K-theory Gysin sequence of a sphere bundle
= K-theory Gysin sequence of a sphere bundle
{c}
The cofibration $S(E)\to D(E)\to\operatorname{Th}(E)$ and the Thom isomorphism give
$$
\cdots\to K^i(X)\xrightarrow{\cdot e^K(E)}K^i(X)
\xrightarrow{p^*}K^i(S(E))
\xrightarrow{p_!}K^{i+1}(X)\to\cdots.
$$