Miller’s theorem [Mil84, Thm 1.5]
implies that the spaces $P_{A}(X)^{\wedge}_{p}$ and
$P_{A}(X^{\wedge}_{p})^{\wedge}_{p}$ are $A$-null. The hypothesis of Lemma 3.9
are then satisfied. ∎

We can also describe a general situation in which the nullification of a mod
$p$ equivalence is so.

######  Corollary 3.12.

Let $A$ be a space such that $\tilde{H}_{*}(A;{\mathbb{Z}}[\frac{1}{p}])=0$.
If $f\colon X\rightarrow Y$ is a mod $p$ equivalence between $1$-connected
spaces then $P_{A}(f)\colon P_{A}(X)\rightarrow P_{A}(Y)$ is a mod $p$
equivalence.

###### Proof.

If $f$ is a mod $p$ equivalence, then $f^{\wedge}_{p}$ is an equivalence. Then
the following diagram commutes

$\textstyle{P_{A}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{A}(\eta_{X})}$$\scriptstyle{P_{A}(f)}$$\textstyle{P_{A}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{A}(\eta_{Y})}$$\textstyle{P_{A}(X^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{A}(f^{\wedge}_{p})}$$\textstyle{P_{A}(Y^{\wedge}_{p})}$

By Corollary 3.11, the two vertical arrows are mod $p$ equivalences and the
bottom horizontal map is an equivalence. Then $P_{A}(f)$ is a mod $p$
equivalence. ∎

######  Remark 3.13.

Note that Corollaries 3.11 and 3.12 hold when $A=B{\mathbb{Z}}/p$. In fact, in
Corollary 3.12, one can relax the assumptions on $1$-connectivity by checking
that both spaces $X$ and $Y$ satisfy the assumptions of Lemma 3.9.

Now we follow the parallelism giving a condition for the analogous equivalence
between cellularizations to hold. According to Proposition 3.6, the hypothesis
of lifting elements in the fundamental group is not a real restriction.

######  Proposition 3.14.

Let $X$ be a space whose fundamental group $\pi_{1}(X)$ is finite and
generated by elements of order $p$ which lift to $X$. Assume that there is a
bijection $[B{\mathbb{Z}}/p,X]=[B{\mathbb{Z}}/p,X^{\wedge}_{p}]$, then the map
induced by the $p$-completion

$CW_{B{\mathbb{Z}}/p}(\eta)\colon CW_{B{\mathbb{Z}}/p}(X)\rightarrow
CW_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})$

is a mod $p$ equivalence.

###### Proof.

Since $\pi_{1}(X)$ is finite, $X$ is $p$-good [BK72, VII.5.1]. There is an
epimorphism $\pi_{1}(X)\rightarrow\pi_{1}(X^{\wedge}_{p})$ and, by assumption,
$[B{\mathbb{Z}}/p,X]\cong[B{\mathbb{Z}}/p,X^{\wedge}_{p}]$. In order to
compute the cellularization, we first analyze Chachólski’s cofibres

$\textstyle{\vee
B{\mathbb{Z}}/p\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\scriptstyle{id}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\vee
B{\mathbb{Z}}/p\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}}$$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D.}$

Since $\pi_{1}(X)$ is finite and generated by elements of order $p$ which lift
to $X$, the maps $h_{1}$ and $h_{2}$ induce epimorphisms on the fundamental
group and then $C$ and $D$ are 1-connected spaces. Moreover $g$ is a mod $p$
equivalence. Now, the cellularization fits in the following diagram of
fibrations:

$\textstyle{CW_{B{\mathbb{Z}}/p}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{CW_{B{\mathbb{Z}}/p}(\eta)}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{\Sigma
B{\mathbb{Z}}/p}(g)}$$\textstyle{CW_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}(D),}$

where $P_{\Sigma B{\mathbb{Z}}/p}(C)$ and $P_{\Sigma B{\mathbb{Z}}/p}(D)$ are
$1$-connected. All the spaces in the previous diagram are $p$-good. Therefore,
to show that $CW_{B{\mathbb{Z}}/p}(\eta)$ is a mod $p$ equivalence, it is
enough to prove that $P_{\Sigma B{\mathbb{Z}}/p}(g)$ is so. This follows from
the previous Corollary 3.12 since $g$ is a mod $p$-equivalence. ∎

######  Remark 3.15.

We note in Example 3.4 that $CW_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})$ does not
need to be $p$-complete and a condition for this to be true was stated in
Lemma 3.3. If $X$ satisfies the hypothesis of Proposition 3.14, we see from
the proof that $CW_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})$ is $p$-complete if
$P_{\Sigma B{\mathbb{Z}}/p}(D)$ is so. This last space is $1$-connected and,
using an arithmetic Sullivan square argument, we see that this is the case if
$X^{\wedge}_{p}\rightarrow P_{\Sigma B{\mathbb{Z}}/p}(D)^{\wedge}_{p}$ is a
rational equivalence. Examples of this situation are provided by classifying
spaces of finite groups, since $(BG^{\wedge}_{p})_{\mathbb{Q}}\simeq*$ and
$P_{\Sigma B{\mathbb{Z}}/p}(D)^{\wedge}_{p}\simeq P_{\Sigma
B{\mathbb{Z}}/p}(C)^{\wedge}_{p}$ is homotopic to the $p$-completion of the
classifying space of a finite group by [FS07, Proposition 5.5] and [FF,
Theorem 4.3].

######  Remark 3.16.

The hypothesis of Proposition 3.14 are satisfied if $\pi_{1}(X)$ is a finite
$p$-group generated by elements of order $p$ which lift to $X$ (see [DZ87,
Proof of 3.1]).

######  Remark 3.17.

Let $P$ be a $p$-toral group. Then $P$ is an extension of a finite $p$-group
$\pi$ by a torus $(S^{1})^{n}$. Assume that $\pi$ is generated by elements of
order $p$ which lift to $BP$. The arguments of [DZ87, proof of 3.1] applied to
the fibration $B(S^{1})^{n}\rightarrow BP\rightarrow B\pi$ show that
$[B{\mathbb{Z}}/p,BP]=[B{\mathbb{Z}}/p,BP^{\wedge}_{p}]$. Then
$BP^{\wedge}_{p}$ is a $p$-compact toral group, and
$CW_{B{\mathbb{Z}}/p}(\eta)\colon CW_{B{\mathbb{Z}}/p}(BP)\rightarrow
CW_{B{\mathbb{Z}}/p}(BP^{\wedge}_{p})$ is a mod $p$ equivalence.

By [DW94, Proposition 6.9], there exists a discrete $p$-toral group
$P_{\infty}$, that is, an extension of a finite $p$-group $\pi$ by a finite
sum of Prüfer groups $({\mathbb{Z}}/p^{\infty})^{n}$, such that
$BP_{\infty}\rightarrow BP^{\wedge}_{p}$ is a mod $p$ equivalence. Moreover,
$[B{\mathbb{Z}}/p,BP_{\infty}]\cong[B{\mathbb{Z}}/p,(BP_{\infty})^{\wedge}_{p}]$
by [DW94, Remark 6.12], so we should study, up to $p$-completion, the
$B{\mathbb{Z}}/p$-cellullarization of discrete $p$-toral groups. See Example
6.16.

##  4 $B{\mathbb{Z}}/p$-nullification of classifying spaces

In this section, we are concerned with $B{\mathbb{Z}}/p$-nullification. The
original motivating example for our study were classifying spaces of compact
Lie groups, for which Dwyer computed in [Dwy96] the value of
$P_{B{\mathbb{Z}}/p}BG$ for the case in which $\pi_{0}(G)$ is a (finite)
$p$-group. For this sake, he used an induction principle based on the
centralizer decomposition of $BG$, a method that also solve the problem when
we take a $p$-compact group $X$ instead of $G$. However, the hypothesis over
the fundamental group is essential and cannot be removed from his proof, so we
need to follow a completely different path to solve the general case. In fact,
our new strategy was useful to describe $P_{B{\mathbb{Z}}/p}X$ for a bigger
family of spaces, which in particular need not to be classifying spaces.

Recall that, if $S$ is a set of primes, the $S$-radical subgroup $T_{S}(G)$ of
a finite group $G$ is the smallest normal subgroup of $G$ which contains all
the $S$-torsion. This is the last ingredient we need to state the main result
of this section.

######  Theorem 4.1.

Let $X$ be a connected space with finite fundamental group and such that
$P_{B{\mathbb{Z}}/p}(X\langle 1\rangle)\simeq*$. Then there is a fibration

$L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})\rightarrow
P_{B{\mathbb{Z}}/p}(X)\rightarrow B(\pi_{1}(X)/T_{p}(\pi_{1}(X)))$

where $X_{p}$ is the covering of $X$ whose fundamental group is
$T_{p}(\pi_{1}(X))$, and $L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})$ denotes the
homological localization of $X_{p}$ in the ring ${\mathbb{Z}}[\frac{1}{p}]$.

Theorem 4.1 will be a consequence of the following result.

######  Theorem 4.2.

Let $X$ be a connected space with finite fundamental group generated by
$p$-torsion elements which lift to $X$ and such that
$P_{B{\mathbb{Z}}/p}(X\langle 1\rangle)\simeq*$. Then there is an equivalence
$P_{B{\mathbb{Z}}/p}(X)\rightarrow L_{{\mathbb{Z}}[\frac{1}{p}]}(X),$ where
$L_{{\mathbb{Z}}[\frac{1}{p}]}(X)$.

Now in order to prove Theorem 4.2 we follow the strategy of the second author
in [Flo07] when dealing with classifying spaces of finite groups, although now
there is rational information that is absent in the finite case. Before,
however, we will be deal with some issues concerning to the fundamental group
of $X$ which will be crucial in the proof.

######  Lemma 4.3.

Let $G$ be a finite group and $S$ a set of primes that divide the order of
$G$. If $G=\emph{T}_{S}G$, then $G$ is $S^{-1}$-perfect. In particular, if $X$
is a space with finite fundamental group such that
$\pi_{1}X=T_{S}(\pi_{1}(X))$, then $L_{\mathbb{Z}[S^{-1}]}(X)$ is simply-
connected.

###### Proof.

The first statement follows from the fact that, since $G$ is generated by
$S$-torsion, $G_{ab}$ is an abelian finite $S$-torsion subgroup, and then
$\mathbb{Z}[S^{-1}]\otimes G^{ab}=0$.

For the second statement, observe that as $G$ is $S^{-1}$-perfect, then $X$ is
a $\mathbb{Z}[S^{-1}]$-good space, and the ${\mathbb{Z}}[S^{-1}]$-completion
of $X$ is $1$-connected, by [BK72, VII.3.2]. But for a connected
${\mathbb{Z}}[\frac{1}{p}]$-good space $X$, the
${\mathbb{Z}}[\frac{1}{p}]$-completion is an
$H_{*}(-;{\mathbb{Z}}[\frac{1}{p}])$-localization (see [BK72, page 205]). ∎

In particular, if $X$ is a connected space such that its fundamental group is
finite and equal to its $\mathbb{Z}/p$-radical, then $L_{\mathbb{Z}[1/p]}X$ is
a simply-connected space.

######  Lemma 4.4.

Let $X$ be a connected space and $p$ a prime. Then the coaugmentation
$X\rightarrow L_{{\mathbb{Z}}[1/p]}X$ is an $\mathbb{F}_{q}$-equivalence and a
${\mathbb{Q}}$-equivalence where $q$ is a prime such that $(q,p)=1$. If
$L_{{\mathbb{Z}}[1/p]}X$ is $1$-connected then $L_{{\mathbb{Z}}[1/p]}X$ is
${\mathbb{F}}_{p}$-acyclic.

