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 : ~X→ X which is characterized by the usual unique lifting criterion and the fact that ~X 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 f*p: f* ~X→Y of such a generalized universal covering p: ~X→X and consider the following question: given a path-component ~Y of f* ~X, when exactly is f* p∣_~: ~Y→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

Primary: 55R65

Secondary: 54B99, 57M10

References

Publication

Received: 14 January 2007
Revised: 29 May 2007
Accepted: 28 August 2007
Published: 24 September 2007

Authors