Spaces whose finite sheeted covers are homeomorphic to a fixed space
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-trivial finite connected covering space of \(Y\) has a total space homeomorphic to \(X\). Then every finite sheeted cover of \(Y\) is an Abelian cover with total space homeomorphic to \(X\), \((\pi_1(Y))'\) is a perfect group, and \(H_1(Y)\neq 1\). Furthermore, either (1) \(Y\) has a unique finite sheeted cover that is a \(p\)-fold cover for some prime \(p\in\mathbb{N}\), and \(H_1(Y,\mathbb{Z}) \simeq\mathbb{Z}/p \mathbb{Z}\), or (2) \(H_1(Y, \mathbb{Z})\) is finitely generated free Abelian. A connected topological space \(X\) has the finite self-cover property (FSP) if every connected finite sheeted covering space of \(X\) has a total space homeomorphic to \(X\). A space with FSP is called trivial if it has no non-trivial finite sheeted covers, and non-trivial otherwise. A space \(Y\) has the constant finite cover property (CFCP) if every non-trivial finite sheeted cover of \(Y\) has total space homeomorphic to a fixed space \(X\), called the finite sheet of \(Y\). Among several interesting corollaries which are given, the following are most suggestive. Corollary 3.3: Let \(K\) be a finite regular 2-complex with CFCP and with a non-trivial finite-sheeted cover \(X\). Then either (1) \(X\) has no non-trivial finite sheeted covers, so that \(K\) has a unique non-trivial finite sheeted cover with total space homoeomorphic to \(X\); \(\pi_1(X)\) has no proper finite index subgroups; there is a prime \(p\) such that \(\pi_1(K)\) is an extension of \(\pi_1(X)\) by \(\mathbb{Z}/p \mathbb{Z}\) and \(H_1(Y,\mathbb{Z}) \simeq\mathbb{Z}/p \mathbb{Z}\), or (2) \(X\) has FSP and is non-trivial and either \(\pi_1(K) \simeq\mathbb{Z}\) and there is a finite tree \(T\) such that \(K\) is homeomorphic to \(S^1\times T\), or \(\pi_1(K) \simeq\mathbb{Z} \oplus \mathbb{Z}\) and \(K\) is homeomorphic to \(S^1\times S^2\). In either case \(K\) is homeomorphic to \(X\). Corollary 3.6. If \(M\) is an orientable Haken manifold with CFCP, then \(M=D^3\), \(S^1\times D^2\), \(S^1\times S^1\times I\) or \(S^1\times S^1 \times S^1\). Corollary 3.8. If \(M\) is a compact 3-manifold that has the FSP and is non-trivial, then \(M\) is homeomorphic to \(S^2\times S^1\), \(D^2\times S^1\), \(S^1 \times I\times S^1\) or \(S^1\times S^1\times S^1\). At the end of the paper the authors formulate two new questions about 3-manifolds with CFCP.
scientific article; zbMATH DE number 1901917