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\define\CW{\bold CW}
\def\P{\bold P}
\def\L{\bold L}
\centerline{\bf Cellular Spaces}
\medskip
\centerline{by}
\centerline{E.\ Dror Farjoun}
\bigskip
\baselineskip=18pt
\centerline{\bf 0.\ \ Introduction and Main Results}
The aim of the present work is to study, in a systematic way,
the possibility of constructing
new spaces out of a given collection of persumably better understood spaces.
Recently there were several developments in this direction: There has been
an intensive
study of the construction of the classifying space of a compcat Lie group
-as a homotopy limit- out of a collection of much smaller subgroups [JMO?].
This was generalized by [DW?] and was used effectively by several authors
to construct new realizations of interesting algebras as cohomology algebras
of spaces.
In a different vein [S?H] introduced 'thick classes'. These classes are closed
under cofibration. In the stable category they are in fact the closure under cofibrations and desuspensions
of a single space, say V(n), constructed by [?] and [?] Thus each one of
these classes is precisely the class of all spectra that can be built form
a single spectra by taking cofibres of arbitrary maps (and desuspensions).
Now unstably the question arrise naturally:
\item {1.} Question: What is the collection of all spaces that can be built
by repeated cofibration fron these generic spaces $V(n)$.?
This question is closely related to a problem posed by F. Adams in 1970:
Classify all $E_*$- acyclic spaces for a give generalized
homology theory $E_*$.
In paticular, under what condition a $K$-acyclic space can actually be
built by a possibly infinite process of repeated cofibration from
the 'elementary' four cells space:e
$V(1)$ first constructed stably by Adams but now [Mis?] [C-N?]
considered as a finite
dimensional p-torsion space[?] [?]?.
Our approach is to start with an arbitrary space $A$ and consider the
class of all spaces gotten from it by repeatedly taking all pointed homotopy
colimits along arbitrary diagrams starting with $A$ itself.
We get a full subcategory denoted here by
${\Cal C}^\cdot (A)$. This class comes with a functor $\CW_A$
[Bou?] that associates to
to every space a member of the class in a natural fashion. Much of the
present work can be done in the category of general topological spaces.
In that context the usual class of CW complexes introduced by Whitehead
is simply the class of
spaces
built from the zero sphere, the class of 1-connected complexes is that
built out of the two-sphere etc. But one can just as well start with
'singular' spaces such as the Hawaian rings and get a full blown homotopy
theory.
Thus our basic concept in the present study is
${\Cal C}^\cdot (A)$ = the smallest full
subcategory of ${\script S}_*$ closed
under arbitrary{\it pointed hocolim}
and weak homotopy
equivalences that contains $A$.
The functor $\CW_A$ starts with an arbitrary space $A$ and associates to any
space $X$ the `best $A$--approximation,' the space in
${\Cal C}^\cdot (A)$ which is the closest space to $X$.
It can be built out of copies of $A$ by gluing them together along
base--point preserving maps in a similar fashion to the construction of
the usual $\CW$--approximation to a given topological space $X$.
The latter is, in fact, equivalent to the space $\CW_{S^0} X$, where $A=S^0$
is the zero sphere.
The functor $\CW_A$, is very closely related
to
the localization (or
periodization) functor $P_A$ with respect to $A$ ?Bou] ?EDF].
By definition whenever a space $X$ is $A$-periodic i.e. the pointed function
complex from $A$ to $X$ is contractible the A-CW approximation to $X$ is
contractible.
In [N] Nofech introduced a model category structure on simplicial sets or topological spaces in which weak equivalences are
maps that induces isomorphisms on the $A$- homotopy groups. In that context the $A$- periodic (or shall we say $A$- trivial ) spaces apear as the fibrant
objects which are weakly equivalent to a point
while the $A$- CW or $A$- cellular appear as the cofibrant objects.
Thus there is a sort of duality between $A$ periodic spaces and
$A$-cellular spaces. In the case where $A$ is the mod-p Moore spaces this
is well established: under mild conditions the homotopy fibre of
the rational localization
and more generally rationally acyclic spaces can be built from these
Moore space Building Blocks,
via cofibrations. But we shall see that
whenever $[A,X]\simeq *$ the space $\CW_A X$ is homotopy equivalent
to the homotopy fibre
of $X\rightarrow P_{\Sigma A} X$ (see 0.7 below).
As an example of application we address the following
\bigskip\noindent
0.1\ \ {\bf Problem}:\ \ Given a $h_*$--acyclic space $X$ for a generalized
homology theory $h_*$,
when is $X$ the limit of its {\it finite} $h_*$--acyclic complexes?
It turns out that for $h_*=K\langle n\rangle$ = Morava's $K$--theories, and
$X=GEM$ a generalized Eilenberg--MacLane space positive answer is
available under some dimensional restriction.
More generally, using some recent work of Bousfield and Thompson one can
show that for the universal theories $S_n$ considered in ??\ \ ] ?\ \ ] the
following holds:
\proclaim{0.2\ \ Theorem} For any of the theories $S(n)$ there exist
$N\geq n$
such that every $N$--connected space $X$, with $\Omega^N X$ a
$S(n)$--acyclic space, is $V(n)$--cellular space and in particular
a limit of its finite
$S(n)$--acyclic subcomplexes.
\endproclaim
Totally unrelated is the following.
\medskip\noindent
\proclaim{0.3\ \ Theorem} Let $A=\Sigma A^\prime$ any suspension and let
$A\rightarrow X @>f>> X\cup CA$ be a
homotopy cofibration sequence.
The homotopy fibre $F$ of $f$ is an $A$--cellular space.
In particular if $\tilde{h}_*(A)\simeq 0$ for some generalized homology
then $\tilde{h}_*(F)\simeq 0$.
\endproclaim
Here is an interesting generalization of a theorem of Serre which follows
quite easily from the present approach:
\proclaim{0.4 Theorem} Let $A$ be any pointed, finite type connected space.
Let $X$ be any finite $\Sigma A$--cellular space, with $\tilde{H}^*(X,
{\Bbb Z}/p{\Bbb Z})\not= 0$ for some $p$.
Then $\pi_i (X,A)=[\Sigma^i A,X]\not= 0$ for infinitely many dimensions
$i\geq 0$.
\endproclaim
One immediate corollary is for $X=\Sigma A$.
\proclaim{0.4.1 Corollary} Let $A$ be any connected space with $\tilde{H}_*
(X,{\Bbb Z}/p{\Bbb Z})\napprox 0$ for some $p$.
There are infinitely many $\ell$'s for which $[\Sigma^\ell A,\Sigma A]
\not= *$.
\endproclaim
For the case $A=S^1$=the circle we get well known theorems of Serre about
the higher homotopy groups of finite simply connected complexes.
A proper context for the present approach
is to define a map $f\colon X\rightarrow Y$ to be weak $A$--equivalence if
map$_*(A,f)$ is a weak homotopy equivalence.
It turns out that for any pointed simplicial set $A$ or any pointed compact
topological space
$A$, the notion of weak $A$--equivalence can be embedded in a model category
in the sense of Quillen ??\ \ ] on the category of all topological spaces or
pointed simplicial sets ?N].
This is discussed briefly in section 3.
For finite $p$--torsion spaces one can classify all the types of equivalences
that arise this way between spaces.
As in the case of the periodization functor $P_A$ the functor $\CW_A$ has
pleasant formal properties that makes it accessible and useful.
Let us summarize some of these properties as follows:
\medskip\noindent
\proclaim{0.5\ \ Theorem} Let the class ${\Cal C}^\cdot (A)\subset {\script S}_*$ be the
closure under arbitrary pointed hocolims and weak equivalences of the
singleton $\{A\}$ containing the pointed space $A$.
\item{1.}There is a homotopy idempotent functor $\CW_A\colon {\script S}_*\rightarrow
{\script S}_*$ which is augmented by $\tau\colon \CW_A X\rightarrow X$ with $\CW_A \tau$
a homotopy equivalence, and therefore $\CW_A$ is a pointed simplicial functor
taking values in ${\Cal C}^\cdot (A)$.
\hfuzz=20pt
\item{2.}The map $\CW_A X\rightarrow X$ is initial among all maps $f\colon Y
\rightarrow X$ with
$$
\text{map}_*(A,f)\colon \text{map}_*(A,Y)\rightarrow \text{map}_*(A,X)
$$
a weak equivalence and terminal among all maps of members of
${\Cal C}^\cdot (A)$ into $X$.
\item{3.}Each of the universal properties (2) above determine
$\CW_A X\rightarrow X$ up to
an equivalence which is itself unique up to homotopy.
\item{4.}Let ${\underset \sim\to X}$ be any $I$--diagram of pointed spaces.
One has a natural homotopy equivalence
$$
\CW_A \text{ ho}\lim\limits_I{\underset \sim \to X} \simeq \CW_A
\text{ ho}\lim\limits_I \CW_A {\underset \sim \to X}.
$$
\item{5.}There is a natural homotopy equivalence $\CW_A(X\times Y)\
@>\simeq >> \ \CW_A(X)\times \CW_A (Y)$.
\item{6.}There is a natural homotopy equivalence $\CW_A\Omega X\
@>\simeq >> \ \Omega \CW_{\Sigma A}X$,
where $\Sigma A$ is the suspension of $A$.
\item{7.}If $Y=GEM$ then $\CW_A Y$ is also a GEM.
If, in addition, $\pi_i Y\simeq 0$ for $i\geq r$,
then $\pi_i \CW_A Y\simeq 0$ for $i\geq 0$.
\item{8.}$\CW_{\Sigma^k A} \CW_{\Sigma^\ell A} X=\CW_{\Sigma^n} X$ where
$n=\max(k,\ell)$.
\item{9.}$P_A \CW_A X\simeq \CW_A P_A X \simeq *$.
\endproclaim
\bigskip
These properties of $\CW_A$ follow in turn from certain closure properties
of ${\Cal C}^\cdot (A)$.
\proclaim{0.5\ \ Theorem} For any fibration $F\rightarrow E\rightarrow B$
we have:
\roster
\item If $F,E\in {\Cal C}^\cdot (A)$ then $B\in {\Cal C}^\cdot (A)$
\item If $E,B\in {\Cal C}^\cdot (\Sigma A)$ then $F\in {\Cal C}^\cdot
(A)$
\item If $F,B\in {\Cal C}^\cdot (\Sigma A)$ then $E\in {\Cal C}^\cdot
(A)$.
\endroster
\endproclaim
\proclaim{0.6\ \ Theorem} For any pointed spaces $A,X,Y$ the product
$X\times Y$ is $A$--cellular if and only if both $X$ and $Y$ are $A$--cellular.
A retract of any $X$ in ${\Cal C}^\cdot (A)$
is also in ${\Cal C}^\cdot (A)$.
\endproclaim
\vskip .25truein
\centerline{\bf Fibrations Associated to $\CW_A$}
The above theorems 0.5--0.6 form the basis on which one can build a reasonable
theory of $\CW_A$ very similar, but not identical, to the theory of $P_A$,
the $A$--localization or $A$--periodization functor as developed in ?Bou]
?EDF] ?DF--S].
The main features of the general theory is to determine to what extent $\CW_A$
preserves fibre sequences, relate $\CW_{\Sigma^k A}$ for various $k$ ,
give practical criteria for a space to belong to ${\Cal C}^\cdot (A)$ and
determine for what $A$ and $B$ one has ${\Cal C}^\cdot (A)$= ${\Cal C}^\cdot (
B)$ .
Here we give a review of some fibration theorems associated to
$\CW_A$.
In particular it often preserves fibrations and commutes with loops
`up to a GEM.'
This close analogy with the results of ?DF--S] about $P_A X$ is perhaps
explained by the following theorem
that presents $\CW_A X$ as the homotopy fibre of $X\rightarrow P_A X$ under
certain circumstances:
\proclaim{0.7\ \ Theorem} Whenever the natural composition
$$
\CW_A X\rightarrow X\rightarrow P_{\Sigma A} X
$$
is null homotopic, the sequence is a homotopy fibration sequence for any
$A,X \in {\script S}_*$.
\endproclaim
\proclaim{0.8\ \ Corollary} In particular if $P_{\Sigma A}X \simeq *$ then
$\CW_A X\rightarrow X$ is a homotopy equivalence and so $X\in {\Cal C}^\cdot (A)$.
\endproclaim
\proclaim{0.9\ \ Corollary} If $[A,X]\simeq *$ then the above sequence is a
fibration sequence.
\endproclaim
\proclaim{0.10\ \ Corollary} If the natural map $\CW_{\Sigma A} X\rightarrow
\CW_A X$ or the natural map $P_{\Sigma A} X @>>> P_A X$ is an equivalence
then the above sequence (1.7) is a fibration.
\endproclaim
The basic tool in analyzing preservation of fibration is the following
theorem which is dual to the implication
$P_{\Sigma A}X\simeq *\Rightarrow P_{\Sigma^2 A} X$ is a GEM (= generalized
Eilenberg MacLane space i.e.\ a possibly infinite product of $K(G,h)$'s for
$G$ abelian group) ?DF-S]:
\medskip
{\smc Preservation of fibrations.}
As in [?] one of our main concerns is to show that $\CW_A$ 'almost' preserves fibration. A moment's
reflection on the example where $A$= the sphere shows that one cannot
expect that it will preserve fibration without some "error term". Thus
our aim is to be able to show that often that error term
is small and manageable. As was seen in [?[? a small error is in
this context a space which is a procuct of (possibly infinite number )
of Eilenberg-MacLane spaces (GEM).
The simplest question in that direction concerns a situation where $\CW_A$
kills two members in a fibre sequence --does it always kill the third?
\smallskip
\proclaim{0.11\ \ Theorem} For any pointed spaces $A,X$
let $f\to E \to B$ be a fibration sequence
if $\CW_{\Sigma A}F\simeq \CW_{\Sigma A} E$ then $\CW_{\Sigma A} B$
is a GEM. Therefore if $\CW_{\Sigma^2 A}
X\simeq *$ then $\CW_{\Sigma A} X$ is a GEM.
\endproclaim
\demo{0.12\ \ Remark} In most cases it is not hard to determine the GEM that
arise in 0.11 since the $A$--cellular $K(G,n)$'s are not hard to understand
in many cases see (5.4) below.
Of course (0.11) is obvious for $A=S^n$ for all it says is that if some
$n$--connected cover of $X$ is contractible then the $(n-1)$--connected cover
is an Eilenberg--MacLane space.
\enddemo
With these techniques we deduce without much difficulties:
\proclaim{0.13\ \ Theorem} Let $A=\Sigma A', X\in {\script S}_*$.
\roster
\item The fibre of
$$
\CW_A \Omega X\rightarrow \Omega \CW_A X
$$
is a polyGEM.
\item If $F\rightarrow E\rightarrow B$ is a {\it fibration then for any}
$A\in {\script S}_*$ {\it there exists a fibration}
$$
\CW_A F\rightarrow \overline{E}\rightarrow \CW_{\Sigma A} B
$$
\endroster
in which $\overline{E}\in {\Cal C}^\cdot (A)$ is a `mixture' of $A-$ and
$\Sigma A$--co--localization of $E$.
