I looked but didn't find any posts or comments about exotic spheres, which are some objects that I've found to be both extremely fascinating and ridiculously difficult to comprehend. I'd like to explain what I've collected about the general concept, but if anyone well-studied on this topic would like to share their more organized thoughts or point out any misconceptions I may've happened to gather, I'd welcome the information/corrections. Questions are appreciated as well, although I myself will likely be unable to offer satisfying answers, but I'll try my best. I hope my way of writing isn't too informal or awkward. Either way, I thank you for your patience. Also, when it comes to the number of exotic spheres in a dimension, I prefer to index by the ambient Euclidean dimension in which the standard sphere is embedded, rather than the one lower dimension of the spheres themselves, although that raises a question to which I'll return shortly.

First, Wikipedia begins its explanation by saying that "an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere." Right away, the bar is set quite high. If anyone knows a more intuitive way to explain "homeomorphic but not diffeomorphic," that'd be swell. Anyways, upon reading a bit more, it becomes obvious that understanding exotic spheres is visually problematic because of the fact that, ignoring dimension 5 for a moment, there aren't any exotic spheres until dimension 8 where there are suddenly 27, in addition to the standard 7-sphere (so 28 total). After that, there's only 1 exotic 8-sphere in dimension 9. Next, in dimension 10, there are 7 exotic 9-spheres. In dimension 11, there are 5 exotic 10-spheres. However, suddenly there are 991 exotic 11-spheres in dimension 12, but followed by no exotic 12-spheres in dimension 13. The list of all these spheres per dimension continues seemingly randomly from there with every 4

^{th}dimension having an increasingly (for the most part) gigantic number (e.g. dimension 24 has 69,524,373,503 exotic ones; dimension 28 has 67,100,671; but then dimension 32 has 7,767,211,311,103; meanwhile, dimension 27 only manages to conjure 11 while dimensions 25 and 29 each only have 1). Dimension 62 is the only other one without any exotic spheres.

This strange pattern is, I think, what really captured my attention.

That being said, apparently these bizarre numbers can be calculated using (1) the homotopy groups of spheres, which "describe how spheres of various dimensions can wrap around each other"; (2) something called the J-homomorphism, which is "a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres"; and (3) Bernoulli numbers, which, well, even though they seem to be related to just about everything, I still have no clear idea what exactly they fundamentally represent. As a brief aside, the homotopy groups of spheres also follow a strange pattern, specifically with respect to the numbers of ways that spheres of higher dimension can "wrap around" spheres of lower dimension (specuation: a pattern that I'm inclined to think emerges from the pattern of exotic spheres, itself simply a result of the inherent structure of the dimensions, a structure that exists because of the fundamental nature of integers, namely prime numbers). Anyways, there are even weird "stabilities" in the pattern. For example, in dimension 23 and higher, the native sphere can wrap around the sphere from 10 dimensions lower in 6 ways (e.g. the 22-sphere can wrap around the 12-sphere 6 ways). When it comes to wrapping around a sphere from 11 dimensions lower, the stable number of ways to do so isn't reached until dimension 25 and with a much larger value of 504; but then, there are no ways for a sphere in dimension 27 and higher to wrap around a sphere from 12 dimensions lower. The next "sphere-wrapping stability numbers" for 13, 14, and 15 dimensions lower are 3 in dimension 29 and up, 4 in dimension 31 and up, and 960 in dimension 33 and up, respectively. Just like the numbers of exotic spheres, these numbers show that peculiar, increasing (for the most part) gigantism in every 4

^{th}value. I couldn't find an OEIS list, but there is a (somewhat confusing) table at the bottom of the Wikipedia page, which, in list form and starting with index 0, reads as {infinity, 2, 2, 24, 0, 0, 2, 240, 4, 8, 6, 504, 0, 3, 4, 960, 4, 16, 16, 528, 24, 4, 4, 3144960, 4, 4, 12, 24, 2, 3, 6, 65280, 16, 32, 32, 114912, 6, 12, 120, 3801600, 384, 32, 96, 552, 8, 5760, 48, 12579840, 64, 12, 24, 384, 24, 16, 8, 20880, 4, 16, 4, 687456, 4, 0, 48, 261120, ...} and means, for example, that a wrapping with higher-to-lower difference of 0 (so same-sphere-wrapping) produces an infinity of unique options, while a wrapping with higher-to-lower difference of 23 produces only 3,144,960 unique options. So, to wrap this side note up by restating my earlier, vague speculation, (a) integers are weird becaue of primes; (b) building dimensions of space according to integers leads to strange phenomena such as hard to predict numbers of ways of distinctly orienting spheres and cubes, along with every 4

^{th}dimension suddenly allowing an immense, somehow Bernoulli-based number of these ways; and (c) wrapping a sphere onto a lower dimension sphere creates a further strange phenomenon, perhaps described as ways of distinctly "entangling" the inherent structures found in dimensions because, well, every integer is unique. Oh, and this whole concept of a higher-to-lower wrapping is impossible to visualize because, well, it doesn't occur until dimension 4 with the Hopf fibration, leading to an infinity of ways of wrapping a 3-sphere around a 2-sphere.

Returning to exotic spheres specifically, the closest I did come to a hint of intuitive understanding was with twisted spheres, because the "group Γ

_{n}of twisted spheres is always isomorphic to the group Θ

_{n}" of exotic spheres. "The notations are different because it was not known at first that they were the same for n = 3 or 4; for example, the case n = 3 is equivalent to the Poincaré conjecture." We know now that they are the same for n = 3, but n = 4 is still elusive, which brings me back to the earlier ignored dimension 5 where the standard 4-sphere lives. It is currently unknown whether or not there are any exotic 4-spheres in dimension 5 or even if there are a finite number or not. Why are 4-spheres so weird? My guess is that it has something to do with the fact that Euclidean dimension 4 is weird. R

^{4}is the only R

^{n}, out of the infinty of them all, that has more than one version of itself.

I'll wrap up by asking that question from the beginning, which is about how exotic spheres are embedded in space. Are all exotic n-spheres embeddable in dimension n+1 like the standard sphere (or do they require even higher dimensions, like, for example, how a Clifford n-torus requires dimension 2n)? If "yes," then I'll feel better about my original preference for indexing numbers of exotic spheres according to the ambient dimension of the standard sphere. If "no," then can different exotic n-spheres sharing the same n have different embedding dimensions, and, either way, just what are all of these higher than usual embedding dimensions?

Thank you for indulging my interest,

d023n.