Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 142 / 4 / Solution

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 $4$.