Fibrations, unique path lifting, and continuous monodromy

Abstract

Given a path-connected space X and H≤π1(X,x0), there is essentially only one construction of a map pH:(X˜H,x˜0)→(X,x0) with connected and locally path-connected domain that can possibly have the following two properties: (pH)#π1(X˜H,x˜0)=H and pH has the unique lifting property. X˜H consists of equivalence classes of paths starting at x0, appropriately topologized, and pH is the endpoint projection. For pH to have these two properties, T1 fibers are necessary and unique path lifting is sufficient. However, pH always admits the standard lifts of paths.We show that pH has unique path lifting if it has continuous (standard) monodromies toward a T1 fiber over x0. 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 pH is a fibration if X is locally path connected, H is locally quasinormal, and all (standard) monodromies are continuous.