Torus-like continua which are not self-covering spaces

Katsuya Eda; Joško Mandić; Vlasta Matijević

doi:10.1016/j.topol.2003.06.006

Torus-like continua which are not self-covering spaces

Katsuya Eda^*, Joško Mandić, Vlasta Matijević

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

For each non-quadratic p-adic integer, p > 2, we give an example of a torus-like continuum Y (i.e., inverse limit of an inverse sequence, where each term is the 2-torus T² and each bonding map is a surjective homomorphism), which admits three 4-sheeted covering maps f₀: X₀ → Y, f₁: X₁ → Y, f₂: X₂ → Y such that the total spaces and X₀ = Y, X₂ are pair-wise non-homeomorphic. Furthermore, Y admits a 2p-sheeted covering map f₃: X₃ → Y such that X₃ and Y are non-homeomorphic. In particular, Y is not a self-covering space. This example shows that the class of self-covering spaces is not closed under the operation of forming inverse limits with open surjective bonding maps.

Original language	English
Pages (from-to)	359-369
Number of pages	11
Journal	Topology and its Applications
Volume	153
Issue number	2-3 SPEC. ISS.
DOIs	https://doi.org/10.1016/j.topol.2003.06.006
Publication status	Published - 2005 Sept 1

Keywords

Covering mapping
Direct system
H-connected space
Inverse system
p-adic number
Quadratic number
Torsion-free group of rank 2
Torus-like continuum

ASJC Scopus subject areas

Geometry and Topology

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10.1016/j.topol.2003.06.006

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@article{009d2958467f4b479886eb66aaa82a0a,

title = "Torus-like continua which are not self-covering spaces",

abstract = "For each non-quadratic p-adic integer, p > 2, we give an example of a torus-like continuum Y (i.e., inverse limit of an inverse sequence, where each term is the 2-torus T2 and each bonding map is a surjective homomorphism), which admits three 4-sheeted covering maps f0: X0 → Y, f1: X1 → Y, f2: X2 → Y such that the total spaces and X0 = Y, X2 are pair-wise non-homeomorphic. Furthermore, Y admits a 2p-sheeted covering map f3: X3 → Y such that X3 and Y are non-homeomorphic. In particular, Y is not a self-covering space. This example shows that the class of self-covering spaces is not closed under the operation of forming inverse limits with open surjective bonding maps.",

keywords = "Covering mapping, Direct system, H-connected space, Inverse system, p-adic number, Quadratic number, Torsion-free group of rank 2, Torus-like continuum",

author = "Katsuya Eda and Jo{\v s}ko Mandi{\'c} and Vlasta Matijevi{\'c}",

year = "2005",

month = sep,

day = "1",

doi = "10.1016/j.topol.2003.06.006",

language = "English",

volume = "153",

pages = "359--369",

journal = "Topology and its Applications",

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}

TY - JOUR

T1 - Torus-like continua which are not self-covering spaces

AU - Eda, Katsuya

AU - Mandić, Joško

AU - Matijević, Vlasta

PY - 2005/9/1

Y1 - 2005/9/1

N2 - For each non-quadratic p-adic integer, p > 2, we give an example of a torus-like continuum Y (i.e., inverse limit of an inverse sequence, where each term is the 2-torus T2 and each bonding map is a surjective homomorphism), which admits three 4-sheeted covering maps f0: X0 → Y, f1: X1 → Y, f2: X2 → Y such that the total spaces and X0 = Y, X2 are pair-wise non-homeomorphic. Furthermore, Y admits a 2p-sheeted covering map f3: X3 → Y such that X3 and Y are non-homeomorphic. In particular, Y is not a self-covering space. This example shows that the class of self-covering spaces is not closed under the operation of forming inverse limits with open surjective bonding maps.

AB - For each non-quadratic p-adic integer, p > 2, we give an example of a torus-like continuum Y (i.e., inverse limit of an inverse sequence, where each term is the 2-torus T2 and each bonding map is a surjective homomorphism), which admits three 4-sheeted covering maps f0: X0 → Y, f1: X1 → Y, f2: X2 → Y such that the total spaces and X0 = Y, X2 are pair-wise non-homeomorphic. Furthermore, Y admits a 2p-sheeted covering map f3: X3 → Y such that X3 and Y are non-homeomorphic. In particular, Y is not a self-covering space. This example shows that the class of self-covering spaces is not closed under the operation of forming inverse limits with open surjective bonding maps.

KW - Covering mapping

KW - Direct system

KW - H-connected space

KW - Inverse system

KW - p-adic number

KW - Quadratic number

KW - Torsion-free group of rank 2

KW - Torus-like continuum

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DO - 10.1016/j.topol.2003.06.006

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SN - 0166-8641

VL - 153

SP - 359

EP - 369

JO - Topology and its Applications

JF - Topology and its Applications

IS - 2-3 SPEC. ISS.

ER -

Torus-like continua which are not self-covering spaces

Abstract

Keywords

ASJC Scopus subject areas

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