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.