Complexity and Good Spaces
               M. Intermont and J. Strom



                       May 27, 2004

                          Abstract

This paper is an exploration of two ideas in the study of closed classes:
the A-complexity of_a_space_X and the notion of good spaces (spaces
A for which C(A) = C(A)). A variety of formulae for the computation
of complexity are given, along with some calculations.  Good spaces
are characterized in terms of the functors CWA and PA .  The main
result is a countable upper bound for  A-complexity when A is a good
space.


MSC Classification  55Q05

Keywords  Closed Class, Complexity, Homotopy Colimit


Introduction


Our purpose in this paper is to explore and bring together two dis-
tinct strands in the study of closed classes, namely the complexity
of a space X with repect to another space A and the notion of good
spaces.
   A closed class is a class C of pointed topological spaces which is
closed under weak equivalences and pointed homotopy colimits. The
main examples of closed classes are those of the form C(A), the smallest
closed class containing the space A; in fact, every known example of a
closed class has this form. The spaces X 2 C(A) are called A-cellular
spaces; such spaces can be `built' from A by repeatedly performing
homotopy colimits, and this relation is often denoted A   X. For an
arbitrary space X and a fixed A, while X may not be in C(A), there is
a closest A-cellular approximation to X, which is denoted CWA (X).



                              1

   The A-complexity of X, denoted ~A (X), is the minimum (or-
dinal) number of homotopy colimit operations needed to construct
CWA (X) starting with wedges of copies of A [CDI ].  Thus ~A (X)
measures how difficult it is to build X from A. In principle, this in-
variant can take on arbitrarily large ordinal number values. However,
a theorem of Stover [S] implies that when A = S1, the S1-complexity
of any space X is at most 1.  More generally, the argument can be
extended to show that ~Sn(X)   2 for every space X and every n   0
[CDI ]. It is shown in the same paper that there are spaces X for which
~M(Z=p,n)(X) = !, the first infinite ordinal number.
   If C is a closed class, then we may add just enough spaces to it in
order to_ensure that it is also closed under extensions by fibrations; we
write C to denote_the resulting strongly closed class. If it happens
that C(A) = C(A), then we say that A is a good space. For example,
C(Sn) is easily seen to consist of all (n - 1)-connected spaces, which
is evidently closed under extensions by fibrations; hence Sn is a good
space. Furthermore, a recent result of Chach'olski, Parent and Stanley
[CPS_] shows that for every space A, there is a space B such that
C(A) = C(B); thus there are many good spaces. On the other hand it
is known that the Moore space M(Z=p, n) is not a good space. Good
spaces are not very well understood:  for example, it is not known
whether the wedge of two good spaces is also good.
   There are three main sections:  section 2 is concerned with A-
complexity; section 3 is devoted to the study of good spaces; and
in section 4 we bring these two themes together and prove our main
theorem on complexity with respect to good spaces.
   In x2, we develop formulae which facilitate the calculation, or at
least the estimation, of A-complexity.  We offer two results here as
examples.

Theorem 16  If A   X, then

     (a)  A    X (due to Chach'olski [C2 ]), and

     (b) ~ A ( X)   1 + ~A (X).


Theorem 18   If X ~  Y for some space Y and  A    X, then

     (a) A   X (due to Chach'olski [C3 ]), and

     (b) ~ A ( X)   ~A (X)   ~ A ( X) + 2.


We use our formulae to make some explicit computations, including,
for example, that ~K(G,n)(K(G, n + m))   2m for Eilenberg-MacLane
spaces.



                              2

   Then in x3 we turn to the study of good spaces. There are three
main results in this section. First, we show that if A is a good space,
then so is  A.  Next we characterize good spaces in terms of the
A-cellularization functor CWA and the A-nullification functor PA .

Theorem 23  A space A is good if and only if the natural sequence


                    CWA (X) ! X ! PA (X)


is a fibration for every space X.

Finally, we use these results to give an efficient construction of the
functor PA under the assumption that A is a good space.
   In the fourth and final section we bring together results from sec-
tions 2 and 3 in order to prove our main result: a countable upper
bound for the complexity of spaces with respect to the suspension of
a good space.

Theorem 25  If A is a good space, then


             ~ A (X)   ! + 1  for every space X.
As a corollary, we show that if A is good and A    Y , then ~A ( Y )
! + 1.

Acknowledgement. We are grateful to Wojciech Chach'olski for helpful
advice at many stages in the development and preparation of this
paper.
1    Preliminaries


In this section we establish our notation and conventions and recall
the basic definitions which we use throughout the paper.



