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 ← 1 , which is, however, notoriously difficult to compute when the fundamental group, π 1 (X) , is uncountable. To circumvent this difficulty, we express the set of path-components of the inverse limit X ˜ of a sequence of covering spaces in terms of the functors lim ← and lim ← 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 ˜ , implies that π 1 (X ˜) supplements π 1 (X) in π ˇ 1 (X) where π ˇ 1 (X) is the inverse limit of fundamental groups of polyhedral approximations of X. As an application we show that G ⋅ Ker Z (F ˆ) = F ˆ ≠ G ⋅ Ker B (1 , n) (F ˆ) , where F ˆ 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 ˆ) is the intersection of kernels of homomorphisms from F ˆ to A. [ABSTRACT FROM AUTHOR]