What is...the fundamental group?

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Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What is...the fundamental group? Or: How not to hang pictures on walls.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
These videos are concerned with algebraic topology, and not general topology. (These two are not to be confused.) I assume that you know bits and pieces about general topology, but not too much, I hope.
Slides.
www.dtubbenhauer.com/youtube.html
Website with exercises.
www.dtubbenhauer.com/lecture-algtop-2021.html
Material used.
Hatcher, Chapter 1
en.wikipedia.org/wiki/Fundamental_group
Awesome fundamental group animations (and other great stuff).
math.unice.fr/~indira/Mygifs.html
Circle pictures.
en.wikipedia.org/wiki/Fundamental_group#/media/File:Fundamental_group_of_the_circle.svg
Spivak’s hanging-pictures-puzzle (I stole some pictures from there).
arxiv.org/pdf/1203.3602.pdf
www.i-programmer.info/news/112-theory/3951-picture-hanging-puzzles.html
Hatcher’s book (I sometimes steal some pictures from there).
pi.math.cornell.edu/~hatcher/AT/AT.pdf
Always useful.
en.wikipedia.org/wiki/Counterexamples_in_Topology
#algebraictopology
#topology
#mathematics

Опубликовано:

3 авг 2021

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Комментарии : 23

@erjio983 5 месяцев назад

This concept has come up a lot while learning Lie groups for physics and this video is a very nice introduction that covers most of what I needed to know. Thanks 🙂👍

@VisualMath 5 месяцев назад

Yes, the concept is fundamental for many fields 😂 Just kidding, thanks for the feedback. That is very much appreciated.

@rimelius 6 дней назад

These videos are really interesting! Thanks for taking the time to share knowledge about this fascinating topic.

@VisualMath 6 дней назад

Thanks for the comment, I am glad that you liked the topic ☺

@Achrononmaster 6 месяцев назад

@12:15 wait a minute... in the case when the π_1 are equivalent (and that's all you know) you can still say a lot about the spaces, you just can't say anything about how they might be different. To wit, you can list all the properties implied by π_1 equivalence. Ex. if I am looking at particle physics I might not care for now what type of quark I've got, but might want to know just that they're not leptons.

@VisualMath 6 месяцев назад

Ah, nitpicking 🤣 No worries, you are right, of course. What I mean is that “we cannot distinguish X and Y”, but of course there is something one can say about X and Y ☺

@Achrononmaster 6 месяцев назад

@16:00 it's also nice to go the other way. Take the F2 idea, x1.x2.inv(x1).inv(x2) then draw pairs of over crossings, then connect them below the holes so the orientations go through, and voila! chances are you drew a solution easy pz. (Worked first time for me.)

@VisualMath 6 месяцев назад

Well, that didn’t work for me 😅 Maybe to clarify: What is the operation you describe on F2? Thanks!

@evebodnia7881 Год назад

at 3:00 min you say that these are not the same. However you can fold the one on the right (hole to hole) and stretch a bit that edge. See what I mean? This would technically be no cutting and no gluing .. continuous map. Thank you a lot by the way for your videos! You make RU-vid a better place for sure. Also your ability to explain hard things in a such easy way is really a form of art. Great job

@VisualMath Год назад

- Let me try to repeat your question, so that we can check whether we are talking about the same. You want to think of the right-hand picture with two holes to have a little folding line in between the two holes. Then you want to fold the picture to get the left-hand picture with one hole. You then wonder about the associated map from right to left. That is a great question! Note that this map is not a bijection: folding identifies points. In other words, unless you are on the image of the folding line, every point in the left picture has two preimages in the right. So the associated map cannot identify left and right. - Thank you for the kind feedback: My way of explaining things certainly does not work for everyone, it is simply what works for me, i.e. how I think about these beasts. So I am very glad to hear that this style works for you as well. I hope you enjoy the videos and learn something interesting and useful!

@evebodnia7881 Год назад

@@VisualMath yes that was exactly my question! Thank you very much again, especially for taking your time to reply comments:) I've been watching your videos nonstop and looking forward for more! Do you have anything on hopfions by any chance? or homotopy of spheres? These are hot topics in theoretical physics these days

@VisualMath Год назад

Glad that it was helpful! Hopfions and homotopy of spheres are hot topics? That is fascinating, I would be happy to know more! I have a brief comparison of the homotopy of spheres with Eilenberg-MacLane spaces in the video about the latter at the end of the AT playlist. Nothing really deep, but there is a nice picture. I really want to cover more about homotopy of spheres eventually; maybe I touch Hopfions also, but I can't tell right now as I am not really familiar with them.

@evebodnia7881 Год назад

@@VisualMath ah that would be wonderful! I am not familiar with them either, that's why learning your videos first before digging into that :)

@yidaweng9153 2 года назад

Thank you for your explanation. It’s very helpful.

@VisualMath 2 года назад

The fundamental group is such a great concept: Glad that you liked it. And I am also glad that the video was helpful ;-)

@josh34578 Год назад

It can be difficult to tell if two group presentations define the same group. Does that difficulty come up often in practice when comparing the fundamental group of two spaces?

@VisualMath Год назад

Absolutely, that happens all the time. The most common calculation tool is the Seifert-van Kampen theorem, and it gives you pi_1 via generators-relations. Already for something like the Klein bottle it is not trivial to answer simple question about pi_1 from this presentation, like whether the group is finite. And that is completely ignoring the isomorphism question! Now imagine a much more complicated manifold - pi_1 will not be easy to identify.

@amoghdadhich9318 10 месяцев назад

How do we conclude if two groups are not isomorphic in this particular example? Can you please elaborate how non-commutativity results in this?

@VisualMath 10 месяцев назад

A commutative group and a noncommutative group cannot be isomorphic since any isomorphism would preserve the property of being commutative: f(g)f(h)=f(gh)=f(hg)=f(h)f(g). I hope that clarifies why Z is not F2.

@amoghdadhich9318 10 месяцев назад

@@VisualMath awesome! thanks for the reply. I'm a computer science student working on computer graphics this semester and I find your videos really helpful in helping me build the background for applications of topology in graphics.

@VisualMath 10 месяцев назад

@@amoghdadhich9318 Welcome, I am glad that you came to pay a visit ;-)

@diktakt1187 2 года назад

06 46 Ok

@VisualMath 2 года назад

The space Ok is a bit tricky since it is not connected. But one component has the fundamental group of the circle, so I can't quite tell whether Ok is a circle or not. Just kidding of course ;-)