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 Gw be the
isotropy subgroup of w in G. Although Gw is not a normal subgroup of G, it
follows from Effros’ theorem and Theorem 2 of this paper that the coset
space G∕Gw is a solenoid homeomorphic to M and, therefore, a topological
group.
