CellularSpaces


                                      by
                                E. Dror Farjoun



                      0.  Introduction and Main Results

   The aim of the present work is to study,in a systematic way, the possibility*
 *of con-

structing new spaces out of a givencollection of persumably better understood s*
 *paces.

Recently there were severaldevelopments in this direction: There has been an in*
 *tensive

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 generaliz*
 *ed by [D-

W?] and was used effectively by several authors to construct new realizations o*
 *f interesting

algebras as cohomology algebras of spaces. In a different vein [S?H]introduced *
 *'thick class-

es'. These classes are closed under cofibration. In the stable category they ar*
 *e in fact the

closure under cofibrations and desuspensionsof a single space, say V(n), constr*
 *ucted by

[?] and [?] Thus each one of these classes is precisely the class of all spectr*
 *a that can be

built form a single spectra by taking cofibres of arbitrary maps (and desuspens*
 *ions). Now

unstably the question arrise naturally:

 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 conditiona K -acyclic space can actually be built b*
 *y 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 asa finite dimens*
 *ional

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 allp ointed homotopy colimits along arbitra*
 *ry diagrams

starting with A itself. Weget a full subcategory denoted here by C (A). This cl*
 *asscomes

with a functor CWA [Bou?] that associates to to every space a member of the cla*
 *ss in a


spaces. In that contextthe usual class of CW complexes intro duced by Whitehead*
 * is simply

the class of spaces built from the zerosphere, the class of 1-connected complex*
 *es 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 C(A) = the smallest full subc*
 *ategory of

S  closed under arbitrarypointed hocolim and weak homotopy equivalences that co*
 *ntains

A. The functor CWA starts with anarbitrary space A and associates to any space *
 *X the

`best A-approximation,'the space in C (A) which is the closest space to X. It c*
 *an 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 usualCW -approximationto a given topological*
 * space

X. The latter is, in fact, equivalent to the space CWS0X, where A = S0 is the z*
 *ero sphere.


   The functor CWA ,is very closely related to the localization (or periodizati*
 *on) functor

PA with respect to A ?Bou] ?EDF]. By definition whenever a space X is A-periodi*
 *c i.e.

the pointed function complex from A to Xis contractible the A-CW approximation *
 *to X

is contractible. In [N]Nofech introduced a model category structure on simplici*
 *al sets or

topological spaces in which weak equivalences are maps that induces isomorphism*
 *s 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 be*
 *tween A

periodic spaces and A-cellular spaces. In the case where A is the mod-p Moore s*
 *paces this

is well established: under mild conditions the homotopy fibre of the rational l*
 *ocalization

and more generally rationally acyclic spaces can be built from these Moore spac*
 *e Building

Blocks, via cofibrations.  But we shall see that whenever [A;X ] ' the space CW*
 *A X is

homotopy equivalent tothe homotopy fibre of X !PA  X (see 0.7b elow). As an exa*
 *mple

of application we address the following



0.1  Problem:  Given a h -acyclic space X for a generalized homology theory h ,*
 * when

is X the limit of its finiteh -acyclic complexes?


   It turns out that for h  =Khni = Morava's K-theories, and X = GEM a generali*
 *zed

Eilenberg-MacLane space positive answer is available under some dimensional res*
 *triction.