K-theory Wang sequence of a mapping torus

For a self-map $f:Z\to Z$, the mapping torus has an exact sequence containingIf $K^{-1}(Z)=0$, then $K^0(T_f)\cong \ker (1-f^{\ast }:K^0(Z)\to K^0(Z))$.

For the factor swap on ${\mathbb{CP}}^2\times {\mathbb{CP}}^2$, the invariant subgroup ofhas basis $1,xy,x^2{y}^2,x+y,x^2+y^2,x{y}^2+x^{2}y$. Hence the mapping torus has $K^0\cong {\mathbb{Z}}^6$.