It is classical to take a division algebra over mathbbR 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 S0 and S1 identifying them as Eilenberg-MacLane spaces - Or if we forget some structure as an Einfty-spaces.
The associative division algebras mathbbH defines an associative product on S3, which is also a Lie-group, but forgetting some structure it is an Ainfty-space.
There division algebra mathbbO defines an A2 structure on S7, which is not Ainfty (is it A3?).
As is well known it is possible to prove that no other spheres has A2 structure.
Question: Is there a heiraki of structures below A2 yet related such that S15 has this structure, but S31 does not?
Remark: A heiraki below A2 could be that A2=Dinfty for some definition of structures Dn, analagous to E1 being Ainfty.
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)?
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