Covering maps for locally path-connected spaces

N. Brodskiy¹ ; J. Dydak¹ ; B. Labuz¹ ; A. Mitra¹

¹Department of Mathematics University of Tennessee Knoxville, TN 37996, U.S.A.

Fundamenta Mathematicae, Tome 218 (2012) no. 1, pp. 13-46

Résumé

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 $\pi_1(X,x_0)\to \check\pi_1(X,x_0)$.

Détail

DOI : 10.4064/fm218-1-2

Keywords: define peano covering maps prove basic properties analogous classical covers their domain always locally path connected range may arbitrary topological space characterizations peano covering maps via uniqueness homotopy lifting property locally path connected spaces regular peano covering maps path connected spaces shown identical generalized regular covering maps introduced fischer zastrow path connected every peano covering map equivalent projection widetilde where subgroup fundamental group widetilde equipped topology introduced spaniers algebraic topology projection widetilde peano covering map only has unique path lifting property define topology widetilde called lasso topology fundamental group subspace widetilde lasso topology becomes topological group has characterization widetilde having unique path lifting property normal subgroup namely closed lasso topology groups include mathcal mathcal being cover kernel natural homomorphism check