K-theory Thom class (${\lambda }_E$)

For a complex vector bundle $E\to X$, the K-theory Thom classgenerates the Thom isomorphism $K^i(X)\cong {\tilde{K}}^i(\mathrm{Th}(E))$.

Pulling the K-theory Thom class back along the zero section gives

The cofibration $S(E)\to D(E)\to \mathrm{Th}(E)$ and the Thom isomorphism give

For $E=\gamma \oplus \gamma$ over ${\mathbb{CP}}^n$ and $t=1-[\overline{\gamma }]$, the K-theory Euler class is $t^2$. Hencefor $n\ge 1$.

For $n=0$, the base is a point, the sphere bundle is $S^3$, and its odd K-theory is $\mathbb{Z}$.