If in addition $A=\Sigma A^\prime$ is a suspension space, then the
natural map $\CW_A E\rightarrow \overline{E}$ has a GEM as a homotopy
fibre, and therefore
$$
\CW_{\Sigma A} F\rightarrow \CW_{\Sigma A}E\rightarrow \CW_{\Sigma A}B
$$
is a fibration, up to a polyGEM.
Namely the homotopy fibre of the natural map
$$
\CW_{\Sigma A}F\rightarrow \text{fibre}(\CW_{\Sigma A}E @>>> \CW_{\Sigma A}B)
$$
is a polyGEM (= a ``generalized Postnikov $n$--stage'').
\endproclaim
{\smc Organization of the paper.}
In the first section we review some basic technical result about localization
theory. We give a full proof of a crucial technical results about preservations of certain fibration by localization functors.
In the second section we discuss in some details general closure properties
of "closed classes" such as ${\Cal C}^\cdot (A)$. In particular we show that they
are closed under cartesian products, half-smash product and to some extend
under taking homotopy fibre. This gives us a useful generalization of an
important lemma[Bou?] of Miller and Zabrodsky.
We prove 0.3 and 0.6 in that section. In the third section we discuss and prove
(0.5) (1,2,3,4,5) as well as formulate the appropriate Whitehead theorem for
detecting homotopy equivalences between $A$-cellular spaces.
The more delicate fibration theorems and (0.5) (6-9) are discussed in section
4 and 5. where we prove the basic 0.7.
The last section is devoted to examples, discussion of $E$-acyclic spaces and a proof of 0.2.
\bigskip\noindent
{\bf 1. A Review of Homotopy Localization with Respect to a Map}
In this paper we will use crucially several properties of $L_f$, the localization functor with respect to a general map $f\colon A @>>> D
$.
Most of the time we will consider $f\colon A @>>> *$ and in that case one
denotes $L_f$ by $P_A$ due to its close formal similarity to the Postnikov
section functor $P_n=P_{s^{n+1}}$.
\bigskip\noindent
{\bf Basic Properties}:
We recall from \cite{\ \ } \cite{\ \ \ } \cite{\ \ \ } several basic
properties of $\L_f$ and $\P_A$ where $A$ is any space and $f:A\to D$ any map.
Given any spaces $A, D, X$ and a map $f:A\to D$ there exists co-augmented
functors $X\to\L_fX$ and $X\to \P_AX$ (where $\P_A$ is a special notation
for $\L_{(A\to*)}$ since its properties are very remenicent of those of the
Postnikov section functor $\P_n$). $X\to\L_fX$ is {\it initial} among all maps
$X\to T$ to $f$-$local\ spaces$ $T$ (or $f$-periodic or $f$-divisible) i.e. spaces
with the function complex map:
$$
\text{map}(f,T):\text{ map}(D,T)\to\text{ map}(A,T)\tag1.1
$$
being a weak equivalence. The co-augmentation $X\to\L_fX$ is also terminal
among all maps $X\to T$ which become equivalences upon taking their function
complexes to any $f$-local space.
The functor $\L_f$ can be defined in either the pointed or unpointed category
of spaces and its value for connected $A$, $D$, $X$ does not depend, up
to homotopy, on the choice of the category in which one works
$(\{{\script S}\}\ \text{ or } \{{\script S}_*\})$.
Being defined on the unpointed category and being homotopy functor it has
an associated fibrewise localization functor that turns any fibration
sequence $F\to E\to B$ into a fibration sequence $\L_fF\to\tilde E\to B$.
In addition $\L_f$ (or $\P_A$) enjoys the following properties.
\bigskip
\item{1.2}
There is a natural equivalence $\L_f(X\times Y)\overset\cong\to\longrightarrow
\L_f(X)\times \L_f(Y)$.
\bigskip
\item{1.3}
Every fibration sequence $F\to E\to B$ with $\L_fF\simeq *$ gives
a homotopy equivalence $\L_fE\overset\simeq\to\longrightarrow \L_f B$.
\bigskip
\item{1.4}
There is a natural equivalence $\L_f\Omega X\simeq \Omega\L_{\Sigma f}X$.
\bigskip
\item{1.5}
If $\overline{\P}_AX$ is the homotopy fibre of $X\to\P_AX$ then $\P_A(
\overline{\P}_AX)\simeq *$. Similarly $\L_f\overline{\L}_{\Sigma f}X\simeq *$.
(Notice $\Sigma f$).
\bigskip
\item{1.6}
If $F\to E\to B$ any fibre map and $B$ is $A$-periodic
(i.e. $\text{map}_*(A,B)
\simeq *)$ or more generally $\P_{\Sigma A}B\simeq \P_AB$
then $\P_AF\to\P_AE\to B$ is also a fibre sequence. Similarly
if $F$ is $A$-periodic that $F\to\P_{\Sigma A}E\to\P_{\Sigma A}B$ is a
fibration sequence.
\bigskip
\item{1.7}
$\L_f\ \underset I\to{\text{hocolim}}\ \underset\sim\to X=
\L_f\ \underset I\to{\text{hocolim}}\ \L_f \underset\sim\to X$, and in
particular $\L_fX\simeq *$ implies $\L_f\Sigma^k X\simeq *$ for all $k$.
\bigskip
\item{1.8}
If $\L_f X\sim *$ and $\L_g Y\simeq *$ (or $\P_W Y\sim *$) then $\L_{f\wedge g}
(X\wedge Y)\simeq *$ (or $\L_{f\wedge W}X\wedge Y\sim *$).
\bigskip
\item{1.9}
If $\P_AB\simeq *$ and $\P_B C\simeq *$ then $\P_A C\simeq *$.
\bigskip
\item{1.10}
If for all $\alpha\in I$ $X(\alpha)$ is $f$-local where $X(\alpha)$ is a
member
of an $I$-diagram $\underset\sim\to X$ indexed by a small category $I$,
then so is $\underset I\to{\text{holim}}\ \underset\sim\to X$.
\bigskip
\item{1.11}
If $Y$ is an $n$-connected GEM then so is $\L_f Y$ for any $f:A\to B$.
\bigskip\noindent
{\bf GEM--Properties}
In addition to the above list the functor $L_f$ the most fundamental property
of $L_f$ is the following
\proclaim{1.12 GEM Theorem} Let $F @>>> E @>>> B$ be a fibration sequence of pointed
connected spaces.
Assume $L_{\Sigma f} B\simeq L_{\Sigma f} E \simeq *$.
Then $\L_{\Sigma f} F$ is a GEM while $\L_f F \simeq *$.
\endproclaim
\proclaim{1.12.1 Corollry:} If $\L_{\Sigma f}X\simeq *$
then $\L_{\Sigma^2 f}X\simeq GEM$.
\endproclaim
\smallskip
\proclaim{1.12.2 Corollry:} The homotopy fibre of $\P_{\Sigma^2 A}X\to \P_{\Sigma A}X$
is a GEM for any $X,A$.
\endproclaim
\demo{Proof}The first corollary follows immediately using
the adjunction (1.4)
and the standard loop space fibration over $X$.
The second corollary follows
from the first using (1.5) and noting that the map
in question is in fact a localization map.
We now turn to the proof of (1.12)
We first show that $L_{\Sigma f} F$ is an $\infty$--loop space
using ?S] ?B--F].
We define a (non--special) $\Gamma$--space as follows.
$$
{\overset\vee\to F}_n=\text{fibre of} (E \vee \ldots \vee E @>>>
B\vee \ldots \vee B).\quad (n-\text{copies of}\ E,B)
$$
We observe that the functor that assigns to a finite pointed set $S$ the
wedge ${\underset s\to\vee} X$ of copies of $X$ for any $X\in {\script S}_*$,
gives a (non--special) $\Gamma$--space:\ \ i.e.\ a functor from $\{$finite
pointed sets$\}$ to spaces.
This functor assigns to every pointed set
its smash product with the given space $X$-a construction that is clearly
natural.
We now consider this construction for $E$ and $B$.
The homotopy--fibre being a functor in ${\script S}_*$ and
$\left\{{\underset n\to\bigvee} E\right\}_{n\geq 0} @>>>
\left\{{\underset n\to\bigvee} B\right\}_{n\geq 0}$
being a map of $\Gamma$--spaces we conclude that ${\overset\vee\to F_\cdot}$
above is a $\Gamma$--space.
\enddemo
We claim:\ \ {\it The natural map (see diagram below)}
$$
f_n \colon {\overset\vee\to F}_n @>>> F \times \ldots \times F
$$
{\it induces an equivalence on} $L_{\Sigma f}(f_n)$.
Since ${\overset \vee\to F}_1=F$ this implies that
$$
\eqalign{
L_{\Sigma f} {\overset \vee\to F}_n=L_{\Sigma f}
(F \times \ldots \times F)&=(L_{\Sigma f}{\overset \vee\to F})^n\cr
&=(F_1)^n\cr
}
$$
Thus $L_{\Sigma f}({\overset \vee\to F}_\cdot)$ is a {\it special}
$\Gamma$--space and therefore $L_{\Sigma f}{\overset \vee\to F}_1=L_{\Sigma f} F$
is an $\infty$--loop space.
Consider the diagram that depicts the above constructions for $n=2$:\ \ This
diagram is built from the lower right square by taking homotopy fibres.
$$
\CD
\Omega (\Omega B * \Omega B) @>>> X @>>> \Omega E*\Omega E @>>> \Omega B *
\Omega B\\
& & @VVV @VVV @VVV\\
& & {\overset \vee\to F}_2 @>>> E\vee E @>>> B\vee B\\
& & @VVV @VVV @VVV\\
& & F\times F @>>> E\times E @>>> B\times B
\endCD
$$
By (1.3) in order to prove the claim it is sufficient to
show that $L_{\Sigma f}X\simeq *$.
First notice $L_{\Sigma f}(\Omega E * \Omega E) \simeq L_{\Sigma f}
(\Sigma (\Omega E \wedge \Omega E)$.
But $L_f (\Omega E \wedge \Omega E)\simeq *$ since $L_f \Omega E \simeq *$ (1.8).
Thus $L_{\Sigma f}\Omega E * \Omega E$ and also $L_{\Sigma f}(\Omega B *
\Omega B)\simeq *$ (1.7).
Now consider $L_{\Sigma f} \Omega (\Omega B * \Omega B)$.
By (1.4)
$$
L_{\Sigma f}\Omega \Sigma (\Omega B \wedge \Omega B)=\Omega L_{\Sigma^2 f}
\Sigma(\Omega B \wedge \Omega B).
$$
But (1.8) $L_{\Sigma f}(\Omega B \wedge \Omega B)\simeq *$ since (1.8)
$L_f \Omega B \simeq *$ and $P_{s^1} \Omega B \simeq *$, thus (1.7)
$L_{\Sigma^2 f} \Sigma(\Omega B \wedge \Omega B)\simeq *$
This proves our claim since it implies (1.3):
$$
L_{\Sigma f} X \simeq L_{\Sigma f} (\Omega E * \Omega E)\simeq *.
$$
Therefore $L_{\Sigma f}F$ is $\infty$--loop space and in particular we
can write:\ \ $L_{\Sigma f} F=\Omega Y$.
$$
\text{claim:\ map}^* (\Sigma^2 F, Y)\simeq *\
\text{i.e.\ map}^* (\Sigma F,\Omega Y)\simeq *.
$$
This follows from universality (1.1):\ \ We have factorization:
But
$$
\CD
\Sigma F & @>>> & \Omega Y=L_{\Sigma f} F\\
@VVV & & @AA\exists ! A\\
L_{\Sigma f} \Sigma F
\endCD
$$
in which:\ \
Claim:\ \ $L_{\Sigma f}\Sigma F \sim *$.
Moreover:\ \ $L_f F \sim *$.
This is clear from (1.3) for the fibration:
$$
\Omega B @>>> F @>>> E
$$
and $L_f \Omega B \simeq \Omega L_{\Sigma f} B \simeq \Omega *\simeq *$.
All the more so $L_{\Sigma f} \Sigma^k F \sim *$ and thus
$(\Sigma^k F @>>> \Omega Y)$ is null.
The claim being proven we can conclude from Bousfield's key lemma (1.13) below that
$F @>>> L_{\Sigma f} F$ factors through the universal GEM asociated with $F$
namely the infinite symmetric product: $SP^\infty F$.
$$
\CD
& & SP^\infty F\\
& & @VVV\\
F @>>> \Omega Y=L_{\Sigma f} F
\endCD
$$
But $SP^\infty F$, the Dold--Thom functor on $F$ is a GEM.
Applying $L_{\Sigma f}$ to the factorization we get that $L_{\Sigma f} F$ is a
retract of a GEM since $L_{\Sigma f}=L_{\Sigma f} L_{\Sigma f}$.
But a retract of a GEM is a GEM.
This concludes the proof.
\proclaim{1.13 Bousfield's Key Lemma}
Let $X$ be a connected, $Y$ a simply-connected spaces. Assume $\text{map}_*
(\Sigma^2 X,Y)\simeq *$. Then $\text{map}_*(X,\Omega Y)\cong \text{map}_*
(SP^kX,\Omega Y)$ for any $k\ge 1$. ?5, 6.9].
\endproclaim
\demo{1.14 Remark}
A way to understand (1.13) is to interpret it as saying that the space
$\Sigma SP^kX$ can be built by successively glueing together copies of
$\Sigma^\ell X$ for $\ell\ge 1$ with precisely one copy for $\ell=1$.
Since the higher suspension $\Sigma^{2+j}X$ $(j\ge 0)$ will not contribute
anything to $\text{map}_*(\Sigma\ SP^\ell X,Y)$ we are left with
$\text{map}_*(\Sigma X,Y)$.
\smallskip
More precisely, it can be easily seen by adjunction that (1.13) is equivalent
to the following:
{\it For any space X the suspension
of the Thom-Dold map t:$\Sigma X\to\Sigma SP^kX$
induces a homotopy equivalence $ \P_{\Sigma^2X}(t)$ upon localization
with respect to the double suspension of $X$}.
\enddemo
\noindent
In fact the same holds for the James functor $J_kX$ and other cases.
This is a correct reformulation because by universality (1.1) a map t induces
a homotopy equivalence on the f-localization iff it becomes an equivalence
upon taking the function complex of t into any f-local space.
In this form (1.13) can be verified using (1.7) and a homotopy colimit presentation of the Dold-Tho
m functors in ?6,6.4],and using the fact discussed above
that the inclusion $\Sigma (X\vee X\to X\times X)$ becomes an equivalence
after localization with respect to the above double suspension.
\vskip .25truein
\centerline{\bf 2.\ \ Closed Classes and $A$--Cellular Spaces}
In this section we discussed certain full subcategories of ${\script S}_*$ called
{\it closed classes}.
