Bockstein basis and resolution theorems in extension theory

TL;DR

This paper generalizes the Edwards-Walsh Resolution Theorem by linking Bockstein basis properties of abelian groups to extension properties of compact metrizable spaces, providing new resolution results in extension theory.

Contribution

It introduces a generalized resolution theorem in extension theory based on Bockstein basis conditions, extending previous results to broader classes of groups and spaces.

Findings

Established a new resolution theorem for spaces with specific Bockstein basis conditions.

Proved existence of cell-like surjective maps reducing dimension while preserving extension properties.

Extended classical resolution results to a wider class of abelian groups and CW-complexes.

Abstract

We prove a generalization of the Edwards-Walsh Resolution Theorem: Theorem: Let G be an abelian group for which $P_G$ equals the set of all primes $\mathbb{P}$, where $P_G=\{p \in \mathbb{P}: \Z_{(p)}\in$ Bockstein Basis $ \sigma(G)\}$. Let n in N and let K be a connected CW-complex with $\pi_n(K)\cong G$, $\pi_k(K)\cong 0$ for $0\leq k< n$. 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 $\pi: Z \to X$ such that (a) $\pi$ is cell-like, (b) $\dim Z \leq n$, and (c) $Z\tau K$.