What is a hole?
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- Опубліковано 28 гру 2023
- An introduction to the fundamental group, a key concept in algebraic topology. This video is sponsored by Brilliant. To try it out for free for 30 days, head to brilliant.org/Aleph0/. The first 200 people to sign up will get 20% off a yearly subscription.
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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.
(a) Video Lectures by Pierre Albin: • Algebraic Topology - P...
(b) “Algebraic Topology” by Allen Hatcher: pi.math.cornell.edu/~hatcher/...
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Twitter: @00aleph00
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MUSIC CREDITS:
Music: www.purple-planet.com
Song: Thinking Ahead
Me: There's a hole in your proof!
Topologist: You're welcome.
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.
Hi, is it possible to extend ideas of homology and cohomology to study infinite dimensional holes on stuff like hilbert manifolds or frechet manifolds (generalizations of manifolds to inf dims)? Is there research being done on this?
@@mastershooter64 Hi, my field of study is singularity theory (more about polynomials and less about groups) but I can tell you that homology and cohomology are defined for any topological space, although there are different "theories" that one uses depending on the context.
All of them are equivalent in the sense that you get the same groups no matter what theory you use, but one important thing is that you have a powerful tool called the "Mayer-Vietoris" sequence, which allows you to break out your space into chunks and recompose the (co) homology groups of the whole by using the (co) homologies of the chunks.
For example: a circumference can be broken into two overlapping curves. Each curve is a segment, and the homology of the segment is trivial to compute. Therefore, the homology of the segment gives you the homology of the circumference!
This is very simplified because this is just a UA-cam comment, but I hope this gives you some insight.
Homotopy groups are valid for any topological space as you are only looking at the mapping of circles and hyper spheres into the space. As for homology, if your manifold is under four dimensional it is triangulable so simplicial homologies and cohomology theories should work, and I don’t see why singular homology and cohomology theories wouldn’t also work in any dimension of these manifolds.
can you point out some literature with the techniques you use? I'm also interested
@@mastershooter64 Fun fact: Infinite dimensional holes are contractable, and infinite dimensional space hence have a trivial topology.
As a wise philosopher once said "The souls meet where the holes meet".
💀
Nice, you're talking about the mouths, right?
😶🌫️
amazing
Docking?
Beautiful! I love the blending of different styles in this video
Thanks Mithuna!
0:12 Bless Poincaré, writing such an important paper at the youthful age of 131 years old.
He died at 58. That is exactly 73 years younger than 131. I don't think *anyone* has lived to that age.
The oldest anyone has lived is 122. That's 9 years (a singular year off from being an entire decade) younger than 131. So no, he did not.
Poincaré didn't even live to his 60s (although he was close), and that's still far, far off from 131.
@@Gordy-io8sb John Smith (Chippewa Indian) claimed to be 137 when he died. Of course, this is disputed but is in Wikipedia...
In the video, the narrator mentioned the date of poincare's paper as being written in the 1980's @@Gordy-io8sb
0:15 bro you legit had me thinking poincare was still alive. i think you meant 1895
I rewatched several times because I heard it too.
Beacause I know Poincare was dead since 1912, I did not hesitate. I presumed he meant 1895.
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!
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 !
Very cool - subscribed. (Microquibble: Analysis Situs was published in 1895, not 1985)
Was about to point this out but then I saw this comment.
Thanks Aleph0, always look forward to your videos
wonderful! it's so awesome to see your iconic style of presentation applied to my favorite subject
Excellent work down, I really enjoy this kind of explanation in the field of pure mathematics :D
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.
Consider two objects in the fundamental group A, B. The group is equipped with an operator * that concatenates the two loops together.
Consider the number of times a given object loops around itself as the number of interest.
Note that the loops that do not go around the deleted point are homotopic to a point, so they go around themselves zero times (let's define this, as "going around yourself" only makes sense in 2D and points are zero-dimensional)
And note that loops that do go around the deleted point are not homotopic to a point, so they must go around themselves at least once (in some direction).
Why does the direction matter? This is far easier as a visual proof. Consider loop A, which goes around the origin CW once; loop B, which goes around the origin CCW once. A * B yields an object that first goes CW around the origin once, then CCW around the origin once. But note that there is a way to trace A * B without ever fully going around the origin! Just double back on B's path after finishing A's path. Indeed, A * B is homotopic to a point.
