Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 142 / 4 / Solution

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