The discussion centers on the group structure of n-spheres, specifically noting that only S1 and S3 are Lie groups. The participants reference John Lee's "Introduction to Smooth Manifolds," particularly Chapter 9, for further insights. They emphasize the connection between spheres and division algebras, stating that division algebras exist only in dimensions 1, 2, 4, and 8, leading to the conclusion that most spheres do not possess group structures. Additionally, the conversation highlights the complexities of extending group operations from Sn-1 to Rn.
PREREQUISITESMathematicians, particularly those specializing in topology, differential geometry, and algebra, as well as students seeking to understand the structure of n-spheres and their group properties.
You kind of have it the other way around. If we can endow ##\mathbb R^n## with some "sufficiently nice" multiplication operation, then ##S^{n-1}## will be a group. But the converse isn't clear - why can't we have a group operation on ##S^{n-1}## that doesn't necessarily extend to a nice multiplication on ##\mathbb R^n##?homeomorphic said:Most spheres are not Lie groups. You could always turn the underlying set into a group, but then there's no point in it being a sphere.
I think this has something to do with division algebras, maybe. I think if it were a Lie group, maybe you could take the group algebra and get a division algebra. Division algebras only exist in certain dimensions: 1, 2, 4, and 8. So you get spheres of dimension 1 less: 1, 3, and 7. And 7 doesn't work because the octonians, the dimension 8 division algebra, are non-associative, so you can't get a group by taking the unit octonians.
So the only spheres that are groups are S^1 and S^3.
Don't quote me on this, though, since I haven't quite thought it through. Just came up with it off the top of my head. I think it's right, though.
The end result is correct, anyway:
http://en.wikipedia.org/wiki/3-sphere
You kind of have it the other way around. If you can endow Rn with some nice "multiplication" operation, then Sn−1 will be a nice group. But the converse isn't clear - why can't we have a nice group operation on Sn−1 that doesn't necessarily extend to a nice multiplication on Rn?
Yes, that result is true: S^1 and S^3 (and S^0) are the only spheres that are Lie groups.homeomorphic said:It isn't clear, but according to wikipedia and probably some other sources I remember, it is true that only S^1 and S^3 are Lie groups. So, my speculation was that given a group operation on S^n-1, you can construct a division algebra of dimension n somehow. But it's not clear how. The wikipedia article also seems to indicate that the proof should proceed along these lines.
Most proofs I know don't mention real division algebras; it typically goes the other way around: once you have this result, you get a restriction on the possible division algebras.