Example 9.3.5 (Complex K-theory of the circle and the two-sphere). There are isomorphisms \[ K^0(S^1)\cong \Z , \qquad K^0(S^2)\cong \Z \times \Z , \] and hence \(\widetilde K^0(S^1)=0\) and \(\widetilde K^0(S^2)\cong \Z \). Identifying \(S^2\) with \(\CP ^1\), the class \([H]-[\ul {\C }]\) of the tautological line bundle generates \(\widetilde K^0(S^2)\).
Proof. The calculation for \(S^1\) follows from Exercise 9.1.6. For \(S^2\), the clutching classification of Exercise 9.1.7 identifies a positive-rank vector bundle with its rank and the winding number of the determinant of its clutching function. These invariants are additive under direct sum, so group completion gives \(K^0(S^2)\cong \Z \times \Z \). The second factor is the reduced group, and the clutching function of \(H\) has winding number \(1\) up to sign. □
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