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Free compact groups. I: Free compact abelian groups. (English) Zbl 0589.22003
The structure of free compact abelian groups is investigated. A compact abelian group G is free compact abelian if and only if \(G\cong K\times {\hat {\mathbb{Q}}}^ a\times \prod_{p\in prime}{\mathbb{Z}}^ b_ p\), where \({\hat {\mathbb{Q}}}\) is the character group of the discrete group of rationals, K is a compact connected abelian group with dense torsion subgroup, and a,b are cardinal numbers such that \(a\geq \max \{2^{\aleph_ 0}\), b, dim \(K\}\) (Theorem 1.7.2). This theorem implies that if G is of the above form then it is the free compact abelian group on the underlying space of the compact abelian group \(X=K\times {\hat {\mathbb{Q}}}^ a\times F\), where F is a discrete abelian group of order b-1, if b is finite, and \({\mathbb{Z}}(2)^ b\), otherwise.
Compact Hausdorff spaces with isomorphic free compact abelian groups are completely characterized by means of their Alexander-Spanier cohomology group and the weight of them. This characterization gives a new information even for compact spaces with isomorphic free abelian topological groups. In particular, if A(X)\(\cong A(Y)\) and the weight of X, w(X), exceeds \(2^{\aleph_ 0}\), then \(w(X)=w(Y)\).
Reviewer: M.G.Tkachenko

MSC:
22C05 Compact groups
20K99 Abelian groups
22B05 General properties and structure of LCA groups
55N05 Čech types
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