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Comment by kiicia | Hacker News Reader

> Mathematicians Weinan Lin, Guozhen Wang, and Zhouli Xu have proven that 126-dimensional space can contain exotic, twisted shapes known as manifolds with a Kervaire invariant of 1—solving a 65-year-old problem in topology. These manifolds, previously known to exist only in dimensions 2, 6, 14, 30, and 62, cannot be smoothed into spheres and were the last possible case under what’s called the “doomsday hypothesis.” Their existence in dimension 126 was confirmed using both theoretical insights and complex computer calculations, marking a major milestone in the study of high-dimensional geometric structures.

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So these are all powers of 2 minus 2, and it looks like from the article that the pattern doesn’t exist in 2^8 - 2 or higher. Is there any description a layperson might understand as to why it stops instead of going on forever!

They’re all double the last dimension plus two, without skipping any in that sequence - but that offers no insight into why it wouldn’t hold for 254.

> They’re all double the last dimension plus two, without skipping any in that sequence - but that offers no insight into why it wouldn’t hold for 254.

Wikipedia at least gives a literature reference and concise explanation for the reason:

> https://en.wikipedia.org/w/index.php?title=Kervaire_invarian...

"Hill, Hopkins & Ravenel (2016) showed that the Kervaire invariant is zero for n-dimensional framed manifolds for n = 2^k− 2 with k ≥ 8. They constructed a cohomology theory Ω with the following properties from which their result follows immediately:

* The coefficient groups Ω^n(point) have period 2^8 = 256 in n

* The coefficient groups Ω^n(point) have a "gap": they vanish for n = -1, -2, and -3

* The coefficient groups Ω^n(point) can detect non-vanishing Kervaire invariants: more precisely if the Kervaire invariant for manifolds of dimension n is nonzero then it has a nonzero image in Ω^{−n}(point)"

Paper:

Hill, Michael A.; Hopkins, Michael J.; Ravenel, Douglas C. (2016). "On the nonexistence of elements of Kervaire invariant one"

* https://arxiv.org/abs/0908.3724

* https://annals.math.princeton.edu/2016/184-1/p01

The proof that it stops instead of going on forever is here: https://arxiv.org/abs/0908.3724

It’s more than 200 pages of pretty technical mathematics, so I’m reasonably confident that there is no description a layperson might understand.

Here is my guess. Number of dimensions is more like a hyperparameter than a parameter. Each time you increase the dimension by 1 you get a new world. You cane expect a simple pattern to go on forever.

What does a 2D manifold with a Kervaire invariant of 1 look like?

I only found a technical article that didn't show any images at all, and 3D looking images in image search

Visually seeing a 2D one might help understand what it is though, and it shouldn't be too bad to render something in 2 dimensions :)

The article shows one, the kind of double torus - though I do not understand how neighbourhoods that intersect the line of contact between the 2 "tubes" can be mapped to R².

Ok I see, it's a 2D shape in 3D space, you actually need the 3th dimension to contain this shape... At least I find that there's s distinction between a 2D shape you can draw on a 2D screen (like a filled rectangle or a disk), and a shape that's a 2D surface itself but requires 3-dimensional space to sit in (like a torus/donut or a non-filled 2-sphere)

So I guess the 126-dimensional shape actually also is in 127-dimensional space then

But the article says "Over the years, mathematicians found that the twisted shapes exist in dimensions 2, 6, 14, 30 and 62.".

To me "Exists in dimension 2" sounds like a shape in 2D space, not in 3D space, but apparently that's not what they mean and the way I understand this language is wrong

> So I guess the 126-dimensional shape actually also is in 127-dimensional space then

Sometimes you need more dimensions to embed the manifold. For a 2-dimencional object, the most famous example is the Klein bottle https://en.wikipedia.org/wiki/Klein_bottle You can construct one of them in 3-dimmension only if you cheat. Yhey look nice and you can buy a few cheating-versions. But you can embed the Klein bottle in 4-dimensions (without cheating).

For the manifold in the article, I'm not sure how many additional dimensions you need. Perhaps 127 (n+1) is enough or perhaps you need 252 (2n) or perhaps something in between. You can always embed an n-dimensional manifold in the 2n space, but that is the worst case. https://en.wikipedia.org/wiki/Whitney_embedding_theorem

Can't wait to read up on volume 127 of the Dimension periodical.

ah yes, now I understand completely

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