BOCKSTEIN THEOREM FOR NILPOTENT GROUPS

We extend the definition of Bockstein basis sigma(G) to nilpotent groups G. A metrizable space X is called a Bockstein space if dim(G)(X) = sup {dim(H)(X)vertical bar H is an element of sigma(G)} for all Abelian groups G. The Bockstein First Theorem says that all compact spaces are Bockstein spaces. Here are the main results of the paper: Theorem 0.1. Let X be a Bockstein space. If G is nilpotent, then dim(G)(X) <= 1 if and only if sup {dim(H)(X)vertical bar H is an element of sigma(C)} <= 1. Theorem 0.2. X is a Bockstein space ifand only if dim(Z(iota)) (X) = dim (Z) over cap ((l)) (X) for all subsets l of prime numbers.