Pullbacks of generalized universal coverings

Fischer, Hanspeter

doi:10.2140/agt.2007.7.1379

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Pullbacks of generalized universal coverings

Hanspeter Fischer

Algebraic & Geometric Topology 7 (2007) 1379–1388

DOI: 10.2140/agt.2007.7.1379

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 $f^{*} p : f^{*} \tilde{X} \to Y$ of such a generalized universal covering $p : \tilde{X} \to X$ and consider the following question: given a path-component $Ỹ$ of $f^{*} \tilde{X}$ , when exactly is $f^{*} p |_{Ỹ} : Ỹ \to Y$ a generalized universal covering? We show that the classical criterion, of $f_{#} : π_{1} (Y) \to π_{1} (X)$ 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

References

Publication

Received: 14 January 2007

Revised: 29 May 2007

Accepted: 28 August 2007

Published: 24 September 2007

Authors

Hanspeter Fischer
Department of Mathematical Sciences Ball State University Muncie IN 47306 U.S.A.
http://www.cs.bsu.edu/~fischer/

Volume 7, issue 3 (2007)