Fred Lunnon writes:

On 1/8/06, Daniel Asimov <dasimov@earthlink.net> wrote:

...

The NY Times just printed an obituary for Raoul Bott, Emeritus of the Harvard math dept. who passed away Dec. 20: <http://www.nytimes.com/2006/01/08/national/08bott.html>. It mentions one of his best-known results, "Bott periodicity", which blew me away when I first learned of it as a grad student. [etc etc]

I have to admit I'd kinda assumed that "Bott periodicity" referred to the fact that the real Clifford algebra Cl(n+8,0) generated by n+8 square roots of +1 is isomorphic to 16x16 matrices over Cl(n,0), on no better grounds than that the relevant ur-text is M.F.Atiyah, R.Bott, A.Shapiro "Clifford Modules" Topology vol.3 (1964). Was this conclusion entirely unjustified, or is there some connection?

I've just learned that this purely algebraic fact is sometimes *also* called "Bott periodicity" -- at least by John Baez in his article on octonions, section 2.3: <http://math.ucr.edu/home/baez/octonions/node6.html>. I believe that with some sophisticated algbraic-topological reasoning, one can show that the dependence of the homotopy group pi_n(O(oo)) only on n mod 8 is closely connected wiith this algebraic periodicity. But Bott's original discovery is the topological fact that pi_(n+8)(O(oo)) = pi_n(O(oo)) for all n = 1,2,3,.... (And similar periodicities for the infinite unitary group U(oo) and thee infinite symplectic group Sp(oo).) --Dan