1.1    Conventions


Unless otherwise stated, all spaces and maps in this paper are pointed;
all basepoints are denoted by *. We use ~ to denote weak equivalence.
The notation map *(X, Y ) indicates the space of pointed maps from
X to Y .
   We write Spaces * to denote the category of pointed spaces and
pointed continuous maps; Spaces  is then the category of unpointed
spaces and unpointed continuous maps.  We will need to refer to



                              3

both pointed and unpointed homotopy colimits:  we write hocolim*
for pointed homotopy colimit and hocolim for unpointed homotopy
colimit.  We refer to Bousfield and Kan [B-K ] for the definition and
basic properties of homotopy colimits.
   Our formulae involve ordinal number arithmetic, for which we refer
to [Fr, Ch. III, x10]. Addition of ordinal numbers is not commutative:
for example 1 + ! = ! 6= ! + 1, where ! is the first infinite ordinal
[Fr, p. 142]. Each ordinal number ff corresponds to a small category
whose objects are the ordinals fi < ff, and whose morphisms fi ! ffi
correspond to inequalites fi   ffi. We will use the same letter to refer to
the ordinal number and the category; thus a functor   : ! ! Spaces *
is a pointed telescope diagram.



1.2    Good Spaces


Definition 1 A class C of pointed spaces is called a closed class if
it is closed under weak equivalences and pointed homotopy colimits;
we say that C is strongly closed if, in addition, it is closed under
extensions by fibrations; that is, if F ! E ! B is a fibration sequence
with F, B 2 C, then E 2 C.


   The main examples of closed classes are the classes C(A), the small-
est closed class containing the space A. The spaces in C(A) are called
A-cellular. The relation B 2 C(A) is often denoted A   B. For any
closed_class C, we denote the smallest strongly closed class containing
C by C.
                                         _____
Definition 2 A space A is good if C(A) = C(A).


   There is an augmented functor CWA : Spaces *! Spaces *which
assigns to a space X its `best approximation' by an A-cellular space
[Fa2, Ch. 2].  In particular, the map CWA (X) ! X induces a weak
equivalence map *(A, CWA (X)) ! map *(A, X); any map which in-
duces a weak equivalence on map *(A, -) is called an A-equivalence.
Related to CWA there is the nullification functor PA which assigns to
X its `best approximation' by a space with map *(A, PA (X)) ~ * [Fa2,
Ch. 1].



1.3    A-Complexity


The A-complexity of a space was defined and studied in the paper
[CDI ].  The definition is made in terms of certain full subcategories



                              4

CAffof the category Spaces *.  Since these subcategories are full, it
suffices to specify their objects, and we begin by setting


     Obj CA0= {X | X ~ a retract of a wedge of copiesAof}.


Having defined Cfifor all ordinal numbers fi < ff, we set

                                                         A
 Obj CAff=  X | X ~ a retract ofhocolim*F where   : D ! C<ff ,

            S
where C<ff=   fi<ffCAfi, D runs over all small categories and   runs
over all functors.


Definition 3 The A-complexity of X is the ordinal number

                                             A
               ~A (X) = min  ff | CWA (X) 2 Cff.


   It follows from the construction of the functor CWA [Fa2, x 2.B]
that C(A) = CAfffor sufficiently large ff [CDI , Prop. 1.5]; hence every
space X has a well-defined A-complexity.
   We point out that we have begun using what will be a consistent
abuse of notation: writing X 2 CAffto mean X 2 Obj CAff.
2    Formulas for Complexity


The paper [CDI ] established a theoretical framework for complexity,
but did not seek to provide methods for its calculation.  Our pur-
pose in this section is to develop formulae that that will allow for the
computation, or at least the estimation, of complexity.
   We begin with two simple but crucial lemmas. The first concerns
complexity with respect to S0.


Lemma 4    For any space X, S0   X and ~S0(X)   1.


Proof  First replace X with a weakly equivalent simplicial complex;
such a replacement necessarily has the same A-complexity.  Let X
denote the simplex category of X; then X is weakly equivalent to the
unpointed homotopy colimit of the functor   : X ! Spaces defined
by  (oe) = * for each simplex oe of X. If we attach disjoint basepoints
throughout, we obtain the space X+ (X with a disjoint basepoint) as
a pointed homotopy colimit of   : X ! Spaces * with  (oe) = S0
for each oe in X.  If x 2 X is the basepoint (which we take to be a



                              5

vertex), then X is the cofiber of natural map {x}+ ! X+ ; thus we
may identify the external basepoint of X+ with the given basepoint
of X by enlarging the diagram slightly.                          2

   Our second lemma is a simple transitivity formula.


