$1/k$-homogeneous long solenoids
We study nonmetric analogues of Vietoris solenoids. Let $Λ$ be an ordered continuum, and let $\vec{p}=\langle p_1,p_2,\dots\rangle$ be a sequence of positive integers. We define a natural inverse limit space $S(Λ,\vec{p})$, 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 $p_1p_2\cdot\dots \cdot p_{n-1}$ copies of $Λ$ laid end to end in a circle. We prove that for every cardinal $κ\geq 1$, there is an ordered continuum $Λ$ such that $S(Λ,\vec{p})$ is $\frac{1}κ$-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-Espinosa from 2005, and with $κ=1$ provides an example of a nonmetric homogeneous circle-like indecomposable continuum. Finally, we employ a cohomology argument to prove that for each ordered continuum $Λ$, as $\vec{p}$ varies there are $2^ω$-many nonhomeomorphic spaces $S(Λ,\vec{p})$.