Complex Bott periodicity theorem

Statement

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 $K(S^{2n})\cong \mathbb {Z} \oplus \mathbb {Z}$