Spaces whose finite sheeted covers are homeomorphic to a fixed space
type: Item
instance of: scholarly article
title: Spaces whose finite sheeted covers are homeomorphic to a fixed space
zbMATH Open document ID: 1018.57001
DOI: 10.1016/S0166-8641(02)00132-3
author: Mathew Timm
author: Alberto Luis Delgado
published in: Topology and its Applications
publication date: 2003-04-28T00:00:00Z
review text: This paper is related to the following question of \textit{S. Miklos} [Questions Answers Gen. Topology (2) 14 (1996)]. Question: Given metric continua \(X\) and \(Y\) having the property that for infinitely many \(n\in \mathbb{N}\), \(n\)-fold covering projections \(f_n:X\to Y\) exist. Does it follow that \(X\) is homeomorphic to \(Y\)? The author of the present paper observes that the answer is no. The Möbius and has \(2k\)-fold covers by the annulus for \(k\in\mathbb{N}\) and the Klein bottle has \(2k\)-fold covers by the torus for all \(k\in\mathbb{N}\). In neither case is the total space of the covering homeomorphic to the base space. The author also indicates a general construction of higher dimensional examples: \(X=T \times X_0\), \(Y=T\times Y_0\), for \(T\) the \(n\)-torus and \(p_0:X_0 \to Y_0\) a non-trivial \(n_0\)-fold covering. However, he considers that even under some restrictions the question is of interest. One sort of condition suggested by the fact that the examples above mentioned are of dimension two or greater, comes from restricting the attention to one-dimensional continua. This case was considered by the second author in two articles submitted for publication. In this paper the authors consider another line of investigation. This line is suggested by the fact that the given examples do have the property that many of the covering spaces of the given \(Y\) have a total space homeomorphic to the given \(X\), but there are some that do not. In this case one wonders if the answer might be yes if one were to add the hypothesis that the total space of every finite sheeted covering space in some sufficiently large subcollection of the collection of all finite sheeted covering spaces \(Y\) have a total space homeomorphic to \(X\). The main result of the paper is the following theorem. Theorem 1.1: Let \(Y\) be a connected finite complex with a non-trivial finite connected cover. Let \(X\) be a connected complex having the property that every non-tri…
reviewed by: Ioan Pop
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 20E34
zbMATH DE Number: 1901917
zbMATH Keywords: Self-cover
zbMATH Keywords: \(h\)-connected
zbMATH Keywords: \(hc\)-group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1901917