A characterization of solenoids
Charles L. Hagopian · Pacific Journal of Mathematics · 1977
Suppose M is a homogeneous continuum and every proper subcontinuum of M is an arc.Using a theorem of E. G. Effros involving topological transformation groups, we prove that M is circle-like.This answers in the affirmative a question raised by R. H. Bing.It follows from this result and a theorem of Bing that M is a solenoid.Hence a continuum is a solenoid if and only if it is homogeneous and all of its proper subcontinua are arcs.The group G of homeomorphisms of M onto M with the topology of uniform convergence has an unusual property.For each point w of M, let G w be the isotropy subgroup of w in G.Although G w is not a normal subgroup of G, it follows from Effros' theorem and Theorem 2 of this paper that the coset space G/G w is a solenoid homeomorphic to M and, therefore, a topological group.