###### Proof.

By universal coefficient theorem (e.g. see [Spa66, 5.2.15]), the
coaugmentation $X\rightarrow L_{{\mathbb{Z}}[1/p]}X$ is a $G$-equivalence for
any ${\mathbb{Z}}[\frac{1}{p}]$-module $G$. The last statement follows form
[Dwy96, Lemma 6.2]. ∎

######  Lemma 4.5.

Let $Z$ be a $B{\mathbb{Z}}/p$-null space and $X$ be a connected space such
that $\pi_{1}(X)$ is a finite group generated by $p$-torsion elements which
lift to $X$. Then for any $f\colon X\rightarrow Z$, the composite
$X\rightarrow Z\rightarrow B\pi_{1}(Z)$ is nullhomotopic.

###### Proof.

Let $f\colon X\rightarrow Z$ be any map. We must check that $\pi_{1}(f)$ is
the trivial morphism. It is enough to show that the map between unpointed
homotopy classes $[S^{1},X]\rightarrow[S^{1},Z]$ is trivial.

Let $x\in\pi_{1}(X)$ be a generator, $\langle
x\rangle\cong{\mathbb{Z}}/p^{n}\subseteq\pi_{1}(X)$, we need to show that the
composite $B{\mathbb{Z}}/p^{n}\rightarrow B\pi_{1}(X)\rightarrow B\pi_{1}(Z)$
is nullhomotopic for any generator $x$.

By hypothesis, there is a lift

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B{\mathbb{Z}}/p^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\pi_{1}(Z).}$

But since $Z$ is $B{\mathbb{Z}}/p$-null and
$P_{B{\mathbb{Z}}/p}(B{\mathbb{Z}}/p^{n})\simeq*$, it follows that the top
composite $B{\mathbb{Z}}/p^{n}\rightarrow X\rightarrow Z$ is nullhomotopic,
and therefore $\pi_{1}(f)(x)=0$. ∎

The hypothesis in Theorem 4.1 concerning the pointed mapping space from the
universal cover of $X$ is also satisfied by connected covers of $X$.

######  Lemma 4.6.

Let $X$ be a connected space.

  1. 1.

Assume that $X$ is $1$-connected. Then
$P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p}\simeq*$ iff ${\rm map\,}_{*}(X,Z)\simeq*$
for any connected $B{\mathbb{Z}}/p$-null $p$-complete space $Z$.

  2. 2.

Assume that $X$ has a finite fundamental group. Let $Y$ be a connected cover
of $X$. If $Z$ is a connected $B{\mathbb{Z}}/p$-null and $p$-complete space,
then the equivalence ${\rm map\,}_{*}(Y,Z)\simeq*$ implies ${\rm
map\,}_{*}(X,Z)\simeq*$.

###### Proof.

  1. 1.

Note that if $Z$ is a connected $B{\mathbb{Z}}/p$-null $p$-complete space $Z$,
there are weak homotopy equivalences

${\rm map\,}_{*}(P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p},Z)\simeq{\rm
map\,}_{*}(P_{B{\mathbb{Z}}/p}(X),Z)\simeq{\rm map\,}(X,Z)$

If $P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p}\simeq*$, then ${\rm
map\,}(X,Z)\simeq*$ for any connected $B{\mathbb{Z}}/p$-null $p$-complete
space $Z$. On the other hand, assume that ${\rm map\,}_{*}(X,Z)\simeq*$ for
any connected $B{\mathbb{Z}}/p$-null $p$-complete space $Z$. Since
$P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p}$ is a $p$-complete $B{\mathbb{Z}}/p$-null
space by Corollary 3.2, ${\rm
map\,}_{*}(P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p},P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p})\simeq*$,
therefore $P_{B{\mathbb{Z}}/p}(X)^{\wedge}_{p}\simeq*$.

  2. 2.

Consider a fibration $Y\rightarrow X\rightarrow BG$ where $G$ is a finite
group. Zabrodsky’s Lemma (see [Mil84, 9.5]) tells us that there is an
equivalence of pointed mapping spaces ${\rm map\,}_{*}(X,Z)\simeq{\rm
map\,}_{*}(BG,Z)$ since ${\rm map\,}_{*}(Y,Z)\simeq*$. Finally, this mapping
space is contractible since we have weak homotopy equivalences ${\rm
map\,}_{*}(BG,Z)\simeq{\rm map\,}_{*}(BG^{\wedge}_{p},Z)$ and
$P_{B{\mathbb{Z}}/p}((BG)^{\wedge}_{p})\simeq*$ by [Flo07, 3.14].

∎

Now we are ready to undertake the proof of Theorem 4.2.

###### Proof of Theorem 4.2.

By hypothesis, $\pi_{1}(X)$ has no quotients whose order is prime to $p$,
which amounts to say that $\pi_{1}(X)$ is equal to its $\mathbb{Z}/p$-radical
$T_{p}(\pi_{1}(X))$.

First of all, notice that $L_{\mathbb{Z}[1/p]}X$ is $B{\mathbb{Z}}/p$-null by
Lemma 4.3 and [Dwy96, Lemma 6.2]. In order to show that
$P_{B{\mathbb{Z}}/p}(X)\rightarrow L_{{\mathbb{Z}}[\frac{1}{p}]}(X)$ is a weak
equivalence, since $L_{\mathbb{Z}[1/p]}X$ is $B{\mathbb{Z}}/p$-null, we must
show that for every $B\mathbb{Z}/p$-null space $Y$ the natural coaugmentation
$X\longrightarrow L_{\mathbb{Z}[1/p]}X$ gives a weak equivalence ${\rm
map\,}_{*}(L_{\mathbb{Z}[1/p]}X,Y)\simeq{\rm map\,}_{*}(X,Y)$.

So let $Y$ be a $B{\mathbb{Z}}/p$-null space. Assume first that $Y$ is simply-
connected. By Miller’s Theorem 3.1, $Y^{\wedge}_{p}$ is also
$B{\mathbb{Z}}/p$-null. According to Bousfield-Kan fracture lemmas ([BK72,
V.6]), we must prove that, for every prime $q$, there is a weak homotopy
equivalence ${\rm map\,}_{*}(L_{\mathbb{Z}[1/p]}(X),Y_{q}^{\wedge})\simeq{\rm
map\,}_{*}(X,Y_{q}^{\wedge})$, and ${\rm
map\,}_{*}(L_{\mathbb{Z}[1/p]}(X),Y_{\mathbb{Q}})\simeq{\rm
map\,}_{*}(X,Y_{\mathbb{Q}})$. By Lemmas 4.4 and 4.6, this is a consequence of
the assumption of the theorem, so we finish the situation in which $Y$ is
simply connected.

Now let $Y$ be a $B\mathbb{Z}/p$-null space and $\tilde{Y}$ its universal
cover. The coaugmentation $X\longrightarrow L_{\mathbb{Z}[1/p]}X$ induces a
diagram of fibrations over the component of the constant map

where ${\rm map\,}_{*}(L_{\mathbb{Z}[1/p]}(X),Y)_{\\{c\\}}$ and ${\rm
map\,}_{*}(X,Y)_{\\{c\\}}$ are those components such that $\rho$ induce the
constant map when composing with $Y\rightarrow B\pi_{1}(Y)$.

The top horizontal map is an equivalence because of the previous argument
since $\tilde{Y}$ is a simply connected $B{\mathbb{Z}}/p$-null space. For any
connected space $A$ and a discrete group $H$, ${\rm map\,}_{*}(A,BH)$ is a
homotopically discrete space, and then ${\rm
map\,}_{*}(L_{\mathbb{Z}[1/p]}(X),B\pi_{1}(Y))_{c}$ and ${\rm
map\,}_{*}(X,B\pi_{1}(Y))_{c}$ are contractible. Thus, the bottom horizontal
arrow in the diagram is also a weak equivalence.

To finish the proof we need to show that there are weak homotopy equivalences
${\rm map\,}_{*}(L_{\mathbb{Z}[1/p]}(X),Y)_{\\{c\\}}\simeq{\rm
map\,}_{*}(L_{\mathbb{Z}[1/p]}(X),Y)$ and ${\rm
map\,}_{*}(X,Y)_{\\{c\\}}\simeq{\rm map\,}_{*}(X,Y)$. The first equivalence
follows now from the fact that $L_{\mathbb{Z}[1/p]}(X)$ is simply connected by
Lemma 4.3, while the second follows from Lemma 4.5. ∎

###### Proof of Theorem 4.1.

Theorem 4.2 applied to the universal cover of $X$ implies that the map in
[Dwy96, 1.6], $P_{B{\mathbb{Z}}/p}(X\langle 1\rangle)\rightarrow
L_{{\mathbb{Z}}[\frac{1}{p}]}(X\langle 1\rangle)$, is an equivalence.

Let $X_{p}$ be the covering space of $X$ with fundamental group
$T_{p}(\pi_{1}(X))$. There is a fibration $X_{p}\rightarrow X\rightarrow
B(\pi_{1}(X)/T_{p}(\pi_{1}(X)))$. Since the base space of this fibration
$B(\pi_{1}(X)/T_{p}(\pi_{1}(X)))$ is $B{\mathbb{Z}}/p$-null, the nullification
functor preserves the fibration by [Far96, 3.D.3] and there is another
fibration

$P_{B{\mathbb{Z}}/p}(X_{p})\rightarrow P_{B{\mathbb{Z}}/p}(X)\rightarrow
B(\pi_{1}(X)/T_{p}(\pi_{1}(X))).$

To prove the theorem we shall show that the natural map
$P_{B{\mathbb{Z}}/p}(X_{p})\rightarrow L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})$,
which exists because $B{\mathbb{Z}}/p$ is $H{\mathbb{Z}}[\frac{1}{p}]$-acyclic
and then $L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})$ is $B{\mathbb{Z}}/p$-null, is
a homotopy equivalence. Note also that $(X_{p})\langle 1\rangle\simeq X\langle
1\rangle$. Therefore $X_{p}$ also satisfies the hypothesis of the theorem.

From now on we assume that $\pi_{1}(X)$ has no quotients whose order is prime
to $p$, which amounts to say that $\pi_{1}(X)$ is equal to its
$\mathbb{Z}/p$-radical $T_{p}(\pi_{1}(X))$.

Consider the fibration $X\langle 1\rangle\rightarrow X\rightarrow B\pi_{1}(X)$
and its fibrewise nullfication (see [Far96, 1.F]) which gives a diagram of
fibrations

$\textstyle{X\langle
1\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\xi}}$$\textstyle{B\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{B{\mathbb{Z}}/p}(X\langle
1\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bar{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\pi_{1}(X).}$

where $\bar{\xi}$ is an equivalence after $B{\mathbb{Z}}/p$-nullification.
Then, by [Dwy96, 1.6], $\bar{\xi}$ is a
${\mathbb{Z}}[\frac{1}{p}]$-equivalence. Note that it is enough to show that
the map $P_{B{\mathbb{Z}}/p}(\bar{X})\rightarrow
L_{{\mathbb{Z}}[\frac{1}{p}]}(\bar{X})$ is an equivalence since there is a
chain

$P_{B{\mathbb{Z}}/p}(X)\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}P_{B{\mathbb{Z}}/p}(\bar{X})\rightarrow
L_{{\mathbb{Z}}[\frac{1}{p}]}(\bar{X})\stackrel{{\scriptstyle\simeq}}{{\leftarrow}}L_{{\mathbb{Z}}[\frac{1}{p}]}({X}).$

Moreover, $\pi_{1}(\bar{X})\cong\pi_{1}(X)$ because the fibre
$P_{B{\mathbb{Z}}/p}(X\langle 1\rangle)$ is $1$-connected, then the universal
cover of $\bar{X}$ is $P_{B{\mathbb{Z}}/p}(X)$ and
$\pi_{1}(\bar{X})=T_{p}(\pi_{1}(\bar{X}))$. For each generator
$x\in\pi_{1}(\bar{X})$, the obstructions to lift the map $B(\langle
x\rangle)\simeq B{\mathbb{Z}}/p^{n}\rightarrow B\pi_{1}(\bar{X})$ to $\bar{X}$
lie in the twisted cohomology groups
$H^{i+1}(B{\mathbb{Z}}/p^{n};\pi_{i}(P_{B{\mathbb{Z}}/p}(X\langle 1\rangle)))$
for $i\geq 1$, and these groups are trivial since the homotopy groups
$\pi_{i}(P_{B{\mathbb{Z}}/p}(X\langle
1\rangle))\cong\pi_{i}(L_{{\mathbb{Z}}[\frac{1}{p}]}(X\langle 1\rangle))$ are
${\mathbb{Z}}[\frac{1}{p}]$-modules. That is, $\bar{X}$ is a connected space
with finite fundamental group generated by $p$-torsion whose generators lift
to $\bar{X}$.

