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Generalized universal covering spaces and the shape group

Hanspeter Fischer, Andreas Zastrow (2007)

Fundamenta Mathematicae

If a paracompact Hausdorff space X admits a (classical) universal covering space, then the natural homomorphism φ: π₁(X) → π̌₁(X) from the fundamental group to its first shape homotopy group is an isomorphism. We present a partial converse to this result: a path-connected topological space X admits a generalized universal covering space if φ: π₁(X) → π̌₁(X) is injective. This generalized notion of universal covering p: X̃ → X enjoys most of the usual properties, with the possible exception of evenly...

Gradient otopies of gradient local maps

Piotr Bartłomiejczyk, Piotr Nowak-Przygodzki (2011)

Fundamenta Mathematicae

We introduce various classes of local maps: gradient, gradient-like, proper etc. We prove Parusiński's theorem for otopy classes of gradient local maps.

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