Lemma 5    If A   X   Y , then ~A (Y )   ~A (X) + ~X (Y ).


Proof Write ~A (X) = ff and ~X (Y ) = fi. Then, since X is A-cellular,
X 2 CAff. It follows that CX0  CAff, and hence that Y 2 CXfi  CAff+fi. 2
2.1    Wedges, Smashes and Products


We begin by giving some formulae for the behavior of complexity with
respect to some basic operations of homotopy theory.
   Wedges are easily dispesnsed with.

                                            W
Proposition 6  For any spaces Xi2 C(A), ~A (  iXi) = sup{~A (Xi)}.


Proof  Since CA0is closed under wedges, so is CAfffor all ff.        2

   We now consider suspensions and, more generally, smash products.


Proposition 7  If A   X, then A    A    X, and

     (a) ~ A ( X)   ~A (X), and

     (b) if ~A (X)   1, then ~A ( X)   ~A (X).


Proof  Part (a) follows instantly from the fact that suspension com-
mutes with homotopy colimits. For part (b), we simply observe that,
if ff   1, then CAffis closed under suspension.                     2

   When ~A (X) = 0, we cannot conclude that ~A ( X) = 0. The best
possible estimate, ~A ( X)   1, follows from a much more general
result about smash products.


Proposition 8  If A   Y , then for any space X,

     (a) A   X ^ Y , and

     (b) ~A (X ^ Y )   ~A (Y ) + 1.


                              6

Proof  Part (a) is well-known [Fa2, Thm. 2.D.8].  For (b), write
~A (Y ) = ff, so CY0   CAff, and hence CYfi  CAff+fifor all fi.  Since

X 2 CS01by Lemma 4, it follows immediately that X ^ Y 2 CS0^Y1
CAff+1, which means that ~A (X ^ Y )   ff + 1.                     2

   We have the following improvement when both spaces are A-cellular.


Corollary 9  If A   X, Y , then

     (a) A   X ^ Y , and

     (b) ~A (X ^ Y )   min{~A (X), ~A (Y )} + 1.


   The fact that closed classes are closed under cartesian products
is one of their most important features [Fa2, 2.D.15]. We finish this
subsection with estimates for the complexity of a product.


Proposition 10  Let A   X, A   Y and let M = max {~A (X), ~A (Y )}.
Then

     (a) A   X x Y ,

     (b) M   ~A (X x Y )   M + !, and

     (c) if A ~  B, then M   ~A (X x Y )   2 + M.


Proof  The cellular inequality (a) is well known.  The lower bound
M   ~A (X x Y ) is valid because X and Y are both retracts of X x Y .
   For (b) we recall the proof of the cellular inequality given in [Fa2,
Thm. 2.D.16].  Replace X and Y  with weakly equivalent CW com-
plexes if necessary. Set P (n) = {*} x Y [ X x Yn and observe that
P (n + 1) is the (homotopy) pushout of a diagram


                 X o Dn+1   X o Sn ! P (n).


Now P (0) = X _ Y  2 CAM, X o Dn+1 ~ X 2 CAM and X o Sn ~
X ^ Sn+ 2 CAM+1 by Proposition 8(b).  It follows by induction that
P (n) 2 CAM+n for all n. The inclusions P (n) ,! P (n + 1) constitute a
telescope   : ! ! Spaces *whose homotopy colimit is X x Y . Thus
~A (X x Y )   M + !, as desired.
   The proof of (c) is an instance of a general approach to products.
Let ~ be the supremum of ~A (P x Q) for P, Q 2 CA0.  We argue by
induction that if X, Y 2 CAff, then ff   ~A (X x Y )   ~ + ff. The case
~A (X) = ~A (Y ) = 0 is taken care of by the definition of ~.  In the
inductive case, we have


