is there a general statement about structures on spheres relating to division algebras? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T19:14:31Z http://mathoverflow.net/feeds/question/19929 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19929/is-there-a-general-statement-about-structures-on-spheres-relating-to-division-alg is there a general statement about structures on spheres relating to division algebras? Thomas Kragh 2010-03-31T10:22:59Z 2010-03-31T16:48:48Z <p>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.</p> <p>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.</p> <p>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.</p> <p>There division algebra $\mathbb{O}$ defines an $A_2$ structure on $S^7$, which is not $A_\infty$ (is it $A_3$?).</p> <p>As is well known it is possible to prove that no other spheres has $A_2$ structure.</p> <p><strong>Question:</strong> Is there a heiraki of structures below $A_2$ yet related such that $S^{15}$ has this structure, but $S^{31}$ does not?</p> <p>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 $A_\infty$. </p> <p><strong>Question:</strong> 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)?</p> http://mathoverflow.net/questions/19929/is-there-a-general-statement-about-structures-on-spheres-relating-to-division-alg/19975#19975 Answer by Tilman for is there a general statement about structures on spheres relating to division algebras? Tilman 2010-03-31T16:48:48Z 2010-03-31T16:48:48Z <p>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 <a href="http://math.ucr.edu/home/baez/octonions/oct.pdf" rel="nofollow">Baez's article on the octonions</a> 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.</p> <p>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. </p>