Free Abelian topological groups and the Pontryagin-van Kampen duality. (English) Zbl 0841.22001
The paper is an enlarged English version of the author’s article published ten years ago in [Mosc. Univ. Math. Bull. 41, 1-4 (1986); translation from Vestn. Mosk. Univ., Ser. I 1986, 3-5 (1986; Zbl 0599.22003)]. The class \({\mathcal K}\) of completely regular spaces \(X\) is studied such that the free Abelian topological group \(A(X)\) over \(X\) is reflexive, that is, satisfies the Pontryagin-van Kampen duality. It is shown that every \(X \in {\mathcal K}\) is totally path-disconnected and furthermore, if \(X\) is pseudocompact, then the first cohomotopy group \(\pi^1 (X)\) is trivial. On the other hand, if \(X\) is compact or metrizable and satisfies \(\dim X = 0\), then \(X\) belongs to \({\mathcal K}\).
Reviewer: M.G.Tkachenko (Mexico)
MSC:
| 22A05 | Structure of general topological groups |
| 43A40 | Character groups and dual objects |
| 22B05 | General properties and structure of LCA groups |
| 54H11 | Topological groups (topological aspects) |
Citations:
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