AlgTop28: Covering spaces and fundamental groups
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- čas přidán 28. 09. 2012
- We illustrate the ideas from the last lectures by giving some more examples of covering spaces: of the torus, and the two-holed torus. Then we begin to explore the relationship between the fundamental groups of a covering space X and a base space B under a covering map p:X to B.
For this we need two important Lemmas: the Lifting Path lemma, and the Lifting Homotopy lemma. Then we obtain the basic result that the fundamental group of X can be viewed as a subgroup of the fundamental group of the base B, via the induced homomorphism of p. So the possibility emerges of studying covering spaces of B by studying subgroups of the fundamental group of B.
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What I really like about this is that he uses a lot of drawings. Most mathematicians seem to be either unwilling or incapable to do this. Instead they use too much formalism and it puts me to sleep. Granted this becomes hard to do when the spaces are more abstract or of higher dimension, for example S^3, needs some more tricks, like stereo graphic projection. But I’m usually lost when I can’t have at least some half ass geometric intuition. Norman is a great teacher.
He is indeed a great teacher and yes, pictures are important especially when your teaching a highly geometric subject like algebraic topology. But you also have to understand that mathematics is an axiomatic subject. Rigorous proofs are necessary, otherwise it's easy to make false assertions.
You need everything to fully understand a subject. I think of formalism as computer software. You implement the concepts in the mathematical language of logic. But try designing software without the concepts.
I agree. A magnificent and talented teacher.
At 23:55 the image of path alpha can be covered by finite number of such neigborhoods since it's compact set. A path is a continuous map from the unit interval wich is compact set and a continuous image of compact set is compact set.
Boris Petkovic perfect explanation!
Dear professor thank you so much for these videos. I hope one day we all can follow in your footsteps.
Also regarding the guy's question at 24:00, a sufficient condition is for the red path to be compact. You can just take a neighborhood at every point on the red path, and that's an open cover. Any open cover of a compact space has a finite subcover.
Lifting lemma assumes that from any open cover of path I can get finite cover. In other words it means that path is compact set. But it follows from definition of path because path is continues function of compact set [0,1].
Presenting the main idea of the 'lifting path lemma' without set-theoretic topology (in particular, the definition of compactness with open covers) is an interesting pedagogic choice. It is what I was looking for, an exposition of the main ideas of algebraic topology without too much technical details (which are exposed in full length in many books anyway, if someone wants to go deeper).
Please start a series on Algebraic Geometry!
Right.
Very clear lecture
I'm having trouble clarifying my thinking. At around 40 minutes you say gamma is an element of the fundamental group of (X,x). So gamma is not a path, but rather an equivalence class of paths? Then when we say there is a homotopy between p* of gamma and the constant loop based at b, isn't that an abuse of notation? Shouldn't we pick some path alpha in the equiv. class of gamma, map it by p, the covering map, and say there is a homotopy between p of alpha and the constant loop based at b? Would that be better? Basically I'm asking if there was a slight abuse of notation in that proof? Thanks for putting up these videos! I use all your playlists to learn. It's really amazing.
+Alex Heaton I know I'm 8 months late, but whatever; formally, you are correct. gamma is both used to denote a path and the equivalence class of paths in the fundamental group it belongs to. This kind of stuff happens all the time when working with equivalence classes though, and I don't really see the harm.
This guy is my kinda guy.
Thank you
What, if any, conditions need to be met by the topological spaces in order for the path lifting lemma to be true?
It would be homotopic to the identity in case the space is contractible, no?