Let's assign zero to the "homotopic to a point" loops, positive numbers to the loops that go CW around the origin, and negative numbers that go CCW (or the other way around, it doesn't matter). We see that if we concatenate two CW loops C, D => C * D, the number of times C * D loops is exactly equal to loops in C + loops in D. If we concatenate two CCW loops E, F => E * F, the same applies. But if we concatenate two loops of opposite orientation G, H => G * H, the number of loops subtracts. Each loop in H "undoes" a loop in G until (potentially) no more loops exist; then the rest of the loops (if there are any) will come from finishing the path of H.
Well, isn't this just like adding a positive and negative number?
Ex. 3 + (-7), the -7 "undoes" 3, and what's left is -4.
A CW loop with 3 loops concatenated to a CCW loop with 7 loops yields a CCW loop with 4 loops.
@@wikiPika Thank you for your answer! Going "around" a point does seem to formally require an interior, with the loop forming a boundary and "around" being the part of space where the point also is (point p is subset of some set A which is subset of whole space). I guess the point could also lie on on the loop and with the homotopic property a loop which contains the point fits the above pondering about the interior.
@@eemilwallin3347 If the loop has p ON its boundary, then... it actually does not exist! This is because the loop is therefore not within the space X \ {p} (well, no shit, the path contains p). Therefore we can ignore this edge case.
Here's another explanation of it that doesn't use the concept of fundamental group (in my topology class we didn't get to see the fundamental group, and we saw this at the start of the homotopy lectures). The idea is that you can consider, given a path f from an interval L to the unit circle U, another path g from L to R, the real numbers, such that given the function p from R to U such that p(t) = (cos(2•pi•t), sen(2•pi•t)), you have that f is equal to the composition of g and p. That is just considering g and kind of "projecting it" to the circle. It turns out that not only you can always do this, but also given two homotopic paths from L to U, their respective paths from L to R are also homotopic. The demonstration relies on differential geometry, as you use the existence of an angle function locally and extend it's domain both to find the path from L to R and to have the homotopy property I mentioned previously.
Having this fact, it is sufficient to consider two paths (cos(2pi•n•t), sen(2pi•n•t)) and (cos(2pi•m•t), sen(2pi•m•t)) from L = [0,1] to U, these are closed paths on U that do n and m loops around the circle respectively. It is clear that their respective paths from L to R are 2pi•n•t and 2pi•m•t respectively, and those paths can't be homotopic because their ending points aren't the same.
Because every two paths with the same starting and ending points in R are homotopic, given two paths f and g from L to U with the same number of loops around the circle and given their respective paths from L to R, f' and g', the last two will be homotopic. As p is continuous, the composition of p with f' and the composition of p with g', that is, f and g, will be homotopic.
The redaction isn't the best but I hope it was understandable :)
The formal reasoning isn't really in this video at all, and I would argue that this isn't the most visually / geometrically apparent fact. The fact that the fundamental group of the punctured plane is the integers is typically proven by reducing the problem to computing the fundamental group of a circle (the rigorous details are a bit lengthy and typically use even more machinery, but it's a lot easier to intuitively convince yourself of this).
The idea behind the reduction is that we can continuously map each point in the punctured plane to a unique point on the unit circle (the map is dividing a point x by its norm) while fixing the entire unit circle. Then, we can construct a homotopy between that mapping and the identity map on the punctured plane (each point moves towards the unit circle along the unique line connecting it to the origin). Using this homotopy, we can show that our map from the punctured plane to the circle is "homotopy equivalent" to the inclusion map from the circle to the punctured plane (this means if you compose the two maps in either order, the result is homotopic to the identity map in the image space). Using some algebraic / categorical tools we can easily show that homotopy equivalent spaces have isomorphic fundamental groups. Thus we can conclude that the fundamental group of the punctured plane is isomorphic to the fundamental group of the circle, which is Z.
This was absolutely wonderful!😊 Thank you for the clarity.🌱
Love the content, thanks so much
This was the best explanation that helped me understand the poincare conjecture!
Amazing video as usual!
Your videos are always precise, while digestible, this video is no exception
So I can eat the vid?????????????????????????? *proceeds to eat part of my memory card*
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!
Fascinating video! 😊
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 :)
Thanks for this!
*Every Set that has those 3 properties is referred to as a “Groupoid” (A Category with isomorphisms). A “Group” restricts the Signature of the System to a single operation (sometimes called the group “action”).
