What is a hole?

An equally valuable form of support is to simply share the videos.

A HUGE thank you to Waleed Qaisar for working with me to make the script and storyboard for this video. You rock Waleed! And thank you to Davide Radaelli for patiently listening to the script and offering helpful feedback!

CORRECTIONS:

At the start of the video, I said that Poincare’s paper Analysis Situs was published in 1985. This is a typo - the paper was published in 1895. Thanks to those who spotted the error.

SOURCES and REFERENCES for Further Reading:

This video is a quick-and-dirty introduction to the fundamental group. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below.

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MUSIC CREDITS:

Song: Thinking Ahead

[logankennedy]

As a Ph.D. Student in algebraic topology it is hard to explain to the average person what I study, so it is cool to finally see a very approachable explanation. I personally study how homotopy and homology/cohomology groups change when taking some kind of product of two or more topological spaces, specifically wild topological spaces. It is interesting how the average person might think the fundamental group is trivial to compute, but we still have researchers like me trying to find ways to actually find ways to compute these fundamental groups for some pretty simple spaces.

[RyeedAglan]

Me: There's a hole in your proof!
Topologist: You're welcome.

[LookingGlassUniverse]

Beautiful! I love the blending of different styles in this video

[tracyh]

0:12 Bless Poincaré, writing such an important paper at the youthful age of 131 years old.

[rentristandelacruz]

As a wise philosopher once said "The souls meet where the holes meet".

[kikivoorburg]

I was just thinking about SO(3) having a hole bc it doesn’t allow for the contraction of 2π-rotation loops, then you used it as an example — fun!

[eemilwallin]

This was a really nice video because I've often seen the symbol for the fundamental group but I have not grasped the concept well enough. Esp. I do know that R^2 minus a point is not simply-connected (or 1-connected), but the fact that the fundamental group is integers is mind blowing. While intuitively the direction of a loop matters, I did not grasp the formal reasoning for it. I'd be delighted if anyone could point it out for me.

[maynardtrendle]

This was absolutely wonderful!😊 Thank you for the clarity.🌱

[StratosFair]

I am into analysis and statistics so I know nothing about this stuff, but I was always interested to learn it. So thank you for giving a super accessible intro, and happy new year !

[celkat]

Very cool - subscribed. (Microquibble: Analysis Situs was published in 1895, not 1985)

[Npvsp]

Adding a resource: I studied algebraic topology from the book of Spanier (algebraic topology, Springer), and to me this is the best reference possible. Hatcher is fine, but Spanier book is perfect.😊great video as always!

[luquest]

Thanks Aleph0, always look forward to your videos

[cephassvosve]

This was the best explanation that helped me understand the poincare conjecture!

[languafranter]

Excellent work down, I really enjoy this kind of explanation in the field of pure mathematics :D

[UJ-ntoo]

0:15 bro you legit had me thinking poincare was still alive. i think you meant 1895

[rhodesmusicofficial]

wonderful! it's so awesome to see your iconic style of presentation applied to my favorite subject

[Rubikorigami]

Really nice and clear video ! I gotta say, I'm more of a fan of the all-paper look of your previous videos as it feels a lot more cohesive design-wise, but I get the need for 3d stuff and CG anims. Also, I think there may be a tiny issue with the exporting of the video where you play with the plate - it looks super pixelated for me even though the rest of the vid was nice and crisp. Anyway, nice work as always ! Keep it up :)

[BriggsProgrammingDevelopment]

Your videos are always precise, while digestible, this video is no exception

[estebanguerrero]

Love the content, thanks so much <3 Really enjoy watching snippets and some deep dives for math topics and even proofs, thanks

[emilmullerv]

Some obersevartions are needed.

1. The fundamental group doesnt necessarily encode our intuitive idea of a hole ina. Space. Take the topologist circle, which has fundamental group trivial, but clearly has a hole. Or a sphere.

2. The fundsmental group also requires to fix a point in space, however for path connected spaces, the point you choose doesnt chsnge the structure of the group, so it can be avoided to simplify the notstion.
3. While I like your explanation for the fundamental group of the circle, I've liked if you had mentioned hiw extremely hsrd is to justify that intuition, and how in general computing htje fundamentsl group is an extremely complcated task