I was wondering if anybody knows an intuitive way to think about the equivalence that only ${ S }^{ 0 },{ S }^{ 1 },{ S }^{ 3 }$ and ${ S }^{ 7 }$ are parallelisable and the existence of only 4 normed division algebras ($\mathbb{R,C,H,O}$). I know that one can use the normed division algebras to define n linear independent sections (on ${ S }^{ 0 },{ S }^{ 1 },{ S }^{ 3 },{ S }^{ 7 }$) but that doesn't tell me why there are exactly 4 of them. I first thought that the answer to the question boils down to lie groups, but ${ S }^{ 7 }$ is not a lie group.
I guess my question involves several sub questions:
- Is there another way to think about a parallelisable manifold than that it has $n$ linear independent vectorfields or that the tangent bundle is isomorphic to the trivial bundle. (Both of them don't feel very intuitive to me.)
- When you want to define a normed division algebra on ${ S }^{ 2 }$ e.g., which part of the definition does not work out? (Why can't one use the antipodal point as inverse elements.)
- What is an intuitive/visual way to think about the equivalence above.