Russel set / generalized set is implied here if that’s important to anyone.
Just a heads up, the fundamental group doesn't encode holes precisely: for example it doesn't distinguish a ball from a sphere (which, arguably, has a "hole", that being the "missing interior"), they both have fundamental group 1. The correct mathematical structure to model holes is called homology, and in some cases it is linked to the fundamental group (for example in the case of compact 2-surfaces without boundary they are actually equivalent), but generally the homology groups contain more information than just the fundamental group. You could consider higher homotopy groups but those turn out to be unmanageable in practice.
Came here to comment this. Homology would've been more appropriate here I think. Homotopy groups aren't really about holes as they are about maps of spheres into your space.
@@persistenthomology it perplexes me because this channel also has an excellent video about homology in which the guy defines holes in the correct way, so I'm not sure what the deal is
Homology also doesn't define what a hole is. "Hole" is not a well-defined mathematical concept.
Homology groups actually lose information about the space in order to more conveniently describe the structure of the holes in the space.
Fundamental Group and Homology don't coincide for all compact 2-manifolds, just for the Sphere, the Torus, and the Real Projective Plane. The others don't have Abelian Fundamental Group.
Really nice video, I learned a lot, thanks for sharing
What we can all agree on is that each of these holes is indeed a goal.
I am so sorry to be specific, but you are defining the Fundamental Groupiod, if you choose not to specify a basepoint.
You run into problems, e.g. if you have more than one component
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
What are you taking as topologist sine? The curve glued like a strange circle?
It is quite easy to prove R/x has a hole.
@@gabrielvieira3026 mistake on my part, I meant the topologist circle.
@@Aesthetycs it's not lol, the intuition is easy. Show me an "easy" proof
@@emilmullerv3519 Basically for any missing point in R, for any pair of path connected points, and for any region of space you can define having the missing point within and the pair on its boundary, the two pieces of boundary connecting the pair do not have the inner region as their path homotopy since a homotopy function should have the entire region as its domain yet the function is undefined on the missing point.
this is gold
at 4:20 , 1 should be the order of the group pi(R) to be precise (or pi(R) the group that only has the neutral element, depending on what you were going for here)
Thank you!
Good video!
One of the best math channels.
Just to add to the resources: Hatcher tends to handwave quite a bit in his book and is sometimes difficult to follow. If you're more algebraically minded Rotman has an excellent book on the subject
I wouldn't call it handwaving, but his sentences are very dense and often times hard to follow if you don't already know where he's going with all of it. From personal experience, I'd say his book is a lot better as a second look to deepen your understanding of the topic (his examples are often gorgeous) but is really bad to get through the first time around without a guide (like a lecture series)
Thanks!
Really awesome video, I’m definitely going to check out some of those resources you listed!
Can't wait for the homology one
Great stuff!
This is a great video, although there is a detail that should not be ignored. The fundamental group depends on a point in the topological space X. It is NOT the set of all loops with concatenation as binary operation. It is the set of all loops starting in a specific point p of X with the concatenation as binary operation of loops. This is important for two reasons:
1) The space X can have more than one connection component. If the space X is path connected, then one can show easily that the fundamental group is up to isomorphy uniquely determined, i.e. not point dependent. But if this is not the case, then it can depend on the connection component. An easy example would be the disjoint union of two topological spaces, one with a trivial fundamental group and one with a nontrivial one.
2) The fundamental group would not be a group without this point dependence, because the concatenation of two loops, starting at different points is not defined.
Without these two remarks the video could be a bit misleading, cause for example at minute 5:11 the "multiplication" (concatenation) wouldnt even be defined, but since R\{0} is path connected, everything works out nicely.
This is exactly what I'm studying right now 🤩
I think this video could have benefited from a little more explanation on the procedures. Like the beginning example looks like you could just shrink the loop further over the hole because its 3d. Or the fact that all loops in the plane are the homotopic to a point. Or the whole plate and arm thing: you are saying the plate ends at the same point but are showing it moving on the screen with two different positions. Im sure if you know all these concepts it makes sense but stuff like this makes the learning inaccessible. I‘d encourage just a little more exposition when doing „experiments“ with the viewer
This! I had a hard time understanding the arm trick because of the lack of context. One needs to firmly establish the analogy.
it is so intresting how so mant things are groups
The fundamental group doesn't count all holes. Simply consider the 3-dimemsional Euclidean space with the origin removed and call it X. The map f(x,t)=x/((1-t) + t |x|) is a homotopy from X to the 2-sphere. Since the 2-sphere is simply connected, X has trivial fundamental group. However, the unit sphere in X cannot be contracted, so X has a hole.
