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)


 * 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)