### 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 language | English |
---|---|

Pages (from-to) | 69-82 |

Number of pages | 14 |

Journal | Fundamenta Mathematicae |

Volume | 221 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Fundamenta Mathematicae*,

*221*(1), 69-82. https://doi.org/10.4064/fm221-1-3

**Covering maps over solenoids which are not covering homomorphisms.** / Eda, Katsuya; Matijević, Vlasta.

Research output: Contribution to journal › Article

*Fundamenta Mathematicae*, vol. 221, no. 1, pp. 69-82. https://doi.org/10.4064/fm221-1-3

}

TY - JOUR

T1 - Covering maps over solenoids which are not covering homomorphisms

AU - Eda, Katsuya

AU - Matijević, Vlasta

PY - 2013

Y1 - 2013

N2 - 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.

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, we construct a connected space X and an infinite-sheeted covering map f : X →, which provides a negative answer to the question.

KW - Compact connected group

KW - Cover-ing homomorphism

KW - Covering map

KW - Infinite-sheeted

KW - Overlay

KW - Solenoid

UR - http://www.scopus.com/inward/record.url?scp=84877644278&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877644278&partnerID=8YFLogxK

U2 - 10.4064/fm221-1-3

DO - 10.4064/fm221-1-3

M3 - Article

VL - 221

SP - 69

EP - 82

JO - Fundamenta Mathematicae

JF - Fundamenta Mathematicae

SN - 0016-2736

IS - 1

ER -