# Fibrations, unique path lifting, and continuous monodromy

## TL;DR

This paper investigates the conditions under which certain fibrations with specific properties exist in topological spaces, emphasizing the roles of path lifting, monodromy continuity, and local quasinormality.

## Contribution

It establishes new criteria linking continuous monodromies and local quasinormality to the existence of fibrations with unique path lifting.

## Key findings

- Unique path lifting is guaranteed by continuous monodromies in $T_1$ fibers.
- $X$ is homotopically path Hausdorff relative to $H$ under local quasinormality.
- Fibrations exist when $X$ is locally path connected, $H$ is locally quasinormal, and monodromies are continuous.

## Abstract

Given a path-connected space $X$ and $H\leq\pi_1(X,x_0)$, there is essentially only one construction of a map $p_H:(\widetilde{X}_H,\widetilde{x}_0)\rightarrow(X,x_0)$ with connected and locally path-connected domain that can possibly have the following two properties: $(p_{H})_{\#}\pi_1(\widetilde{X}_H,\widetilde{x}_0)=H$ and $p_H$ has the unique lifting property. $\widetilde{X}_H$ 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.