Bockstein Basis and Resolution Theorems in Extension Theory

Abstract

Resolution refers to a map (a continuous function) between topological spaces, where the domain is in some way better than the range, and the fibers (point preimages) meet certain requirements. We will be interested in the relationship between covering dimension and cohomological dimension, so the resolution we obtain will be between a domain of finite covering dimension, and a range of finite cohomological dimension, with cell-like or G-acyclic fibers. Both domain and range will be compact metrizable spaces.

A useful tool in investigating dimension of spaces is extension of maps. An indispensable tool in cohomological dimension theory are results of M. F. Bockstein, usually referred to as Bockstein theory. Extending maps and Bockstein theory will be extensively used in this work, as well as the theory of inverse sequences and limits.

We will look at standard resolution theorems in extension theory by R. Edwards-J. Walsh, A. Dranishnikov and M. Levin. Also, we will mention how they generalize to the L. Rubin-P. Schapiro resolution theorem, and we will focus on the proof of the case that the Rubin-Schapiro proof did not cover, namely:

Theorem: Let G be an abelian group with PG equal to set of all primes, where PG = { p in primes : Z(p) in Bockstein Basis &sigma(G) }. Let n be a positive integer, and let K be a connected CW-complex with homotopy groups such that &pin(K) = G, and &pik(K) = 0 for 0 &le k < n. Then for every compact metrizable space X with X &tau K (i.e., with K an absolute extensor for X), there exists a compact metrizable space Z and a surjective map &pi : Z &rarr X such that &pi is cell-like,

dim Z &le n and Z &tau K.