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On maps with continuous path lifting

Volume 261 / 2023

Jeremy Brazas, Atish Mitra Fundamenta Mathematicae 261 (2023), 201-234 MSC: Primary 55R65; Secondary 55Q52, 57M10, 57M05. DOI: 10.4064/fm977-3-2023 Published online: 13 March 2023

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 $\pi _1$: 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 $H\leq \pi _1(X,x_0)$ with totally path-disconnected coset space $\pi _1(X,x_0)/H$. Here, weak equivalence refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.

Authors

  • Jeremy BrazasDepartment of Mathematics
    West Chester University
    West Chester, PA 19383, USA
    e-mail
  • Atish MitraDepartment of Mathematical Sciences
    Montana Technological University
    Butte, MT 59701, USA
    e-mail

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