Uncountable groups and the geometry of inverse limits of coverings

Abstract

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 ${\underleftarrow{\lim }}^1$, which is, however, notoriously difficult to compute when the ${\pi }_1(X)$ is uncountable.To circumvent this difficulty, we express the set of path-components of the inverse limit, $\hat{X}$, in terms of the functors $\underleftarrow{\lim }$ and ${\underleftarrow{\lim }}^1$ applied to sequences of countable groups arising from polyhedral approximations of $X$. A consequence of our computation is that path-connectedness of lifting space implies that ${\pi }_1(\tilde{X})$ supplements ${\pi }_1(X)$ in ${\check{\pi }}_1(X)$ where ${\check{\pi }}_1(X)$ is the inverse limit of fundamental groups of polyhedral approximations of $X$. As an application we show that $\mathcal{G}\cdot {\ker }_{\mathbb{Z}}(\hat{F})=\hat{F}\ne \mathcal{G}\cdot {\ker }_{B(1,n)}(\hat{F})$, where $\hat{F}$ is the canonical inverse limit of finite rank free groups, $\mathcal{G}$ is the fundamental group of the Hawaiian Earring, and ${\ker }_A(\hat{F})$ is the intersection of kernels of homomorphisms from $\hat{F}$ to $A$.