4 ms·

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 laypers

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!

codebje 2y ago
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.
- aleph_minus_one 2y ago
  > 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_invariant&oldid=1282708795#History_2 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://arxiv.org/abs/0908.3724 * https://annals.math.princeton.edu/2016/184-1/p01 https://annals.math.princeton.edu/2016/184-1/p01
robinhouston 2y ago
The proof that it stops instead of going on forever is here: https://arxiv.org/abs/0908.3724 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.
foobahhhhh 2y ago
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.

codebje 2y ago

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.

aleph_minus_one 2y ago
> 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_invariant&oldid=1282708795#History_2 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://arxiv.org/abs/0908.3724 * https://annals.math.princeton.edu/2016/184-1/p01 https://annals.math.princeton.edu/2016/184-1/p01

robinhouston 2y ago

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

foobahhhhh 2y ago

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.