Complexes that Arise in Cohomological Dimension Theory: A Unified Approach

Let π: P ˜ → P be a combinatorial map (that is, π −1 ( L ) is a subcomplex of P ˜ whenever L is a subcomplex of P ) between CW‐complexes. A map f : X → P is said to approximately lift with respect to π provided that there is a map f ˜ : X → P ˜ such that, for each x € X , there is a cell in P containing both πo f ˜ ( x ) and f(x) . A characteristic property of a compact metric space X having covering dimension dim X ⩽ n is that each map from X to a CW‐complex P has an approximate lift with respect to the inclusion P( n ) ↪ P . An analogous characterization of compacta X having integral cohomological dimension dim z X ⩽ n emerged from work of R. D. Edwards [ 12 ] and was introduced in [ 19 ]. Complexes and maps π: EWz ( P,n ) → P are associated to each simplicial complex P so that a compactum X has dim z X ⩾ n if and only if every map f : X → P to a simplicial complex P can be approximately lifted to EW z ( P,n ). These complexes provide a l‘combinatorial’ approach to cohomological dimension theory that has supported many of the recent developments in the area. Historically, cohomological dimension theory with respect to groups other than Z has provided computational machinery for determining covering dimension. Hence, it is not surprising that it has been useful to consider comparable complexes EW G ( L, n ) for other groups G . The goal of this paper is to present a unified exposition of these complexes. As an application, they are used to provide an alternative construction to that of Dranishnikov [ 4, 6 ] of compact metric spaces realizing the Bockstein functions.