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- W4287367087 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 $checkpi_1(X)$ where $checkpi_1(X)$ is the inverse limit of fundamental groups of polyhedral approximations of $X$. As an application we show that $mathcal Gcdot ker_{mathbb Z}(widehat F)= widehat Fnemathcal Gcdot ker_{B(1,n)}(widehat F)$, 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 $A$." @default.
- W4287367087 created "2022-07-25" @default.
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- W4287367087 date "2021-01-27" @default.
- W4287367087 modified "2023-09-27" @default.
- W4287367087 title "Uncountable groups and the geometry of inverse limits of coverings" @default.
- W4287367087 doi "https://doi.org/10.48550/arxiv.2101.11457" @default.
- W4287367087 hasPublicationYear "2021" @default.
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