            X ~ hocolim* X   and  Y ~ hocolim* Y



                              7

where  X : I ! Spaces *and  Y : J ! Spaces *with  X (i),  Y (j) 2
CA<fffor all i 2 I and j 2 J . (Actually, X and Y could be retracts of
such spaces, but then X x Y is a retract of a product of such spaces,
so we suffer no loss of generality.) Define   : I x J ! Spaces *by
the formula  (i, j) =  X (i) x  Y (j). For each i 2 I and j 2 J there
is a fi < ff with  X (i),  Y (j) 2 CAfi.  It follows from the inductive

hypothesis that  (i, j) 2 CA~+fi  CA<~+ffbecause ~ + fi < ~ + ff [Fr,

p. 208]. Hence X x Y ~ hocolim*  2 CA~+ff.
   In view of the previous paragraph, the proof of (c) will be complete
if we simply verify the upper bound in the initial case, ff = 0. Here
X and Y are both (retracts of) wedges of suspensions of B. Thus [A ]
the product can be obtained by the cofiber sequence


                -1X *  -1Y ! X _ Y ! X x Y.


Since  -1X *  -1Y  ~=X ^ ( -1Y ), we have ~A ( -1X *  -1Y )
~A (X) + 1 = 1 by Proposition 8(b). Since ~A (X _ Y ) = 0, it follows
that ~A (X x Y )   2.                                          2
2.2    E.  Dror  Farjoun's  Theorem  and  some  of

its Consequences


Many of the most striking cellular inequalities are proved using the
following theorem of E. Dror Farjoun [Fa2, Thm. 9.A.8] (see also [Fa1,
Thm. 8.1]). The only novelty in our statement of the theorem is the
inclusion of a complexity estimate.


Theorem 11   Let K be a small category, and let


            E : K ! Spaces   and  B : K ! Spaces


be two diagrams whose (unpointed) homotopy colimits are the spaces
E and B, respectively.  Let f : E ! B be a natural transformation
which induces the map f : E ! B. ForWeach k 2 K, let Wk denote the
homotopy fiber of fk, and write W =   ~2KWk. If F is the homotopy
fiber of f, then

     (a) W   F , and

     (b) ~W (F )   1.


                              8

Proof  As mentioned above, part (a) is already known [Fa1].  To
prove part (b), we recall that (a) is proved by constructing an explicit
diagram   : L ! Spaces *whose homotopy colimit is weakly equiva-
lent to F and whose pointwise values are the spaces Wk. Since each
Wk 2 CW0, we have F 2 CW1.                                   2

   This is the general outline of many of the proofs in this section:
simply examine the proof of a known cellular inequality and count the
number of homotopy colimits that are used.
   The following is a useful instant corollary of Theorem 11.


Corollary 12  Let A ! B ! C be a cofiber sequence and let F be the
homotopy fiber of the map B ! C. Then

     (a) A   F , and

     (b) ~A (F )   1.


   Next we consider relations between the fiber and the cofiber of a
map; the cellular inequality is from E. Dror Farjoun's paper [Fa1].


Theorem 13   Let f : X ! Y  be a map with Y  path connected. De-
note the homotopy fiber and cofiber of f by F and C, respecitvely.
Then

     (a)  F   C, and

     (b) ~ F (C)   1.


Proof The proof the cellular inequality given in [C2 , Prop. 10.5] uses
Theorem 11 once.                                             2



Theorem 14   For any path connected space X,

     (a) X     X, and ~X (  X)   1.

     (b)   X   X, and ~  X (X)   1.


Proof The cellular inequality of part (a) appeared in [Fa1]; the proof
given in [C2 , Prop. 10.7] uses Theorem 11 to present   X as a pointed
homotopy colimit of a diagram whose pointwise values are the spaces
X and *. Part (b) follows on applying Theorem 13 to the map * ! X.
2
                              9

2.3    A Replacement for CA0


In order to obtain formulae which control the behavior of loop spaces,
we use a modified definition of complexity.  Rather than beginning
with CA0, we begin with the full subcategory whose objects are


                       Obj(CeA0) = {A, *}.


We denote the resulting notion of complexity by e~A. It is not hard to
verify that the results of Sections 2.2 and 2.3 are valid for the invariant
e~A(with the exception Proposition 6 in the case e~A(Xi) = 0 for all
i).


Proposition 15  For any space X, ~A (X)   e~A(X)   1 + ~A (X).


Proof This follows from the obvious containments CA0  eCA1  CA1. 2

   In [C2 ], Chach'olski proved the surprising result that cellular in-
equalities are preserved by the loop space operation.  We use our
modified invariant to estimate the complexity ~ A ( X).


Theorem 16   If A is connected and A   X, then

     (a)  A    X, and

     (b) ~ A ( X)   1 + ~A (X).


