Lemma 9.4.9. For every \(k \in \Z \) there is a natural isomorphism \[ \pi _k\KU \cong \colim \big (\pi _k\ku \xrightarrow {b_*} \pi _{k+2}\ku \xrightarrow {b_*} \pi _{k+4}\ku \to \cdots \big ). \]
Proof. By Lemma 4.4.28, the functor \(\pi _k\) preserves the filtered colimit defining \(\KU \). Moreover \(\pi _k(\ku [-2j])=\pi _{k+2j}\ku \), and the transition maps are induced by the Bott map. □
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