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On a covering group theorem and its applications. (English) Zbl 1010.22007
This is mainly an announcement of results and proofs are either absent or sketched. The authors first outline how to introduce a group structure on an $$n$$-fold covering of a compact connected group in such a way that the covering map becomes a group homomorphism. This is done by approximating compact connected groups by compact Lie groups in the usual way, subsequently using the covering group theory for path connected and locally path connected topological groups already present in Pontryagin’s 1938 classical monograph and finally putting everything together in an adequate way.
The algebraicity of all coverings of a compact Abelian group and a criterion for the triviality of all $$n$$-fold coverings of a compact connected Abelian group are then announced as applications. This is further applied to give a criterion for the solvability of algebraic equations with functional coefficients.

##### MSC:
 22A10 Analysis on general topological groups 14E20 Coverings in algebraic geometry 43A77 Harmonic analysis on general compact groups 43A10 Measure algebras on groups, semigroups, etc. 43A40 Character groups and dual objects 46J10 Banach algebras of continuous functions, function algebras 20J06 Cohomology of groups 22C05 Compact groups 54H11 Topological groups (topological aspects) 57M10 Covering spaces and low-dimensional topology
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