Covering maps over solenoids which are not covering homomorphisms

Katsuya Eda, and Vlasta Matijević. "Covering maps over solenoids which are not covering homomorphisms." Fundamenta Mathematicae 221.1 (2013): 69-82. <http://eudml.org/doc/286532>.

@article{KatsuyaEda2013,
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 Σ, we construct a connected space X and an infinite-sheeted covering map f: X → Σ, which provides a negative answer to the question.},
author = {Katsuya Eda, Vlasta Matijević},
journal = {Fundamenta Mathematicae},
keywords = {solenoid; compact connected group; covering map; overlay; covering homomorphism; infinite-sheeted},
language = {eng},
number = {1},
pages = {69-82},
title = {Covering maps over solenoids which are not covering homomorphisms},
url = {http://eudml.org/doc/286532},
volume = {221},
year = {2013},
}

TY - JOUR
AU - Katsuya Eda
AU - Vlasta Matijević
TI - Covering maps over solenoids which are not covering homomorphisms
JO - Fundamenta Mathematicae
PY - 2013
VL - 221
IS - 1
SP - 69
EP - 82
AB - 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 Σ, we construct a connected space X and an infinite-sheeted covering map f: X → Σ, which provides a negative answer to the question.
LA - eng
KW - solenoid; compact connected group; covering map; overlay; covering homomorphism; infinite-sheeted
UR - http://eudml.org/doc/286532
ER -