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 → 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 π1: for any path-connected Hausdorff space X, maps E → X with the continuous path-covering property are classified up to weak equivalence by subgroups H ≤ π1(X, x0) with totally path-disconnected coset space π1(X, x0)/H. Here, weak equivalence refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.