Uncountable groups and the geometry of inverse limits of coverings

In this paper we develop a new approach to the study of uncountable fundamental groups by using Hurewicz fibrations with the unique path-lifting property (lifting spaces for short) as a replacement for covering spaces. In particular, we consider the inverse limit of a sequence of covering spaces of X. It is known that the path-connectivity of the inverse limit can be expressed by means of the derived inverse limit functor (lim) under left arrow (1), which is, however, notoriously difficult to compute when the fundamental group, pi(1)(X), is uncountable. To circumvent this difficulty, we express the set of path-components of the inverse limit (X) over tilde of a sequence of covering spaces in terms of the functors (lim) under left arrow and (lim) under left arrow (1) applied to sequences of countable groups arising from polyhedral approximations of X. A consequence of our computation is that path-connectedness of a lifting space, (X) over tilde, implies that pi(1)(X) supplements pi(1)(X) in pi(1)(X) where pi(1)(X) is the inverse limit of fundamental groups of polyhedral approximations of X. As an application we show that G.Ker(Z)((F) over cap) = (F) over cap not equal G.Ker(B(1,n)) ((F) over cap), where (F) over cap is the canonical inverse limit of finite rank free groups, G is the fundamental group of the Hawaiian Earring, B(1, n) is the Baumslag-Solitar group, and Ker(A)((F) over cap) is the intersection of kernels of homomorphisms from (F) over cap to A. (C) 2021 Elsevier B.V. All rights reserved.