K-theory of the unit tangent sphere bundle of complex projective space
= K-theory of the unit tangent sphere bundle of complex projective space
{c}
{title2=$K^*(S(T\mathbb{CP}^n))$}
The <K-theory Gysin sequence of a sphere bundle> and $e^K(T\mathbb{CP}^n)=(n+1)t^n$ give
$$
K^0(S(T\mathbb{CP}^n))
\cong\mathbb Z[t]/(t^{n+1},(n+1)t^n)
\cong\mathbb Z^n\oplus\mathbb Z/(n+1),
$$
and
$$
K^{-1}(S(T\mathbb{CP}^n))
\cong\ker((n+1)t^n\cdot-)
=(t)\cong\mathbb Z^n.
$$