On maps with continuous path lifting

Abstract

We study a natural generalization of covering projections defined in terms of unique lifting properties. A map $p:E\to X$ has the "continuous path-covering property" if all paths in $X$ lift uniquely and continuously (rel. basepoint) with respect to the compact-open topology. We show that maps with this property are closely related to fibrations with totally path-disconnected fibers and to the natural quotient topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering projections to a classification of maps with the continuous path-covering property in terms of topological $π_1$: for any path-connected Hausdorff space $X$, maps $E\to X$ with the continuous path-covering property are classified up to weak equivalence by subgroups $H\leq π_1(X,x_0)$ with totally path-disconnected coset space $π_1(X,x_0)/H$. Here, "weak equivalence" refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.

Figures (1)

• Figure 1: The map $p:X_1\to X_2$ has the continuous path-covering property but is not a Hurewicz fibration.

Theorems & Definitions (86)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• Definition 3.1
• Remark 3.2
• Proposition 3.3