Proof  We observe first that, in view of Proposition 15, it suffices to
prove that e~ A( X)   e~A(X).
   If e~A(X) = 0, then X ~ A or X ~ *; in either case e~ A( X) = 0.
Let us assume inductively that e~ A( X)   e~A(X) for all X 2 eCA<ff.
   The proof of statement (a) given in [C2 , Cor. 10.4] uses Theorem
11 to show that if X ~ hocolim* , where   : K ! Spaces * then
 X ~ hocolim* , where   : L ! Spaces * is a diagram with the
property that, for each l 2 L there is a k 2 K such that  (l) ~   (k).
   If X 2 eCAff, then X is a retract of hocolim* , where   : K ! eCA<ff,
and so  X is a retract of hocolim* , where  (l) ~   (k) for some
 (k) 2 Ce<Aff.  Hence  (l) 2 Ce<Affby the inductive hypothesis, and

 X 2 eCfAf.                                                  2

   Here again we estimate the complexity of the spaces appearing in
cellular inequalities due to Chach'olski [C2 ].


Theorem 17   If X is path connected, then the statements  A   X
and A    X are equivalent, and



                              10

     (a) ~A ( X)   2 + ~ A (X)

     (b) ~ A (X)   ~A ( X) + 1.


Proof  When A is not path connected, the result is trivial.
   When A is path-connected, the equivalence of the two cellular in-
equalities is proved in [C2 , Thm. 10.8]. First we consider the implica-
tion  A   X  ) A    X. From  A   X, we have the sequence of
cellular inequalities
                       A     A    X.

We now estimate the complexity by the computations


              ~A ( X)      ~A (  A) + ~  A ( X)
                           1 + (1 + ~ A (X))
                           2 + ~ A (X)


using Lemma 5 and Theorem 16(b). For the implication A    X  )
 A   X, we have the sequence of inequalities  A     X   X, and
the corresponding complexity estimate


         ~ A (X)   ~ A (  X) + ~  X (X)   ~A ( X) + 1


using Lemma 5 and Proposition 7(a).                           2


Remark  If we use e~Ainstead of ~A , the proof given above shows
that |e~ A(X) - e~A( X)|   1.

   We turn now to two more results of Chach'olski, those concerning
the delooping and desuspending of cellular inequalities.  Note that,
while the theorems of [C3 ] require only that C(A) = C( B) or that
C(X) = C( Y ), our complexity estimate requires the stronger hy-
potheses A ~  B or X ~  Y .


Theorem 18     1. If A ~  B, then

     (a)  A    X implies A   X, and

     (b) ~A (X)   1 + ~ A ( X) + 1.

  2. If X ~  Y , then

     (a)  A    X implies A   X, and

     (b) ~A (X)   2 + ~ A ( X) + 1.



                              11

Proof The proof of the cellular inequality in (a) given in [C3 , Thm. 9.7]
goes as follows: we have


      B     B ~  A    X,   so  A ~  B     X   X.


It follows from the first sequence of inequalities, Lemma 5 and Theo-
rem 14(a) that


        ~B ( X)   ~B (  B) + ~  B ( X)   1 + ~ A ( X).


We now compute


               ~A (X)     ~ B (  X) + ~  X (X)
                          ~B ( X) + ~  X (X)
                          1 + ~ A ( X) + 1


using Lemma 5, Proposition 7(a) and Theorem 14(b).  The proof of
statement (2) is dual to the proof of (1).                        2



Corollary 19  If A ~  B for some space B, then


   max {~A (X), ~A (Y )}   ~A (X x Y )   4 + max {~A (X), ~A (Y )}.


Proof  As mentioned in the proof of Proposition 10(c), it suffices to
estimate sup{~A (X), ~A (Y )} for ~A (X) = ~A (Y ) = 0. If X and Y are
retracts of wedges of copies of A, then X x Y is a retract of a product
of such wedges; therefore it is enough to prove that if X and Y  are
wedges of copies of A, then ~A (X x Y )   4. Using the splitting of a
product after one suspension, we have


                      (X x Y ) =  A ^ Z,


for some space Z, and so ~ A ( (X x Y ))   1 by Proposition 8(b).
Since A ~  B, we may use Theorem 18(b) to desuspend the inequality
and thereby obtain the estimate


            ~A (X x Y )   2 + ~ A ( (X x Y )) + 1   4.


                                                            2


                              12

2.4    Some Examples


Before moving on to our study of good spaces, we apply our re-
sults to obtain some estimates of complexity concerning spheres and
Eilenberg-MacLane spaces.

