What is the real K-theory of the spheres?

Question

In many sources one finds a computation of the complex topological K-theory of the spheres $S^n$, but the real theory $KO(S^n)$ is usually not computed. Maybe it can be read off some more advanced stuff, but I am unable to do so. I would like to see a computation of, say $KO(S^2)$. Can somebody give me a reference?

Answer 1

The computation of these groups is an immediate consequence of real Bott periodicity. Namely, by representability you have $$ \widetilde{KO}(S^n) \cong [S^n, BO \times \mathbb{Z}]_* \cong \begin{cases} [S^n, BO]_* = \pi_n(BO) & \text{if }n > 0 \\ \mathbb{Z} & \text{if } n = 0 \end{cases} $$ and since $\pi_n(BO) \cong \pi_{n - 1}(O)$ you get to read off that \begin{align} \widetilde{KO}(S^0) &\cong \mathbb{Z} \\ \widetilde{KO}(S^1) &\cong \mathbb{Z}/2 \\ \widetilde{KO}(S^2) &\cong \mathbb{Z}/2 \\ \widetilde{KO}(S^3) &\cong 0 \\ \widetilde{KO}(S^4) &\cong \mathbb{Z} \\ \widetilde{KO}(S^5) &\cong 0 \\ \widetilde{KO}(S^6) &\cong 0 \\ \widetilde{KO}(S^7) &\cong 0 \\ \end{align} with $\widetilde{KO}(S^{8k + i}) \cong \widetilde{KO}(S^i)$ accounting for all remaining groups. This table you can also find here, and the Wikipedia pages for topological $K$-theory and Bott periodicity, too, are pretty useful. For a proper mathematical text, Karoubi's "$K$-Theory―An Introduction" gives a proof of the periodicity theorem, but be warned that it's neither easy nor short.