Also, forgetting the base-point of the fundamental group doesn't help the exposition in my opinion.
New Aleph 0 video dropped :3
By the way, your videos make me think that perhaps, if I worked hard enough, I could understand some higher math. Keep it up!
Start with reading a book about mathematical logic. It's the most important subject for understanding math at all.
@@stevenfallinge7149 The playlist Proof Writing by MathMajor (second channel of Michael Penn) might be a good intro?
Lovely!
My background is physics and my intended applications are in physics and engineering. During my studies my mathematics books had the addition "for physicists", like "Group Theory for Physicists".
Can you recommend a resource covering the topic of holes but "for physicists"?
In case you are interested: I am currently doing a project which involves moving holes around using solid bodies. A single hole is trivial, simply have a single hole in a single body and move the body. The non-trivial problem is: take three holes in the plane. Let their motion perform a braid (yes, a braid theoretical braid), find a set of solid bodies which accomplishes this.
Why? In engineering you need to actuate motion. Using solid bodies to hold a tube and move in in the plane is a more elegant way (my physics background) of actuating motions than gears and sticks and sticks and gear and....linearized inelegant simplicity.
2:30 how is the green loop different from the red loop? we can just slide it ower..or maybe im looking at it wrong
Upon rewatching, I realize I have a question: no matter how small you shrink a loop it will always be missing its interior. I seem to understand how a contracting loop will become caught on a hole of finite size, but from the plane minus the origin, perhaps P\O I might say, how might it get caught on that hole? If I idealize my loop to a disk centered on the origin and shrink it down, it will always be a disk, so how could it know if I only delete a single point?
It's less about "getting caught" on a hole and more about the loop itself. The loop needs to continuously change into a point. And there is no place for that to happen.
I might as well add that a key theme in higher level algebra is that you can learn alot about a space by studying functions on that space (it feels this has become synonomous with geometry now). Formally, a loop is just a function from the interval to your space. So it makes sense that you want to think from the loops POV to study holes.
Thank you, this is an excellent reply and I appreciate your thoughtfulness. I admit that I'm still a little puzzled--for the function on this interval, if the interval is [a, b] is f(a) required to be f(b), such that the function intuitively forms a closed loop in the space? Secondly, how could this function detect if there's a deleted point on the interior of the loop if it doesn't need to get "caught" on it? My questions make not make any sense, but if you can attempt to see what I'm getting caught up on, it would be appreciated. Also, the comment on algebra becoming like geometry is interesting! @@LoveFalastin4034
@@kaidenschmidt157 Indeed, a loop is a _continuous_ function f from [a, b] to the space X in question, with f(a) = f(b). Alternatively, you can define a loop as a continuous function from S^1 to X, where S^1 is the unit circle in R^2 centered at the origin.
The loop itself tells you nothing about holes. What you need is to talk about a homotopy H between two loops f and g.
2:30 Aren't green and red loops homotopic? If you fix upper point of a green loop and stretch the bottom part to the left, you can move green loop next to the red ones.
If I take R^2\{0}, can’t I also contract any loop into the origin or is this excluded by definition?
I don't see a link to Pierre Albin's (sp?) videos...?
Every hole's a goal
New alpeh 0 vid. the sun is shining the birds are singing
*Fire in the hole!!!*
Can anyone explain how for SO(3) travelling around the non-constant loop twice yields the constant loop? Struggling to visualise this!
look up dirac's belt trick
@@cubing7276 thanks for this!
His illustration of the plate trick is not well done in the video. What should have been done is holding a plate in your hand, and imagine that there is food on the plate and that you don't want to spill it. Now try to perform a 360 degrees rotation of the plate (around an imaginary vertical rotation axis, without spilling). You can do it but your own arm is twisted. However, if you continue another rotation in the same direction, your arm untwists.
One of the rotation is done with the plate above your arm, the second one with the plate below your arm. It's complex to explain through text.
