K-theory of spheres

Example

Let $S^n$ be the $n$-sphere, $n \ge 1$, with a base point. Its reduced complex K-theory is $$\tilde{K}(S^n) \cong \begin{cases} \Z & n \text{ even}, \\ 0 & n \text{ odd}. \end{cases}$$ Its reduced real K-theory $\widetilde{KO}(S^n)$ depends on $n \bmod 8$: for $n \equiv 0, 1, 2, 3, 4, 5, 6, 7$ it is $\Z, \Z/2, \Z/2, 0, \Z, 0, 0, 0$.

Check: by the clutching construction, complex bundles of rank $m$ over $S^n$ correspond to homotopy classes of maps $S^{n-1} \to GL_m(\C)$, and $GL_m(\C)$ deforms onto $U(m)$. For $m$ large, this gives $\tilde{K}(S^n) \cong \pi_{n-1}(U)$, where $U$ is the infinite unitary group. Bott periodicity gives $\pi_{n-1}(U) \cong \Z$ when $n - 1$ is odd and $0$ when $n - 1$ is even. In the same way $\widetilde{KO}(S^n) \cong \pi_{n-1}(O)$, which gives the table above.

For $n = 2$, $\tilde{K}(S^2)$ is generated by $[H] - 1$, where $H$ is the tautological line bundle over $\C P^1 = S^2$, and $([H] - 1)^2 = 0$. For $n = 1$, $\widetilde{KO}(S^1) \cong \Z/2$ is generated by $[M] - 1$, where $M$ is the Möbius line bundle.