Corollary 9.4.6 (Bott periodicity for \(\ku \)). The Bott map induces an isomorphism \[ b_*\colon \pi _n\ku \xrightarrow {\cong } \pi _{n+2}\ku \] for every \(n \geq 0\).
Proof. Apply Theorem 9.3.10 to \(X=S^n\). Under the comparison above, \[ \pi _n\ku \cong \widetilde \ku ^0(S^n) \cong \widetilde K^0(S^n), \] the map \(b_*\) is external multiplication by the Bott class. The claim therefore follows from classical Bott periodicity. □
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