Volume 7, issue 3 (2007)

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Pullbacks of generalized universal coverings

Hanspeter Fischer

Algebraic & Geometric Topology 7 (2007) 1379–1388
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 $p:\stackrel{̃}{X}\to X$ which is characterized by the usual unique lifting criterion and the fact that $\stackrel{̃}{X}$ 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 ${f}^{\ast }p:{f}^{\ast }\stackrel{̃}{X}\to Y$ of such a generalized universal covering $p:\stackrel{̃}{X}\to X$ and consider the following question: given a path-component $Ỹ$ of ${f}^{\ast }\stackrel{̃}{X}$, 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.

Keywords
generalized covering space, pullback, fibered product
Mathematical Subject Classification 2000
Primary: 55R65
Secondary: 57M10, 54B99