Reduced complex K-theory

Definition 9.3.4 (Reduced K-theory). Let \(X\) be a compact Hausdorff space. We define the abelian group \(\widetilde {K}^0(X)\) as the cokernel of the map \(\Z \cong K^0(\pt ) \to K^0(X)\). This is called the reduced K-theory group of \(X\).

When \(X\) has a basepoint \(x\), this is also equivalent to the kernel of \(K^0(X) \to K^0(\{x\}) \cong \Z \).