Covering maps for locally path-connected spaces
Volume 218 / 2012
Abstract
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 topology introduced in Spanier's Algebraic 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 $\widetilde X$ called the lasso topology. Then the fundamental group $\pi_1(X)$ as a subspace of $\widetilde X$ with the lasso topology becomes a topological group. Also, 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 $\pi_1(X)$ with the lasso topology. Such groups include $\pi(\mathcal{U},x_0)$ ($\mathcal{U}$ being an open cover of $X$) and the kernel of the natural homomorphism $\pi_1(X,x_0)\to \check\pi_1(X,x_0)$.
