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

Let $\gamma$ be the tautological line bundle and put $t=1-[\gamma ]$. The Euler sequence on complex projective space givesHenceand evaluating at $s=-1$ in $K^0({\mathbb{CP}}^n)=\mathbb{Z}[t]/(t^{n+1})$ gives

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

Let $L$ be the complexification of a real vector bundle of the tautological bundle over Real projective space ${\mathbb{RP}}^{2n+1}$. For $k\ge 1$, the K-theory Euler class of $L^{\oplus k}$ is $-2^{k-1}([L]-1)$, andThe K-theory Gysin sequence of a sphere bundle gives an extension by $\mathbb{Z}$, which splits as an exact sequence of abelian groups.

The K-theory Gysin sequence of a sphere bundle and $e^K(T{\mathbb{CP}}^n)=(n+1)t^n$ giveand

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 $\mathbb{Z}$.