Algebraic Topology 3: Fundamental Group is a Group!
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- Опубликовано: 17 сен 2024
- Playlist: • Algebraic Topology
We recall the definition of the fundamental group develop in the previous lecture then prove that it is indeed a group. Finally, we show that the fundamental group of the circle is isomorphic to Z, the integers.
Presented by Anthony Bosman, PhD.
Learn more about math at Andrews University: www.andrews.ed...
In this course we are following Hatcher, Algebraic Topology: pi.math.cornel...
The isomorphism at 31:00 is independent of the path if the fundamental group is abelian. Not in general.
Yoooo banger video!! good explanations and examples. Have been using as a supplemental for my topology class to try to look over key ideas before lectures.
Eres el mejor tío ❤
Great
Also, does this way of mathematising spaces exhaust the possibilities. ie. Is there anything sensible in working topology beyond rationals?
Amazing
59:55
You make wonders of projection possible, like how you make your parameters irrational and what is the meaning of shape when they all approach infinity. Also how many types of "n" are there?
May i ask why do we need three pages of math to justify lifting paths and homotopies? I thought it follows from compositions of continuous maps are continuous, right? (in that composing paths and homotopies with lifts or projections preserves continuity.)
Didn't watch the video but path lifting lemmas are incredibly important foundational facts. It's not just that the maps are continuous which matters, but that you can uniquely extend them and they exist.
22:00 how do we know that [f] * ([g] * [h] ) even exists though? We should probably prove it.
I think we already proved that the binary operation is well defined and it is clearly closed given the definition so then it should exist
Suggestion to the camera operator for future videos... Stop zooming in. Its extremely useful to see more of the blackboard at once.
I personally like the camera work