Covering maps over solenoids which are not covering homomorphisms

Katsuya Eda¹ ; Vlasta Matijević²

¹ Department of Mathematics Waseda University Okubo 3-4-1, Shinjuku-ku Tokyo 169-8555, Japan
² Department of Mathematics University of Split Teslina 12 21000 Split, Croatia

Fundamenta Mathematicae, Tome 221 (2013) no. 1, pp. 69-82

Résumé

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.

Détail

DOI : 10.4064/fm221-1-3

Keywords: connected group rightarrow covering map total space being connected consider following question possible define topological group structure becomes homomorphism topological groups holds particular cases pathwise connected locally pathwise connected group nbsp finite sheeted covering map compact connected group however using shape theoretic techniques foxs notion overlay map answer question negative consider infinite sheeted covering maps solenoids compact connected dimensional abelian groups first infinite sheeted covering map rightarrow varsigma total space being connected solenoid varsigma does admit topological group structure becomes homomorphism arbitrary solenoid varsigma construct connected space infinite sheeted covering map rightarrow varsigma which provides negative answer question