Covering groups and their applications: A survey
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Covering groups and their applications: A survey

R.N. Gumerov

Kazan Federal University, Kazan, 420008 Russia

E-mail: Renat.Gumerov@kpfu.ru

Received February 1, 2022


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DOI: 10.26907/2541-7746.2022.1.5-42

For citation: Gumerov R.N. Covering groups and their applications: A survey. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2022, vol. 164, no. 1, pp. 5–42. doi: 10.26907/2541-7746.2022.1.5-42. (In Russian)

Abstract

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.

Keywords: algebraic equation with continuous coefficients, Weierstrass variety, Weierstrass polynomial, covering group, covering mapping onto topological group, covering homomorphism, overlay mapping, P -adic solenoid, polynomial covering, covering group theorem

Acknowledgments. I would like to sincerely thank Professor V. Matijevi´c who kindly provided copies of the articles essential for this research, as well as Professor V.L. Hansen for the valuable information he gave about polynomial coverings in our correspondence.

This study was supported by the Kazan Federal University Strategic Academic Leadership Program (“PRIORITY-2030”).

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For other references see (total number of references: 99) 


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