The main example of such classes is ${\Cal C}^\cdot (A)$ for a given
pointed $A$, but also the class of spaces that map trivially to all finite
dimensional spaces is closed.
\proclaim{2.1\ \ Definition} A full subcategory of pointed spaces
${\Cal C}^\cdot \subset {\script S}_*$ is called ``closed'' if it is closed under weak
equivalences and arbitrary
pointed homotopy colimits:\ \ Namely for any diagram of space in
${\Cal C}^\cdot$
(i.e.\ a functor ${\underset \sim\to X}\colon I\rightarrow {\Cal C}^\cdot$) the
space hocolim ${\underset \sim\to X}$ is also in ${\Cal C}^\cdot$.
\endproclaim
We prove several closure theorems for any closed class ${\Cal C}^\cdot$, the
most important ones being:
\item{1.}${\Cal C}^\cdot$ is closed under finite product.
\item{2.}If $X\in {\Cal C}^\cdot$ and $Y$ any (unpointed) space then $X\rtimes Y=
(X\times Y)/*\times Y$ is in ${\Cal C}^\cdot$.
\item{3.}If $F\rightarrow E\rightarrow B$ a fibration sequence and $F, E$ in
${\Cal C}^\cdot$ then so is $B$.
\item{4.}If $\Sigma A\rightarrow X @>i>> X\cup C\Sigma A$
is any cofibration sequences and $ A$ is in ${\Cal C}^\cdot$ then so is the
fibre of $i$.
\bigskip\noindent
2.2\ \ {\bf Examples of Closed Classes}:
\medskip\noindent
1.\ \ {\bf The class} ${\Cal C}^\cdot(A)$:\ \ This is the smallest closed class
that contains a given pointed space $A$.
It can be built by a process of transfinite induction by starting with the full
subcategory containing the single space $A$ and closing it repeatedly under
arbitrary pointed hocolim.
In section (3) below we give a `cellular' description of spaces in
${\Cal C}^\cdot(A)$.
We refer to members of ${\Cal C}^\cdot (A)$ as $A$--cellular spaces.
\noindent
2.\ \ {\bf The class} ${\Cal C}^\cdot (A @>f>> B)
={\Cal C}^\cdot (f)$
here we start with any map (or a class of maps) $f\in {\script S}_*$ of pointed spaces and
consider all spaces $X$ such that the induced map on pointed function complexes
$$
\text{map}_* (X,A)\rightarrow \text{map}_*(X,B)
$$
is a (weak) homotopy equivalence of simplicial sets.
Since map$_*(\underset{I}\to{\text{hocolim}} X_\alpha,A)$\hfill\break
$=\underset{I^{op}}\to
{\text{holim}}\ \text{map}_*(X_\alpha,A)$, it is immediate that
${\Cal C}^\cdot(f)$ is a closed class.
{\it This class is often empty. }
\noindent
3.\ \ {\bf The class} of spaces that map trivially to all finite dimensional
spaces.
This includes by Miller's theorem $K(\pi,1)$ for a finite group $\pi$.
\bigskip\noindent
2.3\ \ {\bf Pointed and Unpointed Homotopy Colimits}
Let $A$ be a pointed space.
We have considered ${\Cal C}^\cdot (A)$ the smallest class of pointed spaces
closed
under arbitrary \underbar{pointed} hocolim, and homotopy equivalences, which
contains the space $A$.
Notice that if we consider classes closed under arbitrary \underbar{non--pointed} hocolim we get only two classes the empty class and the class of all unpointed
space.
This is true since a class closed under unpointed hocolim that contains a
contractible space, contains all weak homotopy types, since evey space is
the free hocolim of its simplices.
Notice also that if $A$ is not empty then ${\Cal C}^\cdot (A)$ contains the
one--point space $*\simeq P$--hocolim $(A\rightarrow A\rightarrow A\rightarrow
\ldots)$ where all the maps in this infinite telescope are the trivial maps
into the base point $*\in X$.
In general given a pointed $I$--diagram ${\underset \sim\to X}$ we can
consider its homotopy (inverse) limit in either the pointed or unpointed
category.
By definition, these two spaces have the same (pointed or unpointed)
homotopy type they have in fact the same underlying space.
On the other hand, the homotopy colimits of ${\underset \sim\to X}$ will
generally have a different homotopy type when taken in the pointed or unpointed
category:\ \ If ${\underset \sim\to *}\in {\underset \sim\to X}$ is the
$I$--diagram of base points in ${\underset \sim\to X}$ then we have a
\underbar{cofibration}; with NI = the classifying space (or the nerve) of the
category I.
$$
NI\rightarrow \text{free--hocolim } {\underset\sim\to X}\rightarrow
\text{pointed--hocolim } {\underset\sim\to X}.
$$
Since by definition we have a cofibration:
$$
(I\backslash-\otimes {\underset\sim\to *})\rightarrow (I\backslash-\otimes
{\underset\sim\to X})\rightarrow (I\backslash-|\otimes
{\underset\sim\to X})
$$
where $\otimes$ is the ``tensor product'' of an $I$--diagram with $I^{op}$--diagram and $|\otimes$ is the `pointed tensor product.''
?B--K p.\ 327 \& p.\ 333]
\proclaim{ Corollary} If the classifying space of the indexing category
$I$ is contractible then for any $I$--diagram pointed diagram
${\underset\sim\to Y}$ we
have a homotopy equivalence
free--$\underset{I}\to{\text{hocolim}}{\underset\sim\to Y}
\simeq$ pointed--$\underset{I}\to{\text{hocolim}} \tilde{Y}$.
\endproclaim
\demo{Remark} Thus over the usual pushout diagram $\cdot @<<< \cdot @>>> \cdot$
and over infinite tower $\cdot @>>> \cdot @>>> \ldots \ldots$ hocolim takes
the same value in the pointed and unpointed categories.
But not e.g.\ over a discrete group.
In the present paper unless explicitly expressed otherwise hocolim mean
{\it pointed} hocolim over pointed diagram.
Thus for any small category $I$ we have $\underset{I}\to{\text{hocolim}}\{*\}=
\{*\}$.
Otherwise we use the notation {\it free}--hocolim, thus
free--$\underset{I}\to{\text{hocolim}}\{*\}=BI=NI$ the nerve of $I$ for
every $I$.
\noindent
{\bf 2.4\ \ Half smashes and products in closed classes}:\ \ We now show that
a closed
class ${\Cal C}^\cdot$ is closed in an appropriate sense under half smash with an arbitrary
unpointed space (i.e.\ ${\Cal C}^\cdot$ is an {\it ideal} in ${\script S}_*$ under the
operation ${\Cal C}^\cdot @>>> {\Cal C}^\cdot \rtimes Y)$, and under internal
finite Cartesian products (see 2.1 above).
But first
\noindent
{\it 2.5\ \ Generalities about half--smash}:\ \ Recall the notation
$$
X\rtimes Y=(X\times Y)/*\times Y
$$
where $X$ is pointed and $Y$ is unpointed space.
This gives a bifunctor ${\script S}_*\times {\script S} @>>> {\script S}_*$.
There is another bifunctor ${\script S}\times {\script S}_* @>>> {\script S}_*$ given by
$\widetilde{\text{map}}
(Y,X)$ where $Y$ is unpointed and $X$ pointed and where $\widetilde{\text{map}}
(Y,X)$ is the space of {\it all} maps equipped with the base point $Y @>>> *
@>>> X$.
Thus the underlying space of $\widetilde{\text{map}}(Y,X)$ is the same as that
of the {\it free} maps while the underlying space of $X\rtimes Y$ is
different in general from that of the base point free product $Y^\prime
\times Y$.
There are obvious adjunction identities
\item{(i)}map$_*(A\rtimes Y,X)=\text{map}_*(A,\widetilde{\text{map}}^*
(Y,X))$
\item{(ii)}map$_*(A\rtimes Y,X)=\widetilde{\text{map}}(Y,\text{map}_*(A,X))$
The first identity (i) says that for each $Y\in S$ the functor $-\rtimes Y
\colon {\script S}_* @>>> {\script S}_*$ is left adjoint to
$\widetilde{\text{map}}(Y,-)$.
Whereas identity (ii) says that for each $A\in {\script S}_*$ the functor
$A\rtimes -\colon S @>>> {\script S}_*$ is left adjoint to map$_*(A,-)$,
where the latter is the space of pointed maps {\it as an unpointed space},
i.e.\ forgetting its base point.
In particular we conclude
\proclaim{2.6\ \ Proposition} For each $A\in {\script S}_*$ and $Y\in S$ the
functors $-\rtimes
Y$ and $A\rtimes -$ commute with colimits and hocolimits.
\endproclaim
{\it Notice} that to say that $A\rtimes - \colon S @>>> {\script S}_*$ commutes
with hocolim involves commuting {\it pointed} hocolim i.e.\ the hocolim in
${\script S}_*$ with {\it unpointed} hocolim in $S$.
Explicitly:\ \ For any base point free diagram of space
${\underset\sim\to Y}\colon I @>>> S$ we have an equivalence:
$$
A \rtimes (\text{free}-\underset{I}\to{\text{hocolim}}\underset\sim\to Y)=
\text{pointed}-\underset{I}\to{\text{hocolim}} (A\rtimes\underset\sim\to Y).
$$
\proclaim{2.7\ \ Lemma} If $Y$ is any unpointed space then for any indexing
diagram $I$ the functor\hfill\break
$-\rtimes Y\colon {\script S}_*\rightarrow {\script S}_*$ commutes with
hoco$\lim\limits_I$, and if $X$ is any {\it pointed} space the
functors\hfill\break
$X\rtimes-\colon S @>>> {\script S}_*$ and $-\wedge X\colon {\script S}_*\rightarrow {\script S}_*$ commute with
$\underset{I}\to{\text{hocolim}}$.
\endproclaim
\demo{Proof} We have just considered $X \rtimes -$.
Similarly $-\wedge Y$ is left adjoint to map$_*(Y,-)$ and again commutes with
colim and hocolim.
\enddemo
\proclaim{2.8\ \ Theorem} If $X$ is any closed class ${\Cal C}^\cdot$ space then:
\roster
\item For any (unpointed) space $Y$ the half--smash $X\rtimes Y$ is in
${\Cal C}$.
\item For any (pointed) $B$ cellular--space $Y$ the smash $X\wedge Y$ is an
$(A\wedge B)$ cellular--space and an $A$--cellular space.
\endroster
\endproclaim
\demo{Proof} To prove (1) we start with an example showing that
$X\rtimes S^1$ is an $X$--cellular space.
In fact it can be gotten directly as a pointed hocolim of the push--out
diagram:
\enddemo
$$
\CD
X\vee X @>\text{fold}>> X\\
@V\text{fold}VV @VVV\\
X @>>> X\rtimes S^1
\endCD
$$
This diagram is gotten simply by half--smashing $X$ with the diagram that
presents $S^1$ as {\it free}--hocolim of discrete sets:
$$
\CD
\{0,1\} @>>> \{0\}\\
@VVV @VVV\\
\{1\} @>>> S^1
\endCD
$$
By inducation we present $S^{n+1}$ as a pushout $* @<<< S^n @>>> *$ which
gives by induction $X\rtimes S^{n+1}$ as a pushout along $X @<<< X\rtimes
S^n @>>> X$, that arise since (2.5) $(X\rtimes -)$ commutes with
free--hocolim on the right (smashed) side.
Since the filtration of $Y$ by skeleton $Y_0 \subset Y_1 C\ldots$ presents
$Y_{n+1}=Y_n\cup (C \coprod S^n)$ we get upon half smashing with $X$ a
presentation of $Y\rtimes X$ as a pointed--hocolim.
\demo{Remark} Here is a `global' formulation of the above proof using (2.6):\ \
Present the space $Y$ as free--$\underset{DY}\to{\text{hocolim}}\{*\}$ where
$DY$ is any small category whose nerve is equivalent to $Y$, and
$\{{\underset\sim\to *}\}$ is
the $DY$--diagram consisting of the one--point space for each object of $DY$.
Now by (2.6) above:
\enddemo
$$
X\rtimes Y=X\rtimes \text{free}-\underset{DY}\to{\text{hocolim}}
\{{\underset\sim\to *}\}=
\text{pointed}-\underset{DY}\to{\text{hocolim}} X\rtimes
\{{\underset\sim\to *}\}
$$
Thus $X\rtimes Y$ is directly presented as a pointed hocolim of a pointed
diagram consisting solely of many copies of the space $X$ itself.
Now to prove (2) one just notices that $X\wedge Y=(X\rtimes Y)/X\times\{pt\}$
so $X\wedge Y$ is certainly an $X$--cellular space.
Now since pointed--hocolim commutes with smash--product we get by induction
on the presentation of $Y$ as a $B$ space that $X\wedge Y$ is a $A\wedge B$
space as needed.
\proclaim{2.9\ \ Theorem} Let $F\rightarrow E\rightarrow B$ be any
fibration of pointed spaces.
If $F$ and $E$ are members of some closed class ${\Cal C}^\cdot$ then so is
$B$.
\endproclaim
\proclaim{corollary} We shall see later (4.9) that this implies that if the base and total spaces are $\Sigma A$-cellular for any $A$ then the fibre is $A$-cellular.
\endproclaim
\proclaim{corollry}(This is a generalization of [millerZab?][Bo?]): let
$F \to E \to B$ be any fibration sequence. If both the fibre and total space
have a trivial function complex to a given pointed space $Y$ then so does
the base space $B$.
\endproclaim
The second corollary follows immediately by observing (2.2) (2) that the class
of spaces with a trivial function complex to a given space is closed.
\medskip
{\bf Proof}:\ \ We define a sequence of fibrations $F_i\rightarrow E_i
\rightarrow B$ by $E_0=E,F_0=F,\ E_{i+1}=E_i \cup C F_i$ and $F_{i+1}$ is the
homotopy fibre of obvious map $E_{i+1}\rightarrow B$.
All $E_i,\ F_i$ are naturally pointed spaces.
\vfill\eject
$$
\CD
F@>>> E@>>> B\\
& & @VVV\\
F_1 @>>> E\cup CF=E_1 @>>> B\\
& & @VVV\\
F_2 @>>> E_1\cup C F_1=E_2 @>>> B\\
& & @VVV\\
& & \ldots\\
& & E_\infty=B
\endCD
$$
By Ganea's theorem ? ?] $F_{i+1}\simeq F_i * \Omega B \simeq \Sigma(F_i\wedge
\Omega B)$
and therefore connectivity of $F_{i+1}$ is at least $i$, since $F_0$ is
$(-1)$--connected.
Notice that {\it by definition} since $E_0,\ F_0$ are in ${\Cal C}^\cdot$
spaces so are $E_i,\ F_i$ for all $i$.
But since conn $F_i\rightarrow \infty$, we deduce that hocolim $E_i=B$.
Therefore $B$ is also in ${\Cal C}^\cdot$, as needed.
We now turn to the somewhat surprising closure property of closed classes (2.1) (4.)