In order to apply Theorem 4.2, it remains to check that ${\rm
map\,}_{*}(\bar{X}\langle 1\rangle,Z)\simeq*$ for any connected
$B{\mathbb{Z}}/p$-null $p$-complete space $Z$. Recall that $\bar{X}\langle
1\rangle\simeq P_{B{\mathbb{Z}}/p}({X}\langle 1\rangle)$. Then ${\rm
map\,}_{*}(P_{B{\mathbb{Z}}/p}(X\langle 1\rangle),Z)\simeq{\rm
map\,}_{*}({X}\langle 1\rangle,Z)\simeq*$ where the last equivalence follows
by hypothesis. ∎

######  Remark 4.7.

The proof of Theorem 4.1 also holds if we replace the analysis at one prime
$p$ for a set of primes $S$ and imposing that the hypothesis on pointed
mapping spaces are satisfied for any prime $p$ in the set $S$. In that case we
have to replace $L_{{\mathbb{Z}}[\frac{1}{p}]}$ by $L_{{\mathbb{Z}}[S^{-1}]}$,
and $P_{B{\mathbb{Z}}/p}$ by $P_{W}$ where $W=\vee_{p\in S}B{\mathbb{Z}}/p$.

###  4.1 Examples

We want to explore the implications of these results on classifying spaces of
Lie groups, which was the original motivation for our work. For this sake we
need the following Lemma, which was proved by Dwyer [Dwy96, Theorem 1.2] using
an induction. We include here a shorter proof, based on the homology
decomposition of $BG$ via $p$-toral subgroups. The key point here is that this
decomposition is indexed over an mod $p$ _acyclic_ category, and this opens
the way for computing $P_{B{\mathbb{Z}}/p}$ for a more general class of
$p$-good spaces (see Corollary 4.11).

######  Lemma 4.8.

Let $Z$ be a connected $B{\mathbb{Z}}/p$-null and $p$-complete space. Let
$F\colon\mathcal{C}\rightarrow Top$ be a functor such that for each object
$c\in\mathcal{C}$, $F(c)$ is connected and
$P_{B{\mathbb{Z}}/p}(F(c)^{\wedge}_{p})$ is mod $p$ acyclic. If
$|\mathcal{C}|^{\wedge}_{p}\simeq*$, then ${\rm
map\,}_{*}(\operatornamewithlimits{hocolim}_{\mathcal{C}}F(c),Z)\simeq*$.

###### Proof.

The statement follows from a sequence of equivalences:

${\rm
map\,}(\operatornamewithlimits{hocolim}_{\mathcal{C}}F(c),Z)\simeq\operatornamewithlimits{holim}_{\mathcal{C}}{\rm
map\,}(F(c),Z)\simeq\operatornamewithlimits{holim}_{\mathcal{C}}{\rm
map\,}(P_{B{\mathbb{Z}}/p}(F(c)^{\wedge}_{p}),Z).$

Under the hypothesis of the lemma, this last mapping space is homotopy
equivalent to ${\rm map\,}(|\mathcal{C}|,Z)\simeq Z$ if
$|\mathcal{C}|^{\wedge}_{p}\simeq*$. ∎

######  Corollary 4.9.

Let $p$ be a prime. If $G$ is a compact Lie group and $X$ is a connected
$p$-complete $B{\mathbb{Z}}/p$-null space, then ${\rm
map\,}_{*}(BG^{\wedge}_{p},X)$ is weakly contractible.

###### Proof.

The proof is divided into two steps. In the first one we assume that $G$ is a
$p$-toral group, and then we use the existence of mod $p$ homology
decompositions of $BG$ with respect to certain families of $p$-toral subgroups
of $G$, see [JMO90], to undertake the general case.

Consider first when $G=T=(S^{1})^{n}$. In this case, $BT^{\wedge}_{p}\simeq
K({\mathbb{Z}}^{\wedge}_{p},2)^{n}\simeq(B({\mathbb{Z}}/{p^{\infty}})^{n})^{\wedge}_{p}$.
As $X$ is a $p$-complete space, we have the weak homotopy equivalence ${\rm
map\,}_{*}(BT^{\wedge}_{p},X)\simeq{\rm
map\,}_{*}(B({\mathbb{Z}}/p^{\infty})^{n},X)$. This mapping space is
contractible because ${\mathbb{Z}}/p^{\infty}\cong\lim{\mathbb{Z}}/p^{r}$
defined by inclusions ${\mathbb{Z}}/p^{n}\subset{\mathbb{Z}}/p^{n+1}$, and
since ${\mathbb{Z}}/p^{r}$ is a $p$-group, $B{\mathbb{Z}}/p^{r}$ is
$B{\mathbb{Z}}/p$-acyclic and ${\rm map\,}_{*}(B{\mathbb{Z}}/p^{r},X)\simeq*$,
and we can apply Lemma 4.8. Now, if $G=P$ is a $p$-toral group given by a
group extension $T^{n}\hookrightarrow P\mbox{$\to$$\to$}\pi$, Dwyer and
Wilkerson show that $BG$ admits a $p$-discrete approximation [DW94, Prop 6.9].
There is a sequence of finite $p$-groups $P_{0}\subset P_{1}\subset\ldots$
such that $BP\simeq\operatornamewithlimits{hocolim}BP_{n}$. Again, by Lemma
4.8, we obtain that ${\rm map\,}_{*}(BP,X)\simeq*$.

Let us go now through the general case. Our goal will be to prove that the
inclusion of constant maps induces an equivalence $X\simeq{\rm
map\,}(BG^{\wedge}_{p},X)$. By work of Jackowski-McClure-Oliver ([JMO90, Thm
4]), the space $BG$ is mod $p$ equivalent to
$\operatornamewithlimits{hocolim}_{\mathcal{O}_{p}G}F$, where
$\mathcal{O}_{p}G$ is the orbit category of stubborn $p$-toral subgroups of
$G$ and $F$ is a functor whose values have the homotopy type of classifying
spaces of stubborn $p$-toral subgroups of $G$. Since the statement holds for
$p$-toral groups, by Lemma 4.8 it is enough to observe that $\mathcal{O}_{p}G$
is $\mathbb{F}_{p}$-acyclic, see [JMO90, Prop 6.1], and we are done. ∎

Now we are ready to prove the desired result, which was previously known for
finite groups ([Flo07, Theorem 3.5]).

######  Theorem 4.10.

Let $G$ be a compact Lie group and $\pi$ its group of components. Let $G_{p}$
be the subgroup of $G$ whose group of components is $T_{p}(\pi)$. Then the
$B\mathbb{Z}/p$-nullification of $BG$ fits in the following covering
fibration:

$L_{\mathbb{Z}[1/p]}BG_{p}\longrightarrow P_{B\mathbb{Z}/p}BG\longrightarrow
B(\pi/\emph{T}_{p}\pi)$

###### Proof.

We have to check that the assumptions on Theorem 4.1 are satisfied when
$X=BG$. Since the universal cover of $BG$ is $BG_{0}$ is again the classifying
space of a compact Lie group, by Corollary 4.9 the hypothesis of Theorem 4.2
are satisfied. ∎

The proof of Corollary 4.9 applies to other type of spaces which admits mod
$p$ homology decompositions, for example $p$-compact groups (see [CLN07]). The
theory of $p$-local compact groups introduced by Broto, Levi and Oliver in
[BLO07] includes both the theory of $p$-compact groups [DW94] and $p$-local
finite groups [BLO03]. Roughly speaking, a $p$-local compact is a triple
$(S,\mathcal{F},\mathcal{L})$ where $S$ is a discrete $p$-toral group and
$\mathcal{F}$ and $\mathcal{L}$ are categories which model conjugacy relations
among subgroups of $S$. The classifying space of a $p$-local compact group is
$|\mathcal{L}|^{\wedge}_{p}$, and one of the main features of $p$-local
compact groups is that this space admits mod $p$-homology decompositions in
terms of classifying spaces of $p$-compact toral subgroups over mod $p$
acyclic orbit categories (see [BLO07, Proposition 4.6] and [BLO07, Corollary
5.6]).

######  Proposition 4.11.

Let $p$ be a prime, and $(S,\mathcal{F},\mathcal{L})$ a $p$-local compact
group. Then there is an equivalence
$L_{\mathbb{Z}[1/p]}(|\mathcal{L}|^{\wedge}_{p})\simeq{P}_{\emph{B}\mathbb{Z}/p}(|\mathcal{L}|^{\wedge}_{p})$.

###### Proof.

First of all, $\pi_{1}(|\mathcal{L}|^{\wedge}_{p})$ is a finite $p$-group by
[BLO07, Proposition 4.4], therefore
$T_{p}(\pi_{1}(|\mathcal{L}|^{\wedge}_{p}))=\pi_{1}(|\mathcal{L}|^{\wedge}_{p})$.
Then, we only need to check the hypothesis in Theorem 4.2. That is, ${\rm
map\,}_{*}(X\langle 1\rangle,Z)\simeq*$ for any connected
$B{\mathbb{Z}}/p$-null $p$-complete space $Z$.

The same argument used in the proof of Corollary 4.9 using mod $p$ homology
decompositions can be applied and it shows that ${\rm
map\,}_{*}(|\mathcal{L}|^{\wedge}_{p},Z)\simeq*$ for any connected
$B{\mathbb{Z}}/p$-null $p$-complete space $Z$. But it is not known in general
if the universal cover $|\mathcal{L}|^{\wedge}_{p}\langle 1\rangle$ is the
classifying space of a $p$-local compact group. Instead, we will check that
the proof of Corollary 4.9 applies by showing that
$|\mathcal{L}|^{\wedge}_{p}\langle 1\rangle$ admits a description, up to
$p$-completion, as a homotopy colimit of $B{\mathbb{Z}}/p$-acyclic spaces over
a mod $p$-acyclic category.

Let $P\leq S$ be an object in $\mathcal{O}(\mathcal{F}_{0})$ (see [BLO07,
Proposition 4.6]) and let $E_{P\leq S}$ be the pullback of
$|\mathcal{L}|^{\wedge}_{p}\langle
1\rangle\rightarrow|\mathcal{L}|^{\wedge}_{p}$ along $\tilde{B}P\rightarrow
BS$. Then, by naturality there is a map
$\operatornamewithlimits{hocolim}_{\mathcal{O}(\mathcal{F}_{0})}E_{P\leq
S}\rightarrow|\mathcal{L}|^{\wedge}_{p}\langle 1\rangle$ which fits in a
diagram of fibrations by Puppe’s theorem (e.g. [Far96, Appendix]),

$\textstyle{|\mathcal{L}|^{\wedge}_{p}\langle
1\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{|\mathcal{L}|^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\pi_{1}(|\mathcal{L}|^{\wedge}_{p})}$$\textstyle{\operatornamewithlimits{hocolim}_{\mathcal{O}(\mathcal{F}_{0})}E_{P\leq
S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatornamewithlimits{hocolim}_{\mathcal{O}(\mathcal{F}_{0})}\tilde{B}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\pi_{1}(|\mathcal{L}|^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{id}$

Since the middle vertical arrow is a mod $p$-equivalence, it follows that the
left vertical arrow is also a mod $p$-equivalence. Moreover,
$(\tilde{B}P)^{\wedge}_{p}$ is the classifying space of a $p$-compact toral
group, so it follows from the fibration $E_{P\leq
S}\rightarrow\tilde{B}P\rightarrow B\pi_{1}(|\mathcal{L}|^{\wedge}_{p})$ that
$(E_{P\leq S})^{\wedge}_{p}$ is also the classifying space of a $p$-compact
toral group, and therefore $B{\mathbb{Z}}/p$-acyclic. Then the proof of
Corollary 4.9 applies. ∎

A finite loop space is a triple $(X,BX,e)$ where $e\colon X\rightarrow\Omega
BX$ is a weak equivalence and $H^{*}(X;{\mathbb{Z}})$ is finite. Note that if
$X$ is a finite loop space, then $BX\langle 1\rangle^{\wedge}_{p}$ is the
classifying space of connected $p$-compact group since $H^{*}(\Omega BX\langle
1\rangle;{\mathbb{F}}_{p})$ is finite dimensional (see [DW94]).

