1/κ-Homogeneous long solenoids

We study nonmetric analogues of Vietoris solenoids. Let Λ be an ordered continuum, and let p → = ⟨ p 1 , p 2 , … ⟩ be a sequence of positive integers. We define a natural inverse limit space S ( Λ , p → ) , where the first factor space is the nonmetric “circle” obtained by identifying the endpoints of Λ , and the n th factor space, n > 1 , consists of p 1 p 2 … p n - 1 copies of Λ laid end to end in a circle. We prove that for every cardinal κ ≥ 1 , there is an ordered continuum Λ such that S ( Λ , p → ) is 1 κ -homogeneous; for κ > 1 , Λ is built from copies of the long line. Our example with κ = 2 provides a nonmetric answer to a question of Neumann-Lara, Pellicer-Covarrubias and Puga from 2005, and with κ = 1 provides an example of a nonmetric homogeneous circle-like indecomposable continuum. We also show that for each uncountable cardinal κ and for each fixed p → , there are 2 κ -many 1 κ -homogeneous solenoids of the form S ( Λ , p → ) as Λ varies over ordered continua of weight κ . Finally, we show that for every ordered continuum Λ the shape of S ( Λ , p → ) depends only on the equivalence class of p → for a relation similar to one used to classify the additive subgroups of Q . Consequently, for each fixed Λ , as p → varies, there are exactly c -many different shapes, where c = 2 ℵ 0 , (and there are also exactly that many homeomorphism types) represented by S ( Λ , p → ) .