{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>.


 Complex K-theory of odd-dimensional real projective space

ID: complex-k-theory-of-odd-dimensional-real-projective-space