Universal covering spaces | Algebraic Topology 29 | NJ Wildberger

Описание: We begin by giving some examples of the main theorem from the last lecture: that the associated homomorphism of fundamental groups associated to a covering space p:X to B injects pi(X) as a subgroup of pi(B). We look at helical coverings of a circle, and also a two-fold covering of the wedge of two circles.
So a main idea is that covering spaces of a space B are associated to subgroups of pi(B). The covering space associated to the identity subgroup is called the universal covering space of B; it has the distinguishing property that it is simply connected: any loop on it is homotopic to the constant loop.

To construct the universal cover of a space B, we proceed in an indirect fashion, considering paths in B from a fixed base point b, up to homotopy. Any such path can be mapped to its endpoint: this is the covering map. The universal covering space of a sphere or projective plane is the sphere, that of the torus or Klein bottle is the Euclidean plane, while all surfaces of negative Euler characteristic, like a two holed torus, has universal cover consisting of the Hyperbolic plane. To describe this completely would be a long story, we give just an initial orientation to this important connection between geometry and topology.

Finally we discuss how other covering spaces may be created from a universal covering space.

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