Pro-groups and generalizations of a theorem of Bing

Abstract

A matchbox manifold is a generalized lamination, which is a continuum whose path components define the leaves of a foliation of the space. A matchbox manifold is M -like if it has the shape of a fixed topological space M . When M is a closed manifold, in a previous work, the authors have shown that if M is a matchbox manifold which is M -like, then it is homeomorphic to a weak solenoid. In this work, we associate to a weak solenoid a pro-group, whose pro-isomorphism class is an invariant of the homeomorphism class of M. We then show that an M -like matchbox manifold is homeomorphic to a weak solenoid whose base manifold has fundamental group which is non co-Hopfian; that is, it admits a non-trivial self-embedding of finite index. We include a collection of examples illustrating this conclusion.