Example 20        (a) For any path-connected space X,  X   X,
         and
                              ~ X (X)   2.

         To see this, simply apply Lemma 5, Proposition 7 and Theo-
         rem 14(b) to the sequence of inequalities  X     X   X.

     (b) Since K(G, n) = K(G, n + 1) for any abelian group G, a
         simple induction using part (a) yields the estimate

                      ~K(G,n)(K(G, n + m))   2m.

     (c) More generally, let X0, X1, . .,.Xn, . .b.e an infinite loop
         space, so  Xn+1 ~ Xn. If we write XO to denote the base-
         point component of X, then Xn =  Xn+1 =  ((Xn+1)O).
         Therefore Xn   (Xn+m )O and ~Xn ((Xn+m )O)   2m.

   We finish this subsection with an application to complexity with
respect to spheres. Stover has shown (using the Bousfield-Friedlander
spectral sequence) that every connected space X is a pointed homo-
topy colimit of a diagram of wedges of spheres [S]. In our language,
~S1(X)   1; it was observed in [CDI , 9.6] that a simple modification
of the argument shows that ~Sn(X)   2 for every space X and every
n.

Example 21   We show here that the weaker bound ~Sn(X)   n
can be easily proved by elementary methods (i.e., no spectral se-
quences).  By definition, the Sn-complexity of X is the same as the
Sn-complexity of CWSn(X), the (n - 1)-connected cover of X.  We
will show that if X is (n - 1)-connected, then ~Sn(X)   n + 1.
   The inductive proof begins with Lemma 4.  Let us assume then
that ~Sn(X)   n + 1 for every (n - 1)-connected space X.  If Y  is
n-connected, then so is   Y , and

             ~Sn+1(Y )     ~Sn+1(  Y ) + ~  Y (Y )
                           ~Sn( Y ) + 1
                           n + 2

by Lemma 5, Proposition 7(a), Theorem 14(b), and the inductive
hypothesis.



                              13

3    Properties of Good Spaces


We now turn our attention to the study of good spaces. This section
has three main results: first we show that if A is a good space, then
so is  A; next we characterize good spaces; and finally we give an
efficient construction of the functor PA under the assumption that A
is a good space.



3.1    Suspension of Good Spaces


The following fact will play a key role in the proof of our main theorem
in x4.


Theorem 22   If A is a good space, then  A is also a good space.


Proof  If A is not path connected, then C( A) = C(S1) is strongly
closed, so we may assume that A is path connected.
   We have to show that C( A) is closed under extensions by fibra-
tions. So let F ! E ! B be a fibration sequence with  A   F and
 A   B. Since A is path-connected, so are F and B, and so A    F
and A    B by Theorem 17. Looping the given fibration once yields
the fibration sequence  F !  E !  B.  Since A is a good space,
A    E and hence  A   E.                                  2
3.2    A Characterization of Good Spaces


We show here that when A is a good space, the fiber of the augmenta-
tion X ! PA (X) is naturally equivalent to CWA (X) in the homotopy
category. In fact, this condition characterizes good spaces.


Theorem 23   For every space A the following are equivalent.
                _____
     (a) C(A) = C(A) (i.e., A is a good space).

                                                           j
     (b) For every space X, the sequence CWA (X) _i___//X____//PA (X)
         is a fibration sequence._

     (c) For each space X 2 C(A)  there is a space YX  such that
         [A, YX ] = * and CWA (X) ! X ! YX  is a fibration se-
         quence.
                              14

Proof  To prove that (a) implies (b), we begin with the fibration
         __      l       j
sequence P A(X) ____//_X___//_PA_(X)and show that there is a weak

equivalence ` : CWA (X) ! PA (X) such that the diagram


                                      __
                 CWA (X)  _____~`_____//PA(X)
                       II             xx
                         III        xxx
                          iIII$$I--xxlxx

                               X


commutes up to homotopy. Since i : CWA (X) ! X is characterized
up to weak equivalence by the properties (1) CWA (X) is A-cellular,
and (2) i is a A-equivalence [Fa2], it is_enough to verify that these
properties are_satisfied_by_the map l : P A(X) ! X.  Condition (1)
follows from P A(X) 2 C(A) = C(A) [C2 , Thm. 17.1]. For (2), use the
                   __       l     j
fibration sequence P A(X) ____//_X___//PA (X)to obtain the fibration
sequence


