What is...the fundamental group?

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.

Website with exercises.

Material used.
Hatcher, Chapter 1

Awesome fundamental group animations (and other great stuff).

Circle pictures.

Spivak’s hanging-pictures-puzzle (I stole some pictures from there).

Hatcher’s book (I sometimes steal some pictures from there).

Always useful.

#algebraictopology
#topology
#mathematics

[erjio]

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 🙂👍

[rimelius]

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

[Achrononmaster]

@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.)

[evebodnia]

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 Youtube 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

[Achrononmaster]

@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.

[amoghdadhich]

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

[yidaweng]

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

[josh]

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?