Bott periodicity theorem

Remarks

In the complex case, the isomorphism of part 3 is the external product with the class $b = [H] - 1 \in \tilde{K}(S^2)$, where $H$ is the tautological line bundle over $\C P^1 = S^2$. Bott's original proof used Morse theory on spaces of paths in Lie groups; see Bott's Morse-theoretic proof of periodicity. Atiyah and Bott later gave an elementary proof of the complex case through the clutching construction: they show that $K(X \times S^2) \cong K(X) \otimes K(S^2)$ and $K(S^2) \cong \Z[H]/(H - 1)^2$.

Example: applied to $X = S^0$, part 3 gives $\tilde{K}(S^{2n}) \cong \tilde{K}(S^0) \cong \Z$ and $\tilde{K}(S^{2n+1}) \cong \tilde{K}(S^1) = 0$ for all $n \ge 0$.

The infinite groups matter: for finite $n$ the groups $\pi_i(U(n))$ are not periodic, but they agree with $\pi_i(U)$ for $i < 2n$.

Going deeper

For a student

Bott periodicity is what makes complex K-theory a cohomology theory defined in all degrees: one sets $K^{n} = K^{n-2}$ for all integers $n$. It gives the K-theory of spheres, and through it the Thom isomorphism in K-theory, the Atiyah–Singer index theorem and Adams' solution of the Hopf invariant one problem. The period $8$ of real K-theory matches the period $8$ of real Clifford algebras, which is the content of the Atiyah–Bott–Shapiro construction.