= Solution
The <Bott isomorphism> is multiplication by the <Bott element> $\beta\in\widetilde K^0(S^2)$:
$$
K^i(X)\xrightarrow{\ \cong\ }
\widetilde K^i(S^2\wedge X)
\cong K^{i-2}(X).
$$
Together with the <suspension isomorphism> and
$$
K^0(\mathrm{pt})=\mathbb Z,
\qquad
K^{-1}(\mathrm{pt})=0,
$$
it gives the <Complex K-theory of a sphere>
$$
\widetilde K^i(S^d)\cong K^{i-d}(\mathrm{pt})
\cong
\begin{cases}
\mathbb Z,&i-d\text{ even},\\
0,&i-d\text{ odd}.
\end{cases}
$$
Let
$$
\varnothing=Y_{-1}\subset Y_0\subset\cdots\subset Y_m=Y
$$
be a <CW filtration> in which each quotient $Y_r/Y_{r-1}$ is a wedge of even-dimensional spheres. The six-term exact sequence in <Topological K-theory>, the sphere calculation, and induction give
$$
K^{-1}(Y_r)=0
$$
and a short exact sequence whose new summand in $K^0(Y_r)$ is free abelian on the newly attached cells. Every such extension splits as an extension of <free abelian groups>, so $K^0(Y)$ is free, with one generator for each cell. This proves the <Complex K-theory of an even-cell complex> result.
The exterior product defines
$$
K^0(Y)\otimes K^i(X)\longrightarrow K^i(Y\times X).
$$
For a point it is the identity. Attaching one layer of even cells gives corresponding exact sequences on the source and target; the sphere case is the <suspension isomorphism>, and induction with the <Five lemma> proves that the product map remains an isomorphism. This is the <Künneth theorem for complex K-theory with an even-cell factor>.
For a <mapping torus> $T_f$, the <K-theory Wang sequence of a mapping torus> contains
$$
K^{-1}(Z)\xrightarrow{1-f^*}K^{-1}(Z)
\longrightarrow K^0(T_f)
\longrightarrow K^0(Z)\xrightarrow{1-f^*}K^0(Z).
$$
When $K^{-1}(Z)=0$, exactness gives
$$
K^0(T_f)\cong\ker(1-f^*:K^0(Z)\to K^0(Z)).
$$
For $Z=\mathbb{CP}^2\times\mathbb{CP}^2$, the <Complex K-theory of complex projective space> and the K-theory Künneth isomorphism give
$$
K^0(Z)\cong\mathbb Z[x,y]/(x^3,y^3).
$$
The factor swap interchanges $x$ and $y$. Its invariant subgroup has the basis
$$
1,\quad xy,\quad x^2y^2,\quad x+y,\quad x^2+y^2,\quad xy^2+x^2y.
$$
It follows that the <K-theory of the mapping torus of the factor swap on two complex projective planes> is
$$
K^0(T_f)\cong\mathbb Z^6.
$$
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