######  Corollary 4.12.

Let $X$ be a finite loop space. Then there is a fibration

$L_{\mathbb{Z}[1/p]}BX_{p}\longrightarrow P_{B\mathbb{Z}/p}BX\longrightarrow
B(\pi/\emph{T}_{p}\pi)$

where $\pi=\pi_{0}(X)$ and $BX_{p}$ is the covering of $X$ whose fundamental
group is $T_{p}(\pi_{0}(X))$.

###### Proof.

Since $BX\langle 1\rangle^{\wedge}_{p}$ is the classifying space of a
$p$-compact group, Proposition 4.11 shows that $P_{B{\mathbb{Z}}/p}(BX\langle
1\rangle)^{\wedge}_{p}\simeq*$. Therefore we can apply Theorem 4.1. ∎

We finish with a somewhat two different examples.

######  Corollary 4.13.

Let $p$ be a prime. Let $K$ be a Kač-Moody group with a finite group of
components. Then there is a fibration

$L_{\mathbb{Z}[1/p]}BK_{p}\longrightarrow
P_{\emph{B}\mathbb{Z}/p}BK\longrightarrow\emph{B}(\pi_{1}(BK)/{T}_{p}(\pi_{1}(BK))$

where $K_{p}$ is the subgroup of $K$ whose group of components is
$T_{p}(\pi_{1}(BK)$

###### Proof.

First of all, the universal cover of $BK$ is $BK_{0}$ where $K_{0}$ is the
connected component of the unit in $K$. It is shown in Nitu Kitchloo thesis
(see [BK02]) that $BK$ is homotopy equivalent to a colimit over a contractible
category of classifying spaces of compact Lie groups. Then Corollary 4.9 and
its proof apply to show that ${\rm map\,}_{*}(BK,Z)\simeq*$ for any connected
$B{\mathbb{Z}}/p$-null $p$-complete space $Z$. ∎

A different kind of example arises from the theory of infinite loop spaces and
it is a consequence of Theorem $2$ in [McG].

######  Corollary 4.14.

Let $X$ be a connected infinite loop space with finite fundamental group. Then
there is a fibration

$L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})\rightarrow
P_{B{\mathbb{Z}}/p}(X)\rightarrow B(\pi_{1}(X)/T_{p}(\pi_{1}(X)))$

where $X_{p}$ is the covering of $X$ whose fundamental group is
$T_{p}(\pi_{1}(X))$.

##  5 Relation of nullification functors with other idempotent functors

In this section we compare the effect of nullification $P_{B{\mathbb{Z}}/p}$
on spaces which satisfy the hypothesis of Theorem 4.1 with the effect of some
completions or localizations on it. We analyse both functors that are supposed
to kill the $p$-torsion, like $L_{{\mathbb{Z}}[1/p]}$ or
${\mathbb{Z}}[1/p]_{\infty}$, and functors that usually preserve it, as
$L_{{\mathbb{Z}}[1/q]}$ and $p$-completion do.

######  Lemma 5.1.

Let $X$ be a connected space with finite fundamental group, $p$ and $q$
different primes. Then $(X^{\wedge}_{p})^{\wedge}_{q}$ is contractible.

###### Proof.

If $X$ is $1$-connected the case of $\mathbb{F}_{q}$-completion is described
in [BK72, VI.5.1]. If $X$ is not simply-connected, consider the fibration
$X^{\wedge}_{p}\langle 1\rangle\rightarrow X^{\wedge}_{p}\rightarrow
B\pi_{1}(X^{\wedge}_{p})$ and its fibrewise $q$-completion,

$(X^{\wedge}_{p}\langle 1\rangle)^{\wedge}_{q}\rightarrow Y\rightarrow
B\pi_{1}(X^{\wedge}_{p}).$

Since the fibre is a $1$-connected $p$-complete space completed at $q$ it is
contractible, then $Y\simeq B\pi_{1}(X^{\wedge}_{p})$. But then
$(X^{\wedge}_{p})^{\wedge}_{q}\simeq Y^{\wedge}_{q}\simeq
B\pi_{1}(X^{\wedge}_{p})^{\wedge}_{q}$ which is contractible since
$B\pi_{1}(X^{\wedge}_{p})$ is the classifying space of a finite $p$-group. ∎

We start by showing some direct direct consequences of Theorem 4.1.

######  Proposition 5.2.

Let $X$ be a space which satisfies the hypothesis of Theorem 4.1. Then
$\pi_{1}(P_{B{\mathbb{Z}}/p}(X))=\pi_{1}(X)/T_{p}(\pi_{1}(X))$ and
$(P_{B{\mathbb{Z}}/p}(X))^{\wedge}_{p}\simeq{*}$. Moreover
$P_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})\simeq
L_{{\mathbb{Z}}[\frac{1}{p}]}(X^{\wedge}_{p})\simeq(X^{\wedge}_{p})_{\mathbb{Q}}$
is $1$-connected.

###### Proof.

Since $L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})$ is $1$-connected by Lemma 4.3, it
is clear from the fibration in Theorem 4.1 that
$\pi_{1}(P_{B{\mathbb{Z}}/p}(X))=\pi_{1}(X)/T_{p}(\pi_{1}(X))$. The space
$L_{{\mathbb{Z}}[\frac{1}{p}]}(X_{p})$ is mod $p$ acyclic by [Dwy96, Lemma
6.2] and the order of $\pi_{1}(X)/T_{p}(\pi_{1}(X))$ is prime to $p$, it
follows that $(P_{B{\mathbb{Z}}/p}(X))^{\wedge}_{p}$ is weakly contractible.

The second statement follows from applying Theorem 4.1 to $X^{\wedge}_{p}$.
Observe that if $X$ satisfies the hypothesis of the theorem, then
$X^{\wedge}_{p}$ also does. Moreover, $\pi_{1}(X^{\wedge}_{p})$ is a finite
$p$-group, then $P_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})\simeq
L_{{\mathbb{Z}}[\frac{1}{p}]}(X^{\wedge}_{p})$. It remains to prove that they
are equivalent to $(X^{\wedge}_{p})_{\mathbb{Q}}$. Since they are
$1$-connected we can apply Sullivan’s arithmetic square.

We have proved that $(P_{B{\mathbb{Z}}/p}(X^{\wedge}_{p}))^{\wedge}_{p}$ is
weakly contractible. Moreover, if $q\neq p$ then
$(P_{B{\mathbb{Z}}/p}(X^{\wedge}_{p}))^{\wedge}_{q}\simeq(X^{\wedge}_{p})^{\wedge}_{q}$
which is weakly contractible by Lemma 5.1. Then
$P_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})\simeq
P_{B{\mathbb{Z}}/p}(X^{\wedge}_{p})_{\mathbb{Q}}\simeq(X^{\wedge}_{p})_{\mathbb{Q}}$.
∎

We start by showing that $B{\mathbb{Z}}/p$-nullification and $p$-completion
behave like opposite functors in this context.

######  Remark 5.3.

If we complete in one prime $q$ and $B{\mathbb{Z}}/p$-nullify with regard to a
different prime $p$, then $X^{\wedge}_{q}$ is $B{\mathbb{Z}}/p$-null and the
coaugmentation $X\rightarrow P_{B{\mathbb{Z}}/p}X$ is an equivalence after
$q$-completion if $X$ satisfies the hypothesis of Lemma 3.9.

######  Remark 5.4.

Note that in general a connected space $X$ could be ${\mathbb{Z}}[1/p]$-bad if
$X$ is not $1$-connected, and then it is not possible in general to replace
completion by localization in the previous results. If we know in advance that
$X$ is ${\mathbb{Z}}[1/p]$-good (this happens, for example, if its fundamental
group is ${\mathbb{Z}}[1/p]$-perfect) then we can do the replacement, and
moreover ${\mathbb{Z}}[1/p]_{\infty}X\simeq L_{{\mathbb{Z}}[1/p]}X$. See for
example ([Far96, 1.E]) for more information about the relation between
$R$-localization and $R$-completion.

######  Proposition 5.5.

Let $X$ be a space which satisfies the hypothesis of Theorem 4.1 and such that
$\pi_{1}(X)\cong T_{q}(\pi_{1}(X))$. Then there are homotopy equivalences

$P_{B\mathbb{Z}/p}L_{\mathbb{Z}[1/q]}X\simeq L_{\mathbb{Z}[1/p,1/q]}X\simeq
L_{\mathbb{Z}[1/p]}P_{B\mathbb{Z}/q}X.$

###### Proof.

Since $\pi_{1}(X)\cong T_{q}(\pi_{1}(X))$,
$L_{{\mathbb{Z}}[\frac{1}{q}]}(X)\simeq P_{B{\mathbb{Z}}/q}(X)$ is
$1$-connected. Then, by Theorem 4.1, we have
$P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X))\simeq
P_{B{\mathbb{Z}}/p}(L_{{\mathbb{Z}}[\frac{1}{q}]}(X))\simeq
L_{{\mathbb{Z}}[\frac{1}{p}]}(L_{{\mathbb{Z}}[\frac{1}{q}]}(X))\simeq
L_{{\mathbb{Z}}[\frac{1}{p}]}(P_{B{\mathbb{Z}}/q}(X))$. ∎

We finish by establishing the commutativity of the functors
$P_{B{\mathbb{Z}}/p}$ and $P_{B{\mathbb{Z}}/q}$. The problem of commutation of
localization functors was extensively studied in [RS00].

######  Proposition 5.6.

Let $X$ be a connected space, $p$ and $q$ two different primes. Assume that
$X$ satisfies the hypothesis of Theorem 4.1 for both primes $p$ and $q$. Then
there are homotopy equivalences

$P_{B\mathbb{Z}/p}P_{B\mathbb{Z}/q}X\simeq P_{B\mathbb{Z}/p\vee
B\mathbb{Z}/q}X\simeq P_{B\mathbb{Z}/q}P_{B\mathbb{Z}/p}X.$

###### Proof.

It is enough to show the first equivalence since the other one will follow by
symmetry.

