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, 2n 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 n ~ 1 - 2n (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. 3
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