Uncountable groups and the geometry of inverse limits of coverings

TL;DR

This paper introduces a novel approach to studying uncountable fundamental groups using lifting spaces and inverse limits, providing new insights into their structure and connectivity properties.

Contribution

It develops a new framework replacing covering spaces with lifting spaces to analyze uncountable fundamental groups via inverse limits and derived functors.

Findings

Path-connectedness of inverse limits relates to fundamental group inclusions.

Expresses the set of path-components using inverse limit functors.

Shows specific algebraic properties of inverse limits of free groups.

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 $\varprojlim^1$, 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, $\widehat X$, in terms of the functors $\varprojlim$ and $\varprojlim^1$ 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 $\mathcal G\cdot \ker_{\mathbb Z}(\widehat F)= \widehat F\ne\mathcal G\cdot \ker_{B(1,n)}(\widehat F)$, where $\widehat F$ is the canonical inverse limit of finite rank free groups, $\mathcal G$ is the fundamental group of the Hawaiian Earring, and $\ker_A(\widehat F)$ is the intersection of kernels of homomorphisms from $\widehat{F}$ to $A$.