Covering maps over solenoids which are not covering homomorphisms
Volume 221 / 2013
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.