Complexes that Arise in Cohomological Dimension Theory: A Unified Approach

Abstract

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) andf(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 dimz 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 dimz X ⩾ n if and only if every map f : X → P to a simplicial complex P can be approximately lifted to EWz(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 EWG(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.