Eine Verallgemeinerung derCW-Komplexe

Abstract

In the definition ofCW-complexes, the one-point spaceP, respectively the spaceP∪* with basepoint *, play the roll of the only “building-stone”. Let\(\mathfrak{W}\) be a family of compact spaces. Then the definition of a generalizedCW-complex over\(\mathfrak{W}\) is obtained from the definition of aCW-complex by replacingP by the spaces of\(\mathfrak{W}\) and formation of the mapping cone by a slightly modified construction. LetCW \((\mathfrak{W})\) * denote the category of all pointed spaces which have the homotopy type of a generalizedCW-complex over\(\mathfrak{W}\). If\(\mathfrak{W} = \left\{ P \right\}\), thenCW \((\mathfrak{W})\) * is the category of all pointedCW-spaces.CW \((\mathfrak{W})\) * is closed under the formation of direct sums and of mapping cones, cylinders and tori, and is formally characterized as the smallest such subcategory of Top * containing the spaces W∪*,\(W \cup * , W \in \mathfrak{W}\). Following the methods of E. H. Brown, it is proved, that any half exact homotopy functor onCW \((\mathfrak{W})\) * is representable, and any cohomology theory onCW \((\mathfrak{W})^2 \) is naturally equivalent to the cohomology theory of an Ω-spectrum; for example, the singular cohomo logy is representable onCW \((\mathfrak{W})^2 \) for any family\(\mathfrak{W}\) of compact spaces.