Complex K-theory groups

Definition 9.3.1 (Complex K-theory). Let \(X\) be a compact Hausdorff space. We define the complex K-theory group \(K^0(X)\) as the Grothendieck group of the abelian monoid \(\pi _0\Vect (X)\): \[ K^0(X) := (\pi _0\Vect (X))^{\grp }. \] In other words, \(K^0(X)\) consists of equivalence classes of formal differences \(E- E'\) of complex vector bundles over \(X\), where we say that \(E_1 - E_1' = E_2 - E_2'\) whenever there is an isomorphism \(E_1 \oplus E_2' \oplus E_3 \cong E_1' \oplus E_2\oplus E_3\) of vector bundles over \(X\) for some \(E_3 \in \pi _0\Vect (X)\).