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.
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3$\begingroup$ The forgetful map $KU\rightarrow KO$ is compatible with the Bott periodicities, which answers your question. As to reference, I don't know a good one, but one can deduce the above using what you can find in sections 1.2 and 1.4 of Neil Strickland's thesis neil-strickland.staff.shef.ac.uk/research/thesis.pdf $\endgroup$– user43326Commented Aug 20, 2015 at 15:47
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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.
answered Aug 20, 2015 at 13:22