I was showing off my blog to a nonmathematician recently and realised that I haven’t posted anything designed to be accessible to that audience for a while. Around the same time, an undergraduate asked me to read a class paper they wrote on the Hopf fibration, which made me decide to write a post on spheres.
The classification of spheres that are also Lie groups (or Lie loops, if you relax the need for associativity) is one of my favourite stories. I love it when questions like this with no expectation of finitude in the statement turn out to have very finite answers. I am personally of the view that such finitude is typically for a good reason that is worth exploring. In the question at hand, the only spheres which are H-spaces (more general than Lie loops and groups) are…
which correspond to the only real division algebras
In each case, the sphere is the set of unit length elements in the corresponding division algebra under the usual norm. Since the octonions are not associative, one must allow loops to include . This result is closely related to the ‘Hopf invariant one problem’ that was believed for many years until Adams finally completed the intricate details of the proof inside -theory. This completely classifies the spheres that are -spaces, remarkably giving the same list as for spheres that are Lie loops or parallelisable.
For a recent exhibition of finitude, Papanikolopoulos and Siksek have proved that every cubic hypersurface of dimension at least is ‘fake-cyclic’. There is a similar secant construction to that for elliptic curves allows one to produce new rational points from a pair of old ones, and fake-cyclic means that it suffices to start with a single point and repeatedly apply the construction to it to obtain all rational points on the cubic hypersurface. It’s fake because the construction doesn’t give a group structure. Cubic is needed to get a unique point out by Bezout. Fake-cyclicity of course fails radically for elliptic curves, so there must be a nontrivial threshold after which all cubic hypersurfaces are fake-cyclic. I haven’t read the paper yet, so maybe this is established there or perhaps their proof just doesn’t work in dimensions lower than .