What is a hole? 

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Nice, you're talking about the mouths, right?

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amazing

Docking?

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 po​incare's paper as being written in the 1980's @@Gordy-io8sb

Thanks Mithuna!

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 RU-vid 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.

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.

Was about to point this out but then I saw this comment.

This! I had a hard time understanding the arm trick because of the lack of context. One needs to firmly establish the analogy.

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.

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.

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.

it's literally like every other Aleph 0 video

@@dehnsurgeon The other videos in this channel aren't this flawed (although you can probably nitpick all of them). This one has some very major errors in it.

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)

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?

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.

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 - ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-rC0jAICfNwc.htmlsi=1Ct7Pn4rC7y9bb1L

So I can eat the vid?????????????????????????? *proceeds to eat part of my memory card*

She said im going too fast im exhausted

This isn't a characteristic of a group. It's implied by the very definition of what an operation is.

*Holes the absence of a surface.* Ergo, holes do exist lol.

The definition of a loop does not include the concept of a hole, so this is just incorrect.

Obviously a topological space. What the space is doesn't really matter for the average viewer. You don't need to rigorously define what a topological space is to get across the idea of the fundamental group. If you know enough about other spaces like manifolds and vector spaces, you should know enough to understand that he is obviously talking about topological spaces.

Lol this comment is someone pretending to know more than they do

Because the name was given to them back when no one understood what they were. We know what they are now, but it is too late to change this century-old name.

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.

Ah you touch on a beautiful point; you can't really define what is not there, and your definition should not depend on an arbitrary embedding of your object. You should be delighted to know topologists don't really think of the torus as it's embedding in R^3 but any thing with the connectedness properties of the torus (formally: search up homeomorphism). The fact that the torus has a "2-dimensional hole" is technically DEFINED by the fact that a higher homotopy group is non-trivial! "Hole" is really just a word for intuition. When you apply topology to wacky spaces, the inability to contract a loop is just called a "hole". Eventually you drop the word hole and realize homology is just an algebraic invariant of an object (google: Eilenberg Steenrod axioms).

All you did was make a claim. It doesn't even really mean anything.

What about a liquid or semisolid without clearly visible holes?? You can put your finger through water. You can put your finger through honey. Doesn't mean they're holes.