= Complex K-theory of odd-dimensional real projective space
{title2=$K^*(\mathbb{RP}^{2n+1})$}
If $L$ is the <complexification of a real vector bundle> of the <tautological bundle> and $\mu=[L]-1$, then
$$
K^0(\mathbb{RP}^{2n+1})\cong\mathbb Z[\mu]/(\mu^2+2\mu,2^n\mu),\qquad K^{-1}(\mathbb{RP}^{2n+1})\cong\mathbb Z.
$$
Realizing <Real projective space> as the <circle bundle> of the square of the <tautological bundle> on <Complex projective space> gives this through the <K-theory Gysin sequence of a sphere bundle>.
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