= K-theory of the sphere bundle of copies of a complexified real tautological line
{c}
{title2=$K^0(S(L^{\oplus k}))$}
Let $L$ be the <complexification of a real vector bundle> of the <tautological bundle> over <Real projective space> $\mathbb{RP}^{2n+1}$. For $k\geq1$, the <K-theory Euler class> of $L^{\oplus k}$ is $-2^{k-1}([L]-1)$, and
$$
K^0(S(L^{\oplus k}))\cong\mathbb Z^2\oplus\mathbb Z/2^{\min(n,k-1)}.
$$
The <K-theory Gysin sequence of a sphere bundle> gives an extension by $\mathbb Z$, which splits as an <exact sequence> of <abelian groups>.
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