Bockstein theorem for nilpotent groups

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 ${\dim }_G(X)\le 1$ if and only if $\sup \{{\dim }_H(X)|H\in \sigma (G)\}\le 1$. $X$ is a Bockstein space if and only if ${\dim }_{{\mathbb{Z}}_{(l)}}(X)={\dim }_{{\hat{Z}}_{(l)}}(X)$ for all subsets $l$ of prime numbers.

10 pages

References in corpus (1)

• Cohomological dimension theory of compact metric spaces