Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 142 / 3

For a complex vector bundle $E\to X$, the K-theory Euler class is the zero-section pullback of its K-theory Thom class:The cofibration of the disk and sphere bundles gives the K-theory Gysin sequence of a sphere bundle

Let $\gamma ={\gamma }_{\mathbb{C}}^{1,n+1}$ be the tautological bundle. The Euler sequence on complex projective space gives the bundle isomorphismPut $t=1-[\gamma ]$, so $K^0({\mathbb{CP}}^n)=\mathbb{Z}[t]/(t^{n+1})$. Fromand evaluation at $s=-1$, or polynomial division followed by differentiation at the removable root, we obtain

Because $K^{-1}({\mathbb{CP}}^n)=0$, the Gysin sequence identifies even K-theory with the cokernel and odd K-theory with the kernel of multiplication by $(n+1)t^n$. Thereforeand, since multiplication only detects the constant coefficient,