Algebraic Topology 8: Properties of Covering Spaces

[@IDontKnow-vh3no]

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

[@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" .

[@MathatAndrews]

That's right, each of the open sets in the preimage is isomorphic to the open set in the base space. Good catch!

[@超長数学雑談]

​@@MathatAndrews Thanks for the reply. As others have commented, your lectures are easy to follow with lots of examples! I am watching this Algebraic Topology lecture video from the first one.

[@joefuentes2977]

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

[@-minushyphen1two379]

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

[@MathatAndrews]

Excellent! 👍🏼

[@-minushyphen1two379]

@@MathatAndrews How do you make the homotopy part rigorous? Thanks for the course btw! It really helped with intuition for this stuff. Can’t wait to learn what homology actually means

[@depressedguy9467]

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

[@MathatAndrews]

You're several weeks ahead of us! We'll get there - eventually.

[@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.

[@DDranks]

Ohh, now I got it. I missed that the tilde was there on both sides: p*(π1(X~)) ≈ π1(X~). So all we are saying that the image of p* is isomorphic to the domain, not to the fundamental group of the base space!

[@ashwinis6164]

Lifting and covering space both are same

[@ompatel9017]

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

[@MathatAndrews]

We're following Hatcher's text, linked in description. Releasing one video per week, typically Wed evening or Thursday morning.

[@ompatel9017]

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

[@MathatAndrews]

No mention of functors here. Though, as you learn more about the fundamental group, it will help you get ready to understand what a functor is.

[@ompatel9017]

Thanks

[@ompatel9017]

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

[@MathatAndrews]

I don't anticipate to upload any analysis lectures anytime soon, but we do have a series of lectures on metric spaces that may be of interest!

[@yixiangsun9559]

Thank you. You just saved my final.

[@ompatel9017]

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

[@MathatAndrews]

It is rather abstract. I work out lots of examples (and special cases) while reading to help make it a bit more concrete.

[@ompatel9017]

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

[@MathatAndrews]

You may enjoy Geometric Group Theory!

[@ompatel9017]

Please cover the entire hatcher book

[@MathatAndrews]

Working on it! Plan is to at least get into homology and cohomology.