Corollary 9.4.14. There are isomorphisms of graded rings \[ \pi _*\ku \cong \Z [\beta ] \qquadtext { and } \qquad \pi _*\KU \cong \Z [\beta ^{\pm 1}], \] with \(\abs {\beta }=2\). Equivalently, \[ \pi _k(\KU ) \cong \begin {cases} \Z & k \text { even},\\ 0 & k \text { odd}. \end {cases} \]
Proof. The description \(\BUP \simeq \Z \times \bB U\) of Corollary 9.4.3 gives \(\pi _0\ku \cong \Z \) and \(\pi _1\ku =0\). Multiplication by \(\beta \) induces isomorphisms \(\pi _n\ku \to \pi _{n+2}\ku \) for \(n\geq 0\) by Corollary 9.4.6. Since \(\ku \) is connective, this proves \(\pi _*\ku \cong \Z [\beta ]\). The calculation for \(\KU =\ku [\beta ^{-1}]\) now follows from Lemma 7.2.4. □
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