Covering groups and their applications: A survey

This article is a survey of covering group theorems and their applications. For a given covering mapping from the topological space onto a topological group, it is natural to pose the following question on the lifting of the group structure from the base of the covering mapping to its covering space: do there exist group operations on the covering space that turn this space into a topological group and the original covering mapping into a morphism of topological groups? Each statement giving the positive answer to this question for any class of covering mappings is called a covering group theorem. Here the main stages in the proof of the covering group theorem for finite-sheeted covering mappings from connected topological spaces onto arbitrary compact connected groups are explored. This theorem and the method of its proof have a number of interesting applications in analysis, topology, and topological algebra. The results on the coverings of topological groups obtained by applying this theorem or by using an approximation construction which is built in its proof are discussed. The theorems under study are those establishing a close relationship between the finite-sheeted coverings of compact connected Abelian groups and the polynomials over Banach algebras of continuous functions, i.e., Weierstrass polynomials. Informally speaking, all finite-sheeted coverings of compact connected Abelian groups are defined by zero sets of simple Weierstrass polynomials. Connected coverings of $P$ -adic solenoids are considered. A complete description of such finite-sheeted coverings is provided by using the above-mentioned approximation construction. Applications of the covering group theorems are specified, as well as their corollaries to the study of the structure of coverings and to the problem with the existence of generalized means on topological groups. Particular attention is paid to the applications related to the properties of the solutions of algebraic equations with continuous coefficients.