What is it that makes 4d the hardest dimension in topology?
Only case in which codimension 2 is dimension 2. Codim 2 means “complement generates topology”. Dim2 is “can avoid points using an arc”. I guess this means a lot.
Tadayuki Watanabe disproved a major conjecture about spheres
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Re: Tadayuki Watanabe disproved a major conjecture about spheres
#12What is it that makes 4d the hardest dimension in topology?
Not a topologist, but my understanding is that high dimensions can be handled by one set of methods because there's so much freedom, while low dimensions can be handled by another set of methods because they're so constrained. And so there ends up being a nasty point in the middle that isn't constrained enough to be handled by low-dimensional techniques or free enough to be handled by high-dimensional techniques. Not…
Also not a topologist, but thought it was interesting after watching the video from the current top comment, https://youtu.be/mceaM2_zQd8 , where if you look at the size of the "contained sphere", 4D is the only place where the contained sphere is exactly tangent to the containing box. Lower than that, and it's easy to visualize how the contained sphere is smaller, higher than that, the contained sphere is always bigger. So seems like it might be a natural consequence that techniques that work at higher or lower dimensions don't work for the case in 4D where the sphere is exactly tangent.
Re: Tadayuki Watanabe disproved a major conjecture about spheres
#13What is it that makes 4d the hardest dimension in topology?
Not a topologist, but my understanding is that high dimensions can be handled by one set of methods because there's so much freedom, while low dimensions can be handled by another set of methods because they're so constrained. And so there ends up being a nasty point in the middle that isn't constrained enough to be handled by low-dimensional techniques or free enough to be handled by high-dimensional techniques. Not…
The topologist R. H. Bing described it using more colorful language.
> Dimension 4 is the most difficult dimension. It is too old to spank, the way we might deal with the little dimensions 1, 2, and 3; but it is also too young to reason with, the way we deal with the grown-up dimensions 5 and higher.
Re: Tadayuki Watanabe disproved a major conjecture about spheres
#14Higher dimensional spheres are weird, especially 10+ dimensional: https://youtu.be/mceaM2_zQd8
It has to be said that Matt's explanation in that video is wrong. N-spheres are just spheres, it's the cubes which get "spiky", like sea-urchins. (It's a rare case where reading the Youtube comments is a good idea.)
Re: Tadayuki Watanabe disproved a major conjecture about spheres
#15One view of homotopy theory is that it replaces the relation "=" (equality) with a looser (but fairly intuitive) notion of path-connectedness. Equality of two points on a topological space is replaced with existence of a path leading from one to the other. Then equality of two continuous functions is replaced with the existence of a path between the two functions lying on a space in which the two functions are points (note that if two functions are equivalent in this sense they are referred to as homotopic, and the exact path is a homotopy). Likewise, the notion of isomorphism, which is usually defined by f o g = id_A and g o f = id_B, is replaced with f o g being homotopic to id_A and g o f being homotopic to id_B. If two topological spaces are isomorphic in this looser sense, then they are said to be homotopy equivalent and the functions f and g are each referred to as homotopy equivalences.
A table comparing equals-based terminology with path-connectedness terminology can help:
- Equality of points : existence of path
- Two functions are equal : two functions are homotopic
- Isomorphism : Homotopy equivalence
- Isomorphic : Homotopy equivalent
There is a formal system called Homotopy Type Theory which exploits the above analogy in a really nice way. It does this by treating path-connectedness as a more fundamental notion than equality, with the former becoming the latter if the topological space is discrete (and therefore essentially a set). HoTT actually can't see equality.
Re: Tadayuki Watanabe disproved a major conjecture about spheres
#16Higher dimensional spheres are weird, especially 10+ dimensional: https://youtu.be/mceaM2_zQd8
I wrote a thing about this and have submitted it in the past. Actually, I've realised I submitted it more than once. Each time there was a small amount of conversation that discussed this point. If you You might be interested.
https://news.ycombinator.com/item?id=12998899 : 45 comments
https://news.ycombinator.com/item?id=3995615 : 59 comments
I think this pre-dates Matt talking about this fun, and he may have got it from me, but he may also have got it from the same place I got it, which was (probably) Martin Gardner.
Re: Tadayuki Watanabe disproved a major conjecture about spheres
#17Higher dimensional spheres are weird, especially 10+ dimensional: https://youtu.be/mceaM2_zQd8
It has to be said that Matt's explanation in that video is wrong. N-spheres are just spheres, it's the cubes which get "spiky", like sea-urchins. (It's a rare case where reading the Youtube comments is a good idea.)
I agree that high dimensional cubes are "spikey", but consider this.
If you stand on a high-dimensional sphere, then slice off the "cap" where you are standing, then in high dimensions the resulting cap has virtually no volume, unless your cut is a long way towards the centre of the sphere. In other words, the piece you are standing on has a very, very small volume, unless you cut off a lot of it.
That's the same intuition as if you are standing on a spike. Cutting off the spike results in a solid with nearly no volume. So thinking of a sphere as being "spikey" with a spike at every location does give a better intuition for some things.
But not everything ... high-dimensional stuff is just generally weird.
Re: Tadayuki Watanabe disproved a major conjecture about spheres
#18Earlier quoted context omitted.
It has to be said that Matt's explanation in that video is wrong. N-spheres are just spheres, it's the cubes which get "spiky", like sea-urchins. (It's a rare case where reading the Youtube comments is a good idea.)
I agree that high dimensional cubes are "spikey", but consider this. If you stand on a high-dimensional sphere, then slice off the "cap" where you are standing, then in high dimensions the resulting cap has virtually no volume, unless your cut is a long way towards the centre of the sphere. In other words, the piece you are standing on has a very, very small volume, unless you cut off a lot of it. That's the same int…
The problem with slicing is that here in 3d we would slice with 2d plane, but when you are in 4d, should you slice with 2d plane or 3d something? (3-1=2 so 4-1=3) or is slicing always 2d plane, says who?
Re: Tadayuki Watanabe disproved a major conjecture about spheres
#19From what I understand [1] the existence of exotic spheres in n>5 implies => the Smale conjecture is false in n>5 but does the Smale conjecture being false in n=4 implies nothing => about the existence of exotic spheres in n=4? [2]
[1] https://en.wikipedia.org/wiki/Smale_conjecture
[2] https://en.wikipedia.org/wiki/Exotic_sphere#4-dimensional_ex...
Re: Tadayuki Watanabe disproved a major conjecture about spheres
#20Earlier quoted context omitted.
Not a topologist, but my understanding is that high dimensions can be handled by one set of methods because there's so much freedom, while low dimensions can be handled by another set of methods because they're so constrained. And so there ends up being a nasty point in the middle that isn't constrained enough to be handled by low-dimensional techniques or free enough to be handled by high-dimensional techniques. Not…
Also not a topologist, but thought it was interesting after watching the video from the current top comment, https://youtu.be/mceaM2_zQd8 , where if you look at the size of the "contained sphere", 4D is the only place where the contained sphere is exactly tangent to the containing box. Lower than that, and it's easy to visualize how the contained sphere is smaller, higher than that, the contained sphere is always big…
No, that's almost certainly not related. The relative volumes of the cube and the sphere is a geometric matter, not a topological one. Remember that topology uses much looser equivalences that do not respect such information.