Consider the set of primes $S=\\{p,q\\}$. By pulling back the universal
fibration, there is a fibration $X_{S}\rightarrow X\rightarrow
B(\pi_{1}(X)/(T_{S}(\pi_{1}(X)))$, where $\pi_{1}(X_{S})=T_{S}(\pi_{1}(X))$.
Since the order of $\pi_{1}(X)/T_{S}(\pi_{1}(X))$ is prime to both $p$ and
$q$, the space $B(\pi_{1}(X)/(T_{S}(\pi_{1}(X)))$ is both
$B{\mathbb{Z}}/p$-null and $B{\mathbb{Z}}/q$-null (in particular it is also
$B{\mathbb{Z}}/p\vee B{\mathbb{Z}}/q$-null) the composite of functors
$P_{B{\mathbb{Z}}/p}\circ P_{B{\mathbb{Z}}/q}$ and $P_{B{\mathbb{Z}}/p\vee
B{\mathbb{Z}}/q}$ preserve the fibration [Far96, 3.D.3], and there is a
diagram of fibrations

$\textstyle{P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X_{S}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(\pi_{1}(X)/(T_{S}(\pi_{1}(X)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{P_{B{\mathbb{Z}}/p\vee
B{\mathbb{Z}}/q}(X_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{B{\mathbb{Z}}/p\vee
B{\mathbb{Z}}/q}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(\pi_{1}(X)/(T_{S}(\pi_{1}(X))).}$

where the first two vertical maps exist because if a space $Y$ is
$B{\mathbb{Z}}/p\vee B{\mathbb{Z}}/q$-null, and then it is also
$B{\mathbb{Z}}/p$-null and $B{\mathbb{Z}}/q$-null. Then we can assume that the
group $\pi_{1}(X)=T_{S}(\pi_{1}(X))$, which we simply denote by $\pi$ in the
sequel, is generated by $p$ and $q$ torsion.

By [RS00, Prop 1.1], we need to show that
$P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X))$ is $B{\mathbb{Z}}/q$-null and
conversely $P_{B{\mathbb{Z}}/q}(P_{B{\mathbb{Z}}/p}(X))$ is
$B{\mathbb{Z}}/p$-null. In our situation, by symmetry, it is enough to check
one of the two conditions.

Let us see first that $P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X))$ is
$1$-connected. We can apply the fibrewise $B{\mathbb{Z}}/p$-nullification to
the fibration in Theorem 4.1,

$\textstyle{L_{{\mathbb{Z}}[\frac{1}{q}]}(X_{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{B{\mathbb{Z}}/q}(X_{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(\pi/T_{q}(\pi))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{P_{B{\mathbb{Z}}/p}(L_{{\mathbb{Z}}[\frac{1}{q}]}(X_{q}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bar{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(\pi/T_{q}(\pi)),}$

where $L_{{\mathbb{Z}}[\frac{1}{q}]}(X_{q})$ is $1$-connected and
$P_{B{\mathbb{Z}}/p}(\bar{P})\simeq
P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X_{q}))$;ˇ then $\bar{P}$ has
fundamental group $\pi/T_{q}(\pi)$ which is generated by $p$ torsion. If
$\bar{P}$ satisfies the hypothesis of Theorem 4.1 for the prime $p$ then
$P_{B{\mathbb{Z}}/p}(\bar{P})\simeq L_{{\mathbb{Z}}[\frac{1}{p}]}(\bar{P})$ is
$1$-connected. We need to check that for any connected space $Z$ which is
$p$-complete and $B{\mathbb{Z}}/p$-null, ${\rm map\,}_{*}(\bar{P}\langle
1\rangle,Z)$ is weakly contractible. Note that $\bar{P}\langle 1\rangle\simeq
P_{B{\mathbb{Z}}/p}(L_{{\mathbb{Z}}[\frac{1}{q}]}(X_{q}))$, and then

$\textrm{map}_{*}(P_{B{\mathbb{Z}}/p}(L_{{\mathbb{Z}}[\frac{1}{q}]}(X_{q})),Z)\simeq\textrm{map}_{*}(L_{{\mathbb{Z}}[\frac{1}{q}]}(X_{q}),Z)\simeq\textrm{map}_{*}(P_{B{\mathbb{Z}}/q}(X_{q}),Z)\simeq$
$\simeq\textrm{map}_{*}(X_{q},Z),$

where the last equivalence follows because
$(P_{B{\mathbb{Z}}/q}(X_{q}))^{\wedge}_{p}\simeq(X_{q})^{\wedge}_{p}$ and $Z$
is $p$-complete. Finally Lemma 4.6 tells us that this last mapping space is
weakly contractible.

We denote by $Y$ the space $P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X))$, and
we finally should check that it is $B{\mathbb{Z}}/q$-null. Since it is a
$1$-connected space, we can use Sullivan’s arithmetic square and check that
the mapping spaces ${\rm map\,}_{*}(B{\mathbb{Z}}/q,Y_{\mathbb{Q}})$ and ${\rm
map\,}_{*}(B{\mathbb{Z}}/q,Y^{\wedge}_{r})$ are weakly contractible for any
prime $r$.

If $r\neq q$, ${\rm map\,}_{*}(B{\mathbb{Z}}/q,Y^{\wedge}_{r})\simeq*$ because
$(B{\mathbb{Z}}/q)^{\wedge}_{r}\simeq*$. Also, since
$(B{\mathbb{Z}}/q)_{\mathbb{Q}}\simeq*$, ${\rm
map\,}_{*}(B{\mathbb{Z}}/q,Y_{\mathbb{Q}})\simeq*$. We are left to the case
$r=q$, and
$Y^{\wedge}_{q}=(P_{B{\mathbb{Z}}/p}(P_{B{\mathbb{Z}}/q}(X)))^{\wedge}_{q}\simeq(P_{B{\mathbb{Z}}/q}(X))^{\wedge}_{q}\simeq*$
by Proposition 5.2. So we are done. ∎

For example, given a compact Lie group, $BG$ satisfies the hypothesis of
Theorem 4.1 for any prime $p$.

######  Remark 5.7.

The same proof remains valid if we apply in succession over $X$ a finite
number of $B{\mathbb{Z}}/p$-nullification functors for different primes
assuming $X$ satisifes the hypothesis of Theorem 4.1 for each prime. On the
other hand, it is likely that that the nullification of $X$ with regard to the
wedge of the classifying spaces of _all_ primes is homotopy equivalent to the
rational localization of $X$. See [Flo07, Section 3.2] for details.

##  6 $B{\mathbb{Z}}/p$-cellularization of classifying spaces

In this section we will give a Serre-type general dichotomy theorem (Theorem
6.1), which is very much in the spirit of [FS07]. Then, we will use this
statement to describe several examples concerning the
$B{\mathbb{Z}}/p$-cellularization of some families of classifying spaces of
remarkable groups, such as $p$-toral groups, finite groups with a $p$-subgroup
of $p^{\prime}$-index, $BS^{3}$ or $BSO(3)$ (at the prime $2$). Our
considerations are also based in the results of the previous sections relating
cellularization, nullification and completion.

###  6.1 The dichotomy theorem

The main result of this section is:

######  Theorem 6.1.

Let $X$ be a connected nilpotent $\Sigma^{n}B{\mathbb{Z}}/p$-null space for
some $n\geq 0$. Then the $B{\mathbb{Z}}/p$-cellullarization of $X$ has the
homotopy type of a Postnikov piece with homotopy groups are concentrated in
degrees $1$ to $n$ , or else it has infinitely many nonzero homotopy groups.
Moreover, if $X$ is $1$-connected of finite type, then the fundamental group
$\pi_{1}(CW_{B{\mathbb{Z}}/p}(X))$ is a finite elementary abelian $p$-group.

The rest of the subsection is devoted to the proof of Theorem 6.1. Even if the
statement is similar to the one in [FS07, Proposition 2.3], the authors deal
with the situation in which the space is torsion, and this is not the case for
$BG$ where $G$ is a compact connected Lie group. The strategy used in [FS07]
for classifying spaces of finite groups can be summarized as follows.

######  Proposition 6.2.

Let $X$ be a torsion Postnikov piece whose fundamental group is generated by
elements of order $p$ which lift to $X$. Assume there exists a prime $q\neq p$
such that $X^{\wedge}_{q}$ is torsion and it has infinitely many non-trivial
homotopy groups. Then $CW_{B{\mathbb{Z}}/p}(X)$ also has infinitely many non-
trivial homotopy groups.

###### Proof.

Consider the fibration $CW_{B{\mathbb{Z}}/p}(X)\rightarrow X\rightarrow
P_{\Sigma B{\mathbb{Z}}/p}(C)$ from Theorem 2.3. Note that $P_{\Sigma
B{\mathbb{Z}}/p}(C)$ is $1$-connected since $C$ is so. To prove the statement,
we will show that $P_{\Sigma B{\mathbb{Z}}/p}(C)$ has infinitely many non-
trivial homotopy groups. We apply Sullivan’s arithmetic square to $P_{\Sigma
B{\mathbb{Z}}/p}(C)$ to obtain a pullback diagram

$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\prod_{r\neq
p}X^{\wedge}_{r})\times(P_{\Sigma
B{\mathbb{Z}}/p}(C))^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$*$\textstyle{(\prod_{r\neq
q,p}X^{\wedge}_{r})_{\mathbb{Q}}\times((P_{\Sigma
B{\mathbb{Z}}/p}(C))^{\wedge}_{p})_{\mathbb{Q}}.}$

which allow us to construct a map $s\colon X^{\wedge}_{q}\rightarrow P_{\Sigma
B{\mathbb{Z}}/p}(C)$ such that $\eta\circ s\simeq id$. That is $s$ is a
section of the $q$-completion. Then for $n\geq 2$ we have that
$\pi_{n}(X^{\wedge}_{q})$ is a direct summand of $\pi_{n}(P_{\Sigma
B{\mathbb{Z}}/p}(C))$. ∎

For example, by Levi’s work in [Lev95], the previous theorem applies when $X$
is the classifying space of a finite group.

Now we need to state some general results concerning to the cellularization of
$\Sigma B{\mathbb{Z}}/p$-null spaces, that deal with the consequences of
imposing that $CW_{B{\mathbb{Z}}/p}(X)$ is a Postnikov piece for a certain
space $X$. Note that this is the “forbidden” case in Theorem 6.1.

######  Lemma 6.3.

Let $P[n]$ be a connected $\Sigma B{\mathbb{Z}}/p$-null Postnikov piece with
$n\geq 3$ then $p$ is invertible in $\pi_{n}(P[n])$.

###### Proof.

Note that if $P[n]$ is $\Sigma B{\mathbb{Z}}/p$-null then $\Omega P[n]$ is
$B{\mathbb{Z}}/p$-null, and also $\Omega^{n-1}P[n]$ is so. Since the connected
component of the constant in $\Omega^{n-1}P[n]$ is an Eilenberg-MacLane space
$K(\pi_{n}(P[n]),1)$, we see that $K(\pi_{n}(P[n]),1)$ is also
$B{\mathbb{Z}}/p$-null.

Similarly, the connected component $E$ of $\Omega^{n-2}P[n]$ is
$B{\mathbb{Z}}/p$-null. There is a fibration $K(\pi_{n}(P[n]),2)\rightarrow
E\rightarrow K(\pi_{n-1}(P[n]),1)$. Since the pointed mapping spaces ${\rm
map\,}_{*}(B{\mathbb{Z}}/p,K(\pi_{n-1}(P[n]),1))_{c}$ and ${\rm
map\,}_{*}(B{\mathbb{Z}}/p,E)$ are weakly contractible, we obtain from the
previous fibration that ${\rm map\,}(B{\mathbb{Z}}/p,K(\pi_{n}(P[n]),2))$ is
also weakly contractible. That is $K(\pi_{n}(P[n]),2)$ is
$B{\mathbb{Z}}/p$-null. The conclusion now follows from Lemma 2.7. ∎

######  Proposition 6.4.

Let $X$ be a connected nilpotent $\Sigma B{\mathbb{Z}}/p$-null space such that
${\mathbb{Z}}[\frac{1}{p}]_{\infty}(X)\simeq*$. Then $X$ has the homotopy type
of a $K(G,1)$ or it has infinitely many nonzero homotopy groups.

