Pullbacks of generalized universal coverings

Abstract

It is known that there is a wide class of path-connected topological spaces $X$, which are not semilocally simply-connected but have a generalized universal covering, that is, a surjective map $p:\tilde{X}\to X$ which is characterized by the usual unique lifting criterion and the fact that $\tilde{X}$ is path-connected, locally path-connected and simply-connected.

For a path-connected topological space $Y$ and a map $f:Y\to X$, we form the pullback $f^{\ast }p:f^{\ast }\tilde{X}\to Y$ of such a generalized universal covering $p:\tilde{X}\to X$ and consider the following question: given a path-component $Ỹ$ of $f^{\ast }\tilde{X}$, when exactly is $f^{\ast }p{|}_{Ỹ}:Ỹ\to Y$ a generalized universal covering? We show that the classical criterion, of $f_{\#}:{\pi }_1\left(Y\right)\to {\pi }_1\left(X\right)$ being injective, is too coarse a notion to be sufficient in this context and present its appropriate (necessary and sufficient) refinement.