What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle but not vice versa. You get a map $$ K(X)\to KO(X). $$ What can be said about that?
2 Answers
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There is a long exact sequence of (reduced) K-groups $$ K^{n-1}(X) \to KO^{n+1}(X) \to^\eta KO^n(X) \to^c K^n(X) \to^f KO^{n+2}(X) \to \cdots $$ The map $c$ is induced by complexification, sending a real vector bundle $\xi$ to the associated complex vector bundle. The map $\eta$ is multiplication by the Hopf element (the Mobius band bundle) in $$ KO^{-1}(S^0) = KO^0(S^1) = \pi_1(BO). $$ The map $f$ is induced by the forgetful map from complex vector bundles to real vector bundles, together with the Bott periodicity isomorphism: $$ K^n(X) \cong K^{n+2} \to KO^{n+2}(X) $$ The maps $c$, $f$, and Bott periodicity can be used to produce the splitting of $KO(X)$ off $K(X)$ after inverting the prime 2 that Andrew mentioned.
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Every real bundle can be complexified so there's a natural transformation $KO(X) \to K(X)$. Going all the way around multiplies by $2$ each time so if you localise at the prime $2$, you get a nice splitting.
More generally, these fit into a long exact sequence. I don't remember if the third term has a special name or not. (As this is a community wiki question, someone should feel free to edit this answer to add it in.) I've seen its classifying space written as $B(U/O)$ which suggests not.