\proclaim{2.10\ \ Theorem} For any map $A @>>> X$ of pointed space the homotopy
fibre $F$ of $X @>>> X\cup CA$ satisfies $P_A F\simeq *$.
\endproclaim
\proclaim{2.10.1\ \ Corollary} If $A=\Sigma A^\prime$ then above fibre $F$ is an
$A$--cellular space.
\endproclaim
\demo{Proof} The proof uses the following diagram:
$$
\CD
& & A\\
& & @Vi VV\\
F @>>> X @>c>> X\cup CA\\
@VVV @VVV @V=VV\\
P_A F @>>> \overline{X} @>\overline{c}>> X\cup CA
\endCD
$$
\enddemo
Where the solid arrows are given by the fibrewise localization (p.\ 4) of the
top row.
Thus the fibre map $\overline{c}$ is induced from the composition $X\cup
CA\rightarrow B$ aut $F@>>> B$ out $P_A F$.
Here as in [???\ ] [\ \ \ ] we use the fact that the fibration $Y @>>> B$
out $^\cdot Y@>>> B$ and $Y$, where out $Y$ (out $\cdot Y)$ is the space of
un--pointed (res.\ pointed) self equivalences of $Y$, classifies all fibrations
with fibre $Y$.
Taking $F$ to be the usual path space we have a well defined map $i^\prime$
into the homotopy fibre.
Since map$(A,P_A F)\simeq *$ by construction of $P_A F$ the composition
$A @>>> X @>>> \overline{X}$ factorizing through $P_A F$ is null--homotopic,
where the null homotopy comes from the cone $A @>>> F @>>> F\cup CA @>>>
P_A F$ that defines $P_A F$.
This null homotopy gives a well defined map $c^\prime : X\cup CA @>>>
\overline{X}$ rendering the diagram strictly commutative.
Therefore the fibration $\overline{c}$ is a split fibration having $c^\prime$:
as a section.
Also since $F @>>> \overline{X}$ factors through $X\cup CA$ it is null
homotopic map.
But the splitting of $\overline{c}$ implies from the long exact sequence of
the fibration that the map $P_A F @>>> \overline{X}$ is injective on
pointed homotopy class $[W,-]_*$ for any $W\in {\script S}_*$.
And since $F @>>> P_A F @>>> \overline{X}$ is null homotopic we conclude
that $F @>>> P_A F$ is null.
Now idempotency of $P_A$ implies $P_A F\sim *$ as needed.
\proclaim{Proof of Corollary} We know from (1.8) that $P_{\Sigma A^\prime}F
\simeq *$ implies that $F$ is a $A^\prime$--cellular space,
this is proved in (4.5) below without of course using the present corollary.
\endproclaim
\
\vskip .25truein
\noindent
{\bf Closure under Products}
Many of the pleasant properties of $\CW_A$ depend on its commutation with finite
products.
This commutation rest on the following basic closure properly of any closed
class.
\proclaim{2.11\ \ Theorem} Any closed class ${\Cal C}^\cdot$ is closed under
any finite product:\ \ If $X,Y\in {\Cal C}^\cdot$ then so is $X\times Y$.
\endproclaim
\demo{Remark} It is well known that an infinite product of $S^1$'s does
not have the homotopy type of a $\CW$--complex i.e.\ the class of all
$\CW$--complexes in $Top_*$ is not closed under arbitrary products.
\enddemo
\demo{Remark} If $A=\Sigma A^\prime$ and $B=\Sigma B^\prime$ where $A,B\in
{\Cal C}^\cdot$, then $A\times B$ is easily seen to be in ${\Cal C}^\cdot$
via the cofibration
$$
A^\prime * B^\prime \rightarrow \Sigma A^\prime \vee \Sigma B^\prime
\rightarrow \Sigma A^\prime\times \Sigma B^\prime.
$$
Since $A^\prime * B^\prime \cong \Sigma A^\prime \wedge B^\prime$ one uses
(2.8).
\enddemo
We owe the proof to W.\ Dwyer.
An independent proof can be extracted from [Bou? $-$1].
\demo{Proof} We filter $Y$ by its usual skeleton filtration $Y_{n+1}=Y_n
\cup e^{n+1}\ldots$.
\enddemo
We may assume $X,Y$ are connected.
For brevity of notation we add one {\it pointed} cell at a time but the proof
works verbatim for an arbitrary number of cells.
Let $P(n)$ be the subspace of $X\times Y$ given by
$$
P(n)=\{*\}\times Y \cup X \times Y_n.
$$
Clearly the tower $P(n)\hookrightarrow P(n+1)$ is ``cofibrant'' and its colimit
$X\times Y$ is equivalent to its homotopy colimit.
Since ${\Cal C}^\cdot$ is closed under hocolim it is sufficient to show, by
induction that $ P(n)\in {\Cal C}^\cdot$ for all $n \geq 0$ .
For $n=0$, we have $P(0)=X\vee Y$ clearly in ${\Cal C}^\cdot$.
Now $P(n)$ is given as homotopy pushout diagram:
$$
\CD
X\times S^{n-1}\cup *\times D^n @>\simeq>> X\rtimes S^{n-1} @>>>
\{*\}\times Y\cup X\times Y_{n-1}\\
@VVV @VVV @VVV\\
X\times D^n @>\simeq>> X\rtimes D^n @>>> P(n)
\endCD
$$
coming from the presentation of $X_n$ as a pushout over {\it pointed} diagram:
$X_{n-1} @<<< S^{n-1} @>>> D^n$.
Since the upper--left corner is the half smash $X\rtimes S^n$ it is in
${\Cal C}^\cdot$ by lemma 2.5 above.
Notice that all the maps are pointed.
Therefore $P(n)$ is a homotopy pushout of members of ${\Cal C}^\cdot$ as needed.
\proclaim{2.12\ \ Corollary} For any two $A$--cellular spaces $X,Y$ their
product $X\times Y$ is an $A$--cellular space.
\endproclaim
\demo{Proof} Consider the class ${\Cal C}^\cdot (A)$.
By the theorem just proved it is closed under finite product, therefore the
product of any two $A$--cellular spaces is $A$--cellular.
\enddemo
\vskip .25truein
\centerline{\bf 3.\ \ $A$--Homotopy Theory and the Construction of $\CW_A X$}
In this section we describe some initial elements of $A$--homotopy theory.
This framework replaces the usual sphere $S^0$ in usual homotopy theory of
$\CW$--complexes or simplicial sets by an arbitrary space $A$.
It can be considered in the framework of general compactly generated spaces
where $A$ can be chosen to be any such space.
We will however restrict our discussion to $A\in {\script S}_*$ a pointed space.
In particular there is a model category structure on ${\script S}_*$ denoted by ${\script S}_*^A$
where a weak equivalence $f\colon X\rightarrow Y$ is a map that induces a
usual weak equivalence
$$
\text{map}_*(A,f)\colon \text{map}_* (A,X)\rightarrow \text{map}_*(A,Y)
$$
or function complexes, and $A$--fibre maps are defined similarly.
Cofibrations are then determined by lifting property ?N].
The analog of a cofibrant object i.e.\ $\CW$--complex is an $A$--cellular space.
The natural homotopy groups in this framework are $A$--homotopy groups
$$
\eqalign{
\pi_i(X,A)&=[\Sigma^i A,X]_*\cr
&=\pi_i\text{map}_*(A,X,\text{null})\cr
&=[A,\Omega^i X]_*.\cr
}
$$
The classical Whitehead theorem about $\CW$--complexes takes here the form:
\medskip\noindent
3.1\ \ {\bf $A$--Whitehead theorem}:\ \ {\it A map $f\colon X\rightarrow Y$
between two pointed connected $A$--cellular spaces has a homotopy inverse}
(in the usual sense) {\it if and only if it induces a homotopy equivalence
on pointed function complexes
$$
\text{map}_*(A,X)@>\simeq >> \text{map}_*(A,Y).
\leqno(*)
$$
or equivalently, iff $f$ induces an isomorphism on the pointed homotopy classes:
$$
[A\rtimes S^n, X]_* @>\cong >> [A\rtimes S^n,Y]_*
\leqno(**)
$$
for all $n\geq 0$.
If the two pointed function complexes are connected i.e.} $\pi_0 (X,A)\simeq
\pi_0 (Y,A)\simeq *$ {\it or if $A=\Sigma A^\prime$ is a suspension
then a necessary and sufficient condition is that
it induces an isomorphism on $A$--homotopy groups}:
$$
\pi_*(X;A) @>\cong >> \pi_* (Y;A).
$$
\demo{Proof} (compare [D-?-Z]).
It is sufficient to show that under $(*)$ for every $W\in {\Cal C}^\cdot(A)$ we
have map$_*(W,X)@>\simeq>>\text{map}(W,Y)$ is a homotopy equivalence.
This can be easily shown by a transfinite induction on the presentation of
$W$ as a hocolim of spaces in ${\Cal C}^\cdot (A)$.
Namely one need only show that the class of spaces $W$ for which map$_*(Y,f)$
is a homotopy equivalence is a closed class.
But this is the content of (2.1).
Since by assumption it contains $A$, it follows that it contains also
${\Cal C}^\cdot (A)$ and therefore by our assumption it contains both $X$ and
$Y$.
Thus we get a homotopy inverse to $X\rightarrow Y$ by taking $Y=W$.
This completes the proof.
\enddemo
\centerline{3.2\ \ {\bf An elementary construction of $\CW_A X$}}
Given $A,X\in {\script S}_*$ we construct, in a natural way, a map $\CW_A X\rightarrow X$.
It will be clear from the construction that $\CW_A(-)$ is a functor ${\script S}_*
\longrightarrow {\script S}_*$.
Compare [B?ou--1].
\medskip\noindent
{\bf 3.3\ \ Half--suspensions} $\tilde\Sigma^n X$:\ \ A basic building block for
$\CW_A$ is the
half--smash $S^n\rtimes A=S^n\times A \cup D^{n+1}\times\{*\}$ with the
base point $\{*\}\times \{*\}$.
We denote this space by $\tilde{\Sigma}^n A$, and call them half
$n$--suspensions.
Just as an homotopy class $\alpha\in\pi_n \text{map}_*(A,X,\text{null})$ in
the null component is
respresented by a pointed map $\Sigma^n A\rightarrow X$ so a does a map
$\tilde{\alpha}\colon \tilde{\Sigma}^n A\rightarrow X$ represents an element
in $\pi_n \text{map}_*(A,X;f)$ of the $f$--component where
$f\colon A\rightarrow X$ is any map.
The map $f$ is gotten from $\tilde\alpha$ by restricting $\tilde\alpha$ to
$*\times A\subseteq \tilde{\Sigma}^n A$.
Notice that if $A$ itself is a suspension $A=\Sigma B$ then $\tilde{\Sigma}^n
A\cong \Sigma^n A\vee A$ ?E.D.F] {\it but in general such a decomposition
does not hold}.
Thus for suspension $A=\Sigma B$ an element $\tilde\alpha$ as above is given
simply by a pair $(\alpha\vee f)\colon \Sigma^n A\vee A\rightarrow X$.
In that case of course all the components of map$_*(A,X)$ has the same
homotopy type.
\medskip\noindent
{\bf 3.4\ \ Construction} of $\CW_A X$.
Let $c_0\colon C_0 X={\underset\alpha\in I\to V}\tilde{\Sigma}^i A
@>\alpha >> X$ be the wedge of all the pointed maps
$\tilde\Sigma^i A \rightarrow X$ from all half--suspensions $\tilde\Sigma^i A$
to $X$.
Clearly the map $c_0$ induces a surjection on the homotopy classes
$[\tilde\Sigma^i A,-]$ for every $i\geq 0$.
We now proceed to add enough `$A$--cells' to $C_0$, so as to get an isomorphism
on these classes.
We take the first (transfinite) limit ordinal $\lambda=\lambda(A)$
bigger than the
cardinality of $A$ itself (= cardinality of the simplicies or cells or points
in $A$).
The ordinal $\lambda=\lambda(A)$ clearly has the {\it limit property:\ \ Given
any transfinite tower of spaces of length} $\lambda$
$$
Y_0\rightarrow Y_i \ldots Y_n \rightarrow Y_w \rightarrow
Y_{w+1}\ldots Y_\alpha \rightarrow \ldots (\alpha < \lambda)
$$
{\it every map $\tilde\Sigma^i A\rightarrow \lim\limits_{\rightarrow\atop
\alpha < \lambda}Y_\alpha$ factors through $\tilde\Sigma^i A\rightarrow Y_\beta$
for some ordinal $\beta < \lambda$}.
\demo{Proof} This is clear for every individual cell of $\tilde\Sigma^i A$
and since the number of this cells is strictly smaller than the cardinality
of $\lambda$, it is true for $\tilde\Sigma^i A$.
We proceed to construct a $\lambda$--tower of correction
$C_0=C_0 X @>>> C_1 X @>>> C_2 X @>>> \ldots C_\beta X \ldots$ to our original
map $C_0 @>>> X$:
\enddemo
$$
\CD
D_0={\underset K_0\to \bigvee}\tilde\Sigma^i A @. {\underset K_1 \to \bigvee}
\tilde\Sigma^i
A=D_1 @. D_\beta={\underset K_\beta\to \bigvee}\tilde\Sigma^i A\\
@VV k_1 V @VVV @VVV\\
C_0={\underset I_0\to \bigvee}\tilde\Sigma^i A @>>> C_1 X\rightarrow C_2 X\ldots\quad @.
C_\beta X\rightarrow \ldots (\beta \leq \lambda)\\
@VV c_0 V @VVV @VV c_\beta V\\
\qquad X=X=\quad @.X=X= @. \ldots=X\ldots
\endCD
$$
Since $C_0\rightarrow X$ is surjective on the $A$--homotopy of all components
of map$_*(A,X)$ we proceed to kill the kernel in a functorial fashion.
In order to preserve functoriality we kill it over and over again:\ \ First
notice that any element $\tilde\alpha\colon\tilde\Sigma^n A\rightarrow X$
representing an $A$--homotopy class in the component $\tilde\alpha|\{*\}\times
A=f\colon A\rightarrow X$ is null homotopic {\it in that component} iff
$\tilde\alpha$ can be extended along the map
$$
\tilde\Sigma^n A=S^n\times A\cup D^{n+1}\times\{*\}\hookrightarrow
D^{n+1}\times A.
\leqno(e)
$$
Now let $k_0\colon D_0\rightarrow C_0$ be the wedge of all maps $g\colon
\tilde{\Sigma}^i A\rightarrow C_0$ {\it with a given} extension as (e) of
$c_0\circ g$ (the space $D_0$ being a point if there are no such extensions).
Thus $D_0\rightarrow C_0$ captures every null homotopic map $\tilde\Sigma^i A
\rightarrow C_0\rightarrow X$ many times.
The map $D_0\rightarrow C_0$ is given by $g$.