###### Proof.

Assume that $X\simeq P[n]$ is a finite Postnikov piece. First we show that
$n\leq 2$. Since ${\mathbb{Z}}[\frac{1}{p}]_{\infty}(X)$ is weakly
contractible (see Lemma 2.8), then
$\pi_{i}(P[n])\otimes{\mathbb{Z}}[\frac{1}{p}]=0$ for all $i>0$ ([BK72,
V.4.1]). But if $n\geq 3$, $p$ is invertible in $\pi_{n}(P[n])$ by the
previous Proposition 6.3, then $\pi_{n}(P[n])=0$. We can apply this argument
as long as $n\geq 3$.

From now on we assume $n=2$. Next we prove that $\pi_{2}(P[2])$ has no
$p$-torsion. There is a fibration $K(\pi_{2}(P[2]),2)\rightarrow
P[2]\rightarrow K(\pi_{1}(P[2]),1)$, which induces a covering
$K(\pi_{1}(P[2]),0)\rightarrow K(\pi_{2}(P[2]),2)\rightarrow P[2]$. If
$\pi_{2}(P[2])$ had $p$-torsion, then a non-trivial homomorphism
${\mathbb{Z}}/p\rightarrow\pi_{2}(P[2])$ would induce a nontrivial map
$f\colon\Sigma B{\mathbb{Z}}/p\rightarrow K(\pi_{2}(P[2]),2)$ which is
nullhomotopic when composed with $K(\pi_{2}(P[2]),2)\rightarrow P[2]$ since
$P[2]$ is $\Sigma B{\mathbb{Z}}/p$-null (see Lemma 2.6). Then we obtain a
contradiction since $f$ must be nullhomotopic. So $\pi_{2}(P[2])$ has no
$p$-torsion, and this is again a contradiction since
$\pi_{2}(P[2])\otimes{\mathbb{Z}}[\frac{1}{p}]=0$. ∎

We now state our dichotomy theorem for nilpotent $\Sigma B{\mathbb{Z}}/p$-null
spaces.

######  Theorem 6.5.

Let $X$ be a connected nilpotent $\Sigma B{\mathbb{Z}}/p$-null space. Then the
$B{\mathbb{Z}}/p$-cellulla-rization of $X$ has the homotopy type of a $K(G,1)$
or it has infinitely many nontrivial homotopy groups. Moreover, if $X$ is
$1$-connected of finite type, then $\pi_{1}(CW_{B{\mathbb{Z}}/p}(X))$ is a
finite elementary abelian $p$-group.

###### Proof.

By Lemma 2.5, $CW_{B{\mathbb{Z}}/p}(X)$ is also nilpotent. Moreover, by Lemma
2.8 we have an equivalence
${\mathbb{Z}}[\frac{1}{p}]_{\infty}(CW_{B{\mathbb{Z}}/p}(X))\simeq*$ , and
then we can apply Proposition 6.4 to $CW_{B{\mathbb{Z}}/p}(X)$. ∎

###### Proof of Theorem 6.1.

Assume that $n\geq 2$, we can assume that $X$ is $B{\mathbb{Z}}/p$-cellular
and $\Sigma^{n}B{\mathbb{Z}}/p$-null. By [Bou94, Theorem 7.2], there is a
principal fibration

$K(P,n)\rightarrow X\rightarrow P_{\Sigma^{n-1}}(X)$

where $P$ is a $p$-torsion grup. Since $K(P,n)$ is $B{\mathbb{Z}}/p$-cellular
([CCS07-2, Lemma 3.3]) and $X$ too by assumption, then $P_{\Sigma^{n-1}}(X)$
is also $B{\mathbb{Z}}/p$-cellular by [Cha96, Theorem 4.7]. By induction we
reduce to the case in which $n=1$, which is proved in Theorem 6.5. ∎

The next question we need to refer concerning Theorem 6.1 is when the
cellularization of a classifying space is again a classifying space, not
necessarily of a discrete group. This is important to understand the first
part of the previous dichotomy.

######  Proposition 6.6.

Let $X$ be a space. If $CW_{B{\mathbb{Z}}/p}(X)\simeq BH$ for some compact Lie
group $H$, then it must be a finite $p$-group generated by order $p$ elements.

###### Proof.

Since the pointed homotopy colimit of acyclics is acyclic for any cohomology
theory ([Far96, 2.D.2.5]), it is clear that
$\tilde{H}^{*}(BH;{\mathbb{Q}})=\tilde{H}^{*}(CW_{B{\mathbb{Z}}/p}X;\mathbb{Q})=0$.
On the other hand, it is well-known that the rational cohomology of $BH$ are
the invariants of the rational cohomology of the classifying space of the
maximal torus $T$ under the action of the Weyl group $W$. In fact
$\tilde{H}^{*}(BH;{\mathbb{Q}})=0$ if and only if $H$ is a finite group.
Finally, the functor $CW$ is idempotent, so $BH$ must be
$B{\mathbb{Z}}/p$-cellular. Thus, we can apply [Flo07, Prop 4.14 ] to finish
the proof. ∎

######  Remark 6.7.

The arguments in Proposition 6.6 also work if
$CW_{B{\mathbb{Z}}/p}(X)\simeq(BH)^{\wedge}_{p}$ where $H$ is a compact Lie
group. It is clear then that
$\tilde{H}^{*}(BH;{\mathbb{Z}}^{\wedge}_{p})\otimes{\mathbb{Q}}=0$. But again
this is only possible if $H$ is discrete. If $H$ is in particular finite,
conditions are known (see [FS07, Corollary 3.3]) under which
$(BH)^{\wedge}_{p}$ is $B{\mathbb{Z}}/p$-cellular. See Example 6.14 below.

######  Remark 6.8.

When $X$ is an $H$-space satisfying the hypothesis of Theorem 6.1, Castellana,
Crespo and Scherer proved in [CCS07-2] that the
$B{\mathbb{Z}}/p$-cellularization of $X$ is always a Postnikov piece. Examples
of such spaces are given by $H$-spaces whose mod $p$ cohomology is finitely
generated as an algebra over the Steenrod algebra (see [CCS07-1]).

###  6.2 Examples

In this subsection we concentrate in the description of the
$B{\mathbb{Z}}/p$-cellularization of classifying spaces of compact Lie groups,
generalizing to the continuous case work of the second author in the finite
case ([Flo07] and [FS07]). In the study of the homotopy type of classifying
spaces of Lie groups, a very useful strategy is to isolate the information at
every prime.

Theorem 6.1 implies automatically the following dichotomy theorem for
classifying spaces of compact Lie groups. We say that an element $g\in G$ is
$p$-cohomologically central if the map induced by the inclusion
$BC_{G}(x)\rightarrow BG$ is a mod $p$ homology isomorphism. Mislin in [Mis92]
shows that there is a natural bijection between the set of conjugacy classes
of $p$-cohomologically central elements of order $p$ in $G$ with
$pZ(G/O_{p^{\prime}}(G))$ where $O_{p^{\prime}}(G)$ is the largest normal
$p^{\prime}$-subgroup of $G$ and $pZ(G)$ are the elements of order $p$ in
$Z(G)$.

######  Theorem 6.9.

Let $G$ be a compact connected Lie group. If there exists a non
$p$-cohomologically central element of order $p$, then the
$B{\mathbb{Z}}/p$-cellullarization of $BG$ has infinitely many nonzero
homotopy groups. Otherwise, it has the homotopy type of a $K(V,1)$, where $V$
is a finite elementary abelian $p$-group.

###### Proof.

Since $G$ is assumed to be connected, $BG$ is simply connected. Moreover it is
of finite type, and $\Sigma B{\mathbb{Z}}/p$-null because of Miller’s solution
of the Sullivan conjecture because $\Omega BG\simeq G$ is a finite complex.
Now, we apply Theorem 6.1. If $CW_{BZ/p}(BG)$ is an Eilenberg-MacLane space
$K(V,1)$, then ${\rm map\,}_{*}(B{\mathbb{Z}}/p,BG)$ is homotopically
discrete. Since $BG$ is simply connected,
$[B{\mathbb{Z}}/p,BG]_{*}\cong[B{\mathbb{Z}}/p,BG]\cong{\rm
Rep\,}({\mathbb{Z}}/p,G)$. If ${\rm map\,}_{*}(B{\mathbb{Z}}/p,BG)$ is
homotopically discrete, then for each $\rho\in{\rm Rep\,}({\mathbb{Z}}/p,G)$,
the evaluation ${\rm map\,}(B{\mathbb{Z}}/p,BG)_{B\rho}\rightarrow BG$ induces
an ${\mathbb{F}}_{p}$-homology equivalence
$BC_{G}(\rho({\mathbb{Z}}/p))\rightarrow BG$ by [DZ87]. And this only happens
if all the elements of order $p$ are $p$-cohomologically central. ∎

In the continuous case, there are paradigmatic examples of $BG$ whose
cellularization is again a classifying space.

######  Example 6.10.

If $X=BS^{1}=K({\mathbb{Z}},2)$, it is clear comparing pointed mapping spaces
that $CW_{B{\mathbb{Z}}/p}BS^{1}=B{\mathbb{Z}}/p$ since ${\rm
map\,}_{*}(B{\mathbb{Z}}/p,BS^{1})$ is homotopically discrete with components
${\rm Hom\,}({\mathbb{Z}}/p,S^{1})$. Let us now consider $BS^{3}$ the
classifying space of the $3$-sphere. Lemma 3.2 reduces the computation of
$CW_{B{\mathbb{Z}}/p}(BS^{3})$ to that of
$CW_{B{\mathbb{Z}}/p}((BS^{3})^{\wedge}_{p})$. The mapping space from
$B{\mathbb{Z}}/p$ into $(BS^{3})^{\wedge}_{p}$ has been well studied. If
$p=2$, the inclusion of the centre $B{\mathbb{Z}}/2\rightarrow BS^{3}$ induces
a homotopy equivalence ${\rm
map\,}(B{\mathbb{Z}}/2,B{\mathbb{Z}}/2)\rightarrow{\rm
map\,}(B{\mathbb{Z}}/2,(BS^{3})^{\wedge}_{2})$ since ${\rm
map\,}(B{\mathbb{Z}}/2,(BS^{3})^{\wedge}_{2})_{f}\simeq(BC_{S^{3}}(f))^{\wedge}_{2}$
(see [DMW87]), and therefore $CW_{B{\mathbb{Z}}/2}(BS^{3})\simeq
B{\mathbb{Z}}/2$. If $p$ is odd, then $(BS^{3})^{\wedge}_{p}\simeq
BN(T)^{\wedge}_{p}$, and this case will be studied in Example 6.15.

Sometimes, if we are unable to describe $CW_{B{\mathbb{Z}}/p}BG$, we can at
least identify it with another classifying space at a prime.

######  Example 6.11.

Let $BO(2)$ be the classifying space of the orthogonal group $O(2)$. There is
a mod $2$ equivalence $BD_{2^{\infty}}\rightarrow BO(2)$ where
$D_{2^{\infty}}=colim_{n}D_{2^{n}}$. Moreover $BD_{2^{\infty}}$ is
$B{\mathbb{Z}}/2$-cellular by [Flo07, Example 5.1]. Since
$\pi_{1}(BO(2))={\mathbb{Z}}/2$ is generated by an element of order $2$ which
lifts to $BO(2)$, we are in the situation of Remark 3.17. This will be used in
particular in Proposition 6.17.

We devote the remaining of the section to study some families of Lie groups
which show different and interesting features in this context. We begin with
extensions of elementary abelian $p$-groups by a finite group of order prime
to $p$, which provide an example in which Proposition 3.14 does not hold.
Compare with [FS07]. We start with a situation which deals with fibrations.

######  Proposition 6.12.

Let $F\rightarrow E\rightarrow B$ be a fibration of $p$-good connected spaces
such that $F$ is $B{\mathbb{Z}}/p$-cellular, $B$ is $B{\mathbb{Z}}/p$-null
and, $B^{\wedge}_{p}$ is $\Sigma B{\mathbb{Z}}/p$-null. Assume that
$[B{\mathbb{Z}}/p,E]\rightarrow[B{\mathbb{Z}}/p,E^{\wedge}_{p}]$ is exhaustive
and $\pi_{1}(F)\rightarrow\pi_{1}(E^{\wedge}_{p})$ is an epimorphism. Then
$(CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p}))^{\wedge}_{p}$ is the homotopy fiber of
$E^{\wedge}_{p}\rightarrow B^{\wedge}_{p}$.

