Bockstein theorem for nilpotent groups

TL;DR

This paper extends Bockstein basis concepts to nilpotent groups, establishing conditions under which spaces are Bockstein spaces and relating their cohomological dimensions across different groups.

Contribution

It generalizes Bockstein theory to nilpotent groups and characterizes Bockstein spaces via cohomological dimension equalities for specific group classes.

Findings

All compact spaces are Bockstein spaces.

For Bockstein spaces, nilpotent group dimensions are bounded by 1 iff certain supremum conditions hold.

Bockstein space condition characterized by dimension equality for specific group pairs.

Abstract

We extend the definition of Bockstein basis $\sigma(G)$ to nilpotent groups $G$. A metrizable space $X$ is called a {\it Bockstein space} if $\dim_G(X) = \sup\{\dim_H(X) | H\in \sigma(G)\}$ for all Abelian groups $G$. Bockstein First Theorem says that all compact spaces are Bockstein spaces. Here are the main results of the paper: Let $X$ be a Bockstein space. If $G$ is nilpotent, then $\dim_G(X) \leq 1$ if and only if $\sup\{\dim_H(X) | H\in\sigma(G)\}\leq 1$. $X$ is a Bockstein space if and only if $\dim_{\Z_{(l)}} (X) = \dim_{\hat{Z}_{(l)}}(X)$ for all subsets $l$ of prime numbers.