              __      l*              j*
     map *(A, PA(X))_____//map*(A, X)____//_map*(A, PA (X)).


The map l* is a weak equivalence because map *(A, PA (X)) ~ * by
definition [Fa2, 1.A.4].
   Clearly (b) implies (c): simply take YX = PA (X).
   Now assume that (c) holds._By [CPS , Thm. 1.5], there is a good
space G such that C(A)   C(A)  = C(G).  It remains to prove the
reverse containment. Let X 2 C(G); we'll show that X 2 C(A). The
basepoint component of the space map *(A, YX ) is contractible because
CWA (X) ! X is an A-equivalence. Since [A, YX ] = *, map *(A, YX )
is path-connected, so YX  is A-null.  Thus we have the commutative
diagram
               X = CWG (X)  __________//_CWA (X)

                     |                     |
                     |                     |
                     fflffl|               fflffl|
                    X  ____________________X
                     |                     |
                     |                     |
                     fflffl|               fflffl|
                  PA (X) ________________//YX ,


which shows that X is a homotopy retract of CWA (X) 2 C(A). Since
C(A) is closed under retracts [Fa2, 2.E.11], this implies that X 2 C(A),
verifying (a).                                                 2



                              15




                                                        _____
Remark  The argument given above can be generalized:_if C(A) =
C(B), then there is a weak equivalence ` : CWB  ! P A which is
natural in the homotopy category. It is not clear whether or not ` can
be defined in a natural way on the point-set level.



3.3    An Easy Construction of PA (X)


In this section we derive an efficient construction of PA (X) under the
assumption that A is a good space. This construction plays a key role
in the proof of the main theorem in Section 4.
   We set X(0)= X and inductively define X(k+1)as the cofiber of
the evaluation map

                 W               k    ffl
                   f2[ kA,X(k)](  A)f___//_X(k)


given by ffl|( kA)f= f. Starting with the inclusion i1 : X ,! X(1), we

let ik : X ! X(k)be the unique map making the diagram


                               X D
                        ik-1wwww  DDDik
                          ww        DDD
                        --www         D!!
                   X(k-1)!!           X(k)


commutative. Let X(1) be the telescope of the sequence . .!.X(k)!
X(k+1)! . .a.nd write i : X ! X(1) for the map induced by the
maps ik.


Theorem 24   For connected CW complexes A and X, there is a ho-
motopy equivalence OE : X(1) ! PA (X) making the diagram


                            yX  FF
                         iyyyy   FFjFFF

                       __yyy  ffi   FF##
                   X(1) ______~_____//PA (X)


homotopy commutative.


Proof  We claim that the composite map

                             c1
                        _______________________________________________________*
 *____________________________________________________________@
                   ______________________((____________________________________*
 *____________________________________________________________@
              PA (X)_____//PA (X(1))__//_P A (X(1))



                              16

is a weak equivalence. Consider the commutative diagram


          PA (X) ______//_CWA (X)____//X______//_PA (X)
                                      ||          |
          c1||             ||||       ||||        c1|
            |fflffl        ||         ||          fflffl|
        P A (X(1))_____//CWA (X)_____//X____//_P A (X(1)).


Here the top row is a fibration sequence by Theorem 23 since A is
a good space, and the bottom row is a fibration sequence by [C2 ,
Thm. 20.3]. From the diagram we see that  c1 :  PA (X) !  P A (X(1))
is a weak equivalence. Since A is path connected, so are PA (X) and
P A (X(1)).  This implies that c1 : PA (X) ! P A (X(1)) is a weak
equivalence.
   Now a simple induction using Theorem 22 shows that each of the
natural maps P kA (X(k)) ! P k+1A(X(k+1)) is a weak equivalence.
These maps together constitute a telescope diagram   : ! ! Spaces *.
On taking the homotopy colimit, we obtain a weak equivalence


                   c : PA (X)___//_hocolim* .


By construction of the functor P kA , the map lk : X(k)! P kA X(k)
is a (k + 1)-equivalence. It follows that the limit map


                    l : X(1)____//hocolim*


is a weak equivalence. Since all spaces are CW complexes, c is invert-
ible, and the map OE = c-1 O l is the desired homotopy equivalence.
2


Remark  Without the hypothesis that A and X are CW complexes,
we could only conclude that PA (X) and X(1) were weakly equivalent.
4    Complexity and Good Spaces


In this final section we bring together much of our previous work in
order to prove an upper bound for ~ A (X) when A is a good space.
This upper bound is independent of A and X.


Theorem 25   If A is a good space, then


            ~ A (X)   ! + 1    for every space X.



