Covering maps for locally path-connected spaces

Details Covering maps for locally path-connected spaces Covering maps for locally path-connected spaces

2008-01-31

arxiv.org/abs/0801.4967v3

Mathematics

Geometric Topology

25 pages, several references added plus Proposition 4.12 and Example 4.13

Scientific paper

We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces. Regular Peano covering maps over path-connected spaces are shown to be identical with generalized regular covering maps introduced by Fischer and Zastrow. If $X$ is path-connected, then every Peano covering map is equivalent to the projection $\widetilde X/H\to X$, where $H$ is a subgroup of the fundamental group of $X$ and $\widetilde X$ equipped with the basic topology. The projection $\widetilde X/H\to X$ is a Peano covering map if and only if it has the unique path lifting property. We define a new topology on $\widetilde X$ for which one has a characterization of $\widetilde X/H\to X$ having the unique path lifting property if $H$ is a normal subgroup of $\pi_1(X)$. Namely, $H$ must be closed in $\pi_1(X)$. Such groups include $\pi(\mathcal{U},x_0)$ ($\mathcal{U}$ being an open cover of $X$) and the kernel of the natural homomorphism from the fundamental group to the Cech fundamental group.

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