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 : X̃ X which is characterized by the usual unique lifting criterion and the fact that X̃ is path-connected, locally path-connected and simply-connected.

For a path-connected topological space Y and a map f : Y X, we form the pullback fp : fX̃ Y of such a generalized universal covering p : X̃ X and consider the following question: given a path-component of fX̃, when exactly is fp| : Y a generalized universal covering? We show that the classical criterion, of f# : π1(Y ) π1(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/