On a covering group theorem and its applications. (English) Zbl 1010.22007

Grigorian, S. A.; Gumerov, R. N.

Lobachevskii J. Math. 10, 9-16 (2002).

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.

Reviewer: Jorge Galindo (Castellon)

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

Keywords:

\(n\)-fold covering; covering group; algebraic covering; triviality of \(n\)-fold coverings; algebraic equation with functional coefficients; dual group; one dimensional Čech cohomology group