                              17

Proof  If A is not path connected, then S1 is a retract of  A, so
~A (X)   ~S1(X)   1 for all X.  Thus we restrict our attention to
path connected spaces A. It follows, of course that all spaces that are
A-cellular, or  A-cellular, are also path connected.
   We will use the notation of Section 3.3. Assume first that X ~  Y
and that A   Y . Then Y (1)~ PA (Y ) ~ *. Now consider the diagram


            *________________//Y________________//Y
            |                 |                  |
            |                 |ik                ik+1|
         W  fflffl|           fflffl|            fflffl|
            kAf ____________//Y (k)__________//_Y (k+1)

            |                 |                  |
            |                 |                  |
         W  fflffl|           fflffl|            fflffl|
            kAf __________//_Y (k)=Y________//_Y (k+1)=Y


whose rows and columns are cofiber sequences.  In particular, the
quotient Y (k+1)=Y is the homotopy pushout of the diagram

                         W
                  * oo___   kAf _____//Y (k)=Y



forWk   0. Clearly Y = Y (0)implies Y (0)=Y ~ *, and hence Y (1)=Y ~
 (  Af) 2 C0A .  We conclude by induction that Y (k)=Y  2 CkA  for
each 0 < k < ! and hence that X ~  Y ~ Y (1)=Y 2 C!A .
   Now we handle the general case. The complexity of X is defined
to be ~ A (X) = ~ A (CW A (X)), so we replace X with CW A (X);
that is, we assume that  A   X to begin with. Then we have


                  A   X  implies  A    X.


Therefore  A     X and ~ A (  X)   ! by the special case dis-
cussed in the previous paragraph. Now we apply Lemma 5 and The-
orem 14(b) to conclude that


           ~ A (X)   ~ A (  X) + ~  X (X)   ! + 1,


which completes the proof.                                    2



Corollary 26  If A is a good space and A    Y , then ~A ( Y )
! + 1.



                              18

Proof  If A is not path connected, the result is trivial. If A is path
connected, then so is  Y because A    Y . Then we have  A   Y ,
and by Theorem 25 we have ~ A (Y )   ! + 1. Now we apply Theorem
17 to obtain

           ~A ( Y )   2 + ~ A (Y )   2 + ! + 1 = ! + 1,

using ordinal arithmetic for the final step.                       2

References

[A]     M. Arkowitz, The generalized Whitehead product, Pacific J. Math.
        12 (1962), 7-23.

[B-K]   A. K. Bousfield and D. M. Kan, Homotopy limits, completions and
        localizations, SLNM 304 Springer, Berlin (1972).

[B]     A. K. Bousfield, Localization and periodicity in unstable homotopy
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[C1]    W. Chach'olski, Closed classes, Algebraic topology: new trends in
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        Progr. Math., 136, Birkhäuser, Basel, 1996.
[C2]    W. Chach'olski, On the functors CWA and PA, Duke Math. J. 84
        (1996), 599-631.

[C3]    W. Chach'olski, Desuspending and delooping cellular inequalities,
        Invent. Math. 129 (1997), 37-62.

[CDI]   W. Chach'olski, W. G. Dwyer and M. Intermont, The A-complexity
        of a space, J. London Math. Soc. (2) 65 (2002), 204-222.

[CPS]   W. Chach'olski, P.-E. Parent and D. Stanley, Cellular generators,
        Preprint.

[Fa1]   E.  Dror  Farjoun,  Cellular  Inequalities,  The  ~Cech  Centennial
        (Boston, MA, 1993), 159-181, Contemp. Math., 181, AMS, Prov-
        idence, RI, 1995.
[Fa2]   E. Dror Farjoun, Cellular Sapces, Null Spaces, and Homotopy Lo-
        calization, Springer-Verlag, Berlin, 1996.
[Fr]    Fraenkel, A. A. Abstract set theory, North-Holland Publishing Co.,
        Amsterdam (1966).

[S]     C. R. Stover, A van Kampen spectral sequence for higher homotopy
        groups, Topology 29 (1990), 9-26.

[W]     G. W. Whitehead, Elements of Homotopy Theory, Graduate Texts
        in Mathematics 61, Springer-Verlag, 1978.


Kalamazoo College, Kalamazoo, MI, 49006 USA
E-mail Address:  intermon@kzoo.edu

Western Michigan University, Kalamazoo, MI, 49008 USA
E-mail Address:  Jeff.Strom@wmich.edu



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