User talk:Fropuff/Spheres

Why stop at 8-sphere? More fully, why go up to 8-spehre, and why not go beyond it? (I'm came across this as a result of Homotopy groups of spheres, which has recently been re-worked) Tompw 00:26, 16 December 2005 (UTC)[reply]

The same reason we have articles on the numbers 1, 2, 3, ... but none on 2391. The low-dimensionality gives these spheres special properties not shared by general spheres. For example:

$S^{0},S^{1},S^{3}$ and $S^{7}$ are all admit algebraic structures rendering them parallelizable. They are also the only spheres which are H-spaces. That these spheres are interesting is directly related to the fact that there are only 4 normed division algebras: R, C, H, and O.
The generalized Hopf bundles all involve these low dimensional spheres. (I should really have the 15-sphere on this list as well).
$S^{2}$ and $S^{6}$ are the only spheres admitting almost complex structures
Many of these spheres admit transitive actions by the spin groups (other than the standard orthogonal action)
See Milnor sphere

Ideally I would like to have articles on all these spheres. -- Fropuff 05:51, 16 December 2005 (UTC)[reply]