COMPLEXES THAT ARISE IN COHOMOLOGICAL DIMENSION THEORY: A UNIFIED APPROACH JERZY DYDAK AND JOHN J. WALSH ABSTRACT Let n: P-+P be a combinatorial map (that is, n~\L) is a subcomplex of P whenever L is a subcomplex of P) between CW-complexes. A m a p / : X -* P is said to approximately lift with respect to n provided that there is a map/: X-* P such that, for each xe X, there is a cell in P containing both noj(x) andf{x). A characteristic property of a compact metric space Shaving 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) c» 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 n: EW z (/ > ,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 /: X-+ P to a simplicial complex P can be approximately lifted to EW z (P,n). These complexes provide a '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. 0. Introduction Recall that a space X is said to have cohomological dimension at most n with respect to an abelian group G, written dim G X ^ n, provided that the Cech cohomology groups H q (A,B;G) ~ 0 for all q^n + l and for all pairs of closed subsets B cz A of X. A formal consequence of the axioms of a cohomology theory leads to the restatement dim G X ^ n, provided that the inclusion A c> X induces a surjection H n (X; G) -*• H n (A; G) for every closed subset A <= X. Stating the latter in terms of classifying spaces leads to the formulation that best serves the purposes of this paper. Namely, dim G X ^ n provided that every map/: A -• K(G, n) from a closed subset Ac: X to the Eilenberg-MacLane space extends to a map F: X-> K(G,n). An equivalent reformulation that has been at the center of many of the recent advances in the theory involves associating to each simplicial complex L a ' combinatorial resolution' n: EW G (L, n) -> L specified so that dim G X ^ n if and only if, for every simplicial complex L and map/: X-+L, there is an approximate lift /: X-+ EW G (L,«). This framework was originally introduced in [12,19] for G = Z to establish the equivalence between the problems of whether there is a compactum with infinite covering dimension but finite integral cohomological dimension and whether there is a cell-like dimension-raising map. Subsequently, Dranishnikov [5], further exploiting the complexes EW 2 (L, n), provided examples of the former. While perhaps best suited for producing examples, these complexes were used by Rubin [16] to establish the inequality dim z (y4 U B) ^ dim z ^4 + dim z i?+1, the latter a Received 25 October 1991. 1991 Mathematics Subject Classification 55M10. /. London Math. Soc. (2) 48 (1993) 329-347