Given a path-connected space $X$ and $Hleqpi_1(X,x_0)$, 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.
Download