0. Basic Homotopy Notions
First-time readers may omit this chapter, proceeding directly to §i.
We will make much use of the standard notions of CW complex and CW pair
(consisting of a complex and subcomplex [W44]). We also need more compli-
cated arrangements of spaces. CW lattices in general are discussed in several
papers by E. H. Spanier and J. H. C. Whitehead: see Vol. IV pp. 104-227 of
the latter's collected works. We confine ourselves here to CW n-ads. A CW
(n+ l)-ad consists, by definition, of a CW complex and n subcomplexes thereof.
In studying such an object, we are forced to consider the intersections of various
families of subcomplexes: there are, of course,
such. It is desirable to intro-
duce a systematic notation. We must index all these complexes by reference to
a standard model.
Consider an (n + l)-ad in general as a set (the 'total' set) with n preferred
subsets. We can specify the intersections by a function S on the set of subsets
of {1,2,..., n}, whose values are sets, and which preserves intersections (and
hence, we note, S is compatible with inclusion relations). Then 5 { l , . . . , n }
is the set, and the n preferred subsets are the values of S on the subsets of
{ 1 , . . . , n} obtained by deleting one of its members. We denote 5 { 1 , . . . , n} by
\S\. If \S\ is a topological space, the subsets inherit topologies, and we speak
of a topological (n + l)-ad. For a CW (n + l)-ad we require not merely that
\S\ be a CW complex, but that the subsets be subcomplexes. We speak of a
finite CW (n + l)-ad if |5| is a finite complex. We can also regard the lattice
of subsets of { 1 , . . . , n) as a category 2 n (the morphisms are inclusion maps):
S is then an intersection-preserving functor from 2 n to the category of sets or
spaces or CW complexes, and appropriate maps. We thus obtain categories of
(n + l)-ads: in the CW case we permit any continuous maps here.
There are many operations on (n -f l)-ads. The most natural ones arise as
composition with an intersection-preserving functor 2 m 2 n : for example,
(1) Permutations (we introduce no special notation here).
(2) Given an injective map / : {l,...,ra } { l , . . . , n } , take the induced
map of subsets. This includes (1), but we are more interested in the maps
di : 2
(l^i^n) induced by
j *- j U i) j*-+j + i U i)
The corresponding functor from (n+l)-ads to n-ads corresponds to taking
number i of the n subspaces as total space, and using the intersections of
the others with it as subsets.
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