Covering maps over solenoids which are not covering homomorphisms

Katsuya Eda, Vlasta Matijević

    Research output: Contribution to journalArticle

    4 Citations (Scopus)

    Abstract

    Let Y be a connected group and let f : X → Y be a covering map with the total space X being connected. We consider the following question: Is it possible to define a topological group structure on X in such a way that f becomes a homomorphism of topological groups. This holds in some particular cases: if Y is a pathwise connected and locally pathwise connected group or if f is a finite-sheeted covering map over a compact connected group Y . However, using shape-theoretic techniques and Fox's notion of an overlay map, we answer the question in the negative. We consider infinite-sheeted covering maps over solenoids, i.e. compact connected 1-dimensional abelian groups. First we show that an infinite-sheeted covering map f : X → Σ with a total space being connected over a solenoid Σ does not admit a topological group structure on X such that f becomes a homomorphism. Then, for an arbitrary solenoid, we construct a connected space X and an infinite-sheeted covering map f : X →, which provides a negative answer to the question.

    Original languageEnglish
    Pages (from-to)69-82
    Number of pages14
    JournalFundamenta Mathematicae
    Volume221
    Issue number1
    DOIs
    Publication statusPublished - 2013

    Keywords

    • Compact connected group
    • Cover-ing homomorphism
    • Covering map
    • Infinite-sheeted
    • Overlay
    • Solenoid

    ASJC Scopus subject areas

    • Algebra and Number Theory

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