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