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?



Relation between $KO$ and $K$
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.


