Algebraic Topology 8: Properties of Covering Spaces

We continue our study of covering spaces, reviewing the definition and a couple examples and then proving some important properties. In particular, we discuss the unique path lifting property and homotopy lifting property. These let us prove that the induced homomorphism between a covering space and base space is injective. We also discuss the general existence of a universal cover and construct the universal cover for the torus and 2-holed torus.

Presented by Anthony Bosman, PhD.

[joefuentes]

I have no idea what you're talking about but i promise i will

[depressedguy]

i am learning this from hatcher book along with your videos, i am currently at relative homology but old stuff through your videos.

[IDontKnow-vhno]

Thank you so much for uploading these videos, got a topology exam but my lecturer wasnt the best, this has helped so much

[-minushyphentwo]

20:30 - 26:25 Unique Path Lifting Property, and its proof

The idea is to construct the path one neighborhood/open set at a time. It wasn’t mentioned how to ensure that the path could always be covered with open neighborhoods this way, but I think these are the details:

Since that path is a continuous image of the closed unit interval(which is compact), the path is compact. So, get a small enough open set around each point in the closed unit interval so that the open set is homeomorphic to the corresponding open set in the covering space, and consider the image of the small section of the path in that open set. These small open sets form an open cover of the path, since there’s an open set for each point, so by compactness there’s a finite subcover of it. We can then use this finite subcover to creep along the path and construct it bit by bit.

Or maybe I’m wrong and all this isn’t even necessary?

43:15 That space is called the Hawaiian Earring

[超長数学雑談]

Nice to meet you. I am Japanese and my English is poor, but please allow me to comment. At the beginning of the video(around 1:45), it says "p^-1（U_α） is homeomophic to U_α" on the blackboard.
I think it is a mistake for "p^-1（U_α） is the disjoint union of open sets, and each open set is homeomorphic to U_α by the restriction map of p" .

[ompatel]

I have had courses in real and complex analysis but I would love to learn those subjects from you
Also I loved the course on advanced linear algebra

[ompatel]

Your explanations are crystal clear I personally find that hatchers book is a bit too abstract

[ompatel]

Can you give me a schedule for the videos on algebraic topology

[DDranks]

I'm confused about 33:49 . I get that p* of fundamental group of the covering space is injective; after all, it is a subgroup of the fundamental group of the base space. However, I don't see how they could be _isomorphic_. The lifted fundamental group seems smaller than the base? For example in the case of S1 lifted to R, the fundamental group of R is the trivial group 1. How is that supposed to be isomorphic with S1 fundamental group Z? Injective homomorphism yes, but isomorphism? I don't see that.

[ompatel]

I really enjoy algebraic topology could you give me some other topics in math which are closely related to algebraic topology

[ompatel]

I have a question is it necessary to know functors because I don’t know what those are