We define $C_1 X$ as the push--out along the extension to $D^{n+1}\times A$:
$$
\CD
{\underset K_0\to\bigvee}\tilde\Sigma^i A @>>> {\underset K_0\to\bigvee}
D^{n+1}\times A\\
@VVV @VVV\\
C_0 @>>> C_1=C_1 X
\endCD
$$
In this fashion we proceed by induction.
The map $C_1 X\rightarrow X$
is given by the null homotopies in the indexing set of $D_0={\underset k_0\to V}
\tilde\Sigma^i A$.
Taking limits at limits ordinal we define a functorial tower $C_\beta X$
for $\beta \leq \lambda$.
We now define $\CW_A X=C_\lambda X$.
This is the classical small object argument ??Q, p...] [Bou...].
Since $c_0$ induce surjection on $A$--homotopy sets $[\tilde\Sigma^i A,X]$ for
$i\geq 0$, on all components we get immediately that so does $c_\beta$ for all
$\beta \leq \lambda$.
The limit property of $\lambda=\lambda(A)$ now easily implies that
$C_\lambda X\rightarrow
X$ is injective in $\pi_i(\ ,A;f)$ for any $f\colon A @>>> X$.
Since every null homotopic composition
$\Sigma^i A\rightarrow C_\lambda X\rightarrow X$ factors through $\Sigma^i
A\rightarrow C_\beta X\rightarrow X$ for some $\beta$, a composition that is
also null homotopic by commutativity.
Therefore this map is null homotopic in $C_{\beta+1}X$ and thus in $C_\lambda X$
as needed.
\medskip\noindent
3.5\ \ {\it A smaller non--functorial $A$--cellular approximation} can be
built by choosing representatives in the associated homotopy classes.
But it is clear that in general even if $A,X$ are of finite complexes
$\CW_A X$ may not be of finite type since $\CW_{S^2} (S^1 \bigvee S^n)\approx
{\underset \infty\to \bigvee} S^n$ since this construction is just the
universal cover of $S^1 \bigvee S^n$.
\proclaim{3.6 Corollary} Let $A$ be a {\it finite} complex.
Then for any countable space $X$ we have the following form:
$$
\eqalign{
\CW_A X=(\bigvee\tilde\Sigma^i A)&\cup_{\varphi_1} \tilde{C}\tilde\Sigma^{i1}A
\cup_{\varphi_2} \tilde{C} \tilde\Sigma^{i2} A\cr
&\ldots \cup_{\varphi_\ell} \tilde{C}\tilde\Sigma^{i\ell} A \cup \ldots\cr
}
$$
Where the ``characteristic maps'' $\varphi_\ell$ are defined over
$\tilde\Sigma^{i\ell} A$ for $0\leq \ell < \infty$, and therefore $\CW_A X$ is also countable cell
complex.
\endproclaim
\proclaim{3.7 Corollary} In case $A$ a {\it finite} suspension space
$A=\Sigma B$
of {\it pointed} $B$ we have $\tilde\Sigma^i A=\Sigma^i A\vee A$ and
$\tilde{C}\Sigma^i A=C\Sigma^i A\vee A$ and therefore in order to kill the
kernels of $C_\beta\rightarrow X$ it is sufficient to attach cones over the
usual $\Sigma^i A\rightarrow C_\beta$.
Thus in this case the $A$--cellular approximation to $X$ has the usual form
$$
\CW_A X=\left(\bigvee \Sigma^{i_1} A\right)\cup_{\varphi_1}C\Sigma^{i_2}
A\cup_{\varphi_2} C\ldots \Sigma^{i_2} A\ldots
$$
Which is just the usual $\CW$--complex for $A=S^1=\Sigma S^0$, and $X$ any
connected $\CW$ complex.
\endproclaim
As in usual homotopy theory any map $X\rightarrow Y$ can be turned into a
cofibration $X\hookrightarrow X^\prime \rightarrow Y$ where $X\hookrightarrow
X^\prime$ is an $A$--cofibration i.e.\ $X^\prime$ is gotten from $X$ by
adding ``$A$--cells'' and $X^\prime @>>> Y$ is a trivial fibration i.e.\ in
particular it induces an isomorphism on $A$--homotopy groups.
Thus if $Y\simeq *$ we get $X^\prime \simeq P_A X$ since map$_*(A,P_A X)
\simeq \text{map}_*(A,*)$ and $X\longrightarrow P_A X$ is an $A$--cofibration.
If, on the other hand we take $X\simeq *$, the factorization becomes
$*\rightarrow \CW_A Y \rightarrow Y$ where $\CW_A Y$ now appears as the
$A$--cellular approximation to $X$ with the same $A$--homotopy in all
dimensions.
\noindent
{\bf 3.8\ \ Universality properties}
We now show that $r\colon \CW_A X @>>> X$ has two universality properties:
\item{(U1)}(Bou. 7.5) The map $r$ is initial among all maps $f\colon Y @>>> X$
with map$_*(A,f)$ a homotopy equivalence.
Namely for any such map there is a factorization $\tilde{f}$:
$$
\CD
\CW_A X @>r>> X\\
\tilde{f} @VVV\\
& f\\
\quad Y
\endCD
$$
and such $\tilde{f}$ with $f\circ \tilde{f}\sim r$ is unique up to homotopy.
\item{(U2)}The map $r$ is terminal among all map $\omega \colon W @>>> X$ of
spaces $W\in {\Cal C}(A)$ into $X$.
Namely for every $\omega$ there is a $\tilde\omega \colon W @>>>
\CW_A X$ with $r\circ \tilde\omega \sim \omega$ unique up to homotopy.
\demo{Proof} Both (U1) and (U2) are easy consequences of the functoriality of
$\CW_A$ when coupled with the $A$--Whitehead theorem.
Thus to prove (U1) consider $\CW_A(f)\colon \CW_A Y @>>> \CW_A X$.
This map is an $A$--equivalence between two $A$--cellular spaces, therefore
it is a homotopy equivalence.
Uniqueness follows by a simple diagram chase using naturality and idempotency
of $\CW_A$.
To prove $U(2)$:\ \ One gets a map $A @>>> \CW_A X$ by noticing that
$\CW_A W \simeq W$, so $\CW(\omega)$ gives the unique factorization.
Furthermore, uniqueness of factorization implies that each one of these
universality properties determine $\CW_A X$ up to an equivalence which itself
is unique up to homotopy.
This proves 1.4 (1)--(3).
\proclaim{3.13 Proposition} The following conditions or pointed spaces are
equivalent:
\item{(1)}{\it For any space $X$ there is an equivalence} $\CW_A X\simeq
\CW_B X$.
\item{(2)}${\Cal C}^\cdot (A)\simeq {\Cal C}^\cdot (B)$
\item{(3)}{\it A map $f\colon X\rightarrow Y$ is an $A$--equivalence if and
only if it is a $B$--equivalence}.
\item{(4)}$A=\CW_B A$ {\it and} $B=\CW_A B$.
\endproclaim
\demo{Proof} These equivalences follow easily from the universal properties
of $\CW_A X\rightarrow X$.
\enddemo
\item{}(1)$\Leftrightarrow$(2) Since the members of ${\Cal C}^\cdot (A)$ are
precisely
the space $X$ for which $\CW_A X\simeq X$ this is clear from universality.
\item{}(1)$\Leftrightarrow$(3) Clearly map $(B,f)$ is an equivalence $\CW_B f$
is a homotopy equivalence.
But since by (1)$\Leftrightarrow$(2) $\CW_B f\simeq \CW_A f$ we get (3).
\item{}(2)$\Leftrightarrow$(4) One direction is immediate.
If $A=\CW_B A$, then $A\in {\Cal C}^\cdot (B)$ and thus by theorem
$A^n\in {\Cal C}^\cdot (B)$ and
therefore ${\Cal C}^\cdot \subset {\Cal C}^\cdot (B)$.
Thus we get (2).
\proclaim{3.9\ \ Theorem (1.7)} For any $A,X,Y\in {\script S}_*$ there is a
homotopy equivalence
$$
\Psi\colon \CW_A (X\times Y)@>>> \CW_A X \times \CW_A Y.
$$
\endproclaim
\demo{Proof} There is an obvious map
$$
g\colon \CW_A X\times \CW_A Y @>>> X\times Y
$$
It is clear that $g$ induces a homotopy equivalence map$(A,g)$ and therefore
the map $\Psi$ in the theorem induce the same equivalence map$(A,\Psi)$.
But by corollary (2.11) the range of $\Psi$ is an $A$--cellular space.
Thus by the $A$--Whitehead theorem $\Psi$ is a homotopy equivalence.
\smallskip
\proclaim{6.1\ \ Lemma} If $X\simeq \CW_A X$ and $Y$ is a retract of $X$ then
$Y\simeq \CW_A X$.
\endproclaim
\demo{Proof} The retraction $r:X @>\gets>> Y$ implies that the map $\CW_A Y
\rightarrow Y$ is a retract of the homotopy equivalence $\CW_A X\rightarrow X$.
But a retract of an equivalence is an equivalence.
\enddemo
\vskip .25truein
\noindent
3.10\ \ {\bf Finite $\Sigma A$--Cellular Spaces Have Infinite
$\Sigma A$--Homotopy}:\ \ It is well known that 1--connected $\CW$--complex have non--trivial homotopy
groups in infinitely many dimensions.
This has been generalized in many directions --- relaxing the assumption of
finiteness.
In this section we consider a different direction of generalizing.
Instead of considering $[S^n,X]$ we will consider $[\Sigma^n A,X]$ for any
arbitrary space $A$:\ \ Instead of assuming $X$ is a finite simply
connected $\CW$--complex
we assume $X$ is a {\it finite $\Sigma A$--cellular space for any
connected $A$}:\ \ Namely a space
gotten by finite number of steps starting with a finite wedge of copies of
$\Sigma A$ and adding cones along maps from $\Sigma A$ to the earlier step.
$$
X\simeq (\bigvee \Sigma A)\cup C\Sigma^{\ell_1}A\cup C\Sigma^{\ell_2}
A\ldots C \Sigma^{\ell_k} A.\qquad (\ell_i\geq 1)
$$
\proclaim{3.11\ \ Theorem} Let $A$ be any pointed, finite type connected space.
Let $X$ be any finite $\Sigma A$--cellular space, with $\tilde{H}^*(X,
{\Bbb Z}/p{\Bbb Z})\not= 0$ for some $p$.
Then $\pi_i (X,A)=[\Sigma^i A,X]\not= 0$ for infinitely many dimensions
$i\geq 0$.
\endproclaim
One immediate corollary is for $X=\Sigma A$.
\proclaim{3.12\ \ Corollary} Let $A$ be any connected space with $\tilde{H}_*
(X,{\Bbb Z}/p{\Bbb Z})\napprox 0$ for some $p$.
There are infinitely many $\ell$'s for which $[\Sigma^\ell A,\Sigma A]
\not= *$.
\endproclaim
\demo{Proof} First we note that since we consider spaces built from
$\Sigma A$, by a finite number of
cofibration steps we get a space which is conic in the sense of
??HFLT] namely is gotten from a single point by finite number of steps of taking
mapping cones.
Now since $\tilde{H}^*(X,{\Bbb F}_p)\not= 0$ for some $p,\ X$ satisfies
the hypothesis of ??HFLT] and so $X$ does not have a finite generalized
Postnikov decomposition,
i.e.\ $X$ cannot be a polyGEM, since $X$ is of finite type.
On the other hand suppose $\pi_i(X,A)\cong 0$ for $i\geq N$.
Then map$_*(\Sigma^N A,X)\simeq *$ since all the homotopy groups of this space
vanish.
In other words $X$ is $\Sigma^N A$--periodic or $P_{\Sigma^N A} X\simeq X$.
We claim that $P_{\Sigma A}X\simeq *$.
This is true since by assumption $X$ is an $\Sigma A$--cellular space.
(Theorem P--5 above).
But from (??F-S]) we know that the homotopy fibre of $P_{\Sigma^N A}
X\rightarrow P_{\Sigma A}X$ is a poly GEM for any connected $A,X$.
But we just saw that the homotopy fibre of that map is $X$ itself which
cannot be a polyGEM.
This contradiction implies $\pi_i(X,A)\napprox 0$ for infinitely may $i$'s
as needed.
\enddemo
\demo{Remark} Notice that in order to prove the corollary we need not use
the heavy result of ?HFLT] since $H_*\Omega\Sigma Y$ is a tensor algebra and is
not nilpotent.
Therefore $\Sigma A$ cannot be a GEM if
$\tilde{H}_*(\Sigma A,{\Bbb Z}/p{\Bbb Z})\not= 0$ by Moore--Smith ?M--5].
Hence there must be infinitely many maps $\Sigma^\ell A\rightarrow \Sigma A$
for any such $A$.
\enddemo
\bigskip
\centerline{\bf Resolution of $\CW_A X$}
Here we record two simplicial resolutions of any space by $A$--cellular
simplicial space.
The diagonal of that simplicial space is not, in general $\CW_A X$, rather
it is a kind of dual to tot$R^\cdot X=R_\infty X$ in ?B--K].
Its relation to $\CW_A X$ is similar to the relation between $R_\infty X$
the Bousfield--K an $R$--completion functor and $L_{HR}$, the Bousfield
HR--homological localization functor.
Thus this diagonal may prove to be a useful approximation to $\CW_A X$.
If $[A,X]\simeq *$ and A is finite we show that the diagonal of these
resolutions give $\CW_A X$.
\bigskip\noindent
{\bf Blanc--Stover resolution}: For any space $X??\ \ ]$ gives
a simplicial resolution of $X$ by means of
$U_n X\ldots U_2 X {\underset \rightarrow\to\rightarrow} U_1 X\rightarrow X$
where for all $n\geq 1$ the space $U_n X$ is homotopy equivalent to a large
wedge of half--suspensions of $A$, namely
${\underset i\to\bigvee}\tilde\Sigma^{\ell_i} A$.
The realization $\|U_\cdot X\|=\underset{k}\to{\text{hocolim}} U_k X$ is
homotopy equivalent to $\CW_A X$.
Whenever $A$ is finite and $[A,X]\simeq *$ ?B--T].
\medskip\noindent
{\bf Dwyer's resolution}:For any $A,X$ we can associate
a functorial map $T_A X
@>\tau >> X$ from an $A$--cellular space $T_A X$
to $X$ as follows (see 2.5):
$$
\eqalign{
T_A X&=A\rtimes \text{map}_* (A,X)\cr
\tau(a,f)&=f(a)\in X.\cr
}
$$
\proclaim{Proposition} The functor $T_A$ enjoys the following properties:
\item{1.}If $f\colon X\rightarrow Y$ is an $A$--equivalence then $T_A f$ is a
weak homotopy equivalence.
\item{2.}The map $\tau$ is surjective on $A$--homotopy
classes $[\tilde\Sigma^k
A,\underline{*}]$.
\endproclaim
\demo{Proof} The first assertion is clear since $T_A f=A\rtimes \text{map}
(A,f)$ which is an equivalence if $f$ is an equivalence.
For the second assertion consider map$_*(A,\tau)\colon\text{map}_*(A,T_A X)
\rightarrow \text{map}_*(A,X)$.
