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For a complex vector bundle $E$, define the kth Adams operation by the kth Newton polynomial in its exterior powers:After applying the splitting principle for complex vector bundles and writing $E=L_1\oplus \cdots \oplus L_r$, the Newton identities giveThis formula proves additivity. Applied simultaneously to splittings of $E$ and $F$, it also givesThus ${\psi }^k$ extends to a natural ring endomorphism of $K^0(X)$ and, for a line bundle,

Write $X={\mathbb{CP}}_k^{k+2}$ and let $x=[\overline{\gamma }]-1$. Its reduced K-theory is freely generated by $x^k,x^{k+1},x^{k+2}$, with $x^{k+3}=0$, and restriction to the bottom cell sends $x^k$ to a Bott element $u\in {\tilde{K}}^0(S^{2k})$ and the two higher powers to zero. If $r:X\to S^{2k}$ retracts the bottom-cell inclusion, thenfor some integers $a,b$. Naturality and the Adams operation on a Bott class giveSince ${\psi }^2(x)=2x+x^2$, comparison modulo $x^{k+3}$ first in degree $k+1$ and then in degree $k+2$ givesThe first equality makes $k$ even. Then $3k+5$ is odd, so integrality of $b$ forces $8\mid k$; it is also congruent to $2$ modulo $3$, so $3\mid k$. Consequently

The cofibrationgives the long exact sequence of a pair in Topological K-theory. Identify $D(E)$ with $X$ by deformation retraction and use multiplication by the K-theory Thom classto identify the relative term with $K^{\ast }(X)$. Pullback along the zero section sends ${\lambda }_E$ to the K-theory Euler classThe map from the relative term to $K^{\ast }(D(E))$ is therefore multiplication by $e^K(E)$, giving the K-theory Gysin sequence of a sphere bundle

Forover ${\mathbb{CP}}^n$, put $t=1-[\overline{\gamma }]$. The Complex K-theory of complex projective space isandThe Gysin sequence consequently identifiesfor $n\ge 1$. This is the Odd K-theory of the sphere bundle of two tautological lines.

If $n=0$, then the base is a point and $Y=S^3$, so $K^{-1}(Y)\cong \mathbb{Z}$ by Bott periodicity.

The cannibalistic class is defined by the identityfor the Adams operation ${\psi }^k$. The Thom class of a direct sum is the product of the pulled-back Thom classes. Applying the ring homomorphism ${\psi }^k$ givesIf $L$ is a line bundle, restriction along the zero section givesso

Let $\delta :K^{-1}(S(E))\to {\tilde{K}}^0(\mathrm{Th}(E))$ be the boundary map. By definition of $p_{!}$,The natural operation ${\psi }^k$ commutes with $\delta$, and thereforeCancelling the Thom class proves the Adams operation and the boundary pushforward of a sphere bundle formula

Choose the basis $a,b$ of $K^{-1}(Y)$ characterized byFor $E=\gamma \oplus \gamma$,Modulo $t^{n+1}$, this givesSince $p_{!}$ identifies $K^{-1}(Y)$ with this kernel, the Second Adams operation on the odd K-theory of the sphere bundle of two tautological lines isFor $n=0$, the single generator of $K^{-1}(S^3)$ is multiplied by $4$.