Covering maps over solenoids which are not covering homomorphisms

Volume 221 / 2013

Katsuya Eda, Vlasta Matijević Fundamenta Mathematicae 221 (2013), 69-82 MSC: 22C05, 57M10. DOI: 10.4064/fm221-1-3

Abstract

Let $Y$ be a connected group and let $f:X\rightarrow Y$ 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\rightarrow \varSigma $ with a total space being connected over a solenoid $\varSigma $ does not admit a topological group structure on $X$ such that $f$ becomes a homomorphism. Then, for an arbitrary solenoid $\varSigma $, we construct a connected space $X$ and an infinite-sheeted covering map $f:X\rightarrow \varSigma $, which provides a negative answer to the question.

Authors

  • Katsuya EdaDepartment of Mathematics
    Waseda University
    Okubo 3-4-1, Shinjuku-ku
    Tokyo 169-8555, Japan
    e-mail
  • Vlasta MatijevićDepartment of Mathematics
    University of Split
    Teslina 12
    21000 Split, Croatia
    e-mail

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