Exotic Spheres.

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Exotic Spheres.

Postby d023n » Sat Nov 11, 2017 4:32 am

Hello,

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 4th 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 4th 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 4th 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 Γn of twisted spheres is always isomorphic to the group Θn" 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. R4 is the only Rn, 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.
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Re: Exotic Spheres.

Postby wendy » Sat Nov 11, 2017 9:52 am

Welcome.

The vast bulk of what is done here is on polytopes, but we accept all strange things. The torotope subforum is given over to analogues of the torus in higher dimensions.

The special group of the clifford parallels is represented by "lines" (argand diagrams), crossing at a point in complex-eucliedan N-space, or CEn. This has a real projection in E2n. The various clifford-toruses, wherein you can have a torus in 4d, with say 120 holes, all interlinked torus-shapes with each other, is understood.

Hopf fibulation is understood here under many different names. I suppose swirlybobs is a variation. But stuff over eight or nine dimensions is a little high.
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Re: Exotic Spheres.

Postby d023n » Sat Nov 11, 2017 11:21 pm

wendy wrote:Welcome.


Thank you, Wendy. (: I've read a lot of your posts already, so I sort of already "know" you a bit, heh. You're a hyperdimensional celebrity to me. :P

I actually accidentally discovered this site at the beginning of October, and it was exactly what I had been wishing to find for a few years now. Also! Coincidentally, ICN5D is someone I had already seen on Reddit on his r/hypershape subreddit. In fact, it was one of your comments where you said his name that made me realize this. :D

wendy wrote:The special group of the clifford parallels is represented by "lines" (argand diagrams), crossing at a point in complex-eucliedan N-space, or CEn. This has a real projection in E2n.


Would you be so kind as to explain what you mean here?

Thanks!

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Re: Exotic Spheres.

Postby Klitzing » Sun Nov 12, 2017 8:01 am

That one is easy. Rotation takes place in real 2D subspace, i.e. around the corresponding co-dimensional subspace as ist fixed element. Clifford rotations apply to real spaces with 4D or higher, using 2 (perpendicular) simultanuous rotation subspaces (or even more).

Wendy now was refering to complex spaces. Then a real 2D subspace there happens to be a 1D complex subspace (an Argand plane). Therefore indeed: Clifford rotations do refer to (perpendicular) complex "lines" as their rotational subspaces.

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Re: Exotic Spheres.

Postby wendy » Sun Nov 12, 2017 9:50 am

CEn resolves to complex-euclidean n-space. It is implemented as doing analytic geometry in complex numbers. It's also the space where the Coxeter-Shephard 'Regular Complex Polytopes' live. The operation of a particular operator demonstrates the real version of a particular swirlybob operates, and we derive a geometric realisation of how the 3d and 4d spheres interlink.

The first equation is y=ax+b. In real geometry, this is the equation of a line, of slope a, passing through (0,b). We now consider the set of lines where b=0, viz y=ax.

The second variable, is 'w'. This is a representation of the greek letter omega, but it's used to represent a cyclic time-wise rotation, viz w(n) = cis(2 pi n), which leads to a rotation for every integer, and a fraction of a circle, for the fraction part, so w(0.5).x = -x in every case. That is, a central inversion.

Since we have yw = axw for all values of w, and thus a = y/x = yw/xw, this means that the operation of w in CEn means that in even dimensions, there is a rotation where every point circles the origin, on a plane that contains the origin, thus proving the 'hairy ball is combable' in all even dimensions.

The second equation is the perpendicular to y=ax, is ay=-x, ie the slope perpendicular to a is -1/a. For the complex plane as (re,Im,0) represented by the slopes a, we suppose a sphere is placed with the diameter at (0,0,0) to (0,0,1). We draw a line from a point a,0 to 0,1, and the perpendicular is -1/a,0 to 0,1. This means that the intercepts of the sphere of orthogonal lines, are diametric, for all cases.

This particular sphere has no real binding to the plane, and has spherical symmetry. Instead, it is one of the spheres that is 'perpendicular' to a set of clifford parallels. The angle between clifford parallels are doubled on the sphere (so a orthogonal pair appear at 180 deg).

The rotation phase space

This is a space, where each point represents a different mode of rotation without motion,

For 2d, the phase-space is simply a straight line, where 0 is stopped, and +x, -x is the speed of rotation.

For 3d, the phase-space is to set the north-pole, and then +x as the speed. Note that -x runs through the south pole, and the the reversal of rotation at the equator causes the poles to reverse. So the inversion is still -x.

For 4d, the phase-space is more complex. There are two axies of 3d, representing left-clifford and right-clifford rotations. The difference is that in the complex numbers, the Y axis is reversed relative to the X axis, gives a different rotation. The left and right-cliffords represent left and right multiplication of quarterions, but we shall consider the general case of (x,y) representating a left rotation and right rotation.

The point (x+y, x-y) turns out to be the common single rotation present in both of these, adding in the first, and subtracting in the y axis. These are the great-arrow rotations, represented by a rotation in the wx space, and fixed in the yz space. This is the 'wheel-rotation', which means that wx is the space of height (w) and forward (x), the rotation of the wheel. But the cabin does not rotate, so yz has no rotation. The space of wheel-rotations is the only one that has no handedness. The rest, the lesser rotation revolves around the larger one like a helix, left-handed or right-handed.

Planets tend to distribute energy between modes of rotation, viz between wx and yz, so these lead to simple clifford rotations. What we also observe, is that the sphere described here is the 'lattitude' sphere, and the lengths of the lines is the 'longitude' circle. The product of these gives 2pi, r * pi r^2 (the surfaces of a sphere of diam r and a circle of diameter r) gives the surface of a 4sphere 2pi² r³.

In general, in 4d, the speeds are different, so it rotates faster in one space wx, than it does on the other yz, but they still share two great arrows. The effect is a lissajours rotation.

The space for 5d is ten-dimensional, and includes 5 real spaces, and the 4d space such that the six dimensions of that space is reduced to 5 by a shared line. But the rest is poorly understood.

The general swirlbob

The existance of 'w' in all dimensions, is that when any part rotates in complex space, the whole lot does. This creates in CEn, a swirlybob, which is replicated at every point, in a way that says that there is a parallel great arrow at every point, and no two great arrows at any point intersect. This is exactly in the manner of euclidean space, except that every dimension is doubled, and sense of direction is preserved everywhere.

The swirlybob in 4d is understood as above, and we can construct a symmetry based on these. The one for six dimensions isn't, but its space appears to be pi²/2.
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