###### Proof.

First of all, note that since $B$ is $B{\mathbb{Z}}/p$-null, then
$F\rightarrow E$ is a $B{\mathbb{Z}}/p$-equivalence, and thus $F\simeq
CW_{B{\mathbb{Z}}/p}(E)$. To compute the cellularization of $E^{\wedge}_{p}$
we proceed by applying Chachólski’s strategy (Theorem 2.3, see also [Cha96,
Section 7] for the slightly general formulation we use here). Consider the
following diagram of horizontal cofibrations,

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{E^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D}$

where $D$ is $1$-connected since $i$ is an epimorphism on fundamental groups.
Since $E$ is $p$-good, $g$ induces an homotopy equivalence
$C^{\wedge}_{p}\simeq D^{\wedge}_{p}$, and therefore $C^{\wedge}_{p}$ is also
$1$-connected.

Zabrodsky’s Lemma (see [Dwy96, Prop 3.4]) applied to the fibration
$F\rightarrow E\rightarrow B$ and the composite map $E\rightarrow C\rightarrow
P_{\Sigma B{\mathbb{Z}}/p}(C)$ implies that there is a map $B\rightarrow
P_{\Sigma B{\mathbb{Z}}/p}(C)$ which fits in a diagram of fibrations:

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{CW_{B{\mathbb{Z}}/p}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}C.}$

where the first vertical map is a homotopy equivalence,
$CW_{B{\mathbb{Z}}/p}(E)\simeq F$. The long exact sequence for homotopy groups
shows that the last vertical arrow is also a homotopy equivalence. Now
consider the diagram of fibre sequences

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}C\simeq
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}D.}$

The spaces $P_{\Sigma B{\mathbb{Z}}/p}(C)\simeq B$ and $P_{\Sigma
B{\mathbb{Z}}/p}(C^{\wedge}_{p})$ are $p$-good spaces (note that $P_{\Sigma
B{\mathbb{Z}}/p}(C^{\wedge}_{p})$ is $1$-connected by Lemma 2.4) and Miller’s
theorem apply to show that $P_{\Sigma
B{\mathbb{Z}}/p}(C^{\wedge}_{p})^{\wedge}_{p}$ is $\Sigma
B{\mathbb{Z}}/p$-null. Also $P_{\Sigma B{\mathbb{Z}}/p}(C)^{\wedge}_{p}\simeq
B^{\wedge}_{p}$ is $\Sigma B{\mathbb{Z}}/p$-null by hypothesis. Applying Lemma
3.9 and the proof of Corollary 3.12, we obtain that the composite $P_{\Sigma
B{\mathbb{Z}}/p}(C)\simeq B\rightarrow P_{\Sigma B{\mathbb{Z}}/p}(D)$ is a mod
$p$ equivalence, and therefore we conclude that the $p$-completion
$(CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p}))^{\wedge}_{p}$ is the homotopy fiber of
$E^{\wedge}_{p}\rightarrow B^{\wedge}_{p}$ is a mod $p$ equivalence by
$p$-completion of the fibration
$CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p})\rightarrow E^{\wedge}_{p}\rightarrow
P_{\Sigma B{\mathbb{Z}}/p}(D)$. ∎

######  Corollary 6.13.

Let $F\rightarrow E\rightarrow B$ be a fibration of connected spaces such that
$F$ is $B{\mathbb{Z}}/p$-cellular, $B$ is $B{\mathbb{Z}}/p$-null and
$B^{\wedge}_{p}\simeq*$. Assume that
$[B{\mathbb{Z}}/p,E]\rightarrow[B{\mathbb{Z}}/p,E^{\wedge}_{p}]$ is
exhaustive, $\pi_{1}(F)\rightarrow\pi_{1}(E^{\wedge}_{p})$ is an epimorphism
and $\pi_{i}(E)$ are finite groups for all $i\geq 1$. Then $E^{\wedge}_{p}$ is
$B{\mathbb{Z}}/p$-cellular.

###### Proof.

By Proposition 6.12, we know that
$(CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p}))^{\wedge}_{p}\simeq E^{\wedge}_{p}$. We
will prove that $CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p})$ is $p$-complete. Since
$\pi_{i}(E)$ are finite groups for all $i\geq 1$, $\pi_{i}(E^{\wedge}_{p})$
are all finite $p$-groups and $E^{\wedge}_{p}$ is nilpotent ([BK72, VII.4.3]).
Therefore $CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p})$ is nilpotent by Lemma 2.5. A
Sullivan’s arithmetic square argument shows that
$CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p})$ is $p$-complete since
$((CW_{B{\mathbb{Z}}/p}(E^{\wedge}_{p}))^{\wedge}_{p})_{\mathbb{Q}}\simeq(E^{\wedge}_{p})_{\mathbb{Q}}\simeq*$.
∎

######  Example 6.14.

Let $G$ be a finite group which is an extension $H\rightarrow G\rightarrow W$
where $BH$ is $B{\mathbb{Z}}/p$-cellular and $(|W|,p)=1$. Then
$CW_{B{\mathbb{Z}}/p}(BG)\simeq BH$ and $BG^{\wedge}_{p}$ is
$B{\mathbb{Z}}/p$-cellular by the previous result. Note that $G$ does not need
to be generated by elements of order $p$; compare with [FS07, Section 4].
Other examples are provided by nilpotent Postnikov pieces whose fundamental
group is of order prime to $p$ and the $1$-connected cover is $p$-torsion.

######  Example 6.15.

Let $N$ be an extension of a finite group of order prime to $p$ with a torus,
that is, we have a fibration $BT\rightarrow BN\rightarrow BW$ where
$T\cong(S^{1})^{n}$ and $(|W|,p)=1$. From this fibration we see that
$CW_{B{\mathbb{Z}}/p}(BN)\simeq CW_{B{\mathbb{Z}}/p}(BT)\simeq BV$ where
$V\cong({\mathbb{Z}}/p)^{n}$, as $BW$ is $B{\mathbb{Z}}/p$-null and
$BT\rightarrow BN$ is a $B{\mathbb{Z}}/p$-equivalence.

Next we compute the cellularization of $(BN)^{\wedge}_{p}$. First, by [BK02,
Prop. 7.5], there is a bijection
$[B{\mathbb{Z}}/p,BN]\cong[B{\mathbb{Z}}/p,BN^{\wedge}_{p}]$. Consider the
following diagram of horizontal cofibrations,

$\textstyle{BV\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BN\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{BV\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{BN^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D.}$

where $D$ is $1$-connected since $BN^{\wedge}_{p}$ is also $1$-connected.
Therefore $P_{\Sigma B{\mathbb{Z}}/p}(D)$ is also $1$-connected by Lemma 2.4.
Since $\pi_{1}(C)$ is finite, $C$ is $p$-good. Moreover, $g$ is a mod $p$
equivalence, therefore $C^{\wedge}_{p}$ is $1$-connected. Now consider the
following diagram of fibrations:

$\textstyle{BV\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BN\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{CW_{B{\mathbb{Z}}/p}(BN^{\wedge}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{BN^{\wedge}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}D.}$

We will show that $P_{\Sigma B{\mathbb{Z}}/p}(g)\colon P_{\Sigma
B{\mathbb{Z}}/p}C\rightarrow P_{\Sigma B{\mathbb{Z}}/p}D$ is a mod $p$
equivalence. Since $g$ is a mod $p$ equivalence, and also $P_{\Sigma
B{\mathbb{Z}}/p}(\eta_{D})\colon P_{\Sigma B{\mathbb{Z}}/p}(D)\rightarrow
P_{\Sigma B{\mathbb{Z}}/p}(D^{\wedge}_{p})$ by Corollary 3.11, we only need to
prove that $P_{\Sigma B{\mathbb{Z}}/p}(\eta_{C})\colon P_{\Sigma
B{\mathbb{Z}}/p}(C)\rightarrow P_{\Sigma B{\mathbb{Z}}/p}(C^{\wedge}_{p})$ is
also a mod $p$ equivalence by checking that $C$ satisfies the hypothesis of
Lemma 3.9.

Consider the following diagram of fibrations

$\textstyle{T/V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BV\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BT\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E(T/V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BN\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BN\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(T/V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\Sigma
B{\mathbb{Z}}/p}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{BW.}$

where $f$ exists by Zabrodsky’s Lemma (see [Dwy96, Prop 3.4]) applied to the
fibration $BV\rightarrow BN\rightarrow P_{\Sigma B{\mathbb{Z}}/p}(C)$ and the
map $BN\rightarrow BW$. It implies that there is a map $P_{\Sigma
B{\mathbb{Z}}/p}(C)\rightarrow BG$ which fits in a diagram of fibrations. The
bottom fibration shows that $P_{\Sigma B{\mathbb{Z}}/p}(C)$ is homotopy
equivalent to the classifying space of a compact Lie group whose fundamental
group is $G$.

Now we check that $C$ satisfies the hypothesis of Lemma 3.9. First, $C$ and
$P_{\Sigma B{\mathbb{Z}}/p}(C)$ are $p$-good since they have finite
fundamental groups ([BK72, VII.5.1]), $P_{\Sigma
B{\mathbb{Z}}/p}(C^{\wedge}_{p})$ is $1$-connected and therefore it is also
$p$-good. It remains to check that $P_{\Sigma
B{\mathbb{Z}}/p}(C)^{\wedge}_{p}$ and $P_{\Sigma
B{\mathbb{Z}}/p}(C^{\wedge}_{p})^{\wedge}_{p}$ are $\Sigma
B{\mathbb{Z}}/p$-null spaces. Since $P_{\Sigma B{\mathbb{Z}}/p}(C)$ is
homotopy equivalent to the classifying space of a compact Lie group, its
$p$-completion is $\Sigma B{\mathbb{Z}}/p$-null (see e.g. [BK02, Prop 7.5]).
Finally $P_{\Sigma B{\mathbb{Z}}/p}(C^{\wedge}_{p})^{\wedge}_{p}$ is also
$\Sigma B{\mathbb{Z}}/p$-null since $P_{\Sigma
B{\mathbb{Z}}/p}(C^{\wedge}_{p})$ is $1$-connected by Theorem 3.1.

Summarizing, $(CW_{B{\mathbb{Z}}/p}(BN^{\wedge}_{p}))^{\wedge}_{p}$ is the
homotopy fiber of $BN^{\wedge}_{p}\rightarrow BK^{\wedge}_{p}$ where $K$ is an
extension of $W$ by $T/V$ and $V$ is the maximal elementary abelian
$p$-subgroup in the torus $T$.

Our next example concerns $p$-toral groups. Recall that a $p$-toral group is
an extension of a torus by a finite $p$-group. A $p$-compact toral group is an
extension of a $p$-compact torus by a finite $p$-group, and a discrete
$p$-toral group is a group $P$ with normal subgroup $T$ such that $T$ is
isomorphic to a finite product of copies of ${\mathbb{Z}}/p^{\infty}$ and
$P/T$ is a finite $p$-group.

Since $CW_{B{\mathbb{Z}}/p}(BT^{\wedge}_{p})\simeq CW_{B{\mathbb{Z}}/p}(BT)$
by Lemma 3.2 and $CW_{B{\mathbb{Z}}/p}(BT)\simeq BV$ where $V$ is the subgroup
of elements of order $p$, the following is also true for $p$-compact toral
groups.

