Fibrations, unique path lifting, and continuous monodromy

摘要

Given a path-connected space X and H < pi(1)(X, x(0)), there is essentially only one construction of a map p(H) : (<(H)over tilde>(H), (x) over tilde (0)) -> (X, x(0)) with connected and locally path-connected domain that can possibly have the following two properties: (p(H))(#)pi(1)((H) over tilde (H), (x) over tilde (0))) = H and p(H) has the unique lifting property. (H) over tilde (H) consists of equivalence classes of paths starting at x(0), appropriately topologized, and p(H) is the endpoint projection. For p(H) to have these two properties, T-1 fibers are necessary and unique path lifting is sufficient. However, p(H) always admits the standard lifts of paths.We show that p(H) has unique path lifting if it has continuous (standard) monodromies toward a T-1 fiber over x(0). Assuming, in addition, that H is locally quasinormal (e.g., if H is normal) we show that X is homotopically path Hausdorff relative to H. We show that p(H) is a fibration if X is locally path connected, H is locally quasinormal, and all (standard) monodromies are continuous. (C) 2019 Elsevier B.V. All rights reserved.