\(1/\kappa \)-Homogeneous long solenoids

We study nonmetric analogues of Vietoris solenoids. Let \(\Lambda \) be an ordered continuum, and let \(\vec {p}=\langle p_1,p_2,\ldots \rangle \) be a sequence of positive integers. We define a natural inverse limit space \(S(\Lambda ,\vec {p})\), where the first factor space is the nonmetric “circle” obtained by identifying the endpoints of \(\Lambda \), and the nth factor space, \(n>1\), consists of \(p_1p_2 \ldots p_{n-1}\) copies of \(\Lambda \) laid end to end in a circle. We prove that for every cardinal \(\kappa \ge 1\), there is an ordered continuum \(\Lambda \) such that \(S(\Lambda ,\vec {p})\) is \(\frac{1}{\kappa }\)-homogeneous; for \(\kappa >1\), \(\Lambda \) is built from copies of the long line. Our example with \(\kappa =2\) provides a nonmetric answer to a question of Neumann-Lara, Pellicer-Covarrubias and Puga from 2005, and with \(\kappa =1\) provides an example of a nonmetric homogeneous circle-like indecomposable continuum. We also show that for each uncountable cardinal \(\kappa \) and for each fixed \(\vec {p}\), there are \(2^\kappa \)-many \(\frac{1}{\kappa }\)-homogeneous solenoids of the form \(S(\Lambda ,\vec {p})\) as \(\Lambda \) varies over ordered continua of weight \(\kappa \). Finally, we show that for every ordered continuum \(\Lambda \) the shape of \(S(\Lambda ,\vec {p})\) depends only on the equivalence class of \(\vec {p}\) for a relation similar to one used to classify the additive subgroups of \(\mathbb {Q}\). Consequently, for each fixed \(\Lambda \), as \(\vec {p}\) varies, there are exactly \(\mathfrak {c}\)-many different shapes, where \(\mathfrak {c}=2^{\aleph _0}\), (and there are also exactly that many homeomorphism types) represented by \(S(\Lambda ,\vec {p})\).