Let $\overset{\sim}{K}(X)$ and $\overset{\sim}{KO}$ denote the reduced stable isomorphic classes of complex and real bundles over X and $\rho$ be the realization map. We know that $\overset{\sim}{K}(\mathbb{S}^2)\to\overset{\sim}{KO}(\mathbb{S}^2)$ is a map $\mathbb{Z}\to \mathbb{Z}\_2$ and is non-zero since the realization of the Hopf bundle will be non-zero. Is $\overset{\sim}{K}(\mathbb{S}^{8t+2})\to\overset{\sim}{KO}(\mathbb{S}^{8t+2})$ (which is also $\mathbb{Z}\to\mathbb{Z}\_2$) non-zero map for any $t\geq 0$? If it's true, please give a reference.



realization map for K-theory of spheres
Yes. If this map $\pi\_{8t+1}U\to \pi\_{8t+1}O$ were trivial then $\pi\_{8t+1}(O/U)$ would have an element of order $2$. But $O/U$ is homotopy equivalent to $\Omega O$, and $\pi\_{8t+2}O$ is trivial.


I do not know a reference offhand.