This helped better with the visualization - ua-cam.com/video/rC0jAICfNwc/v-deo.htmlsi=1Ct7Pn4rC7y9bb1L
if you use groupoids, your loops can have MULTIPLE basepoints :)
this is absolutely fascinating, im still in Apostol's analysis and baby rudin. About to study topology and abstract algebra nxt year and i cant wait to study algebraic topology.
if pi_1 of the plane minus a point is Z,
what is is minus two points? you can loop around one or the other in complicated knots?
Free group on two elements
Essentially. You have two important loops, call them a and b, where a goes around your first point counterclockwise and b goes around your second point counterclockwise. We also have a^-1 and b^-1 which are the same as a and b but go around clockwise. Now any loop in the plane minus two points will be homotopic to a string of a's, b's, a^-1's, and b^-1's. The way you actually figure this out is using deformation retractions. This simplifies the problem of finding the fundamental group of the plane minus two points to finding the fundamental group of two circles stuck together at a single point.
I've come to think it is homology, not homotopy, that correctly captures the notion of a hole, and homotopy groups carry more (e.g. fibrational) data than this. Hence why higher homotopy groups of spheres are so complicated.
How do you distinguish twisting your arm once more from untwisting it?
Ie, from a double loop to unlooping the first loop?
So if a*a = 1, what is a/a ?
a/a = aa = 1. What you are ignoring is that a^(-1) = a.
@@angelmendez-rivera351 that would imply you could not distinguish the two states completely unraveling the analogy - hence why I asked.
Fire in the hole.
Isn't the fourth characteristic of groups missing? The one about the groups being closed (mixing elements within that group with one onother cannot result in new elements)
This isn't a characteristic of a group. It's implied by the very definition of what an operation is.
What about the following (equivalent) definition?
A space has a hole if there's a loop that can't be filled. By filling, I mean continuously mapping the disk in a way that its boundary is mapped in the given way
This seems equivalent to the property that a loop can be contracted to a point.
A disk is just a cylinder but you have contracted one of the ends to a point (A disk is homeomorphic to a cone). So what you said is the same as the definition in the video because the fact that one of the ends is a point forces the loop to contract to a point.
@@ethanbottomley-mason8447 [removed - I probably misunderstood]. Also, I put the fifth word in the comment there before you made the reply
omg you mentione Pierra-Albin videos
Nice video. A small correction. Poincaré died in 1912. He couldn't have published his "Analysis Situs" in 1985. I think you mean't 1895.
Yup, thanks for the correction! I’ve added this correction to the description.
A goal.
A chance for glory babyeee
Poincare died in the early 20th century. 1885?
Well ... math at andrews university (which is a youtube channel ) team are posting videos conserning algebraic topology course
As long as there's a goal.
When you realize that something that look trivially obvious is NOT trivial at all.
Bro that 720 degree turn must hurt your arm
Nice example with an arm
a goal
Taurus
1:25 “a loop in your space x”
I heard “a Hyperloop in your SpaceX”
A bridge is a hole, and a hole is a bridge
Omg Aleph null!
your mom.
(but in all seriousness, great video man!)
0:17 1895 not 1985 :)
And thank you for your work 😊
So uhh I still have no idea how neither to compute a fundamental group, nor how to count holes.
I missed you:(
0:19 did you mean 1895?
She said I’m in the wrong hole I said I’m lost uh uh
She said im going too fast im exhausted
❤
Nice Video!
You didn't have time to watch it when you commented...
@@RM-xq7gf I have now
I don't see why a² = 1. Spinning around twice seems different than staying still.
You missed a couple key examples but it's alright your mom has a couple invaluable demos and accompanying lectures
Hole is where you put the pole
A hole is a goal, simple as that - A simple man 2023
Nope, I don't get the plate trick.
"A hole on M is when a k-form is closed but not exact on M."
A hole is when ker(d_i) / im(d_(i+1)) is non-trivial. What's the problem?
@@hybmnzz2658 Yeah, thats because ker(d_i) = im(d_(i+1)) is implied by ker/im trivial and thus no. of close k forms = no. of closed exact forms
@@hybmnzz2658 So essentially the same thing
@@habukichandayo yes I was continuing the joke lol
@@hybmnzz2658 Oh lol
Fire in the hole🔥🔥😈😈😈😛👹👹
Where you shall travel hard.
Fire in where?🙂
multiply two loops? bro what?? you can do that?
Were the final two words in your title meant to have been hyphenated?
Fun fact: Infinite dimensional holes are contractable, and infinite dimensional space hence have a trivial topology.
Poincare in 1985?