Universal covering spaces | Algebraic Topology | NJ Wildberger

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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[@njwildberger]

I don't really know at this point. Perhaps I will teach this in a few years again, at which point I will try to post some additional lectures, or I might just do so anyway. But I anticipate around 40 lectures.

[@jonz9813]

thanks for another crystal clear lecture. I enjoyed each and every video of this series.

[@flashbacktim]

Moi aussi

[@njwildberger]

Basically correct: when we were constructing/classifying surfaces, we saw that we could make a torus with four edges: say a,b,a^(-1),b^(-1). For a two holed torus we needed another four edges, say c,d,c^(-1),d^(-1). For a 3-hole torus, another four edges; say e,f,e^(-1),f^(-1) and so on. All the surfaces of genus bigger than one can be covered by gluing suitably sized octagons, or 12-gons, or 16-gons etc together in the right way in the hyperbolic plane.

[@morelli6831]

THANK YOU, the explanations are so clear and smooth that make it really easy to get a hold of the concepts. Thank you again for this

[@loicetienne7570]

Is the action of the fundamental group on the universal covering space a group of isometries (with the standard metric of the sphere resp. the affine plane resp. the hyperbolic plane) ? And is there a 'natural' way to define the distance in the universal covering space from the definition of its points being homotopy-equivalence classes of paths in the base space ?

[@brendawilliams8062]

Thank you

[@elliecottam1590]

thank you