This map clearly splits naturally:\ \ Take a pointed map $g\colon A\rightarrow X$
to the map $A\rightarrow T_A X$ taking at $A$ to the pair $(a,g)$.
A little computation shows that this canonically splits map$(A,\tau)$.
The above natural splitting makes $T_A$ into a natural cotriple with a
map$\colon T_A\rightarrow T_A T_A$ enjoy the properties of triple.
Therefore we extend this triple to a simplical resolution of $X$ by cellular
$A$--space $T_A^\cdot X\rightarrow X$.
\enddemo
\proclaim{Definition} Denote the diagonal of $T_A^\cdot X$ by
$T_A^\infty X$.
\endproclaim
\proclaim{Definition} If $X=V\tilde\Sigma^{n_i} A$ then
$T_A^\infty X\simeq X$.
\endproclaim
\demo{Proof} We first consider $X=\tilde\Sigma^n A$.
For $n=0$ there is a canonical splitting of $A\rtimes\text{map}(A,A)\rightarrow
A$ taking a to $(a,id)$.
This splits the simplicial resolution and yields $T_A^\infty A\simeq A$.
Taking $n\geq 1$ we have a splitting of
$$
A\rtimes \text{map}(A,\tilde\Sigma^n A)\rightarrow \tilde\Sigma^n
A.
$$
Taking $(t,a)$ to $(a,f_t)$ where $f_t$ is the $t$--level identity map
$f_t\colon A\rightarrow \tilde\Sigma^n A$ taking $a\rightarrow (t,a)$; here $t$
varies over $I^n=I\times\ldots\times I$.
\enddemo
Therefore again one gets
$$
T^\infty (\tilde\Sigma^n A)@>\simeq >> \tilde\Sigma^n A.
$$
Now one can define a split for
$X={\underset i\to \bigvee}\tilde\Sigma^{n_i} A$
component--wise, thereby proving the proposition.
\proclaim{Theorem} For any finite space with $[A,X]\simeq *$
$T_A^\infty X\simeq \CW_A X$.
\endproclaim
\demo{Proof} We use a result of Blanc--Thompson according to which there is
a functorial resolution $U^A_\cdot X\rightarrow X$ where $U_n^A X\simeq
V \tilde\Sigma^{n_i} X$ for all $n$, and with the property that
$\|U_\cdot^A X|\simeq \CW_A X$.
$U_\cdot^A X$ is essentially an $A$--version of the Blanc--Stover resolution
??\ \ ].
Now consider the bi--simplicial space $T_a^n U_k^A X$.
We claim that its diagonal $D=\text{diag} T_A^\circ U_\circ^A X$ is
$\CW_A X$.
This is so because the diagonal is equivalent to the realization of the
simplicial space $\|T_A^\infty U_k^A X\|_k$ which by the above is equivalent to
$\|U^A X^U\cong \CW_A X$.
But taking first the realization along the $U_k^A$--direction we get $D_n X
\simeq T_A^n \|U_k^A X\|=T_A^A \CW_A X$ by Thomson--Blanc's result.
But clearly the map $\CW_A X\rightarrow X$ induces an equivalence
on $T_A^n$ and $T_A^\infty$.\hfill
\enddemo
\vskip .25truein
\centerline{\bf 4.\ \ Commuting $\CW_A$ with other functors $(\Omega,\Sigma,
P_A)$}
In this section we consider the relations between $\CW_A X,\ \CW_{\Sigma^k A}
X,\ \CW_A \Omega X$ and $P_A X$.
Technically speaking these are the most useful properties of $\CW_A$ and in
particular these are crucial for understanding the preservation of fibration
under it.
We get four main results:
\medskip
\item{4.1}$\CW_A \Omega X\cong \Omega \CW_{\Sigma A} X$\hfill (1.4)
\item{4.2}If $\CW_A X @>>> X @>>> \CW_{\Sigma A} X$
is null homotopic then the sequence is a fibration.\hfill (1.11)
\item{4.3}If $\CW_{\Sigma^2 A} X\simeq *$ then $\CW_{\Sigma A} X$ is a GEM.
\hfill (1.9)
\item{4.4}The fibre of $\CW_{\Sigma^2 A} X @>>> \CW_{\Sigma A} X$ is a
polyGEM which is also $A$--cellular and $\Sigma^2 A$--periodic.
\medskip
We start with (2) since it exposes the close relationship between $\CW_A X$
and $P_A X$ and in fact directly implies the rest under additional assumptions.
In fact (2) motivated our interest in $\CW_A$.
\medskip
\proclaim{4.5\ \ Theorem (see 1.7 above)} Consider the sequence
$$
\CW_A X @>\ell >> X @>r >> P_{\Sigma A} X
$$
for arbitrary pointed spaces $A,X$.
This sequence is a fibration sequence if (and only if) the composition
$r\circ \ell$ is null homotopic.
In particular, if $[A,X]\simeq *$ or $P_{\Sigma A} X\simeq *$ then this is a
fibration sequence.
\endproclaim
\demo{Proof} With first prove the special case $P_{\Sigma A} X\simeq *$.
\enddemo
\proclaim{4.6\ \ Proposition} For any $\CW$--complex $A,X$ in ${\script S}_*$, if
$P_{\Sigma A} X\simeq
*$ then $\CW_A X @>\simeq >> X$ is a homotopy equivalence.
\endproclaim
\demo{Proof} Consider the fibre sequence:
$$
\Omega X @>>> F @>>> \CW_A X @>>> X.
\leqno(*)
$$
In this sequence we now show that $F\simeq *$.
\enddemo
In order to show that we show:
\item{(1)}map$(A,F)\simeq *$
\item{(2)}$P_A F\simeq *$
\noindent
Clearly any space $Y$ that satifies (1) i.e.\ is $A$--periodic does not change
under $P_A$ thus (1) and (2) imply $F\simeq *$.
The fibration (*) implies that map$(A,F)$ is the homotopy fibre of
map$(A,\CW_A X) @>>> \text{map}(A,X)$ over the trivial component.
But by definition of $\CW_A$ the latter map is a homotopy equivalence thus its
fibre is contractible and (1) holds.
To prove (2) we use theorem $P(4)$ in (0.3) above, with respect to the fibration
sequence $\Omega X @>>> F @>>> \CW_A X$.
First notice that by $P(4)$ $P_A \Omega X\simeq\Omega P_{\Sigma A} X$ which
is by our assumption contractible.
But now Theorem $P(5)$ means that $P_A F @>\cong >> P_A \CW_A X$ is a
homotopy equivalence.
Theorem (1.5) above now implies $P_A F\simeq *$ as claimed in (2).
This completes the proof of the proposition.
\noindent
We now proceed with the proof of the theorem.
Let $Y$ be the fibre of $X @>>> P_{\Sigma A} X$.
By theorem $P(3)$ we deduce that $P_{\Sigma A} Y \simeq *$ and therefore by the
proposition just proved we deduce $\CW_A Y @>\simeq >> Y$ is a homotopy
equivalence.
The following claim now completes the proof:
\demo{Claim} $\CW_A Y \simeq \CW_A X$.
\enddemo
\demo{Proof} The map $Y @>>> X$ gives us a map
$Y\simeq \CW_A Y @>>> \CW_A X$.
Since both spaces are $A$--cellular it suffices by the A--Whitehead theorem
(3.1) to prove that we have a homotopy equivalence:
$$
\text{map}_*(A,\CW_A Y)\simeq \text{map}_*(A,\CW_A X).
$$
\enddemo
Since for all $W$ the map $\CW_A W @>>> W$ is a natural $A$--equivalence it
suffices to show that map$(A,Y)@>>> \text{map}(A,X)$ is a homotopy equivalence
of function complexes.
Consider first the set of components:\ \ By definition of $Y$ as a fibre we
have an exact sequence of pointed sets:
$$
[\Sigma A,P_{\Sigma A}X]@>>> [A,Y]@>>> [A,X]@>>> [A,P_{\Sigma A} X].
\leqno(*)
$$
First notice that by universal property of $P_{\Sigma A}X$ the left--most group
is zero.
Now we claim that it follows from the assumption $r\circ \ell \simeq *$ in
our theorem, that the right--most arrow is null.
This is because by the universal property of $\CW_A$
(Theorem 1.4 (2), 3.8 (U.2))
every map$A@>>> X$ factors (uniquely up to homotopy) through $\CW_A X @>>> X$,
therefore its composition with $r\colon A @>>> X @>r>> P_{\Sigma A} X$ must be
null homotopic.
Therefore the middle arrow in $(*)$ is an isomorphism of sets.
Now consider the pull--back sequence:
$$
\CD
\text{map}_*(A,Y) @>>> \text{map}_*(A,X)\\
@VVV @V\overline{r} VV\\
* @>>> \text{map}_*(A,P_{\Sigma A} X;\text{null})\simeq *
\endCD
$$
We just saw that $\overline{r}=\text{map}_*(A,r)$ carries the whole function
complex to the null component of map$(A,P_{\Sigma A} X)$.
Therefore we can and do restrict the lower right corner of the square to
the null component.
But we claim that the component of the null map in map$(A,P_{\Sigma A}X)$ is
contractible since its loop $\Omega\text{map}(A,P_{\Sigma A} X;\text{null})$
is by adjunction just map$(\Sigma A,P_{\Sigma A}X)\simeq *$, as needed.
Now a pull back square with two lower corners contractible implies an
equivalence of the top arrow as needed.
\demo{4.7\ \ Proof of 4.1} We now turn to the proof of basic adjunction (4.1).
If $[A,X]\sim *$, then this equation was just proved since it follows from
(2) and the corresponding
equation for $P_A$ namely the equivalence (1.4) above:\ \ $P_A \Omega X
\simeq \Omega P_{\Sigma A} X$.
But here we prove it in general:
\enddemo
\proclaim{4.8\ \ Theorem} For any $A,X\in {\script S}_*$ we have a homotopy
equivalence
$$
r\colon \CW_A \Omega X @> \leftarrow >>
\Omega \CW_{\Sigma A} X\colon \ell.
$$
\endproclaim
\demo{Proof} The proof follows very closely the proof for the periodization
functor $X\rightarrow P_A X$.
Let us recall in broad lines the proof.
We use Segal's $\Delta$--space machine[1][?] to recognize $\CW_A\Omega X$ as a
loop space and the map $\CW_A \Omega X @>>> \Omega X$ as a loop map.
For this we need only an augment functor ${\script S}_*\rightarrow {\script S}_*$
that commutes with
products:\ \ This we have in virtue of theorem above:\ \ Thus we can write
$\Omega X$ as a simplicial space $Y_\cdot$ with $Y_1=X,\
Y_2=X\times X\ldots Y_n=X^n$, and
$Y_\cdot$ is a simplicial space which is very special namely the natural map
$Y_n @>\simeq>> Y_1\times \ldots\times Y_1 (n$--times) is a homotopy
equivalence.
Applying $\CW_A$ to $Y_\cdot$ and using $\CW_A (X^n)\simeq (\CW_A X)^n$ we
get immediately that $\Omega X @>>> X \CW_A \Omega X$ is in fact a loop map.
We now use universality properties of $\CW_A$ to get the desired equivalence.
First take
$$
\Omega_{j\Sigma}\colon \Omega \CW_{\Sigma A} X\rightarrow \Omega X
$$
to be the loop of the structure map for $\CW_{\Sigma A}$.
We get a diagram
$$
\CD
& \CW_A\Omega X\\
r j\Omega\\
\Omega \CW_{\Sigma A} X @>\Omega_{j\Sigma}>>\Omega X
\endCD
$$
It is easy to see that $(\Omega_{j\Sigma})$ induce a homotopy equivalence on
map$(A,-)$.
Therefore, by universality of the map $(j\Omega)$ we get the map $r$, which is
unique up to homotopy.
To get the map $\ell$ we first construct a map ``$\overline{W}\ell$''
$$
\ell^\prime \colon \CW_{\Sigma A}X\rightarrow \overline{W} \CW_A\Omega X
$$
Where $\overline{W}$ is the classifying functor otherwise denoted by $B-$.
Here we use crucially the fact proven above that $\CW_A \Omega X @>>> \Omega X$
is a loop map.
One deloop this map to get a map $\overline{W}j\Omega\colon \overline{W}
\CW_A \Omega X @>>> X$.
We lift the structure map $\CW_{\Sigma A} X @>>> X$ to $\overline{W}\CW_A
\Omega X$.
Again this lift exists by universality of $\CW_{\Sigma A}X$ (3.8 U.1)
since $\overline{W}(j\Omega)$ is easily seen by adjunction to induce
homotopy equivalence on the
mapping space from $A\colon$ i.e.\ map$(A,\overline{W}j\Omega)$ is a h.e.
Since these two maps were defined by universality it is easily checked as in
??DF] that these are mutual inverses up to homotopy.
\proclaim{4.9\ \ Corollary (1.5)} In the fibration (2.8) above $B$ and $E$
are in ${\Cal C}^\cdot (\Sigma A)$
then $F$ is in ${\Cal C}^\cdot (A)$.
If $F$ and $B$ are $\Sigma A$ cellular then $E$ is $A$--cellular.
\endproclaim
\demo{Proof} This is immediate from above by considering the fibrations
$\Omega E\rightarrow \Omega B\rightarrow F$ and $\Omega B\rightarrow
F\rightarrow E$ and the fact that $\CW_A \Omega B\simeq\Omega \CW_{\Sigma A}
B$ so that the latter is an $A$--cellular, and so is $\Omega \CW_{\Sigma A}E$.
\hfill
\enddemo
\bigskip
\noindent
{\bf Deviation by GEM and PolyGEM}.
We now turn to consider the relations between $\CW_{\Sigma^k A} X$ for
various $k$'s.
This will pave the way for considering the difference between $\CW_A X$ and
$\CW_A \Omega X$.
The slogan is ``Whenever the function complex map$_*(\Sigma A,X)$ is
homotopically discrete (i.e.\ it is h.e.\ to a discrete space) the space
$\CW_{\Sigma A}X$ is a GEM and thus the above set is an abelian group''
(compare ??Bou--4 6.7].
Recall that (1.19) if $P_{\Sigma A} X \simeq *$ then $P_{\Sigma^2 A} X$ is a
GEM.
Here we have a similar result about $\CW_A X$ (see .??..).
\proclaim{4.10\ \ Theorem} Assume $\CW_{\Sigma^2 A} X\simeq *$.
Then $\CW_{\Sigma A} X$ is a GEM.
\endproclaim
\demo{Proof} Consider the natural square of maps associated to any space:
$$
\CD
j\colon \CW_{\Sigma A} X @>>> X\\
@VVV @V\simeq VV\\
P_{\Sigma^2 A}j\colon P_{\Sigma^2 A} \CW_{\Sigma A} X @>>> P_{\Sigma^2 A}
\endCD
$$
\enddemo
Our assumption is equivalent to map$_*(\Sigma^2 A,X)\simeq *$ i.e.
