Bott periodicity
/ BOTT /
If you computed K-theory degree by degree and the answers kept changing forever, the theory would be useless to compute by hand. Bott periodicity is the astonishing fact that they do not change forever: in complex K-theory the answer repeats every two degrees, and in real K-theory every eight. After you have understood a short stretch, the whole infinite ladder is determined by copying it over and over.
In its bundle form, complex Bott periodicity says K-tilde(X) is naturally isomorphic to K-tilde(X smashed with S^2) — multiplying by the Bott element [H] - 1 from K-tilde(S^2) (H the tautological line bundle) gives an isomorphism K(X) -> K(X times S^2 reduced). Iterated, this says the complex K-groups satisfy K^{n}(X) = K^{n+2}(X) for all n: complex K-theory is 2-periodic, so there are really only two groups, K^0 and K^1. The real theory KO is 8-periodic, KO^n(point) running through the famous pattern Z, Z/2, Z/2, 0, Z, 0, 0, 0 and then repeating. Homotopy-theoretically the same fact reads as periodicity of the homotopy groups of the stable orthogonal and unitary groups, pi_k(U) and pi_k(O), with periods 2 and 8.
This single theorem is why K-theory is computable at all and why it deserves to be called a cohomology theory with a clean structure. It powers the index theorem's bookkeeping, the solution of the vector-fields-on-spheres problem, and the classification of division algebras and parallelizable spheres (only S^1, S^3, S^7). A point worth internalizing: the periods 2 and 8 are not arbitrary decoration — the 8 in the real case is the same 8 that appears in Clifford algebras and the octonions, and the mismatch between 2 and 8 is exactly why real and complex phenomena diverge.
Because complex K-theory is 2-periodic, knowing K^0(point) = Z and K^1(point) = 0 determines K^n(point) for all n: it is Z in even degrees and 0 in odd degrees, forever. No further computation of higher degrees is needed — periodicity copies these two values up and down the whole integer line.
Two values plus periodicity give the entire complex K-theory of a point.
The complex period is 2 and the real period is 8 — do not assume one number works for both. The real KO pattern Z, Z/2, Z/2, 0, Z, 0, 0, 0 includes 2-torsion that complex K-theory never sees; the two theories are genuinely different, related but not the same.