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. So in case anyone's interested, I thought I'd describe it here for those not already familiar with it. (Some technical details are omitted.) Given a topological space X, there are algebraic invariants which describe its holes, which can come in various dimensions. These include the groups pi_n(X) for each n = 1,2,3,.... We consider only spaces X for which any two points can be connected by a continuous path. As a set, pi_n(X) is all continuous maps of the unit n-sphere S^n into X where a preassigned point of S^n is required to map to a preassigned point (call it *) of X . . . and THEN any two such maps f,g: S^n -> X are considered equivalent if there is a continuous *family* of such maps {h_t, 0 <= t <= 1}, with h_0 = f and h_1 = g. This set is given a group structure, so each pi_n(X) is a group. (The "fundamental" group, pi_1(X), is the only one that can be (and often is) non-abelian; any pi_n(X) for n > 1 is abelian.) For some examples, pi_1(figure-8) is the free group on 2 generators. For any n, pi_n(S^n) is the group Z of integers. ------------------------------------------------------------------------------ A convergent thread starts with the fact that any Euclidean space R^n has its own group of rotations, which is also a topological space such that the group operations are continuous. If we also toss in the isometries of R^n preserving the origin which *reverse* orientation, we get the set of all invertible nxn matrices M such that M-transpose = M-inverse. This topological ("Lie") group is denoted by O(n), the orthogonal group of R^n. (O(n) corresponds to the set of all n-tuples of orthonormal vectors in R^n.) O(n) is locally Euclidean -- i.e., a manifold -- of dimension (n^2 - n)/2. Just as R^n naturally includes in R^(n+1) as the first n coordinates, we may form the matural inclusions O(1) in O(2) in . . . O(n) in O(n+1) in . . .. Using these inclusions we can take the union of all the O(n)'s for n >= 1 to get an all-inclusive group that we call O(oo). ------------------------------------------------------------------------------ Embodying all the symmetry of all Euclidean spaces R^n, O(oo) is a very interesting object. Raoul Bott showed that pi_n(O(oo)) depends only on n mod 8. Here is the table: n mod 8 0 1 2 3 4 5 6 7 ---------------------------------------------------------------------- pi_n(O(oo) Z_2 Z_2 0 Z 0 0 0 Z It is not a coincidence that the dimensions 0,1,3,7 of the non-zero groups mod 8 are the same as the dimensions of the unit spheres in the only normed real division algebras: the reals, complexes, quaternions, octonions; these four division algebras give rise to the existence of non-zero homotopy classes in pi_n(O(oo)). There is much more to Bott periodicity, but this will do for now. --Dan