$X$ is $\Sigma^2 A$--periodic and the right vertical map is an equivalence.
Now since $P_{\Sigma A}\CW_{\Sigma A}X\simeq *$ for any $X,A$ by 1.4 (11) above,
we get from ??EDF--S] that $P_{\Sigma^2 A}\CW_{\Sigma A}X$ is a GEM.
Therefore we conclude that the canonical map $\CW_{\Sigma A} X\rightarrow X$
factors up to homotopy through a GEM.
Since by lemma 4.11 below $\CW_A$ (GEM) is always a GEM we conclude that
$\CW_{\Sigma A}X$ is a retract of a GEM, thus a GEM.
\demo{4.11\ \ Lemma} For any space $M$ of the homotopy type of product of
Eilenberg--MacLane spaces $K(G,n)$ with abelian $G$ (i.e.\ $M$ is a GEM),
we have $\CW_A M$ is also a GEM.
\enddemo
\demo{Proof} This follows from the structure map $K({\Bbb Z},n)\times M
\rightarrow M$ for any GEM space $M$, realizing $K({\Bbb Z},n)$ and $M$ as
strictly abelian groups which is always possible.
The above action presents $M$ as a module over $K({\Bbb Z},n)$ for any $n\geq
0$.
Since for any $k\in K({\Bbb Z},n)\ k\cdot 0=0$ for $0\in M$.
Therefore for $k\in K({\Bbb Z},n)$ we get a pointed map $M@>>> M$ and thus we
have an induced map $\CW_A M @>>> \CW_A M$.
This consideration together with the usual machinary [Bou] yield an action
$K({\Bbb Z},n)\times \CW_A M\rightarrow \CW_A M$.
Therefore $\CW_A M$ is presented as a $K({\Bbb Z},n)$--module for any $n\geq 0$,
this it is a GEM.
\enddemo
\proclaim{4.12\ \ Theorem} Let $Y$ be a GEM with $\pi_i Y=0$ for $i\geq 0$.
Then $\pi_i \CW_A Y\simeq 0$ for $i\geq n$.
\endproclaim
\demo{Proof} Assume $\pi_k \CW_A Y\cong G\napprox 0$ for some
$k\geq n$.
Since $\CW_A Y=W$ is also a GEM by theorem above, $K(G,k)$ is a retract of $W$
and therefore by proposition ... $K(G,k)$ is an $A$--cellular space.
Therefore the retraction $\CW_A K(G,k) {\underset \leftarrow\to\rightarrow}
\CW_A Y=W$ is equivalent to the original retraction $K(G,k){\underset
\leftarrow\to\rightarrow} W$.
For dimensional reasons since $k\geq n$, the composition $K(G,k)\rightarrow
W\rightarrow Y$ is null.
Applying $\CW_A$ we get null map, a contradiction.
\enddemo
\noindent
{\it Problem}:\ \ Is it true that for any $A$ and a polyGEM $X$ the space
$\CW_A X$ is also a polyGEM?
(A polyGEM is ``generalized $n$--Postnikov stage'').
\noindent
Finally we turn to (4) which may not be the best possible result:
\proclaim{4.13\ \ Theorem} For any $A,X\in {\script S}_*$ the homotopy fibre $F$ of
$$
j\colon \CW_{\Sigma^2 A} X @>>> \CW_{\Sigma A} X
$$
is a polyGEM.
\endproclaim
\demo{Proof}This follows from (1.12): Notice that $\P_{\Sigma A}$ kills both
the domain and range of $j$ above. Therefore $\P_{\Sigma A}F$ is a GEM.
But map$_*(\Sigma^2 A,j)$ is a
homotopy equivalence.
Therefore map$_*(\Sigma^2 A,F)\simeq *$ and $F$ is $\Sigma^2 A$--periodic.
Therefore by (1.12.2) the fibre of the map form $F$ to $\P_{\Sigma A}F$
is also a GEM and we are done.
\hfuzz=3.3pt
\subheading{4.14 Commuting $\CW_A$ with taking homotopy fibres}
We will now address the question of preservation of filbration by $\CW_A$.
Looking at $A=S^n$ we see immediately that $X\langle n\rangle=\CW_{S^{n+1}}$
being the $n$--connected cover of $X$ does not preserve fibration in general.
However we shall see that when $A$ is a suspension the functor $\CW_A$
``almost'' preserves fibrations, the error term being under control.
In order to measure the extent to which $\CW_A$ preserves fibration we will
now compare the fibre of the $\CW$--approximation with the $\CW$--approximation
of the fibre via a natural map.
\subheading{4.15 $\lambda \colon \CW_A F @>>> \text{Fib} (\CW_A E @>>> \CW_A B)$}
associated to any fibration sequence $F @>>> E {\overset p\to \rightarrow} B$
over a connected $B$.
For $E \simeq *$ we get as a special case a map $\CW_A \Omega B @>>> \Omega
\CW_A X$ for any space $B$.
In order to construct $\lambda$ one notices that the fibre of the map $\CW_A(p)$
denoted here by Fib$\CW_A (p)$ maps naturally to $F$.
This map induces an equivalence on function complexes map$_*(A, \text{Fib}
(\CW_A p)) @>>> \text{map}_*(A,F)$ since map$_*(A,-)$ commutes with taking
homotopy fibres.
Therefore, by the universal property (3.8) U1.there is a factorization
$\CW_A F @>>> \text{fib} (\CW_A (p))$ unique up to
homotopy.
Now in general one shows:
\proclaim{4.16 Proposition} Whenever $A=\Sigma^2 A^\prime$ is a double the
homotopy fibre $\Delta$ of the above natural $\lambda$ is an extension of two
GEM spaces:\ \ (2--GEM).
$$
(\text{GEM})_2 @>>> \Delta @>>> (\text{GEM})_1.
$$
Moreover $\Delta$ is an $A$--periodic, $A^i$--cellular space.
\endproclaim
\demo{Proof} First we notice that by a straightforward argument one shows
that map$_*(A,\Delta)\simeq *$, i.e.\ $\Delta$ is $A$--periodic.
This is because the map map$_*(A,\lambda)$ is a homotopy equivalence.
Since the fibre $J$ of $P_{\Sigma^2 A^\prime} \Delta @>>> P_{\Sigma A^\prime}
\Delta$ is a GEM (by 1.5 $P_{\Sigma A^\prime} J \simeq *$ and $J$ is
$\Sigma^2 A^\prime$--local so use 1.12).
Since
$P_{\Sigma^2 A^\prime} \Delta \simeq \Delta$ it is sufficient to show
$P_{\Sigma A^\prime}\Delta$ is a GEM.
For this we use (1.12).
By (4.9) above both domain and range of $\lambda$ are $\Sigma A^\prime$--cellular and thus (1.7) both are killed by $P_{\Sigma A^\prime}$.
Therefore the condition of 1.12 is satisfied and $P_{\Sigma A^\prime} \Delta$
is a GEM.
This completes the proof.
\enddemo
\proclaim{4.17 Corollary} For $A=\Sigma^2 A^\prime$ the fibre of
$\CW_A \Omega X @>>> \Omega \CW_A X$ is a 2--poly GEM.
\endproclaim
\demo{Proof} Apply the above to $\Omega X @>>> * @>>> X$.
\enddemo
\demo{Remark} Using the adjunction (4.8) (4.17) is a special case of 4.13
albeit with more control on the fibre.
\enddemo
\proclaim{4.18 Proof of 0.11}
This follows directly from (4.10) since by
(4.8) $B$ satisfies the condition: The triviality of the A-CW approximation
is equivalent to the triviality of the pointed function complex from $A$
and this follows directly form the condition in 0.11.
\endproclaim
\bigskip\noindent
{\bf 5. \ \ A Fibration Theorem}
\bigskip\noindent
\proclaim{5.1 Theorem F} Given any fibration of pointed space $F\rightarrow
E\rightarrow B$ one can map the following fibration into it:
$$
\CD
\CW_A F @>>> \overline{E} @>>> \CW_{\Sigma A} B\\
@VfVV @VgVV @VhVV\\
F @>>> E @>>> B
\endCD
$$
where $\overline{E}$ is $A$--cellular and $g$ a weak $\Sigma A$--equivalence.
\endproclaim
\demo{5.2 Remark} Thus although $\overline{E}$ is $A$-cellular it is slightly
removed from being $\CW_A E$
since it is only $\Sigma A$--equivalent to $E$ not $A$--equivalent to it.
We shall soon see that the fibre of a canonical map
$E @>>> \CW_A E @>E>> \rightarrow \CW_A E$
associated to such fibration is a GEM for $A=\Sigma A^\prime$, a suspension
space.
\enddemo
\demo{Proof} This is similar to ?F] and ?Bou], and uses crucially
the equivalence
$\CW_A \Omega X\simeq \Omega \CW_{\Sigma A}X$
and $\CW_A (X\times Y)\simeq \CW_A X\times \CW_A Y$ (3.9) and (4.8)
We consider first the associated principal fibration:
$$
\Omega B\rightarrow F\rightarrow E
$$
where we consider $E$ as the `quotient space' up to homotopy of $F$ under
the action $\Omega B\times F\rightarrow F$.
This action gives rise to a map
$$
\CW_A (\Omega B)\times \CW_A F\rightarrow \CW_A F,
$$
or $\Omega \CW_{\Sigma A}B\times\CW_A F\rightarrow \CW_A F$.
We now use the equivalence (4.8) and the usual arguments ?Bou] show that the original action $\Omega B\times F
\rightarrow F$ gives rise to an action of $\Omega \CW_{\Sigma A} B$ on
$\CW_A F$.
Therefore we get a principal fibration
$$
\Omega \CW_{\Sigma A}B\rightarrow \CW_A F\rightarrow \overline{E}
$$
associated to that action. Classifiying this fibration gives us the
desired sequence.
Since in that last fibration both fibre and total spaces are
$A$--cellular so is
$\overline{E}$ by (2.9).
Further since in the map of fibrations $\Omega h$ is an $A$--equivalence
and $f$ is an $A$--equivalence we get by the usual long exact sequence for
$A$--homotopy groups $[\Sigma^\ell A,-]^*$, that $g$ is an
$\Sigma A$--equivalence.
This completes the proof.
\enddemo
\proclaim{5.3 Corollary} Let $F @>>> E @>p>> B$ be any fibration sequence in
${\script S}_*$ and let $\Sigma^2 A$ be any double suspension in ${\script S}_*$.
If $B$ is $\Sigma^2 A$--cellular and $[\Sigma A,p]\sim *$ then
$$
\CW_{\Sigma A} F @>>> \CW_{\Sigma A} F @>>> B=\CW_{\Sigma A} B
$$
is also a fibration sequence.
\endproclaim
\demo{5.4 Example} Take $A=S^1$.
Then the theorem states the easily checked fact that over a 2--connected
space $B$ one can define {\it fibrewise universal covering} space.
\enddemo
\demo{Proof} We consider the diagram:
$$
\CD
Y= & & Y\\
@VVV @VVV\\
\CW_{\Sigma A} F @>>> \overline{E} @>>> \CW_{\Sigma^2 A} B=\CW_{\Sigma A} B\\
f@VVV g@VVV \|\\
F @>>> E @>>> B
\endCD
$$
This diagram is constructed using the theorem above.
We claim that there is a natural equivalence:\ \ $\overline{e}:\overline{E}
@>>> \CW_{\Sigma A} E$.
First notice that by theorem 2.8 $\overline{E}$ is $\Sigma A$--cellular.
Therefore there is a unique natural factorization $\overline{e}$.
To check that $\overline{e}$ is a homotopy equivalence we use the A--Whitehead
theorem and check that the map
$$
\text{map}_* (\Sigma A,\overline{e})\colon \text{map}_*(\Sigma A,
\overline{E}) @>>> \text{map}(\Sigma A, \CW_A E)
$$
is a homotopy equivalence.
Notice that map$_*(\Sigma A,Y)\simeq *$ since $Y$ is the fibre of $f$
and map$_*(\Sigma A, f)$ is a
homotopy equivalence.
We know from theorem F above(6.2) that G is a $\Sigma^2 A$- equivalence. But
$\text map^*(\Sigma A,Y)\simeq *$ since $Y$ is the fibre of a $\Sigma A$-
equivalence f. Therefore it is enough to check the function complex on
the level of components,since all the components are equivalent to each
other. Our assumption guarantees that it is surjective on componenets and
injectivity follows from the above mentioned propery of $Y$.
\bigskip
\bigskip\noindent
6.\ \ {\bf Examples and $E_*$--acyclic spaces.}
Many well--known constructions in algebraic topology leads to $A$ cellular
spaces.
\proclaim{6.1\ \ James functor $JX$} For any $X$ the space
$JX\simeq \Omega \Sigma X$
is an $X$ cellular space:
$$
\CW_X JX\simeq JX.
$$
\endproclaim
\demo{Proof} We have a filtration
$$
J_n X \subset JX
$$
with homotopy pushouts of pointed spaces:
\enddemo
$$
\CD
J_n X \vee X @>>> J_n X\times X\\
@VVV @VVV\\
J_n X @>>> J_{n+1} X.
\endCD
$$
This gives an inductive definition of $J_n X$.
Since by theorem 2.9 a product of two $X$--spaces is an $X$--space we get by
induction that $J_{n+1}X$ is an $X$--space.
Therefore $JX$ = hocolim $J_n X$ is also an $X$ space.
\noindent
{\bf Hilton--Milnor--James decomposition}:\ \ This theorem provides a decomposition
of $\Sigma \Omega \Sigma X$ for an arbitrary pointed $X$ in as a wedge of
smash--powers of $\Sigma X$.
Thus it gives an explicit description of $\Sigma \Omega \Sigma X$ as an
$\Sigma X$--cellular space:\ \ (Any smash--power of $W$ is $W$--cellular).
Using the adjunction relation (4.8 ) yields immediately that $\Sigma \Omega
\Sigma X$ is in fact a $\Sigma X$--cellular without however saying anything
about the nature of the decomposition.
One computes:
$$
\eqalign{
\CW_{\Sigma X} \Sigma\Omega\Sigma X&=\overline{W} \CW_X \Omega \Sigma \Omega
\Sigma X\cr
&=\overline{W}\Omega\Sigma\Omega\Sigma X=\Sigma\Omega\Sigma X\cr
}
$$
Here we used $\CW_Y \Omega \Sigma Y=\Omega \CW_{\Sigma Y} \Sigma Y=\Sigma Y$.
We get that $\Omega\Sigma(\Omega\Sigma X)$ is $\Omega\Sigma X$--cellular which
is $X$--cellular.
Notice, however, that for non suspension $\CW_Y \Sigma\Omega Y \not= \Sigma
\Omega Y$.
In fact $\Sigma\Omega Y$ is not a $Y$--cellular space, rather the other way
around as we saw $X$ is $\Sigma\Omega X$--cellular.
