is there a general statement about structures on spheres relating to division algebras? - MathOverflow

is there a general statement about structures on spheres relating to division algebras?

2010-03-31T10:22:59Z

It is classical to take a division algebra over $\mathbb{R}$ and defining an H-space structure on the unit spheres by restricting and normalizing.

There are commutative division algebras of dimension 1 and 2 leading to commutative products on $S^0$ and $S^1$ identifying them as Eilenberg-MacLane spaces - Or if we forget some structure as an $E_{\infty}$-spaces.

The associative division algebras $\mathbb{H}$ defines an associative product on $S^3$, which is also a Lie-group, but forgetting some structure it is an $A_\infty$-space.

There division algebra $\mathbb{O}$ defines an $A_2$ structure on $S^7$, which is not $A_\infty$ (is it $A_3$?).

As is well known it is possible to prove that no other spheres has $A_2$ structure.

Question: Is there a heiraki of structures below $A_2$ yet related such that $S^{15}$ has this structure, but $S^{31}$ does not?

Remark: A heiraki below $A_2$ could be that $A_2=D_\infty$ for some definition of structures $D_n$, analagous to $E_1$ being $A_\infty$.

Question: Is there an even more general definition of "lower" structures and a statement about all spheres (including possibly non-trivial structures on even-dimensional spheres)?

Answer by Tilman for is there a general statement about structures on spheres relating to division algebras?

2010-03-31T16:48:48Z

I like your spelling of hierarchy! $S^7$ is not $A_3$ -- if it were, you could construct the projective space $\mathbb{O}P^3$, but that's impossible (some decomposition of Steenrod operations argument). You'll find this and answers to your other question in Baez's article on the octonions in the Bulletin of the AMS. There are higher dimensional algebras in the sequence $\mathbb R$, $\mathbb C$, $\mathbb H$, $\mathbb O$, which you can get by the Cayley-Dickson construction, but they are not normed, so you can't take the unit sphere. The 16-dimensional guy is "power-commutative", meaning that powers of an element x commute with x (not obvious if you're not associative), this starts failing in dimension 32 if I remember correctly. So there's some sort of hierarchy of structure on the algebras themselves, if not on the spheres.

Some of the spheres have an H-space structure or a loop space structure after completing at a prime p ($S^{2p-3}$, for example). Maybe this goes a bit in the direction of your question about some kind of multiplication on higher spheres.