Second Adams operation on the odd K-theory of the sphere bundle of two tautological lines
= Second Adams operation on the odd K-theory of the sphere bundle of two tautological lines
{c}
{title2=$\psi^2:K^{-1}(S(\gamma\oplus\gamma))\to K^{-1}(S(\gamma\oplus\gamma))$}
Choose $a,b$ with $p_!(a)=t^{n-1}$ and $p_!(b)=t^n$, where $t=1-[\overline\gamma]$. Then
$$
\psi^2(a)=2^{n+1}a-(n+1)2^n b,
\qquad
\psi^2(b)=2^{n+2}b.
$$