For example $\Sigma\Omega K({\Bbb Z},3)=\Sigma {\Bbb C} P^\infty$ is not
$K({\Bbb Z},3)$--cellular since any $K({\Bbb Z},3)$--cellular space must have
vanishing reduced complex $K$--theory and $\Sigma {\Bbb C} P^\infty$ is not
$K$--acyclic.
Similarly $\Omega^n S^n X$ is also an $X$--cellular space.
\proclaim{6.2\ \ Theorem} For any $X$ the Dold--Thom functor $SP^\infty X$
is an $X$--space.
\endproclaim
This follows from a much more general observation about arbitrary
``convergent functors'' of ?B--F] [Bou], or $\Gamma$--spaces of
??Segal].
Let $\Gamma^\circ c$ Sets$_*$ be the full subcategory of the objects
$n^+=\{0,\ldots,n\}$ with base point $0\in n^+$, for $n \geq 0$.
A $\Gamma$--space is a functor $\cup\colon \Gamma\rightarrow {\script S}_*$.
It is special if $\cup(n^+)=U(1^+)\times\ldots\times \cup(1^+)$.
Each $\Gamma$--space determines a functor $\cup {\script S}_*\rightarrow {\script S}_*$ with
$\cup X=\text{diag}(\cup X\cdot)\cdot$
where $(\cup X_k)_\cdot$ is the space associate by the $\Gamma$--space
$\cup$ to the set of $k$--simplices $X_k$ of $X$.
Thus every very special $\Gamma$--space $h\colon \Gamma\rightarrow {\script S}_*$
determines a reduced homology theory $\pi_+ hX\equiv h_* X$.
\proclaim{6.3\ \ Proposition} For any $\Gamma$--space $U$ and any
$X\in {\script S}_*$ the space $U X$ is an $X$ cellular--space.
\endproclaim
\demo{Proof} Almost by definition $U$ can be written as the ``tensor product
of $\Gamma^{\circ p}$--spaces with $\Gamma$--space:\ \ ??Bou--4 6.4]
\enddemo
$$
U X\simeq X^\cdot \otimes_\cdot U(\cdot)
$$
Where $\otimes_\cdot$ denotes the ``coend'' coequalizer ??Mac] over $\Gamma$.
Notice that $X^\cdot\colon\Gamma^{\circ p}\rightarrow$ space is
$$
\eqalign{
X^{n^+}&=X\times \ldots \times X,\ (n+1)-\text{times}.\cr
X^{n^+}&=\text{map}(n^+,X)\ \text{this gives a functor}\cr
&\Gamma^{\circ p}\rightarrow\text{spaces}.\cr
}
$$
Now since by definition $X^{n^+}$ is an $X$ space and by lemma 2.3 above
$X^{n^+}\wedge Y$ is an $X$--space for any $Y$ we get that $U X$ is a
hocolim of $X$--spaces and therefore a $\Gamma$--space.
\medskip\noindent
In order to deduce theorem above it is enough to show that
$SP^\infty X$ is equivalent to $UX$ for some $\Gamma$--space $U\colon
\Gamma\rightarrow {\script S}_*$.
But ??Bou 6.2] shows that choosing the discrete $\Gamma$--space $\tilde{\Bbb Z}$
to be $\tilde{\Bbb Z}(n^+)={\Bbb Z}\oplus\ldots\oplus{\Bbb Z}\ n$--times and
regarding $\tilde{\Bbb Z}$ as a discrete $\Gamma$--space, gives $\tilde{\Bbb Z}
X\simeq SP^\infty X$.
Therefore $SP^\infty X$ is an $X$--space.
By the same token $\Omega^\infty S^\infty X$ is also an $X$--space since
$\Omega^\infty S^\infty X$ can be realized as a $\Gamma$--space.
Further examples of $A$ cellular--spaces can be gotten from the following.
\proclaim{6.4\ \ Proposition} Let $n\geq 3$ then for any Morava $K$--theory
$K\langle n\rangle$, there exists an integer $k(n)$ such that $h>h(n)
\Rightarrow K(G,n+k)=\CW_{V(n)} K(G,n+k)$.
In particular $K(G,n+k)$ are homotopy colimits of $K\langle n\rangle$--acyclic
subcomplexes.
\endproclaim
\demo{Proof} See ??EDF].
The point is that one can show that $P_{\Sigma V(n)}K(G,n+k)\simeq *$ for
large $k$.
\enddemo
\bigskip\noindent
{\bf 6.4\ \ Classifying Spaces}: \ \ It is not hard to see directly that Milnor's
classifying space construction leads to a description of $BG$, for any
group--space $G$, as $G$--cellular space i.e.\ $BG\in {\Cal C}^\cdot (G)$.
But this fact is a direct corollary of (??? ) and (?? ) above.
In fact to check $P_{\Sigma G}BG\simeq *$ one use (\ \ \ ) to get
$P_{\Sigma G}BG=\overline{W} P_G\Omega BG\simeq \overline{W} P_G G\simeq
\overline{W}\{*\}=\{*\}$.
Therefore $BG=\CW_G (BG)$ as needed.
In particular $K(G,n+k)$ is a $K(G,n)$--space for any $k\geq 1$.
Moreover $BG$ is always a $\Sigma G$--space since (using 4.1)
$$
\CW_{\Sigma G} BG \simeq B\CW_G \Omega BG=B \CW_G G=BG.
$$
>From this observation we get also
\enddemo
\demo{6.5\ \ Corollary} Any connected space $X$ in ${\script S}_*$, is in
${\Cal C}^\cdot (\Sigma\Omega X)$.
\enddemo
\proclaim{6.6\ \ Proposition} For all $n,k\geq 0\ K(G,n+k)$ is a
$K({\Bbb Z},n)$ cellular--space.
\endproclaim
\demo{Proof} For $n=1$ it is clear since $K({\Bbb Z},1)=S^1$ and
$K(G,n+k)$ is a connected $\CW$--complex.
For $n\geq 2\ K(G,n+k)$ is an abelion Eilenberg--Maclane complex.
If $G$ is abelion then $G$ = dir lim $G_\alpha$ where $G$ is the system
of finitely generated abelion subgroups.
So $BG = P$--hocolim $BG_\alpha$.
Therefore it is sufficient to prove the proposition for $G$ = finitely
generated abelion group.
In that case we have a homotopy fibre sequence
\enddemo
$$
K(F,n)\rightarrow K(F^\prime,n)\rightarrow K(G,n).
$$
which corresponds to the representation of $G\simeq F^\prime/F$ as a quadrant
of two finitely generated free abelion groups.
Now $K(F,n)$ and $K(F^\prime,n)$ are finite products of $K{\Bbb Z},n)$ with
itself and therefore by (3.7) $K({\Bbb Z},n)$ cellular--space.
Now by theorem (2.9) above it follows that $K(G,n)$ is $K({\Bbb Z},n)$--space.
\demo{6.7\ \ Example} $\CW_{K({\Bbb Z}/p {\Bbb Z},1)} K({\Bbb Z}/p^2
{\Bbb Z},1)=K({\Bbb Z}/p{\Bbb Z},1)$.
Proof:
Consider the map $g\colon {\Bbb Z}/p{\Bbb Z}\rightarrow
{\Bbb Z}/p^2 {\Bbb Z}$ of abelian groups $1\rightarrow p$.
This is the generator of Hom$({\Bbb Z}/p {\Bbb Z}, Z/p^2 {\Bbb Z})\simeq
{\Bbb Z}/p{\Bbb Z}$,
$g$ induces $Bg\colon K({\Bbb Z}/p{\Bbb Z},1)\rightarrow K({\Bbb Z}/p^2
{\Bbb Z},1)$.
Since the source is clearly a $K({\Bbb Z}/p{\Bbb Z},1)-\CW$--space it is
sufficient to show that $Bg$ induces a homotopy equivalence on the pointed
function complex map$(K({\Bbb Z}/p {\Bbb Z},1),\ Bg)$.
But the pointed function complex is homotopically discrete with
\enddemo
$$
\text{map}_*(K({\Bbb Z}/p{\Bbb Z},1),\ K(G,1))=\text{Hom}({\Bbb Z}/p{\Bbb Z},G).
$$
Therefore the above map $Bg$ gives us the correct $\CW_A$--approximation for
$A=K({\Bbb Z}/p{\Bbb Z},1)$.
\proclaim{6.8\ \ Corollary} If $A=K({\Bbb Z}/p^k {\Bbb Z},n)$ and
$X=K({\Bbb Z}/p^\ell {\Bbb Z},n)$ then
$$
\CW_A X=\cases A &\text{if $k\leq\ell$}\\
X&\text{if $k\geq \ell$}.\endcases
$$
\endproclaim
\demo{Proof} This is clear using the above together with the fibration
theorem:
\enddemo
\noindent
Thus the fibration
$$
K({\Bbb Z}/p^2 {\Bbb Z},n)@>{\times p}>> K({\Bbb Z}/p^2 {\Bbb Z},n)\rightarrow
K({\Bbb Z}/p{\Bbb Z},n)\times K({\Bbb Z}/p {\Bbb Z} n+1)
$$
by (\ \ \ ) and
(\ \ \ ) above $K({\Bbb Z}/p{\Bbb Z},n)$ as a
$K({\Bbb Z}/p^2 {\Bbb Z},n)-\CW$--space.
\demo{6.9\ \ Example} Let $X=M^{n+1}(p^\ell)$ and $A=M^{n+1}(p)$ be two Moore
spaces, with\hfill\break
$H_n (M^{n+1}(p^\ell),{\Bbb Z})={\Bbb Z}/p^\ell {\Bbb Z}$.
Then $\CW_A X$ is a fibre in:
\enddemo
$$
F\rightarrow X\rightarrow K({\Bbb Z}/p^{\ell-1}{\Bbb Z},n).
$$
while $\Sigma X=\CW_A \Sigma X$.
\demo{Proof} To compute the fibre of the composition
$X\rightarrow K(\pi_n X,n)\rightarrow K({\Bbb Z}/p^\ell {\Bbb Z},n)$ as
$\CW_A X$ we consider the
pointed function complex of $M^{n+1}(p)$ into the fibration.
Since map$(M^{n+1}(p),\ K({\Bbb Z}/p^\ell,{\Bbb Z}))\simeq {\Bbb Z}/p{\Bbb Z}$
the homotopically discrete, by cohomological computation we first notice
that the fibre has the correct function complex from $M^{n+1}(p)$.
We then must show that the fibre is a $M^{n+1}(p)$ cellular--space.
But the fibre is a--$p$--torsion space so it has an Hilton--Eckman cell
decomposition $M^{n+1}(p)\cup C M^{n+2}(H_{n+1} (F),n+1)\cup \ldots$ where
all the attacking maps can be taken to be pointed maps.
This gives direct representation of $F$ as $M^{n+1}(p)$--cellular since clearly
\enddemo
\proclaim{6.10\ \ Lemma} For any $p$--group $G_n$ the Moore space
$M^{n+j}(G,n+j) j\geq 2$ is an $M^{n+1}(p)$--space.
\endproclaim
\demo{Proof} Use theorem 4.5
$$
P_{\Sigma M^{n+1}(p)} M^{n+j} (G,n+j) \simeq *
$$
since the localization is an $n$--connected $p$--torsion space with all
maps from $M^{n+1}(p)$ being null, thus this localization is contractible.
\enddemo
{\bf $E_*$-acyclic spaces}:\ \ The fibration (? ) relating $P_{\Sigma A}X$
and $\CW_A X$ can be used to show that certain $E_*$--acyclic spaces are
$V(n)$--cellular, where $V(n)$ are the spaces introduced by Smith ??Rav] [H--5].
\proclaim{6.11 Lemma} Let $X\simeq \CW_A X$ where $A$ is a finite complex with
$\tilde{E}_* A \cong 0$.
Then $X$ is the limit of its finite $E_*$--acyclic subcomplexes.
\endproclaim
\demo{Proof} Recall from above the construction of $\CW_A X$.
For a finite $A$ the limit ordinal $\lambda (A)$ is the first infinite
ordinal $w$.
Therefore in that case
$$
\CW_A X =\lim_{i < \infty} (X_1 \hookrightarrow X_2 \hookrightarrow X_i
\hookrightarrow )
$$
where $X_i$ are all subcomplexes of $X$.
But now by induction we can show that each $X_i$ is the limit of finite $E_*$--acyclic subcomplex.
Notice that if $A$ is any $E_*$--acyclic space then so is the half--suspension
$\tilde\Sigma^n A=S^n \rtimes A=S^n \times A/S^n \times \{*\}$ by a
Mayer--Vietoris argument.
Now if by induction $X_j=\lim\limits_{\underset \alpha\to\rightarrow} A_2 (i)$
where $A(i)$ are finite $E_*$--acyclic then since $X_{j+1}$ is a push--out
along a collection of maps from $\tilde\Sigma^n A$, $X_{j+1}$ is again
$\lim\limits_{\underset \beta\to\rightarrow} A_\beta (j+1)$ where $A(j+1)$ are all
finite.
This completes the proof.
\enddemo
Our principal tool to detect when is an $E_*$--acyclic complex $X$ the limit
of its finite $E_*$--acyclic subcomplexes is the following.
\proclaim{6.12 Proposition} Let $A$ be an $E_*$--acyclic finite complex.
Then $X$ is the limit of its finite acyclic subcomplexes if $P_{\Sigma A} X
\simeq *$.
\endproclaim
\demo{Proof} This is immediate from theorem (0.8) and the lemma above.
\enddemo
We apply the proposition to the spaces to spaces $V(n)$ of type $n+1$.
Thus $V(0)$ is $S^\prime \cup_p e^2$, and for every prime $p$ and $n\geq 0$
there exists a finite $p$--torsion space $X(n)$ of type $n$.
This means $\tilde{K}\langle m\rangle_* X(n)=0$ for all $m < n$ and
$\tilde{K}(m)_* X(n)
\not= 0$ for all $m \geq n$, where $K\langle n\rangle$ denotes the $n$--th
Morava $K$--theory (compare discussion in ?Rav] ?Thomp] ?H--S] ?Bou].)
We apply this proposition in three interesting cases using ?6, 9.14 and 13.6].
\proclaim{6.13 Theorem} In the following cases every $E_*$--acyclic space is in
${\Cal C}^\cdot (V(n))$ for an appropriate $n \geq 0$.
\item{(1)}For all $n$ there exist $m \geq n$ with $K(G,m+j)\in {\Cal C}^\cdot
V(n)$ for all $j$, and all $p$--torsion groups $G$.
\item{(2)}If $\tilde{K}_{\Bbb C} \Omega^2 X \simeq 0$ then $X\in {\Cal C}^\cdot
(V(n))$ for any $p$--torsion 2--connected $X$.
\item{(3)}For every $n\geq 1$ there exist $N \geq n$ so that if $X$ is
$N$--connected, $p$--torsion and $\tilde{S}(n)_* \Omega^N X=0$ then $X\in
{\Cal C}^\cdot (V(n))$.
\endproclaim
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\hfill May 1992,
\hfill Hebrew University,
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\medskip
\hfill Purdue University
\hfill W. Lafayette
\enddocument