A note on finite-sheeted covering maps from 2-dimensional compact Abelian groups
A (weak) solenoidal space is the limit space of an inverse sequence, where each term is a connected, locally pathwise connected, semilocally 1-connected space and each bounding map is a (covering map) regular covering map [cf. \textit{M. C. McCord}, Trans. Am. Math. Soc. 114, 197--209 (1965; Zbl 0136.43603)]. Each compact connected 2-dimensional abelian group \(X\) is a solenoidal space obtained as the limit of an inverse sequence, where each term is a 2-torus \(\mathbb{T}^2\) and each bounding map is a covering homomorphism; such groups are called toroidal groups. A Klein bottle weak solenoidal space is the limit of an inverse sequence, where each term is a Klein bottle \(K\) and each bounding map is a covering map from \(K\) to \(K\). \textit{C. Tezer} introduced and studied Klein bottle weak solenoidal spaces \(\pmb{\Sigma}(\mathbf{p,q,r})\), where \(p=(p_i),q=(q_i), r=(r_i)\) are sequences of integers, \(p_i\neq 0\) and \(r_i\) odd for each \(i\) in [Q. J. Math., Oxf. II. Ser. 40, No.~158, 225--243 (1989; Zbl 0675.55008)]. Recently, finite-sheeted covering maps over toroidal groups \(X\) were studied by \textit{K. Eda} and \textit{V. Matijević} [Topology Appl. 153, No.~7, 1033--1045 (2006; Zbl 1104.22005)]. It turned out that finite-sheeted covering maps over \(X\) are determined by finite-index torsion free supergroups of the Pontryagin dual \(\widehat{X}\), cf. \textit{K. Eda} and \textit{V. Matijević} [J. Algebra 319, 3567--3587 (2008; Zbl 1225.20046)]. Moreover, using finite index subgroups of \(\widehat{X}\) the authors also presented finite-sheeted covering maps from \(X\) to other compact connected groups. ``The main step in the investigation was the reduction to the case of finite-sheeted covering homomorphisms \(f:X'\rightarrow X\) between two toroidal groups. Since \(T^2\) is a covering space for a Klein bottle \(K\), the next step of investigation was to study finite-sheeted covering maps over Klein bottle weak solenoidal spaces \(\pmb{\Sigma}(\mathbf{p,q,r})\) (see \textit{V. Matijević} [Glas. Mat., III. Ser. 42, No.~1, 19--41 (2007; Zbl 1130.57003)])'' (from the Introduction). If a toroidal group \(X\) admits a finite-sheeted covering map, then the total space is a toroidal group \(X'\), generally non-homeomorphic to \(X\). If a Klein bottle weak solenoidal space \(\pmb{\Sigma}(\mathbf{p,q,r})\) admits a finite-sheeted covering map, then the total space is a toroidal group \(X\) or a Klein bottle weak solenoidal space \(\pmb{\Sigma}(\mathbf{p}',\mathbf{q}',\mathbf{r}')\) homeomorphic to \(\pmb{\Sigma}(\mathbf{p,q,r})\). In the present paper the author considers the following questions: {\parindent4mm \begin{itemize}\item[{\(\bullet\)}] Is every toroidal group \(X\) a covering space over some Klein bottle weak solenoidal space \(\pmb{\Sigma}(\mathbf{p,q,r})\) ? \item[{\(\bullet\)}] If a toroidal group \(X\) covers groups, does it cover Klein bottle weak solenoidal spaces? \item[{\(\bullet\)}] If a toroidal group \(X\) covers Klein bottle weak solenoidal spaces, does it cover groups? \end{itemize}} The author answers the first two question in the negative (Theorem 21, Corollary 22) and the third one in the positive (Theorem 4, Corollary 5). Moreover, the author gives an example of a toroidal group \(X\) which admits an \(s\)-sheeted covering map \(f:X\rightarrow X'\) to groups for each \(s\in \mathbb{N}\), but which does not cover any Klein bottle weak solenoidal space \(\pmb{\Sigma}(\mathbf{p,q,r})\) (Theorem 23, Corollary 24). It turns out that the key object in this investigation is the product \(\pmb{\Sigma}(\mathbf{p})\times \pmb{\Sigma}(\mathbf{r})\) of two solenoids, where at least one of the sequences \(\mathbf{p}\) and \(\mathbf{r}\) consists of odd integers.
scientific article; zbMATH DE number 5808696