Theorem 9.3.10 (Bott periodicity for \(\widetilde {K}^0\), Hatcher (2003), Theorem 2.11). For every pointed compact Hausdorff space \(X\), external multiplication by the Bott class induces an isomorphism \[ \widetilde K^0(X) \xrightarrow {\cong } \widetilde K^0(X\wedge \CP ^1) \cong \widetilde K^0(\Sigma ^2X). \]
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