Tadayuki Watanabe disproved a major conjecture about spheres | Hacker News Reader

Tadayuki Watanabe disproved a major conjecture about spheres

quantamagazine.org

132 pointsbryanrasmussen20 comments

One 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.

Higher 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.

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?

To "slice" something you need a "plane" in one dimension lower. To get some sense of why, think about a ball in three dimensions. You can "slice off" some of its volume with a plane (one dimension less), but if you have a line all you do is have a line through it ... the ball is still connected, and you haven't sliced anything from it.

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.

https://en.wikipedia.org/wiki/Martin_Gardner

I think this is especially interiguing case to casual math enjoyers too, because it feels like the four-dimensional case would still barely be within a grasp of intuitive understanding. So, now that it's proven that they exist, if/when they find examples of the diffeomorphims that don't preserve the rigid structure, I'd love to see (attempts of) explainers about them!

What 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 being much of a topologist, though, I couldn't possibly say why that middle point occurs where it does.

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.

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.

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.

After 4d, dimensions usually generalize. Before 4d, dimensions can be visualized.

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.

Can't visualize and no Whitney trick.

My understanding is that higher dimensions have more “space”/“freedom” to perform moves and operations.

And of course, we can visualize anything 3D or less.

From 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...

Is there an intrinsic relationship between n-dimensional spherical harmonics and the quantum mechanics of atomic electron clouds? Or just hydrogen?

Check out “the orbitron”: https://winter.group.shef.ac.uk/orbitron/

Spherical harmonics arise as the azimuthal and zenithal components of solutions to the Schrodinger equation for any spherically symmetric potential. In nature, such a potential really only arises in atoms for the case of hydrogen (due to the lack of other electrons, which break this symmetry), but as you might expect can also describe certain other limiting cases where the potential is approximately spherically symmetric quite well. An example of this is an alkali atomic species -- lithium, potassium, sodium, and so on -- where the single electron in the valence shell is excited to what is called a Rydberg state. If you're familiar with the usual set of quantum numbers applicable to atoms, a Rydberg state is where the principal quantum number n is quite large, usually around 60 or so. In this limit, the single valence electron is far enough away from all the other electrons that their angular non-uniformities can be neglected, and hence the resulting potential seen by the valence electron is approximately spherically symmetric.

Another thing about the spherical harmonics is that they're also just a generally applicable, linearly independent basis of functions with purely angular dependence, so we can, in principle, use them to decompose any angular wavefunction -- whether the specific harmonic for a given set of quantum numbers is an eigenstate of the angular Hamiltonian or not.

EDIT: For completeness, I should note -- in case anyone's not so familiar, the numbers that index the spherical harmonics are called quantum numbers when used to describe atomic orbitals. Physically, the positive-or-zero index gives the orbital angular momentum of the particle whose wavefunction it applies to, and the positive-or-negative index gives the projection of the orbital angular momentum onto a specific axis.

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