Noncommutative Geometry of Quantized Coverings

Abstract

This research is devoted to the noncommutative generalization of topological coverings. Otherwise since topological coverings are related to the set of geometric constructions one can obtain noncommutative generalizations of these constructions. Here the generalizations of the universal covering space, fundamental group, homotopy theory, Hurewicz homomorphism, covering of the Riemannian manifold, flat connection are explained. The theory gives pure algebraic proof well known results of the topology and the differential geometry. Besides there are applications of the theory to (unbounded) operator spaces and this theme is also discussed here.Comment: 686 pages, 134 reference