######  Example 6.16.

Let $P$ be a $p$-toral group with group of components $\pi$. First of all, by
[CCS07-2, Proposition 2.1], we can assume that $\pi$ is a finite $p$-group
generated by elements of order $p$ which lift to $BP$. By Proposition 3.14 and
Remark 3.17, there is a mod $p$ equivalence
$CW_{B{\mathbb{Z}}/p}(BP)\rightarrow CW_{B{\mathbb{Z}}/p}(BP^{\wedge}_{p})$.
Dwyer and Wilkerson show in [DW94] that there exists a discrete $p$-toral
group $P_{\infty}$ such that $BP_{\infty}\rightarrow BP$ is a mod $p$
equivalence. We are reduced then to study the cellularization of discrete
$p$-toral groups.

Following [FS07, Section 4], we consider $\Omega_{1}(P_{\infty})$, the
subgroup generated by the elements of order $p$. Since a subgroup of a
$p$-toral discrete group is also a $p$-toral discrete group and the map
$B\Omega_{1}(P_{\infty})\rightarrow BP_{\infty}$ is a
$B{\mathbb{Z}}/p$-cellular equivalence (note that ${\rm
map\,}_{*}(B{\mathbb{Z}}/p,BP_{\infty})\simeq
Hom({\mathbb{Z}}/p,P_{\infty})$), we can assume that $P_{\infty}$ is generated
by elements of order $p$. For any $p$-discrete toral group there is an
increasing sequence $P_{0}\leq P_{1}\leq\cdots$ such that $P_{\infty}=\cup
P_{n}$. Take a countable set of generators of order $p$ for $P_{\infty}$,
$\\{g_{i}|i=1,\ldots,n\\}$; then the subgroups $Q_{n}=\langle
g_{1},\ldots,g_{n}\rangle$ satisfy that $P_{\infty}=\cup Q_{n}$ and each
$Q_{n}$ is a finite $p$-group generated by elements of order $p$, so by
[Flo07, Prop 4.14, Prop 4.8], $BQ_{n}$ is $B{\mathbb{Z}}/p$-cellular and
therefore $BP_{\infty}$ is so.

Finally the space $B\Omega_{1}(P_{\infty})$ is $B{\mathbb{Z}}/p$-cellular, so
it remains to check that $B\Omega_{1}(P_{\infty})\rightarrow
CW_{B{\mathbb{Z}}/p}(BP^{\wedge}_{p})$ is a mod $p$ equivalence. Let
$C_{BP_{\infty}}$ and $C_{BP^{\wedge}_{p}}$ be the corresponding Chach-ólski’s
cofibres. Zabrodsky’s Lemma (see [Dwy96, Prop 3.4]) applied to the fibration
$B\Omega_{1}(P_{\infty})\rightarrow BP_{\infty}\rightarrow P_{\Sigma
B{\mathbb{Z}}/p}(C_{BP_{\infty}})$ and the map $BP_{\infty}\rightarrow
B(P_{\infty}/\Omega_{1}(P_{\infty}))$ shows that there is a homotopy
equivalence $P_{\Sigma B{\mathbb{Z}}/p}(C_{BP_{\infty}})\simeq
B(P_{\infty}/\Omega_{1}(P_{\infty}))$. In particular, $C_{BP\infty}$ satisfies
the hypothesis of Lemma 3.9. Moreover, $C_{BP^{\wedge}_{p}}$ is $1$-connected.

The map $g\colon C_{BP\infty}\rightarrow C_{BP^{\wedge}_{p}}$ is a mod $p$
equivalence and, by Remark 3.13 and Corollary 3.12, $P_{\Sigma
B{\mathbb{Z}}/p}(g)$ is a mod $p$ equivalence. Finally, Proposition 3.14
combined with the previous results, show that
$B\Omega_{1}(P_{\infty})\rightarrow CW_{B{\mathbb{Z}}/p}(BP^{\wedge}_{p})$ is
a mod $p$ equivalence.

In particular, from Example 6.11 we obtain that there are mod $2$ equivalences
$BD_{2^{\infty}}\rightarrow CW_{B{\mathbb{Z}}/2}(BO(2))\rightarrow
CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})$, and hence a chain of homotopy
equivalences $CW_{B{\mathbb{Z}}/2}(BO(2))^{\wedge}_{2}\simeq
CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})^{\wedge}_{2}\simeq
BO(2)^{\wedge}_{2}$.

We finish the section with a last example in which we can observe a completely
different pattern, and where the cellularization is obtained by combining in
an adequate way some nice push-out decompositions.

######  Proposition 6.17.

The $\emph{B}{\mathbb{Z}}/2$-cellularization of ${B}SO(3)$ fits in a fibration

$({CW}_{\emph{B}{\mathbb{Z}}/2}{B}SO(3))^{\wedge}_{2}\rightarrow{B}SO(3)^{\wedge}_{2}\rightarrow({B}SO(3)^{\wedge}_{2})_{\mathbb{Q}}.$

###### Proof.

Since $SO(3)$ is connected, by Lemma 3.2 the $p$-completion induces a homotopy
equivalence
${CW}_{B{\mathbb{Z}}/p}BSO(3)\simeq{CW}_{B{\mathbb{Z}}/p}(BSO(3)^{\wedge}_{p})$.
According to [DMW87, Cor 4.2], $BSO(3)$ is equivalent at the prime 2 to the
pushout $X$ of the following diagram:

$\textstyle{BD_{8}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\textstyle{BO(2)^{\wedge}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{(B\Sigma_{4})^{\wedge}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X,}$

where $f_{1}$ is induced by inclusion of the 2-Sylow subgroup, and $f_{2}$ is
given by the map of extensions

$\textstyle{{\mathbb{Z}}/4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{8}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\textstyle{{\mathbb{Z}}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{SO(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{O(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}}/2.}$

Our strategy will be to cellularize the previous diagram, and compare the
respective pushouts. Now, recall that $BD_{8}$ is $B{\mathbb{Z}}/2$-cellular
([Flo07, 4.14]) and moreover the 2-completion of $B\Sigma_{4}$ is so ([FS07,
Thm 4.4]). On the other hand, $D_{2^{\infty}}$ is a $2$-discrete approximation
of $O(2)$ -i.e.$BD_{2^{\infty}}\rightarrow BO(2)$ is a mod $2$ equivalence-,
so the previous Example 6.16 implies $BD_{2^{\infty}}\rightarrow
CW_{B{\mathbb{Z}}/2}(BO(2))$ is a mod $2$ equivalence. Moreover, by
Proposition 3.14 and Remark 3.17, there is also a mod $2$ equivalence
$CW_{B{\mathbb{Z}}/2}(BO(2))\rightarrow
CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})$. So, we can consider another pushout
diagram by applying the functor $CW_{B{\mathbb{Z}}/2}$ to the previous one,

$\textstyle{BD_{8}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{(B\Sigma_{4})^{\wedge}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$

There exists a map $g\colon Y\rightarrow X$ induced by the augmentation map
from the cellularization which is a mod $2$ equivalence since
$CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})\rightarrow BO(2)^{\wedge}_{2}$ is
so.

Now we attempt to compute the $B{\mathbb{Z}}/2$-cellularization of
$X^{\wedge}_{2}$ by using the cofibre of the map $k\colon Y\rightarrow
X^{\wedge}_{2}$. In order to do this, a result of Chachólski [Cha96, Thm 20.3]
together with [FS07, Thm 1.1] tells us that we need to check that
$[B{\mathbb{Z}}/2,Y]\rightarrow[B{\mathbb{Z}}/2,Y^{\wedge}_{2}]\cong[B{\mathbb{Z}}/2,X^{\wedge}_{2}]$
is exhaustive and $Y$ is $B{\mathbb{Z}}/2$-cellular. $Y$ is
$B{\mathbb{Z}}/2$-cellular since it is a pushout of $B{\mathbb{Z}}/2$-cellular
spaces. It remains to check that
$[B{\mathbb{Z}}/2,Y]\rightarrow[B{\mathbb{Z}}/2,Y^{\wedge}_{2}]$ is
exhaustive.

Let $\mathcal{P}$ be the category $1\leftarrow 0\rightarrow 2$ describing a
pushout diagram, and let $F\colon\mathcal{P}\rightarrow Top$ be the functor
describing the pushout for $Y$, that is, $F(1)=(B\Sigma_{4})^{\wedge}_{2}$,
$F(0)=BD_{8}$ and $F(2)=CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})$ with the
corresponding morphisms. There is a commutative diagram of sets

$\textstyle{\operatornamewithlimits{\hbox{$\varinjlim$}}[B{\mathbb{Z}}/2,F]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\eta_{F})_{*}}$$\textstyle{[B{\mathbb{Z}}/2,Y]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{*}}$$\textstyle{\operatornamewithlimits{\hbox{$\varinjlim$}}[B{\mathbb{Z}}/2,F^{\wedge}_{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[B{\mathbb{Z}}/2,Y^{\wedge}_{2}]}$

where the vertical maps are induced by $2$-completion of the target. Since the
spaces ${\rm map\,}(B{\mathbb{Z}}/2,F^{\wedge}_{2})$ are $2$-complete (see
[BK02, Proposition 7.5]), by [BLO03, Lemma 4.2] the bottom horizontal map is a
bijection. To prove that $\eta_{*}$ is exhaustive, it is enough to show that
$(\eta_{F})_{*}$ is so. But then, looking at the diagram, it reduces to check
that
$[B{\mathbb{Z}}/2,CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})]\rightarrow[B{\mathbb{Z}}/2,CW_{B{\mathbb{Z}}/2}(BO(2)^{\wedge}_{2})^{\wedge}_{2}]\cong[B{\mathbb{Z}}/2,BO(2)^{\wedge}_{2}]$
is exhaustive (see Example 6.16) and this follows from Corollary 3.7.

Let $C$ be the cofibre of $k$. We know that $C$ is mod $2$ acyclic and
$1$-connected. Now if $q$ is an odd prime, since $Y$ is
$B{\mathbb{Z}}/2$-cellular, $Y$ is mod $q$ acyclic and
$C^{\wedge}_{q}\simeq(BSO(3)^{\wedge}_{2})^{\wedge}_{q}$ is contractible.
Finally $C_{\mathbb{Q}}\simeq(BSO(3)^{\wedge}_{2})_{\mathbb{Q}}$. Then, by a
Sullivan arithmetic square argument,
$C\simeq(BSO(3)^{\wedge}_{2})_{\mathbb{Q}}$ which is, in turn,
$B{\mathbb{Z}}/2$-null. In particular $C$ is $\Sigma B{\mathbb{Z}}/2$-null.
Therefore, the fibration of the theorem follows from Chachólski’s fibration
describing the cellularization (Theorem 2.3). ∎

######  Remark 6.18.

Note that if $p$ is an odd prime, then $BSO(3)^{\wedge}_{p}\simeq
BN(T)^{\wedge}_{p}$, where $N(T)$ is the normalizer of the maximal torus, and
we analyzed this case in Example 6.15.

It seems natural to ask if the problem of computing $CW_{B{\mathbb{Z}}/p}(BG)$
for any compact Lie group $G$ is accessible at this point. A strategy was
developed for finite groups in [FF], based in the description of the strongly
closed subgroups of $G$, which are classified. Recent research has remarked
the role of the strongly closed subgroups of discrete $p$-toral groups in the
homotopy theory of compact Lie groups and, more generally, $p$-local compact
groups [Gon10], but to our knowledge there is no available classification of
these objects. On the other hand, the nontrivial rational homotopy of $BG$
seems an important obstacle to generalize the arithmetic square arguments of
the strategy. We plan to undertake these issues in subsequent work, and, in
particular, an intriguing question which arises in a natural way from the last
example:

Question: For which class of classifying spaces of compact Lie groups (or
spaces in general) is the $B{\mathbb{Z}}/p$-cellularization equivalent to the
homotopy fibre of the rationalization, up to $p$-completion?

Acknowledgements. We would like to thank Carles Broto and Jérôme Scherer for
interesting conversations on this subject.

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