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Shape fibrations, multivalued maps and shape groups. (English) Zbl 0904.54010
The author defines the notion of shape fibration with the near-lifting of near-multivalued paths property. He studies this extension of the unique path lifting property of Hurewicz fibrations, adapted to shape fibrations. He shows that a shape fibration has this property iff its fibers are totally disconnected. He completely determines its relation with Hurewicz fibrations with the unique path lifting property.
In the last part of this paper the author studies algebraic properties of shape fibration with the near-lifting of near-multivalued path property, related to homotopy groups, shape groups and strong shape groups. The main result of this part is a theorem giving necessary and sufficient algebraic conditions for the existence of lifting of fine multivalued maps defined in Peano continua and with values in the base space of a shape fibration with near-lifting of near-multivalued paths.

54C56 Shape theory in general topology
55P55 Shape theory
55Q05 Homotopy groups, general; sets of homotopy classes
55Q07 Shape groups
55R05 Fiber spaces in algebraic topology
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