Complex Bott periodicity theorem

Simplest form of statement

 * The reduced K-groups of odd-dimensional spheres are 0. In symbols, $$\tilde{K}(S^{2n+1}) = 0$$. Thus, the K-group is $$\mathbb{Z}$$: $$K(S^{2n+1}) = \mathbb{Z}$$
 * The reduced K-groups of even-dimensional spheres are each isomorphic to $$\mathbb{Z}$$. Thus $$\tilde{K}(S^{2n}) = \mathbb{Z}$$ and $$K(S^{2n}) \cong \Z \oplus \Z$$